Liability Driven Investing: Hedging Inflation And Interest ...
A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing,...
Transcript of A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing,...
![Page 1: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/1.jpg)
A Benchmark Approach to Investing,Pricing and Hedging
Eckhard PlatenUniversity of Technology Sydney
Pl. & Heath (2006, 2010), A Benchmark Approach to Quantitative Finance. Springer
Pl. & Bruti-Liberati (2010), Numerical Solution of Stochastic DifferentialEquations with Jumps in Finance. Springer
Baldeaux &Pl. (2013). Functionals of MultidimensionalDiffusions with Applications to Finance. Springer
![Page 2: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/2.jpg)
1 Best Performing Portfolio as Benchmark
• with focus on long term
• if possible pathwise
• should prefer more for less
• diversified portfolio
• robust, almost model independent construction
• sustainable (strictly positive long only)
• criterion independent from denomination
• criterion independent from sequence of drawdown events
⇒maximize logarithm of portfolio
1
![Page 3: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/3.jpg)
1973 1980 1988 1995 2003 20110
3
6
9
12
15x 10
4
EWI114MCI
EWI141 and MCI
2
![Page 4: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/4.jpg)
1973 1980 1988 1995 2003 20111.5
2
2.5
3
3.5
4
4.5
5
5.5
6
log(EWI114)log(MCI)
logarithms of EWI114 and MCI
3
![Page 5: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/5.jpg)
1976 1984 1993 2002 2011
0
0.02
0.04
0.06
0.08
0.10
0.12
long term growth EWI114long term growth MCI
Long term growth of EWI114 and MCI
4
![Page 6: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/6.jpg)
Key idea:
Make the “best” performing portfolio Sδ∗t numeraire or benchmark
5
![Page 7: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/7.jpg)
Portfolio:
Sδt =
d∑j=0
δjtSjt
Sjt - jth constituent (e.g. cum dividend stocks)
6
![Page 8: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/8.jpg)
Long Term Growth:
limt→∞
1
tlog
(Sδt
Sδ0
)≤ lim
t→∞
1
tlog
(Sδ∗t
Sδ∗0
)a.s. pathwise
Sδ∗ is the numeraire portfolio (NP)Long (1990)
7
![Page 9: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/9.jpg)
Main Assumption:
There exists NP Sδ∗t ,
which means
Sδt
Sδ∗t
≥ Et
(Sδs
Sδ∗s
)(∗)
for all 0 ≤ t ≤ s <∞ and all nonnegative portfolios Sδt ,
Et(·) - real world conditional expectation under information given at time tSδ∗t - numeraire portfolio (NP), see Long (1990)
8
![Page 10: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/10.jpg)
Benchmarking:
make NP numeraire and benchmark
Sδt =
Sδt
Sδ∗t
− benchmarked portfolio
Sjt =
Sjt
Sδ∗t
− jth benchmarked security
9
![Page 11: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/11.jpg)
Existence of NP ⇔ Supermartingale Property
Sδt ≥ Et(S
δs) (∗)
for all 0 ≤ t ≤ s <∞ and all nonnegative Sδt ,
• The key property of financial market!
• model independent
• makes also in the short term “best” performance of Sδ∗ precise
• existence of NP, extremely general, Karatzas and Kardaras (2007)
10
![Page 12: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/12.jpg)
Outperforming Long Term Growth
• long term growth rate
gδ = lim supt→∞
1
tln
(Sδt
Sδ0
)
strictly positive portfolio Sδ
pathwise characteristic
11
![Page 13: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/13.jpg)
Theorem The NP Sδ∗ achieves the maximum long term growth rate.For any strictly positive portfolio Sδ
gδ ≤ gδ∗ .
NP outperforms long term growth pathwise!
• Pathwise outperformance asymptotically over time!
• Ideal long term investment if no constraints!
12
![Page 14: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/14.jpg)
Proof: Consider strictly positive portfolio Sδ , with Sδ0 = Sδ∗
0 = x >
0. By supermartingale property (∗) of Sδ and Doob (1953), for any k ∈1, 2, . . . and ε ∈ (0, 1) one has
expε kP(
supk≤t<∞
Sδt > expε k
)≤ E0
(Sδk
)≤ Sδ
0 = 1.
13
![Page 15: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/15.jpg)
For fixed ε ∈ (0, 1)
∞∑k=1
P
(sup
k≤t<∞ln(Sδt
)> εk
)≤
∞∑k=1
exp−ε k <∞.
By Lemma of Borel and Cantelli there exists kε such that for all k ≥ kε
and t ≥ kln(Sδt
)≤ ε k ≤ ε t.
14
![Page 16: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/16.jpg)
Therefore, for all k > kε
supt≥k
1
tln(Sδt
)≤ ε,
which implies
lim supt→∞
1
tln
(Sδt
Sδ0
)≤ lim sup
t→∞
1
tln
(Sδ∗t
Sδ∗0
)+ ε.
Since the inequality holds for all ε ∈ (0, 1), theorem follows.
15
![Page 17: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/17.jpg)
Best Performance of NP also in the Short Term
Outperforming Expected Benchmarked Returns
(∗)⇒ nonnegative Sδ is supermartingale and Sδ∗t = 1
=⇒
• expected benchmarked returns
Et
(Sδt+h − Sδ
t
Sδt
)≤ Et
(Sδ∗t+h − Sδ∗
t
Sδ∗t
)= 0
t > 0, h > 0, Sδ nonnegative
=⇒ No strictly positive expected benchmarked returns possible!
16
![Page 18: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/18.jpg)
Outperforming Expected Growth
• expected growth
gδt,h = Et
(ln
(Sδt+h
Sδt
))for t, h ≥ 0.
17
![Page 19: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/19.jpg)
invest in strictly positive portfolio Sδ and small fraction ε ∈ (0, 112
) insome nonnegative portfolio Sδ ,which yields perturbed portfolio Sδε
• derivative of expected growth in the direction of Sδ:
∂gδε
t,h
∂ε
∣∣∣∣ε=0
= limε→0+
1
ε
(gδε
t,h − gδt,h
)
18
![Page 20: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/20.jpg)
Definition Sδ is called growth optimal if
∂gδε
t,h
∂ε
∣∣∣∣ε=0
≤ 0
for all t and h ≥ 0.
19
![Page 21: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/21.jpg)
• classical characterization: maximization of expected logarithmic utilityfrom terminal wealthKelly (1956), Latane (1959), Breiman (1960), Hakansson (1971a), Thorp(1972), Merton (1973a), Roll (1973) and Markowitz (1976)equivalent
20
![Page 22: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/22.jpg)
Theorem
The NP is growth optimalif its expected growth is finite.
NP outperforms expected growth!
21
![Page 23: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/23.jpg)
Proof: Pl. (2011)For t ≥ 0, h > 0, and a nonnegative portfolioSδ we introduce the portfolio
ratio Aδ,ht =
Sδt+h
Sδt
, which is set to 1 for Sδt = 0. For ε ∈ (0, 1
2); and
a nonnegative portfolio Sδ , with Sδt > 0, consider the perturbed portfolio
Sδε with Sδt = Sδ∗
t , yielding portfolio ratio Aδε
t,h = εAδt,h + (1 −
ε)Aδ∗t,h > 0. By the inequality ln(x) ≤ x− 1 for x ≥ 0,
Gδε
t,h =1
εln
(Aδε
t,h
Aδ∗t,h
)≤
1
ε
(Aδε
t,h
Aδ∗t,h
− 1
)=Aδ
t,h
Aδ∗t,h
− 1
and
Gδε
t,h = −1
εln
(Aδ∗
t,h
Aδε
t,h
)≥ −
1
ε
(Aδ∗
t,h
Aδε
t,h
− 1
)=Aδ
t,h −Aδ∗t,h
Aδε
t,h
.
22
![Page 24: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/24.jpg)
Because ofAδε
t,h > 0 one obtains forAδt,h −A
δ∗t,h ≥ 0
Gδε
t,h ≥ 0,
and forAδt,h −A
δ∗t,h < 0 because of ε ∈ (0, 1
2) andAδ
t,h ≥ 0
Gδε
t,h ≥ −Aδ∗
t,h
Aδε
t,h
= −1
1− ε+ εAδ
t,h
Aδ∗t,h
≥ −1
1− ε≥ −2.
23
![Page 25: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/25.jpg)
Summarizing yields
−2 ≤ Gδε
t,h ≤Aδ
t,h
Aδ∗t,h
− 1,
where by (∗)
Et
(Aδ
t,h
Aδ∗t,h
)≤ 1.
24
![Page 26: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/26.jpg)
By the Dominated Convergence Theorem
∂gδε
t,h
∂ε
∣∣∣∣ε=0
= limε→0+
Et
(Gδε
t,h
)= Et
(lim
ε→0+Gδε
t,h
)
= Et
(∂
∂εln
(Aδε
t,h
Aδ∗t,h
) ∣∣∣∣ε=0
)= Et
(Aδ
t,h
Aδ∗t,h
)− 1.
This proves that NP Sδ∗ is growth optimal.
25
![Page 27: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/27.jpg)
Systematic Outperformance
Definition A nonnegative portfolioSδ systematically outperforms a strictlypositive portfolio Sδ if
(i) Sδt0
= Sδt0
;
(ii) P (Sδt ≥ Sδ
t ) = 1 for some t > 0 and
(iii) P(Sδt > Sδ
t
)> 0.
• also called relative arbitrage in Fernholz & Karatzas (2005)
26
![Page 28: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/28.jpg)
Theorem The NP cannot be systematically outperformedby any nonnegative portfolio.
• NP can systematically outperform other portfolios.
27
![Page 29: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/29.jpg)
Proof: Consider nonnegative Sδ with Sδt = 1 where Sδ
s ≥ 1 almostsurely at s ∈ [t,∞). By the supermartingale property (∗)
0 ≥ Et
(Sδs − S
δt
)= Et
(Sδs − 1
).
Since Sδs ≥ 1 and Et(S
δs) ≤ 1, it can only follow Sδ
s = 1.=⇒ Sδ
s = Sδ∗s .
28
![Page 30: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/30.jpg)
• NP needs shortest expected time to reach a given wealth levelKardaras & Platen (2010)
29
![Page 31: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/31.jpg)
Strong Arbitrage
• only market participants can exploit arbitrage with their total wealth
• limited liability
=⇒ nonnegative total wealth of each market participant
Definition A nonnegative portfolio Sδ is a strong arbitrage if Sδ0 = 0
and for some T > 0
P(SδT > 0
)> 0.
• Loewenstein & Willard (2000) - same notion of arbitragethrough economic arguments
• Pl. (2002)-mathematical motivationthrough supermartingale property (∗)
30
![Page 32: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/32.jpg)
Theorem There is no strong arbitrage.
31
![Page 33: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/33.jpg)
Proof:(∗) =⇒
Sδ nonnegative supermartingale
=⇒0 = Sδ
0 ≥ E(SδT
∣∣A0
)= E(Sδ
T )
=⇒P (Sδ
T > 0) = P (SδT > 0) = 0.
32
![Page 34: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/34.jpg)
Classical notion of arbitrage is too restrictive
• Delbaen & Schachermayer (1998)
no free lunches with vanishing risk , existence of risk neutral measure, APT
• Loewenstein & Willard (2000)
free snacks & cheap thrills , some free lunch with vanishing risk
• exploiting weak forms of classical arbitrage requires to allow negativetotal wealth
• pricing via hedging by avoiding “strong arbitrage” makes no sensesince there is no “strong arbitrage” under (∗).
33
![Page 35: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/35.jpg)
How to approximate the NP in the real market?
• diversification leads asymptotically to the NP (Pl. 2005)
• confirm empirically by visualizing diversification effect
• simplest diversification:
equal value weighting = naive diversification
34
![Page 36: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/36.jpg)
Pl. & Rendek (2012)
• investment universe:
market capitalization weighted indices generating the MSCI
• Industry Classification Benchmark (Datastream, Reuters ...):
54 countries with country index
10 industry indices
19 supersector indices
41 sector indices
114 country subsector indices
all market capitalization weighted
35
![Page 37: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/37.jpg)
Equi-Weighted Index (EWId)
d constituents
SδEWIdtn
= SδEWIdtn−1
1 +d∑
j=1
ℓd,j∑k=1
πj,kδEWId,tn−1
Sj,ktn − S
j,ktn−1
Sj,ktn−1
,
• Naive diversification over ℓd,j countries
and then over d = 10, 19, 41, 114 world industries
• equal value weighting uncertainties of different economic activities
36
![Page 38: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/38.jpg)
1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 20120
50,000
100,000
150,000
MCI
EWI10EWI19
EWI41
EWI114
EWI1
The MCI and five equi-weighted indices: EWI1 (market), EWI10 (industry),EWI19 (supersector), EWI41 (sector), EWI114 (subsector).
37
![Page 39: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/39.jpg)
1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 201210
1
102
103
104
105
106
MCIEWI1
EWI10
EWI19
EWI114
EWI41
The MCI and five equi-weighted indices in log-scale: EWI1 (market), EWI10(industry), EWI19 (supersector), EWI41 (sector), EWI114 (subsector).
38
![Page 40: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/40.jpg)
• all based on the same country industry subsectors of the equity market
• all resulting indices mostly driven by nondiversifiable uncertainty
• Empirically: Better performance through better diversification!
39
![Page 41: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/41.jpg)
Illustrating Diversification Phenomenon:
• return of benchmarked constituent over small period [t, t+ h]
Rj =Sjt+h − S
jt
Sjt
j ∈ 1, 2, . . .
40
![Page 42: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/42.jpg)
Assume for illustration purposes: R1,R2, · · · independent, and
Et(Rj) = 0
Et
((Rj)2
)= σ2h
h > 0, j ∈ 1, 2, . . .
41
![Page 43: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/43.jpg)
• return of portfolio equalssum of weighted returns of constituents
• return of benchmarked EWIℓ with ℓ constituents:
RδEWIℓ =SδEWIℓ
t+h − SδEWIℓt
SδEWIℓt
=1
ℓ
ℓ∑j=1
Rj
42
![Page 44: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/44.jpg)
=⇒
Et(RδEWIℓ) = 0
Et
((RδEWIℓ)2
)=
1
ℓ2
ℓ∑j=1
σ2h =1
ℓσ2h
=⇒
limℓ→∞
Et
((RδEWIℓ)2
)= 0
Strong Law of Large Numbers (Kolmogorov)⇒
limℓ→0
RδEWIℓ = 0
almost surely
43
![Page 45: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/45.jpg)
for ℓ >> 1
=⇒ RδEWIℓ ≈ 0
=⇒ SδEWIℓt ≈ 1
=⇒ SδEWIℓt = SδEWIℓ
t Sδ∗t ≈ Sδ∗
t
a.s. for all t
44
![Page 46: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/46.jpg)
Diversification yields approximation of the NP !
• diversification holds more generally
• model independent phenomenon
• arises naturally
• can be systematically exploited
45
![Page 47: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/47.jpg)
Naive Diversification Theorem
Pl. & Rendek (2012)
• continuous market, Merton (1972)
• benchmarked constituents
dSjt
Sjt
=∑k
σj,kt dW k
t
W 1,W 2, . . . - Brownian motions
46
![Page 48: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/48.jpg)
• fraction of wealth invested
πjδ,t =
δjt Sjt
Sδt
=δjtS
jt
Sδt
47
![Page 49: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/49.jpg)
• benchmarked portfolio
dSδt
Sδt
=∑j
πjδ,t
dSjt
Sjt
=∑j
πjδ,t
∑k
σj,kt dW k
t
48
![Page 50: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/50.jpg)
• return process of a benchmarked portfolio Sδt
dQδt =
1
Sδt
dSδt
t ≥ 0 with Qδ0 = 0
49
![Page 51: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/51.jpg)
• Quadratic Variation of Return Process
⟨Qδℓ⟩t =∫ t
0
∑k
∑j
πjδℓ,sσj,ks
2
ds
• Tracking rate
Tδℓ(t) =d
dt⟨Qδℓ⟩t =
∑k
∑j
πjδℓ,tσj,kt
2
50
![Page 52: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/52.jpg)
• benchmarked NP equals constant onereturn process of benchmarked NP equals zero
dSδ∗t
Sδ∗t
=∑k
∑j
πjδ∗,tσj,kt dW k
t = 0
=⇒ tracking rate
Tδ∗(t) =d
dt⟨Qδ∗⟩t = 0
51
![Page 53: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/53.jpg)
• ℓth equi-weighted index (EWIℓ)
πjδEWIℓ,t
=
1ℓ
for j ∈ 1, 2, . . . , ℓ0 otherwise.
52
![Page 54: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/54.jpg)
Definition:
A sequence (Sδℓ)ℓ∈1,2,... of strictly positive benchmarked portfolios, withinitial value equal to one, is called a sequence of benchmarked approxi-mate NPs if for each ε > 0 and t ≥ 0 one has
limℓ→∞
P
(Tδℓ(t) =
d
dt
⟨Qδℓ
⟩t> ε
)= 0.
53
![Page 55: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/55.jpg)
• benchmark captures general, systematic or non-diversifiable market un-certainty
• benchmarked primary security accounts capture specific or idiosyncraticuncertainty
• different types of economic activity yield naturally different specific un-certainties
• specific uncertainty can be diversified
54
![Page 56: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/56.jpg)
Particular specific uncertainty drives in reality usually only
returns of restricted number of benchmarked primary security accounts:
55
![Page 57: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/57.jpg)
Definition:
A market is well-securitized if there exists q > 0 and square integrableσ2 = σ2
t , t ≥ 0 with finite mean for all ℓ, k ∈ 1, 2, . . . and t ≥ 0
1
ℓ
∣∣∣∣∣∣ℓ∑
j=1
σj,kt
∣∣∣∣∣∣2
≤1
ℓqσ2
t
P-almost surely.
• k-th uncertainty affects not too many constituents.
56
![Page 58: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/58.jpg)
Naive Diversification Theorem:
In a well-securitized market the sequence of benchmarked equi-weightedindices is a sequence of benchmarked approximate NPs.
Pl. & Rendek (2012)
57
![Page 59: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/59.jpg)
• return process of ℓth benchmarked EWI
QδEWIℓt =
ℓ∑j=1
1
ℓ
∑k
∫ t
0
σj,ks dW k
s
58
![Page 60: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/60.jpg)
• quadratic variation of return process
⟨QδEWIℓ
⟩t=
1
ℓ
∫ t
0
∑k
∣∣∣ 1√ℓ
ℓ∑j=1
σj,ks
∣∣∣2ds• tracking rate
TδEWIℓ(t) =d
dt
⟨QδEWIℓ
⟩t=
1
ℓ
∑k
∣∣∣ 1√ℓ
ℓ∑j=1
σj,kt
∣∣∣2
59
![Page 61: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/61.jpg)
Lemma Assume that for all ε > 0 and t ≥ 0
limℓ→∞
P
1
ℓ
ℓ∑k
∣∣∣ 1√ℓ
ℓ∑j=1
σj,kt
∣∣∣2 > ε
= 0,
then the sequence of equi-weighted indices is a sequence of benchmarkedapproximate NPs.
60
![Page 62: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/62.jpg)
Proof:For ε > 0 and t > 0
0 = limℓ→∞
P
1
ℓ
∑k
∣∣∣ 1√ℓ
ℓ∑j=1
σj,kt
∣∣∣2 > ϵ
= lim
ℓ→∞P
(d
dt
⟨QδEWIℓ
⟩t> ε
)= lim
ℓ→∞P (TδEWIℓ(t) > ε)
61
![Page 63: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/63.jpg)
Proof of Naive Diversification Theorem:
By Markov inequality in a well-securitized market
limℓ→∞
P
1
ℓ
∑k
∣∣∣ 1√ℓ
ℓ∑j=1
σj,kt
∣∣∣2 > ε
= lim
ℓ→∞P
(1
ℓqσ2
t > ε
)≤ lim
ℓ→∞
1
ε
1
ℓqE(σ2
t
)= 0,
NDT follows from previous Lemma.
62
![Page 64: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/64.jpg)
1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 201210
1
102
103
104
105
106
EWI114
EWI1145
EWI11480
EWI11440
EWI114200
MCIEWI114240
Logarithms of MCI, EWI114 without transaction cost and EWI114ξ withtransaction costs of 5,40,80,200 and 240 basis points.
63
![Page 65: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/65.jpg)
1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 20120
50,000
100,000
150,000
EWI114
EWI1142
EWI1144
EWI11416
EWI11432
EWI114m reallocated daily and every 2, 4, 8, 16 and 32 days.
64
![Page 66: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/66.jpg)
Transaction cost 0 5 40 80 200 240
Reallocation terms 1
Final value 139338.64 130111.93 80543.07 46555.04 8988.23 5194.46
Annualised average return 0.1979 0.1961 0.1834 0.1689 0.1254 0.1109
Annualised volatility 0.1135 0.1135 0.1135 0.1135 0.1134 0.1134
Sharpe ratio 1.4205 1.4046 1.2930 1.1654 0.7822 0.6544
Reallocation terms 2
Final value 124542.04 119369.00 88697.63 63166.73 22808.64 16240.40
Annualised average return 0.1949 0.1938 0.1859 0.1770 0.1500 0.1411
Annualised volatility 0.1134 0.1134 0.1134 0.1134 0.1135 0.1136
Sharpe ratio 1.3955 1.3856 1.3163 1.2369 0.9987 0.9193
Reallocation terms 4
Final value 111899.82 108230.16 85698.25 65628.82 29467.48 22562.42
Annualised average return 0.1921 0.1912 0.1850 0.1780 0.1568 0.1497
Annualised volatility 0.1135 0.1135 0.1134 0.1134 0.1134 0.1135
Sharpe ratio 1.3699 1.3622 1.3080 1.2459 1.0591 0.9967
65
![Page 67: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/67.jpg)
Transaction cost 0 5 40 80 200 240
Reallocation terms 8
Final value 100505.37 97963.66 81881.57 66705.62 36055.10 29367.02
Annualised average return 0.1892 0.1885 0.1837 0.1783 0.1621 0.1566
Annualised volatility 0.1127 0.1127 0.1127 0.1127 0.1128 0.1128
Sharpe ratio 1.3531 1.3471 1.3051 1.2569 1.1119 1.0634
Reallocation terms 16
Final value 98775.24 96892.29 84677.43 72588.14 45711.97 39177.91
Annualised average return 0.1887 0.1882 0.1847 0.1806 0.1684 0.1643
Annualised volatility 0.1130 0.1130 0.1130 0.1130 0.1131 0.1131
Sharpe ratio 1.3463 1.3418 1.3102 1.2740 1.1647 1.1281
Reallocation terms 32
Final value 114592.50 112929.09 101939.59 90678.85 63804.84 56744.28
Annualised average return 0.1927 0.1923 0.1896 0.1865 0.1772 0.1741
Annualised volatility 0.1131 0.1131 0.1131 0.1131 0.1133 0.1133
Sharpe ratio 1.3797 1.3763 1.3522 1.3245 1.2408 1.2127
66
![Page 68: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/68.jpg)
0 50 100 1500
100
200
300
400
500
600
GOP
EWI
MCI
Simulated GOP, EWI and MCI under the Black-Scholes model
67
![Page 69: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/69.jpg)
0 50 100 1500.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
benchmarked MCI
benchmarked EWI
Simulated benchmarked GOP, EWI and MCI under the Black-Scholesmodel
68
![Page 70: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/70.jpg)
Example of a Multi-Asset BS Model
• jth primary security account
dSjt = Sj
t
[(r + σ2
(1 +
1√d
))dt+
σ√d
d∑k=1
dW kt + σ dW j
t
]
t ∈ [0, T ], j ∈ 1, 2, . . . , d, σ > 0
general market risk, specific market risk
69
![Page 71: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/71.jpg)
• NP fractions
πjδ∗,t
=(√
d(1 +√d))−1
j ∈ 1, 2, . . . , d
π0δ∗,t
=(1 +√d)−1
• NP SDE
dSδ∗t = Sδ∗
t
((r + σ2
)dt+
σ√d
d∑k=1
dW kt
)
• simulate over T = 32 years d = 50 risky primary security accounts
σ = 0.15, r = 0.05
70
![Page 72: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/72.jpg)
• Benchmarked primary security accountBS - model
dSjt = d
(Sjt
Sδ∗t
)= Sj
tσdWjt
martingale
71
![Page 73: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/73.jpg)
0
5
10
15
20
25
0 5 10 15 20 25 30 35
time
Simulated primary security accounts.
72
![Page 74: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/74.jpg)
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30 35
time
GOPB
Simulated GOP and savings account.
73
![Page 75: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/75.jpg)
Equal Weighted Index
SδEWId holds equal fractions
πjδEWId,t
=1
d+ 1
j ∈ 0, 1, . . . , d
• tracking rate for BS example:
T dδEWId
(t) =d∑
k=1
(σ
d+ 1
(1√d− 1
))2
= d
(σ
d+ 1
(1√d− 1
))2
≤σ2
(d+ 1)→ 0
74
![Page 76: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/76.jpg)
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30 35
time
GOPEWI
Simulated NP and EWI.
75
![Page 77: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/77.jpg)
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30 35
time
GOPindex
Simulated diversified accumulation index and NP.
76
![Page 78: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/78.jpg)
Diversification in an MMM Setting
Pl. (2001), minimal market model (MMM)
• savings accountS0(d)(t) = expr t
• discounted GOP drift
αδ∗t = α0 expη t
• jth benchmarked primary security account
Sj(d)(t) =
1
Y jt α
δ∗t
77
![Page 79: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/79.jpg)
• square root process
dY jt =
(1− η Y j
t
)dt+
√Y jt dW
jt
• NP
Sδ∗(d)(t) =
S0(d)(t)
S0(d)(t)
• jth primary security account
Sj(d)(t) = Sj
(d)(t)Sδ∗(d)(t)
78
![Page 80: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/80.jpg)
0
50
100
150
200
250
300
350
400
450
0 5 9 14 18 23 27 32
Primary security accounts under the MMM
79
![Page 81: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/81.jpg)
0
10
20
30
40
50
60
0 5 9 14 18 23 27 32
Benchmarked primary security accounts
80
![Page 82: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/82.jpg)
0
10
20
30
40
50
60
70
80
90
100
0 5 9 14 18 23 27 32
EWI
GOP
NP and EWI
81
![Page 83: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/83.jpg)
0
10
20
30
40
50
60
70
80
90
100
0 5 9 14 18 23 27 32
Market index
GOP
NP and market index
82
![Page 84: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/84.jpg)
Diversified Portfolios
Fundamental phenomenon of diversification leads naturally to NPalso for more general diversified portfolios than the EWI:
Definition A sequence of strictly positive portfolios (Sδd)d∈1,2,... isa sequence of diversified portfolios if∣∣∣πj
δd,t
∣∣∣ ≤
K2
d12+K1
for j ∈ 0, 1, . . . , d
0 otherwise
a.s. for t ∈ [0, T ].
• still small fractions but more general than EWIs
83
![Page 85: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/85.jpg)
• jth benchmarked primary security account
Sjt =
Sjt
Sδ∗t
dSjt = −Sj
t
∑k
sj,kt dW kt
• sj,kt (j, k)th specific volatilitymeasures specific market risk of Sj
with respect toW k diversifiable uncertainty
• volatility of NP measures general market risk, nondiversifiable uncertainty
84
![Page 86: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/86.jpg)
• kth total specific volatility
skt =∑j
|sj,kt |
Definition A market is called regular if
E((skt)2) ≤ K5
for all t ∈ [0,∞), k ∈ 1, 2, . . . .
85
![Page 87: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/87.jpg)
Tracking Rate
Tδ(t) =d
dt⟨Qδ⟩t =
∑k
d∑j=0
πjδ,t s
j,kt
2
Tδ(t) - quantifies “distance” between Sδ and Sδ∗
Tδ∗(t) = 0
86
![Page 88: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/88.jpg)
Approximate NP
Definition A strictly positive portfolio Sδd is an approximate NP if forall t ∈ [0, T ] and ε > 0
limd→∞
P (Tδd(t) > ε) = 0.
87
![Page 89: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/89.jpg)
Diversification Theorem (Pl. 2005)
In a regular market
any diversified portfolio is an approximate NP.
• model independent
88
![Page 90: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/90.jpg)
Proof:
E (Tδd(t)) ≤d∑
k=1
E
d∑
j=0
|πjδ,t| |s
j,kt |
2
≤(K2)
2
d(1+2K1)
d∑k=1
E((skt)2) ≤ (K2)
2K5
d(1+2K1)d→ 0
Markov inequality =⇒
limd→∞
P (Tδd(t) > ε) ≤ limd→∞
1
εE (Tδd(t)) = 0.
89
![Page 91: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/91.jpg)
Summary on Diversification
• extremely robust approach to asset management
• does not rely on estimating some expected returns
• diversification can be refined
• diversified market indices are in reality similar in their dynamics⇒ proxies for the NPe.g. MSCI, S&P 500, EWI114, EWI142 ...
90
![Page 92: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/92.jpg)
Continuous Financial Market
• forces us to be more specific
(Ω,A,A, P ) - filtered probability spaceA = (At)t≥0 - filtrationAt - information at time tEt(·) = E(·|At) - conditional expectation
• d sources of continuous traded uncertainty
W 1,W 2, . . . ,W d, d ∈ 1, 2, . . .
independent standard Brownian motions
91
![Page 93: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/93.jpg)
Primary Security Accounts
Sjt - jth primary security account at time t,
j ∈ 0, 1, . . . , d
cum-dividend share value or savings account value,
dividends or interest are reinvested
• vector process of primary security accounts
S = St = (S0t , . . . , S
dt )
⊤, t ∈ [0, T ]
92
![Page 94: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/94.jpg)
• assume unique strong solution of SDE
dSjt = Sj
t
(ajt dt+
d∑k=1
bj,kt dW kt
)
t ∈ [0,∞), Sj0 > 0, j ∈ 1, 2, . . . , d
• predictable, integrable appreciation rate processes aj
• predictable, square integrable volatility processes bj,k
93
![Page 95: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/95.jpg)
• savings account
S0t = exp
∫ t
0
rs ds
predictable, integrable short rate process
r = rt, t ∈ [0,∞)
94
![Page 96: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/96.jpg)
• Market price of risk
θt = (θ1t , θ2t , . . . , θ
dt )
⊤
unique invariant of the market
solution of relation
bt θt = at − rt 1
whereat = (a1
t , a2t , . . . , a
dt )
⊤
1 = (1, 1, . . . , 1)⊤
95
![Page 97: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/97.jpg)
Assumption:
Volatility matrix bt = [bj,kt ]dj,k=1 is invertible
=⇒
• market price of risk
θt = b−1t (at − rt 1)
96
![Page 98: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/98.jpg)
• can rewrite jth primary security account SDE
dSjt = Sj
t
rt dt+
d∑k=1
bj,kt (θkt dt+ dW kt )
t ∈ [0,∞), j ∈ 1, 2, . . . , d
97
![Page 99: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/99.jpg)
• strategy
δ = δt = (δ0t , . . . , δdt )
⊤, t ∈ [0, T ]
predictable and S-integrable,
δjt number of units of jth primary security account
• portfolio
Sδt =
d∑j=0
δjt Sjt
98
![Page 100: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/100.jpg)
• Sδ self-financing⇐⇒
dSδt =
d∑j=0
δjt dSjt
• changes in the portfolio value only due to gains from trading,
• self-financing in each denomination
99
![Page 101: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/101.jpg)
• jth fraction
πjδ,t = δjt
Sjt
Sδt
j ∈ 0, 1, . . . , d
d∑j=0
πjδ,t = 1
100
![Page 102: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/102.jpg)
• portfolio SDE
dSδt = Sδ
t
(rt dt+
d∑k=1
bkδ,t(θkt dt+ dW k
t
))
with kth portfolio volatility
bkδ,t =d∑
j=0
πjδ,t b
j,kt
101
![Page 103: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/103.jpg)
• logarithm of strictly positive portfolio
d ln(Sδt ) = gδt dt+
d∑k=1
bkδ,t dWkt
with growth rate
gδt = rt +
d∑k=1
bkδ,t
(θkt −
1
2bkδ,t
)
102
![Page 104: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/104.jpg)
=⇒ maximize the growth rate
0 =∂gδt∂bkδ,t
= θkt − bkδ∗,t
=⇒ for each k ∈ 1, 2, . . . , d
θkt = bkδ∗,t=
d∑ℓ=1
πℓδ∗,t
bℓ,kt
=⇒θ⊤t = π⊤
δ∗,tbt
103
![Page 105: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/105.jpg)
Since bt invertible =⇒
πδ∗,t = (π1δ∗,t, . . . , πd
δ∗,t)⊤
= (b−1t )⊤ θt
=⇒ Kelly portfolio, growth optimal portfolio, GOP,log-optimal portfolio, NP
dSδ∗t = Sδ∗
t
([rt +
d∑k=1
(θkt )2
]dt+
d∑k=1
θkt dWkt
)
finite
104
![Page 106: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/106.jpg)
=⇒
• GOP
dSδ∗t = Sδ∗
t
(rt dt+
d∑k=1
θkt(θkt dt+ dW k
t
))
• general portfolio
dSδt = Sδ
t
rt dt+ d∑k=1
d∑j=0
πjδ,t b
j,kt
(θkt dt+ dW k
t
)
105
![Page 107: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/107.jpg)
Benchmarked Portfolio
Sδt =
Sδt
Sδ∗t
Ito formula =⇒
dSδt = −Sδ
t
d∑k=1
d∑j=0
πjδ,t
(θkt − b
j,kt
)dW k
t
driftless =⇒ local martingale,“zero expected return locally”Local Efficient Market Hypothesissimilar to Fama (1965)
106
![Page 108: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/108.jpg)
• nonnegative local martingale is supermartingale
=⇒ Kelly portfolio (GOP) is the NP
107
![Page 109: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/109.jpg)
• NP benchmark for investing
• numeraire for pricing
• captures general market risk in risk management
• central in benchmark approach
108
![Page 110: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/110.jpg)
Benchmark Approach
• provides much wider framework for modeling than classical APT
• can capture more phenomena relevant in real world
• contains as special cases classical approachese.g. APT, actuarial approach, CAPM, ...
109
![Page 111: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/111.jpg)
Capital Asset Pricing Model
• risk premium of security V (t)
pV (t) = limh↓0
1
hEt
(V (t+ h)− V (t)
V (t)
)− rt
excess expected return
• NP
dSδ∗t = Sδ∗
t
((rt + pSδ∗ (t)) dt+
d∑k=1
θkt dWkt
)with volatility
|θt| =
√√√√ d∑k=1
(θkt )2 =
√pSδ∗ (t)
110
![Page 112: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/112.jpg)
• underlying security
dSt = St
(rt dt+
d∑k=1
σkt
(θkt dt+ dW k
t
))
risk premium
pS(t) =d∑
k=1
σkt θ
kt
=d
dt
[ln(S), ln(Sδ∗)
]t
= limh↓0
1
hEt
((St+h − St
St
)(Sδ∗t+h − S
δ∗t
Sδ∗t
))
111
![Page 113: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/113.jpg)
• portfolio
dSδt = Sδ
t
(rt dt+ π1
δ,t
d∑k=1
σkt
(θkt dt+ dW k
t
))
risk premium
pSδ(t) = π1δ,t
d∑k=1
σkt θ
kt
= limh↓0
1
hEt
((Sδt+h − Sδ
t
Sδt
)(Sδ∗t+h − S
δ∗t
Sδ∗t
))
=d
dt
[ln(Sδ), ln(Sδ∗)
]t
112
![Page 114: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/114.jpg)
Assumption: Market portfolio SδMP is diversified and approximates NP
SδMPt ≈ Sδ∗
t .
113
![Page 115: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/115.jpg)
• systematic risk parameter beta
βSδ(t) =ddt
[ln(Sδ), ln(SδMP)
]t
ddt
[ln(SδMP)
]t
=⇒
βSδ(t) ≈ddt
[ln(Sδ), ln(Sδ∗)
]t
ddt
[ln(Sδ∗)
]t
=pSδ(t)
pSδ∗ (t)≈
pSδ(t)
pSδMP (t)
114
![Page 116: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/116.jpg)
=⇒
• fundamental CAPM relation follows directly
pSδ(t) ≈ βSδ(t) pSδMP (t)
115
![Page 117: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/117.jpg)
• CAPM relationship still holds for SδMP with SDE
dSδMPt = SδMP
t
(rt dt+ πt
d∑k=1
θkt(θkt dt+ dW k
t
))
combination of savings account and NP, two fund separation
• βSδ(t) =ddt
[ln(Sδ), ln(SδMP)
]t
ddt
[ln(SδMP)]t
=π1δ,t
∑dk=1 σ
kt θ
kt πt
(πt)2∑d
k=1
(θkt)2
=π1δ,t
∑dk=1 σ
kt θ
kt
πt
∑dk=1
(θkt)2 =
pSδ(t)
pSδMP (t)
116
![Page 118: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/118.jpg)
no utility function involved,no principal agent,no equilibrium assumption
follows simply by Ito calculus and BA
=⇒
• capital asset pricing model (CAPM)
Sharpe (1964), Lintner (1965),
Mossin (1966) and Merton (1973b)
117
![Page 119: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/119.jpg)
Examples for Beta:
• savings accountβS0(t) = 0
• underlying security
βS(t) =
∑dk=1 σ
kt θ
kt
|θt|2
• NPβSδ∗ (t) = 1
118
![Page 120: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/120.jpg)
• portfolio
βSδ(t) =π1δ,t
∑dk=1 σ
kt θ
kt
|θt|2
• If Sδ involves only diversifiable uncertainty, which is not in NP,
then it has zero beta
=⇒
βSδ(t) = 0
119
![Page 121: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/121.jpg)
Portfolio Optimization
• Kelly portfolio (GOP, NP) best long term investment
Kelly (1956), Latane (1959), Breiman (1961),Hakansson (1971b), Thorp (1972)
• optimal portfolio separated into two funds
NP and savings accountTobin (1958), Sharpe (1964)
120
![Page 122: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/122.jpg)
• mean-variance efficient portfolio
Markowitz (1959)
• intertemporal capital asset pricing model
Merton (1973a)
121
![Page 123: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/123.jpg)
Discounted Portfolio
• strictly positive portfolio Sδ ∈ V+
• discounted value
Sδt =
Sδt
S0t
=⇒
• SDE
dSδt =
d∑k=1
ψkδ,t
(θkt dt+ dW k
t
)with kth diffusion coefficient
ψkδ,t =
d∑j=1
δjt Sjt b
j,kt
122
![Page 124: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/124.jpg)
• discounted drift
αδt =
d∑k=1
ψkδ,t θ
kt
link to macro economy
• fluctuations
Mt =
d∑k=1
∫ t
0
ψkδ,s dW
ks
=⇒
Sδt = Sδ
0 +
∫ t
0
αδs ds+ Mt
123
![Page 125: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/125.jpg)
• aggregate diffusion coefficient (deviation)
γδt =
√√√√ d∑k=1
(ψk
δ,t
)2
=⇒ aggregate volatility
bδt =γδt
Sδt
124
![Page 126: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/126.jpg)
Locally Optimal Portfolios
Definition
Sδ ∈ V+ locally optimal, if for all t ∈ [0,∞) and Sδ ∈ V+ with
γδt = γ δ
t
almost surely:
αδt ≤ α
δt .
generalization of mean-variance optimality
Markowitz (1952, 1959)
125
![Page 127: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/127.jpg)
Sharpe Ratio
Sharpe (1964, 1966)
• risk premium
pSδ(t) =αδ
t
Sδt
• aggregate volatility bδt
• Sharpe ratio
sδt =pSδ(t)
bδt=αδ
t
γδt
=mean
deviation
126
![Page 128: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/128.jpg)
• total market price of risk
|θt| =
√√√√ d∑k=1
(θkt)2
Assumption:0 < |θt| <∞
and
π0δ∗,t= 1
almost surely.
127
![Page 129: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/129.jpg)
• Portfolio Selection TheoremFor any Sδ ∈ V+ it follows Sharpe ratio
sδt ≤ |θt|,
where equality when Sδ locally optimal=⇒
dSδt = Sδ
t
bδt|θt|
d∑k=1
θkt(θkt dt+ dW k
t
),
=⇒ fractions
πjδ,t =
bδt|θt|
πjδ∗,t
Markowitz (1959), Sharpe (1964), Merton (1973a)Khanna & Kulldorff (1999), Platen (2002)Sharpe ratio maximization
128
![Page 130: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/130.jpg)
Proof of Portfolio Selection Theorem
Lagrange multiplier λ
consider
L(ψ1δ , . . . , ψ
dδ , λ) =
d∑k=1
ψkδ θ
k + λ
((γ δ)2−
d∑k=1
(ψk
δ
)2)
suppressing time dependence
first-order conditions
∂L(ψ1δ , . . . , ψ
dδ , λ)
∂ψkδ
= θk − 2λψkδ = 0
for all k ∈ 1, 2, . . . , d as well as
129
![Page 131: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/131.jpg)
∂L(ψ1δ , . . . , ψ
dδ , λ)
∂λ=(γ δ)2−
d∑k=1
(ψk
δ
)2= 0
=⇒ locally optimal portfolio S(δ) must satisfy
ψkδ=θk
2λ
for all k ∈ 1, 2, . . . , d.
Thend∑
k=1
(ψk
δ
)2=(γ δ)2
130
![Page 132: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/132.jpg)
=⇒ (γ δ)2
=d∑
k=1
(ψk
δ
)2=
∑dk=1(θ
k)2
4λ2
we have |θ| =√∑d
k=1(θk)2 > 0,
then
ψkδ=γ δ
|θ|θk
for all k ∈ 1, 2, . . . , d
131
![Page 133: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/133.jpg)
=⇒
αδt = γ δ
t
|θt|2
|θt|= γ δ
t |θt|
dSδt = γ δ
t
d∑k=1
θkt|θt|
(θkt dt+ dW kt )
ψkδ,t
=d∑
j=1
δjt Sjt b
j,kt = S
(δ)t
d∑j=1
πj
δ,tbj,kt =
γ δt
|θt|θkt = S
(δ)t bδt
θkt|θt|
invertibility of volatility matrix=⇒
πj
δ,t=
bδt|θt|
d∑k=1
θkt b−1 j,kt
132
![Page 134: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/134.jpg)
Two Fund Separation
• locally optimal portfolio
fraction of wealth in the NP
bδt|θt|
=1− π0
δ,t
1− π0δ∗,t
remainder in savings account
π0δ,t = 1−
bδt|θt|
(1− π0
δ∗,t
)
also known as fractional Kelly strategy:
Kelly (1956), Latane (1959), Thorp (1972), Hakansson & Ziemba (1995)
133
![Page 135: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/135.jpg)
Risk Aversion Coefficient
Jδt =
1− π0δ∗,t
1− π0δ,t
=|θt|bδt
Pratt (1964), Arrow (1965)
• discounted locally optimal portfolio
dSδt = Sδ
t
1
Jδt
|θt| (|θt| dt+ dWt),
where
dWt =
d∑k=1
θkt|θt|
dW kt
1Jδt
fraction in the NP
134
![Page 136: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/136.jpg)
Capital Market Line
• expected rate of return
aδt = rt + pSδ(t)
• for a locally optimal portfolio
aδt = rt + |θt| bδt
• portfolio process at the capital market line
has fraction 1Jδt=
bδt
|θt|in NP
fractional Kelly strategy
135
![Page 137: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/137.jpg)
Markowitz Efficient Frontier
• aggregate volatility
bδt =1− π0
δ,t
1− π0δ∗,t
|θt|
DefinitionMarkowitz efficient portfolio has expected rate of return on efficient frontierif
aδt = rt +
√(bδt )
2 |θt|.
136
![Page 138: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/138.jpg)
Theorem Any locally optimal portfolio Sδ is Markowitz efficient port-folio.
pSδ(t) =1− π0
δ,t
1− π0δ∗,t
|θt|2 = bδt |θt|
Sharpe ratio maximizationrisk attitude expressed via Jδ
t = |θt|bδt
short term view
137
![Page 139: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/139.jpg)
-0.04 0 0.04 0.08 0.12
0.05
0.09
-0.04 0 0.04 0.08 0.12
0
0.05
0.09
Efficient frontier
138
![Page 140: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/140.jpg)
Efficient Growth Rates
• efficient growth rate frontier
of locally optimal portfolio
gδt = rt +
√|bδt |2 |θt| −
1
2|bδt |
2 = rt +|θt|2
Jδt
(1−
1
2 Jδt
)
• maximum
gδ∗t = rt +
1
2|θt|2
• long term view
139
![Page 141: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/141.jpg)
0 0.04 0.08 0.12 0.16
0.05
0 0.04 0.08 0.12 0.16
0
0.05
Efficient growth rates
140
![Page 142: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/142.jpg)
Market Portfolio
SδMPt =
n∑ℓ=1
Sδℓt
portfolio of tradable wealth of ℓth investor Sδℓ
Remark:
If each investor forms a nonnegative, locally optimal portfolio with her or his
total tradable wealth, then MP is locally optimal .
141
![Page 143: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/143.jpg)
dSδMPt =
n∑ℓ=1
dSδℓt
=n∑
ℓ=1
(Sδℓt − δ0ℓ
)(1− π0
δ∗,t
) d∑k=1
θkt (θkt dt+ dW kt )
= SδMPt
(1− π0
δMP,t
)(1− π0
δ∗,t
) d∑k=1
θkt (θkt dt+ dW kt )
• CAPM relationship still holds when MP locally optimal
142
![Page 144: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/144.jpg)
Practical Feasibility ofSample Based Markowitz Mean-Variance Approach:
• NP theoretically known for given model
• Markowitz (1952)
• Best & Grauer (1991)
• DeMiguel, Garlappi & Uppal (2009)
practical applicationrequires for 50 assets observation window of about 500 years
• not realistic to be applied
• theoretically can be reconciled under strong assumptions(MP locally optimal)
143
![Page 145: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/145.jpg)
Maximum Drawdown Constrained Portfolios
X - set of nonnegative continuous discounted portfolios
• running maximumforX = Xt, t ≥ 0 ∈ X
X∗t = sup
u∈[0,t]
Xu
• drawdownX∗
t −Xt
144
![Page 146: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/146.jpg)
• relative drawdownXt
X∗t
• maximum relative drawdown
• express attitude towards risk by restricting toX ∈ αX , where
Xt
X∗t
≥ α ,α ∈ [0, 1)
pathwise criterion
145
![Page 147: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/147.jpg)
• drawdownforX ∈ αXXt ≥ αX∗
t
=⇒X∗
t −Xt ≤ (1− α)X∗t
146
![Page 148: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/148.jpg)
• maximum drawdown constrained portfolioα ∈ [0, 1),X ∈ X
αXt = α(X∗t )
1−α + (1− α)Xt(X∗t )
−α
= 1 +
∫ t
0
(1− α)(X∗s )
−αdXs ∈ [αX∗t , (X
∗t )
1−α]
Grossman & Zhou (1993), Cvitanic & Karatzas (1994),Kardaras, Obloj & Pl. (2012)
147
![Page 149: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/149.jpg)
=⇒ SDEdαXt
αXt
= απt
dXt
Xt
fraction
απt = 1−(1− α)Xt
X∗t
α+ (1− α)Xt
X∗t
model independentdepends only on Xt
X∗t
and ααSδ∗
t - locally optimal portfolio
148
![Page 150: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/150.jpg)
1871 1899 1927 1955 1983 2011−1
0
1
2
3
4
5
6
ln(X)
ln(X*)
logarithm of discounted S&P500 and its running maximum
149
![Page 151: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/151.jpg)
1871 1899 1927 1955 1983 20110
0.2
0.4
0.6
0.8
1
1.2
relative drawdown
Relative drawdown of discounted S&P500
150
![Page 152: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/152.jpg)
1871 1899 1927 1955 1983 2011−0.5
0
0.5
1
1.5
2
2.5
ln(αX)
ln(αX*)
Logarithm of maximum drawdown constrained portfolio, α = 0.6
151
![Page 153: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/153.jpg)
1871 1899 1927 1955 1983 20110
0.2
0.4
0.6
0.8
1
1.2
relative drawdown
Relative drawdown for drawdown constrained portfolio, α = 0.6
152
![Page 154: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/154.jpg)
1871 1899 1927 1955 1983 20110
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
fraction in X
Fraction of wealth in S&P500 for α = 0.6
153
![Page 155: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/155.jpg)
1871 1899 1927 1955 1983 20110
0.002
0.004
0.006
0.008
0.01
0.012
0.014
relative hedge error
Relative hedge error, α = 0.6, monthly hedging
154
![Page 156: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/156.jpg)
• Long term growth rateFor α ∈ [0, 1),X ∈ αX
limt→∞
(log(Xt)
Gt
)≤ 1− α = lim
t→∞
(log(αSδ∗
t )
Gt
)
log(Sδ∗t ) = Gt +
∫ t
0
|θs|dWs
Kardaras, Obloj & Pl. (2012)
155
![Page 157: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/157.jpg)
• asymptotic maximum long term growth rate
limt→∞
1
tlog(αSδ∗
t ) = (1− α) limt→∞
1
tlog(Sδ∗
t )
αg = (1− α)g
restricted drawdown =⇒ reduced maximum growth ratelong term view with short term attitude towards riskrealistic alternative to Markowitz mean-variance approachand utility maximization
156
![Page 158: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/158.jpg)
1871 1900 1928 1956 1984 2012−1
0
1
2
3
4
5
6
Logarithm of drawdown constrained portfolios
157
![Page 159: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/159.jpg)
1879 1906 1932 1959 1985 2012−0.01
0
0.01
0.02
0.03
0.04
0.05
Long term growth of drawdown constrained portfolios
158
![Page 160: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/160.jpg)
Expected Utility Maximization
• utility function U(·), U ′(·) > 0 U ′′(·) < 0
• examples
power utility
U(x) =1
γxγ
for γ = 0 and γ < 1
log-utilityU(x) = ln(x)
159
![Page 161: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/161.jpg)
5 10 15 20x
-15
-10
-5
5
U
Examples for power utility (upper graph) and log-utility (lower graph)
160
![Page 162: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/162.jpg)
• assume S0 is scalar Markov process
• maximize expected utility
vδ = maxSδ∈V+
S0
E(U(SδT
) ∣∣∣A0
)
V+S0
set of strictly positive, discounted, fair portfolios
Sδ0 = S0 > 0
161
![Page 163: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/163.jpg)
Theorem
Benchmarked, expected utility maximizing portfolio:
Sδt = u(t, S0
t ) = E(U ′−1
(λ S0
T
)S0T
∣∣∣At
),
λ s.t. S0 = Sδ0 S
δ∗0 = u(0, S0)S
δ∗0
two fund separation with risk aversion coefficient
J δt =
1
1− S0t
u(t,S0t )
∂u(t,S0t )
∂S0t
.
=⇒ locally optimal portfolio
162
![Page 164: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/164.jpg)
Proof:
vδ = E(U(Sδ
T ))− λ
(E(Sδ
T )− Sδ0
)= E
(F (Sδ
T ))
F (SδT ) = U(Sδ
T )− λ(SδT
Sδ∗T
− Sδ0
)
F ′(SδT ) = U ′(Sδ
T )−λ
Sδ∗T
= 0
163
![Page 165: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/165.jpg)
=⇒
U ′(SδT ) =
λ
Sδ∗T
SδT = U
′−1
(λ
Sδ∗T
)= U
′−1(λS0T )
SδT = U
′−1(λS0T )S
0T
164
![Page 166: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/166.jpg)
• log-utility function U(x) = ln(x)
U ′(x) =1
x
U ′−1(y) =1
y
U ′′(x) = −1
x2
utility concave
u(t, S0t ) = E
(U ′−1
(λ S0
T
)S0T
∣∣∣At
)= E
(1
λ S0T
S0T
∣∣∣At
)=
1
λ
Lagrange multiplier
165
![Page 167: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/167.jpg)
λ =Sδ∗0
S0
risk aversion coefficientJ δt = 1
• expected log-utility
vδ = E(ln(Sδ∗T
) ∣∣∣A0
)= ln(λ) + ln(S0) +
1
2
∫ T
0
E((θ(s, Sδ∗
s ))2 ∣∣∣A0
)ds
if local martingale part forms a martingale
166
![Page 168: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/168.jpg)
• power utility
U(x) =1
γxγ
for γ < 1, γ = 0
U ′(x) = xγ−1
U ′−1(y) = y1
γ−1
U ′′(x) = (γ − 1)xγ−2
concave
u(t, S0t ) = E
( λ
Sδ∗T
) 1γ−1
1
Sδ∗T
∣∣∣∣At
= λ1
γ−1 E
((Sδ∗T
) γ1−γ
∣∣∣At
)
167
![Page 169: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/169.jpg)
Sδ∗ geometric Brownian motion
u(t, S0t ) = λ
1γ−1
(Sδ∗t
) γ1−γ
×E(exp
γ
1− γ
(θ2
2(T − t) + θ (WT −Wt)
) ∣∣∣∣At
)
= λ1
γ−1(Sδ∗t
) γ1−γ exp
θ2
2
γ
(1− γ)2(T − t)
168
![Page 170: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/170.jpg)
Lagrange multiplier
λ = Sγ−10 Sδ∗
0 exp
θ2
2
γ
1− γT
S0t = (Sδ∗
t )−1
∂u(t, S0t )
∂S0=u(t, S0
t )
S0t
γ
γ − 1
risk aversion coefficientJ δt = 1− γ
expected utility
vδ = E
(1
γ
(SδT
)γ ∣∣∣∣A0
)= exp
−θ2
2
γ
1− γT
(S0)
γ
169
![Page 171: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/171.jpg)
Various Approaches to Asset Pricing
• Actuarial Pricing Approach (APA)
• Capital Asset Pricing Model (CAPM)
• Arbitrage Pricing Theory (APT)
• Utility Indifference Pricing
• Benchmark Approach (BA)
170
![Page 172: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/172.jpg)
Classical
Law of One Price
“All replicating portfolios of a payoff have the same price!”
Debreu (1959), Sharpe (1964), Lintner (1965),
Merton (1973a, 1973b), Ross (1976), Harrison & Kreps (1979),
Cochrane (2001), . . .
will be, in general, violated under the benchmark approach.
171
![Page 173: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/173.jpg)
-8
-7
-6
-5
-4
-3
-2
-1
0
1
1930 1940 1950 1960 1970 1980 1990 2000
time
ln(savings bond)ln(fair zero coupon bond)
ln(savings account)
Logarithms of savings bond, fair zero coupon bond and savings account
172
![Page 174: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/174.jpg)
• benchmarked value
Sδt =
Sδt
Sδ∗t
Supermartingale Property
For nonnegative Sδ
Sδt ≥ Et
(Sδs
)0 ≤ t ≤ s <∞
Sδ supermartingale
173
![Page 175: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/175.jpg)
• One needs consistent pricing concept
• All benchmarked nonnegative securities are supermartingales
(no upward trend)
174
![Page 176: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/176.jpg)
Definition Price is fair if, when benchmarked, forms martingale
(no trend)
Sδt = Et
(Sδs
)0 ≤ t ≤ s <∞.
175
![Page 177: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/177.jpg)
Lemma The minimal nonnegative supermartingale that reaches a givenbenchmarked contingent claim is a martingale.
see Revuz & Yor (1999)
176
![Page 178: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/178.jpg)
0
0.02
0.04
0.06
0.08
0.1
0.12
1930 1940 1950 1960 1970 1980 1990 2000
time
benchmarked savings bondbenchmarked fair zero coupon bond
Benchmarked savings bond and benchmarked fair zero coupon bond
177
![Page 179: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/179.jpg)
Benchmark Approach:
Law of the Minimal Price
Pl. (2008)
Theorem If a fair portfolio replicates a nonnegative payoff, then this rep-resents the minimal replicating portfolio.
• least expensive
• minimal possible hedge
• economically correct price in a competitive market
178
![Page 180: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/180.jpg)
• contingent claim
HT
E0
(HT
Sδ∗T
)<∞
• SδHt fair if
SδHt =
SδHt
Sδ∗t
= Et
(HT
Sδ∗T
)
• real world expectationbest performing portfolio as numeraire
=⇒ Direct link with real world and economy !
179
![Page 181: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/181.jpg)
Law of the Minimal Price =⇒
Corollary
Minimal price for replicableHT is fair and given by
real world pricing formula
SδHt = Sδ∗
t Et
(HT
Sδ∗T
).
• Du & Pl. (2013) benchmarked risk minimizationfor nonreplicable claims
180
![Page 182: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/182.jpg)
• most pricing concepts become unified and generalized by real worldpricing
=⇒
actuarial pricingrisk neutral pricingpricing with stochastic discount factorpricing with numeraire changepricing with deflatorpricing with state pricing densitypricing with pricing kernelutility indifference pricingpricing with numeraire portfolioclassical risk minimizationbenchmarked risk minimization
181
![Page 183: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/183.jpg)
• Equivalent to risk neutral pricing are pricing via:
stochastic discount factor, Cochrane (2001);
deflator, Duffie (2001);
pricing kernel, Constantinides (1992);
state price density, Ingersoll (1987);
numeraire portfolio as in Long (1990)
182
![Page 184: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/184.jpg)
Actuarial Pricing
WhenHT independent of Sδ∗T
=⇒ rigorous derivation of
• actuarial pricing formula
SδHt = P (t, T )Et(HT )
with zero coupon bond as discount factor
P (t, T ) = Sδ∗t Et
(1
Sδ∗T
)
183
![Page 185: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/185.jpg)
Risk Neutral Pricing
• real world pricing formula =⇒
SδH0 = E0
(ΛT
S00
S0T
HT
)Radon-Nikodym derivative (density)
Λt =S0t
S00
supermartingale
1 = Λ0 ≥ E0(ΛT )
=⇒
SδH0 ≤
E0
(ΛT
S00
S0THT
)E0(ΛT )
184
![Page 186: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/186.jpg)
similar for any numeraire (here S0 savings account)
185
![Page 187: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/187.jpg)
0
10
20
30
40
50
60
70
80
90
100
1930 1940 1950 1960 1970 1980 1990 2000
time
Discounted S&P500 total return index
186
![Page 188: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/188.jpg)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1930 1940 1950 1960 1970 1980 1990 2000
time
Radon-Nikodym derivativeTotal RN-Mass
Radon-Nikodym derivative and total mass of putative risk neutral measure
187
![Page 189: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/189.jpg)
• special case when savings account is fair:
=⇒ ΛT = dQdP
forms martingale; E0(ΛT ) = 1;
equivalent risk neutral probability measureQ exists;
Bayes’ formula =⇒risk neutral pricing formula
SδH0 = EQ
0
(S00
S0T
HT
)
Harrison & Kreps (1979), Ingersoll (1987),
Constantinides (1992), Duffie (2001), Cochrane (2001), . . .
• otherwise “risk neutral price” ≥ real world price
188
![Page 190: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/190.jpg)
-1
0
1
2
3
4
5
6
7
8
9
1930 1940 1950 1960 1970 1980 1990 2000
time
ln(index)ln(savings account)
ln(S&P500 accumulation index) and ln(savings account)
189
![Page 191: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/191.jpg)
Risk Neutral Pricing under BS Model
θt = θ, at = a, rt = r, σt = σ
Drifted Wiener Process
• drifted Wiener process Wθ
Wθ(t) = Wt + θ t
• market price of risk
θ =a− rσ
190
![Page 192: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/192.jpg)
• underlying security
dSt = (a− σ θ)St dt+ σ St (θ dt+ dWt)
= (a− σ θ)St dt+ σ St dWθ(t)
= r St dt+ σ St dWθ(t)
St = S0 exp
(r −
1
2σ2
)t+ σWθ(t)
Wθ is not a Wiener process under the real world probability P ,
however, will be under some Pθ
191
![Page 193: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/193.jpg)
0
5
10
15
20
25
0 5 10 15 20 25 30 35
time
Simulated primary security accounts.
192
![Page 194: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/194.jpg)
• Radon-Nikodym derivative
Λθ(T ) =dPθ
dP=S0T
S00
• risk neutral measure Pθ
Pθ(A) =
∫Ω
1A dPθ(ω) =
∫A
dPθ(ω)
dP (ω)dP (ω)
=
∫A
Λθ(T ) dP (ω)
for all subsetsA ∈ Ω
193
![Page 195: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/195.jpg)
Definition P is equivalent to P if both have the same sets
of events that have probability zero.
• equivalence of P and Pθ
fundamental for risk neutral pricing
Key Risk Neutral assumption:
Λθ is (A, P ) martingale
194
![Page 196: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/196.jpg)
• under BS model
Radon-Nikodym derivative
Λθ(t) =S0t
S00
is (A, P )-martingale,
=⇒ there exists equivalent risk neutral probability measure Pθ
for BS model
195
![Page 197: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/197.jpg)
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
0 5 10 15 20 25 30 35
time
Benchmarked savings account under BS model, martingale.
196
![Page 198: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/198.jpg)
Risk Neutral Measure Transformation
dPθ
dP
∣∣∣∣At
= Λθ(t) = exp
−
1
2θ2 t− θWt
• total risk neutral measure at t = 0:
dPθ
dP
∣∣∣∣A0
= E(Λθ(T ) | A0) = Λθ(0) = 1
=⇒ Pθ is a probability measure
197
![Page 199: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/199.jpg)
Let us show thatWθ is Wiener process under Pθ:
For fixed y ∈ ℜ,
t ∈ [0, T ] and s ∈ [0, t]
letA be the event
A = ω ∈ Ω : Wθ(t, ω)−Wθ(s, ω) < y
=⇒
A = ω ∈ Ω : Wt(ω)−Ws(ω) < y − θ (t− s)
198
![Page 200: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/200.jpg)
Eθ denoting expectation under Pθ =⇒
Pθ(A) = Eθ(1A|A0) = E
(dPθ
dP
∣∣∣∣AT
1A
∣∣∣∣A0
)
= E(Λθ(T ) 1A|A0) = E
(Λθ(t) 1A
Λθ(T )
Λθ(t)
∣∣∣∣A0
)
= E
(Λθ(t) 1AE
(Λθ(T )
Λθ(t)
∣∣∣∣At
) ∣∣∣∣A0
)
= E (Λθ(t) 1A|A0) = E
(Λθ(s)
Λθ(t)
Λθ(s)1A
∣∣∣∣A0
)
= E (Λθ(s)|A0) E
(Λθ(t)
Λθ(s)1A
∣∣∣∣A0
)
= E
(Λθ(t)
Λθ(s)1A
∣∣∣∣A0
)199
![Page 201: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/201.jpg)
since y = Wt −Ws is Gaussian ∼ N (0, t− s) under P =⇒
Pθ(A) = E
(Λθ(t)
Λθ(s)1A
)
=
∫ y−θ(t−s)
−∞exp
−θ2
2(t− s)− θ y
1√
2π (t− s)
exp
−
y2
2 (t− s)
dy
=
∫ y−θ(t−s)
−∞
1√2π (t− s)
exp
−
(y + θ (t− s))2
2 (t− s)
dy
=
∫ y
−∞
1√2π (t− s)
exp
−
z2
2 (t− s)
dz
for z = y + θ(t− s)
200
![Page 202: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/202.jpg)
=⇒
Theorem (Girsanov) Under the above BS model the process
Wθ = Wθ(t) = Wt + θ t, t ≥ 0
is a standard Wiener process in the filtered probability space (Ω,AT ,A, Pθ).
• Wθ is (A, Pθ)-Wiener process
• simply a transformation of variable
(not always possible, exploits here BS model)
201
![Page 203: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/203.jpg)
Real World Pricing =⇒ Risk Neutral Pricing
V (0, S0) = Sδ∗0 E
(H(ST )
Sδ∗T
∣∣∣∣A0
)
= E
(S0T
S00
H(ST )
S0T
∣∣∣∣A0
)
= E
(Λθ(T )
(H(ST )
S0T
) ∣∣∣∣A0
)
≤E(Λθ(T )
H(ST )S0
T
∣∣A0
)E(Λθ(T )
∣∣A0
)
202
![Page 204: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/204.jpg)
=⇒ Risk Neutral Pricing if Λθ is (A, P )-martingale
V (0, S0) = E
(exp
−θ2
2T − θWT
exp−r TH(ST )
∣∣∣A0
)= exp−r T
∫ ∞
−∞H
(S0 exp
(a−
1
2σ2
)T + σ y
)× exp
−θ2
2T − θ y
1√TN ′(y√T
)dy
= exp−r T∫ ∞
−∞
(H
(S0 exp
(r −
1
2σ2
)T + σ z
))×
1√TN ′(z√T
)dz
= exp−r TEθ
(H(ST )
∣∣A0
)= Eθ
(H(ST )
S0T
∣∣∣A0
)z = y + θ T =⇒ Eθ - expectation with respect to Pθ
203
![Page 205: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/205.jpg)
Risk Neutral SDEs under BS model
• discounted underlying security price
dSt = σ St dWθ(t)
driftless under Pθ, σS ∈ L2T =⇒
(A, Pθ)-martingale
204
![Page 206: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/206.jpg)
• discounted option price
dV (t, St) =∂V (t, St)
∂Sσ St dWθ(t)
driftless under Pθ, ∂V∂SσS ∈ L2
T =⇒
(A, Pθ)-martingale
• discounted savings account S0t = 1 =⇒ (A, Pθ)-martingale
205
![Page 207: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/207.jpg)
• discounted portfolio
dSδt = d(Sδ∗
t Sδt )
= Sδt π
1δ(t)σ (θ dt+ dWt)
= Sδt π
1δ(t)σ dWθ(t)
(A, Pθ)-local martingale
• if Sδ π1δ δ = δ1 S σ ∈ L2
T =⇒ Sδ is an (A, Pθ)-martingale,
=⇒ risk neutral pricing formula holds=⇒
Sδt = Eθ
(SδT
∣∣At
)= Eθ
(SδT
S0T
∣∣∣∣At
)• Sδ∗ is (A, Pθ)-martingale under BS model
206
![Page 208: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/208.jpg)
• strict supermartingale portfolio under BS model
if Zt strict (A, Pθ)-supermartingale, independent of Sδ∗t and
Sδt = Sδ∗
t Zt
=⇒ Sδt strict (A, Pθ)-supermartingale
• risk neutral pricing formula does not hold for such Sδt
Sδt > Eθ
(SδT
∣∣At
)for T > t
=⇒ not all discounted portfolios are (A, Pθ)-martingales
even when Radon-Nikodym derivative Λθ is (A, P )-martingale
207
![Page 209: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/209.jpg)
Change of Variables
• we performed a change of variables fromWt toWθ(t)
W Wiener processes under P
Wθ Wiener processes under Pθ
• relies on existence of an
equivalent risk neutral probability measure Pθ
208
![Page 210: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/210.jpg)
General Girsanov Transformation
• m-dimensional standard Wiener process
W = W t = (W 1t , . . . ,W
mt )⊤, t ∈ [0, T ]
(Ω,AT ,A, P )
• A-adapted, predictablem-dimensional stochastic process
θ = θt = (θ1t , . . . , θmt )⊤, t ∈ [0, T ]
∫ T
0
m∑i=1
(θit)2 dt <∞
209
![Page 211: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/211.jpg)
• assume Radon-Nikodym derivative
Λθ(t) = 1−m∑i=1
∫ t
0
Λθ(s) θis dW
is
= exp
−∫ t
0
θ⊤s dW s −1
2
∫ t
0
θ⊤s θs ds
<∞
forms (A, P )-martingale
210
![Page 212: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/212.jpg)
• Λθ (A, P )-martingale
=⇒E(Λθ(t) | As) = Λθ(s)
for t ∈ [0, T ] and s ∈ [0, t]
=⇒E(Λθ(t) | A0) = Λθ(0) = 1
211
![Page 213: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/213.jpg)
• define a candidate risk neutral measure Pθ with
dPθ
dP
∣∣∣∣AT
= Λθ(T )
by setting
Pθ(A) = E(Λθ(T ) 1A)
= Eθ(1A)
• since Λθ is an (A, P )-martingale
Pθ(Ω) = E(Λθ(T )) = E(Λθ(T )
∣∣A0
)= Λθ(0) = 1
=⇒ Pθ is a probability measure
212
![Page 214: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/214.jpg)
Girsanov Theorem
Theorem (Girsanov) If a given strictly positive Radon-Nikodymderivative process Λθ is an (A, P )-martingale, then the m-dimensionalprocess Wθ = Wθ(t), t ∈ [0, T ], given by
Wθ(t) = W t +
∫ t
0
θs ds
for all t ∈ [0, T ], is an m-dimensional standard Wiener process on thefiltered probability space (Ω,AT ,A, Pθ).
• W is (A, P )-Wiener process
Wθ is (A, Pθ)-Wiener process
213
![Page 215: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/215.jpg)
Novikov Condition
When is the strictly positive (A, P )-local martingale Λθ an (A, P )-martingale?
• a sufficient condition is the Novikov condition
Novikov (1972)
E
(exp
1
2
∫ T
0
|θ⊤s θs| ds)
<∞
(is satisfied for the BS model)
• for other sufficient conditions see Revuz & Yor (1999)
214
![Page 216: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/216.jpg)
Bayes’s Theorem
Theorem (Bayes) Assume that Λθ is an (A, P )-martingale, thenfor any given stopping time τ ∈ [0, T ] and any Aτ -measurable randomvariable Y , satisfying the integrability condition
Eθ(|Y |) <∞,
one has the Bayes rule
Eθ(Y | As) =E(Λθ(τ )Y | As)
E(Λθ(τ ) | As)
for s ∈ [0, τ ].
215
![Page 217: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/217.jpg)
Proof of Bayes’s Theorem (*)
For eachAτ -measurable random variable Y and a setA ∈ As
with s ∈ [0, τ ], τ ≤ T
1AEθ (Y |As) = Eθ
(1A Y
∣∣As
)= E
(1A Y Λθ(T )
∣∣As
)= E (1A Y Λθ(τ )|As) = E (1AE (Y Λθ(τ )|As) |As)
= E
(Λθ(s)
1A
Λθ(s)E (Y Λθ(τ ) | As)
∣∣∣∣As
)
= Eθ
(1A
Λθ(s)E (Y Λθ(τ ) | As)
∣∣∣∣As
)
= Eθ
(1A
E (Λθ(τ )Y | As)
E (Λθ(τ ) | As)
∣∣∣∣As
)
= 1A
E (Λθ(τ )Y | As)
E (Λθ(τ ) | As).
216
![Page 218: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/218.jpg)
Change of Numeraire Technique
Geman, El Karoui & Rochet (1995)
Jamshidian (1997)
• numeraireSδ = Sδ
t , t ∈ [0, T ]
normalizes all other portfolios
• may lead to simplifications in calculations
• relative price of a portfolioSδt
Sδt
217
![Page 219: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/219.jpg)
Self-financing under Numeraire Change
Sδt = δ0t S
0t + δ1t St
dSδt = δ0t dS
0t + δ1t dSt
dSδt = δ0t dS
0t + δ1t dSt
Ito formula =⇒
218
![Page 220: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/220.jpg)
d
(Sδt
Sδt
)=
1
Sδt
dSδt + Sδ
t d
(1
Sδt
)+ d
[1
Sδ, Sδ
]t
= δ0t
(1
Sδt
dS0t + S0
t d
(1
Sδt
))
+ δ1t
(1
Sδt
dSt + St d
(1
Sδt
)+ d
[1
Sδ, S
]t
)
[X,Y ]t ≈it∑
i=0
(Xti+1
−Xti
) (Yti+1
− Yti
)covariation
219
![Page 221: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/221.jpg)
Ito formula for
S0t
Sδt
andSt
Sδt
=⇒
d
(Sδt
Sδt
)= δ0t d
(S0t
Sδt
)+ δ1t d
(St
Sδt
)
=⇒
• in denomination of another numeraire always self-financing
220
![Page 222: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/222.jpg)
Numeraire Pairs
• numeraire pair (Sδ, Pθδ)
real world pricing formula:
St = E
(Sδ∗t
Sδ∗T
ST
∣∣∣∣At
)
existence of GOP
=⇒ numeraire pair (Sδ∗ , P )
221
![Page 223: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/223.jpg)
if Λθ is (A, P )-martingale =⇒
• risk neutral pricing formula:
St = Eθ
(S0t
S0T
ST
∣∣∣∣At
)
=⇒ numeraire pair (S0, Pθ)
222
![Page 224: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/224.jpg)
• for fair price process S
from real world pricing formula
St
Sδ∗t
= E
(ST
Sδ∗T
∣∣∣∣At
)
=⇒ for any strictly positive numeraire Sδ
St
Sδt
= E
(Sδ∗t
Sδt
SδT
Sδ∗T
ST
SδT
∣∣∣∣At
)
= E
(Λθδ
(T )
Λθδ(t)
ST
SδT
∣∣∣∣At
)
223
![Page 225: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/225.jpg)
with benchmarked normalized numeraire
=⇒
Λθδ(t) =
Sδt
Sδ0
dΛθδ(t) = d
(Sδt
Sδ0
)
= Λθδ(t)(π1δ(t)σt − θt
)dWt
(A, P )-local martingale
=⇒ Radon-Nikodym derivative for candidate measure Pθδ
dPθδ
dP= Λθδ
(T )
224
![Page 226: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/226.jpg)
• drifted Wiener process
dWθδ(t) = dWt + θδ(t) dt,
whereθδ(t) = θt − π1
δ(t)σt
• If Λθδis true (A, P )-martingale
=⇒ Pθδ equivalent to P and probability measure
allows to apply Girsanov transformation
=⇒ Wθδstandard Wiener process under Pθδ
=⇒ numeraire pair (Sδ, Pθδ)
225
![Page 227: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/227.jpg)
Some Examples
• GOP Sδ∗ as numeraire =⇒ θδ∗(t) = 0
• savings account S0 as numeraire
π1δ(t) = 0 =⇒ θδ(t) = θt
• underlying security S as numeraire
π1δ(t) = 1 =⇒ θδ(t) = θt − σt
226
![Page 228: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/228.jpg)
• unfair portfolio Sδt = Sδ∗
t Zt as numeraire
with Z an (A, P )-strict supermartingale, independent of Sδ∗
=⇒ Sδ is not an (A, P )-martingale
Girsanov Theorem can not be applied !
=⇒ If savings account is not fair,
then the change of numeraire approach is not applicable !
227
![Page 229: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/229.jpg)
Pricing Formula for General Numeraire Sδ
V (0, S0) = E
(Sδ∗0
Sδ∗T
H(ST )
∣∣∣∣A0
)
= E
(Λθδ
(T )H(ST )
SδT
∣∣∣∣A0
)
the quantityΛθδ
(T )
SδT
=Sδ∗0
Sδ∗T
remains numeraire invariant
228
![Page 230: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/230.jpg)
• if Λθδforms an (A, P )-martingale
Bayes’s Theorem =⇒
E
(Λθδ
(T )H(ST )
SδT
∣∣∣∣A0
)= Eθδ
(H(ST )
SδT
∣∣∣∣A0
)=⇒
pricing formula with general numeraire
V (0, S0) = Eθδ
(H(ST )
SδT
∣∣∣∣A0
)
• represents a transformation of variables in integration
• may provide computational advantages
229
![Page 231: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/231.jpg)
Utility Indifference Pricing
discounted payoff H = HS0
T(not perfectly hedgable)
vδε,V = E
(U
((S0 − ε V )
SδT
S0
+ ε H
) ∣∣∣∣A0
)
• assume S0 is scalar diffusion
Definition Utility indifference price V s.t.
limε→0
vδε,V − vδ0,Vε
= 0
e.g. Davis (1997)
230
![Page 232: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/232.jpg)
• Taylor expansion
vδε,V ≈ E
(U(SδT
)+ εU ′
(SδT
) (H −
V SδT
S0
) ∣∣∣∣A0
)
= vδ0,V + εE(U ′(SδT
)H∣∣∣A0
)− V εE
(U ′(SδT
) SδT
S0
∣∣∣∣A0
)
231
![Page 233: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/233.jpg)
=⇒
• utility indifference price
V =E(U ′(SδT
)H∣∣∣A0
)E
(U ′(SδT
)Sδ
T
S0
∣∣∣A0
) =E(U ′(U ′−1
(λ
Sδ∗T
))H∣∣∣A0
)E
(U ′(U ′−1
(λ
Sδ∗T
))Sδ
T
S0
∣∣∣A0
)
=E(
H
Sδ∗T
∣∣∣A0
)1S0
S0
Sδ∗0
independent of U and independent of particular model =⇒
• real world pricing formula
V = Sδ∗0 E
(H
Sδ∗T
∣∣∣A0
)
232
![Page 234: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/234.jpg)
Pricing via Hedging
• underlying security
S = St, t ∈ [0, T ]
dSt = at St dt+ σt St dWt
t ∈ [0, T ], S0 > 0
233
![Page 235: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/235.jpg)
appropriate stochastic appreciation rate at
and volatility σt
Wiener processW = Wt, t ∈ [0, T ]
• savings account
dS0t = rt S
0t dt
t ∈ [0, T ], S00 = 1
appropriate stochastic short rate
234
![Page 236: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/236.jpg)
• replicate payofff(ST ) ≥ 0
assumingE(f(ST )) <∞
• hedge portfolio
V (t, St) = δ0t S0t + δ1t St
δ0t units of savings account
δ1t units of underlying security
235
![Page 237: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/237.jpg)
• hedging strategy
δ = δt = (δ0t , δ1t )
⊤, t ∈ [0, T ]
δ0 = δ0t , t ∈ [0, T ]
δ1 = δ1t , t ∈ [0, T ]
predictable processes
236
![Page 238: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/238.jpg)
• replicating portfolio
V (T, ST ) = f(ST )
• SDE for hedge portfolio
dV (t, St) = δ0t dS0t + δ1t dSt
+S0t dδ
0t + St dδ
1t + d
[δ1, S
]t
237
![Page 239: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/239.jpg)
• self-financing
portfolio changes are caused by gains from trade
dV (t, St) = δ0t dS0t + δ1t dSt
=⇒
self-financing condition:
S0t dδ
0t + St dδ
1t + d
[δ1, S
]t= 0
238
![Page 240: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/240.jpg)
• discounted underlying security
St =St
S0t
t ∈ [0, T ]
Ito formula =⇒
dSt = (a− r) St dt+ σt St dWt
t ∈ [0, T ], S0 = S0
• discounted value function
V : [0, T ]× [0,∞)→ [0,∞)
V (t, St) =V (t, St)
S0t
239
![Page 241: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/241.jpg)
• profit and loss (P&L)
Ct = price – gains from trade – initial price
• discounted P&L
Ct =Ct
S0t
= V (t, St)− Iδ1,S(t)− V (0, S0)
240
![Page 242: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/242.jpg)
• discounted gains from trade
Iδ1,S(t) =
∫ t
0
δ1u dSu
=
∫ t
0
δ1u (a− r) Su du
+
∫ t
0
δ1u σ Su dWu
Iδ0,S0(t) =
∫ t
0
δ0u dS0u = 0
241
![Page 243: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/243.jpg)
• initial payment
V (0, S0) = V (0, S0),
=⇒
C0 = 0
242
![Page 244: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/244.jpg)
• identify hedging strategy δ
for which discounted P&L zero
Ct = 0
for all t ∈ [0, T ]
=⇒ perfect hedge
V (t, St) - replicating portfolio
243
![Page 245: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/245.jpg)
• increments of discounted P&L
0 = Ct − Cs = V (t, St)− V (s, Ss)−∫ t
s
δ1u dSu
=
∫ t
s
[∂V (u, Su)
∂t+
1
2σ2u S
2u
∂2V (u, Su)
∂S2
+(au − ru) Su
(∂V (u, Su)
∂S− δ1u
)]du
+
∫ t
s
σu Su
(∂V (u, Su)
∂S− δ1u
)dWu
=⇒
244
![Page 246: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/246.jpg)
• hedge ratio
δ1u =∂V (u, Su)
∂S
• discounted PDE
∂V (u, S)
∂t+
1
2σ2u S
2 ∂2V (u, S)
∂S2= 0
u ∈ [0, T ), S ∈ (0,∞)
with terminal condition
V (T, S) =f(S S0
T )
S0T
=f(S)
S0T
245
![Page 247: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/247.jpg)
=⇒
• PDE
∂V (u, S)
∂t+ ru S
∂V (u, S)
∂S
+1
2σ2u S
2 ∂2V (u, S)
∂S2− ru V (u, S) = 0
u ∈ [0, T ), S ∈ (0,∞)
with terminal condition
V (T, S) = f(S)
• special boundary condition not specifiedPDE may not provide minimal possible price
246
![Page 248: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/248.jpg)
=⇒
• units in savings account
δ0t =V (t, St)− δ1t St
S0t
= V (t, St)− δ1t St
• option price
V (t, St) = V (t, St)S0t
= δ0t S0t + δ1t St
t ∈ [0, T ]
247
![Page 249: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/249.jpg)
=⇒ strategyδt = (δ0t , δ
1t )
⊤
such that0 = dCt = dV (t, St)− δ1t dSt
=⇒dV (t, St) = δ0t dS
0t + δ1t dSt
=⇒ self-financing
P&L vanishesCt = Ct S
0t = 0
248
![Page 250: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/250.jpg)
for all t ∈ [0, T ]
If V (t, St) is not fair, then there may exist less expensive hedge portfo-lio!
249
![Page 251: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/251.jpg)
Numeraire Invariance
V self-financing
dV (t, St) = d(V (t, St)S0t )
= S0t dV (t, St) + V (t, St) dS
0t + d[S0, V (·, S·)]t
= S0t δ
1t dSt + (δ0t + δ1t St) dS
0t + δ1t d[S
0, S]t
= δ0t dS0t + δ1t
(S0t dSt + St dS
0t + d[S0, S]t
)= δ0t dS
0t + δ1t d(S
0t St)
= δ0t dS0t + δ1t dSt
=⇒ V also self-financing
250
![Page 252: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/252.jpg)
The Black-Scholes Formula
• solution cT,K(t, S) of PDE
with call payoff function
f(S) = (S −K)+
• at = a, σt = σ, rt = r
Black & Scholes (1973)Merton (1973a)
251
![Page 253: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/253.jpg)
Black-Scholes Formula
cT,K(t, St) = StN(d1(t))−KS0t
S0T
N(d2(t))
with
d1(t) =ln(St
K
)+(r + 1
2σ2)(T − t)
σ√T − t
d2(t) = d1(t)− σ√T − t
t ∈ [0, T )
cT,K(t, St) =cT,K(t, St)
Sδ∗t
(P,A)− martingale
minimal possible price
252
![Page 254: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/254.jpg)
• N(·) standard Gaussian distribution function
with density
N ′(x) =1√2π
exp
−x2
2
253
![Page 255: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/255.jpg)
0
2
4
6
8
t
0.5
1
1.5
2
S
0
0.5
1
0
2
4
6
8
t
0
0.5
1
Black-Scholes European call option price.
254
![Page 256: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/256.jpg)
Delta
• sensitivity with respect to the underlying St
∆ =∂V (t, St)
∂S= δ1t
∆ = N(d1(t))
255
![Page 257: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/257.jpg)
0
2
4
6
8
t
0.5
1
1.5
2
S
0
0.25
0.5
0.75
1
0
2
4
6
8
t
0
0.25
0.5
0.75
Delta as a function of t and St.
256
![Page 258: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/258.jpg)
Gamma
• sensitivity of the hedge ratio with respect to underlying
Γ =∂∆
∂S=∂2V (t, St)
∂S2
Γ = N ′(d1(t))1
St σ√T − t
257
![Page 259: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/259.jpg)
0
2
4
6
8
t
0.5
1
1.5
2
S
0
1
2
3
4
0
2
4
6
8
t
0
1
2
3
Gamma as a function of t and St.
258
![Page 260: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/260.jpg)
Simulation of Delta Hedging
• simulate scenario along an equidistant time discretization
• price of call option
cT,K(t, St) = δ1t St + δ0t S0t
• hedging strategy
δ =δt = (δ0t , δ
1t )
⊤, t ∈ [0, T ]
259
![Page 261: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/261.jpg)
• hedge ratio, units in the underlying
δ1t = N(d1(t))
= N
(ln(St
K
)+(r + 1
2σ2)(T − t)
σ√T − t
)
260
![Page 262: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/262.jpg)
• units in domestic savings account
δ0t = −K
S0T
N(d2(t))
= −K
S0T
N
(ln(St
K
)+(r − 1
2σ2)(T − t)
σ√T − t
)
261
![Page 263: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/263.jpg)
• discounted P&L
Ct = V (t, St)−∫ t
0
δ1s dSs − V (0, S0)
V (t, St) = V (0, S0) +
∫ t
0
δ1s dSs
=⇒ theoretically under continuous hedging
Ct = 0
for all t ∈ [0, T ]
262
![Page 264: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/264.jpg)
In-the-Money Scenario
• security price ends up in-the-money ST > K
=⇒ hedge ratio converges to δ1T = 1
• self-financing strategy δt = (δ0t , δ1t )
⊤
• intrinsic valuef(St) = (St −K)+
• discounted P&L Ct remains almost perfectly zero
263
![Page 265: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/265.jpg)
0.5
1
1.5
2
2.5
0 2 4 6 8 10
time
S(t)Delta
Underlying security and hedge ratio for in-the-money call.
264
![Page 266: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/266.jpg)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 1 2 3 4 5 6 7 8 9 10
time
call priceintrinsic value
P&L
Price, intrinsic value and P&L for in-the-money call.
265
![Page 267: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/267.jpg)
Out-of-the Money Scenario
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10
time
S(t)Delta
Underlying security and hedge ratio for out-of-the-money call. 266
![Page 268: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/268.jpg)
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 1 2 3 4 5 6 7 8 9 10
time
call priceintrinsic value
P&L
Price, intrinsic value and P&L for out-of-the-money call.
267
![Page 269: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/269.jpg)
Hedging a European Call Option on the S&P500
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time
normalized S&P500Delta
Normalized S&P500 and hedge ratio forK = 1.2. 268
![Page 270: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/270.jpg)
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time
Quadratic variation of log-S&P500.
269
![Page 271: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/271.jpg)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time
call priceintrinsic value
P&L
Call price on S&P500, intrinsic value and P&L.
270
![Page 272: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/272.jpg)
Benchmarked Risk Minimization
Risk Minimization
• Follmer and Sondermann (1986)
• Follmer and Schweizer (1989)
• Schweizer (1991, 2001)
• Biagini et al. (1999)
• Biagini and Cretarola (2009)
• Du & Pl. (2012)
271
![Page 273: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/273.jpg)
Financial Market
• d primary security accounts
• Si,jt - jth primary security account in ith security denomination
272
![Page 274: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/274.jpg)
• locally riskless savings account
Si,it = Si,i
0 exp
∫ t
0
risds
• portfolio
Si,δt =
d∑j=1
δjtSi,jt
273
![Page 275: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/275.jpg)
• self-financing portfolio
Si,δt = Si,δ
0 +d∑
j=1
∫ t
0
δjsdSi,js
274
![Page 276: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/276.jpg)
Numeraire Portfolio
• Si,δ∗t numeraire portfolio
Sδt =
Si,δt
Si,δ∗t
≥ Et(Sδs)
0 ≤ t ≤ s <∞
• supermartingale property
275
![Page 277: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/277.jpg)
• benchmarked jth primary security account
Sjt =
Si,jt
Si,δ∗t
local martingale; supermartingale
276
![Page 278: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/278.jpg)
• matrix valued optional covariation process
[S] = [S]t = ([Si, Sj]t)di,j=1, t ∈ [0,∞)
• benchmarked primary security accounts
S = St = (S1t , . . . , S
dt )
⊤, t ∈ [0,∞)
277
![Page 279: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/279.jpg)
Definition A dynamic trading strategy v, initiated at time t = 0, v =
vt = (ηt, ϑ1t , . . . , ϑ
dt )
⊤, t ∈ [0,∞), where ϑ = ϑt = (ϑ1t , . . . ,
ϑdt )
⊤, t ∈ [0,∞) forms benchmarked self-financing partϑ⊤t St of bench-
marked price process
V vt = ϑ⊤
t St + ηt
ϑ is predictable ∫ t
0
ϑ⊤u d[S]uϑu <∞
278
![Page 280: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/280.jpg)
η = ηt, t ∈ [0,∞) adapted with η0 = 0 monitors the benchmarkednon-self-financing part of
V vt = V v
0 +
∫ t
0
ϑ⊤s dSs + ηt
V vt forms a supermartingale.
279
![Page 281: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/281.jpg)
In general, capital has to be added or removed to match desired price
V vt = Sδ
t =d∑
j=1
δjt Sjt
with
δjt = ϑjt + ηtδ
j∗,t .
280
![Page 282: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/282.jpg)
• benchmarked contingent claim HT
dynamic trading strategy v delivers HT if
V vT = Sδ
T = HT
=⇒ replicable if self-financing
281
![Page 283: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/283.jpg)
Proposition If for HT a self-financing benchmarked portfolio SδHT exists,satisfying
SδHTt = E(HT |Ft)
for all t ∈ [0, T ] P -a.s., then this provides least expensive hedge for HT .
282
![Page 284: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/284.jpg)
Real World Pricing
• least expensive replication by a self-financing benchmarked portfolio=⇒real world pricing formula
SδHTt = Et(HT )
minimal possible price process
283
![Page 285: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/285.jpg)
Definition For a dynamic trading strategy v which delivers
Sδt =
d∑j=1
δjt Sjt
benchmarked P&L
Cδt = Sδ
t −d∑
j=1
∫ t
0
ϑjudS
ju − S
δ0
284
![Page 286: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/286.jpg)
Corollary Cδt = ηt for t ∈ [0,∞).
• usually fluctuating benchmarked P&L
• intrinsic risk
285
![Page 287: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/287.jpg)
What criterion would be most natural?
• symmetric view with respect to all primary security accounts, includingthe domestic savings account
• pooling in large trading book=⇒ vanishing total hedge error
286
![Page 288: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/288.jpg)
Proposition HT,l, V vl withCvl independent square integrable martingales with
E
((C
vlt
Vvl0
)2)≤ Kt <∞ for l ∈ 1, 2, . . . , t ∈ [0, T ], T ∈ [0,∞).
at initial time well diversified trading book holds equal fractions
• total benchmarked wealth Ut =U0
m
∑ml=1
Vvlt
Vvl0
• total benchmarked P&L
Rm(t) =U0
m
m∑l=1
Cvlt
V vl0
,
⇒
limm→∞
Rm(t) = 0
P -a.s.
287
![Page 289: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/289.jpg)
Definition A dynamic trading strategy v, initiated at time zero, is calledlocally real-world mean-self-financing if its adapted process ηt = Cδ
t formsan local martingale.
288
![Page 290: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/290.jpg)
• classical risk minimizationFollmer and Sondermann (1986)Follmer and Schweizer (1989)Schweizer (1991, 1995)Schweizer (1999)Follmer-Schweizer decomposition assuming a risk neutral probabilitymeasureBuckdahn (1993) Schweizer (1994) Stricker (1996)Delbaen et al. (1997) Pham et al. (1998)
289
![Page 291: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/291.jpg)
• local risk minimizationBouleau and Lamberton (1989)Duffie and Richardson (1991)Schweizer (1994)Biagini et al. (1999)Schweizer (2001)
290
![Page 292: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/292.jpg)
• Benchmarked P&L is orthogonal if
ηtSt
is vector local martingale.
- above product has no trend- benchmarked hedge error orthogonal to any benchmarked traded wealth
291
![Page 293: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/293.jpg)
• VHTset of locally real-world mean-self-financing dynamic trading strate-
gies v initiated at time zero with
V vt = Sδ
t , t ≥ 0
which deliver HT
with orthogonal benchmarked P&L
292
![Page 294: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/294.jpg)
• market participants prefer more for less
⇒ Benchmarked Risk Minimization
For HT strategy v ∈ VHT
benchmarked risk minimizing (BRM) if for all v ∈ VHTwith V v
t = Sδt
price Sδt is minimal
Sδt ≤ S
δt
P -a.s. for all t ∈ [0, T ].
293
![Page 295: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/295.jpg)
Regular Benchmarked Contingent Claims
HT is called regular if
HT = Et(HT ) +
d∑j=1
∫ T
t
ϑj
HT(s)dSj
s + ηHT(T )− ηHT
(t)
ϑHT- predictable
ηHT- local martingale, orthogonal to S
294
![Page 296: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/296.jpg)
Theorem A regular HT has a BRM strategy v = vt = (ηt, ϑ1t , . . . , ϑ
dt )
⊤, t ∈[0, T ] ∈ VHT
with
Sδt = V v
t = Et(HT ) ,
SδT = HT ,
and orthogonal benchmarked P&L: Cδt = ηHT
(t)
295
![Page 297: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/297.jpg)
=⇒
Cδt - local martingale
Cδt St - vector local martingale
orthogonal
296
![Page 298: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/298.jpg)
hedgeable part:
Et(HT )− Cδt = E0(HT ) +
∫ t
0
ϑ⊤HT
(s)dSs .
local martingale
297
![Page 299: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/299.jpg)
• benchmarked price V vt = Et(HT ) minimal
• do not request square integrability
• no equivalent risk neutral probability measure required
• need only to check zero drift of ηHT
• and ηHT(t), St orthogonality
298
![Page 300: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/300.jpg)
Hedging a Regular Claim
• benchmarked contingent claim HT driven by continuous local mar-tingalesW 1,W 2, . . . ,W d orthogonal to each other
• benchmarked primary security account Sjt , j ∈ 1, . . . , d
dSjt = −Sj
t
d−1∑k=1
θj,kt dW kt
• d× d matrix Φt = [Φi,kt ]di,k=1
Φi,kt =
θi,kt for k ∈ 1, . . . , d− 11 for k = d
for t ∈ [0, T ].
299
![Page 301: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/301.jpg)
Proposition assume Φt invertible and Vt = E(HT |Ft) has representa-tion
Vt = V0 +d∑
k=1
∫ t
0
xksdW
ks +
∫ t
0
xdsdW
ds .
Then HT is a regular benchmarked contingent claim
VvHTt = Vt for all t ∈ [0, T ]
ϑHT(t) = diag(St)
−1(Φ⊤t )−1ξt
ηHT(t) =
∫ t
0
xdsdW
ds
with ξt = (x1t , . . . , x
d−1t , V0 +
∑d−1k=1
∫ t
0xksdW
ks )
⊤ .
300
![Page 302: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/302.jpg)
Si,jt =
Sjt
Sit
dSi,jt = Si,j
t
d−1∑k=1
(θi,kt − θj,kt )(θi,kt dt+ dW k
t )
bi,j,kt = (θi,kt − θj,kt ) .
bdt = [bd,j,kt ]d−1,d−1j,k=1
301
![Page 303: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/303.jpg)
Proposition The matrix Φt is invertible if and only if bdt is invertible.
Proof: elementary transform of bdt
Φt ←→
bdt 0
θd,1t . . . θd,d−1t 1
Φt has full rank if and only if bdt has full rank
302
![Page 304: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/304.jpg)
Comparison with Quadratic Criterion
•∫ t
0ϑ⊤HT
(s)dSs and ηHT(t) independent, square integrable martingales
• S1t , . . . , S
dt and ηHT
(t) independent, square integrable martingales
• ηHTorthogonal to benchmarked primary security accounts
• HT square integrable
• V vt = Sδ
t delivers HT
303
![Page 305: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/305.jpg)
Quadratic Criterion
E((Cδ
T )2)= E
((HT −
∫ T
0
ϑ⊤s dSs − Sδ
0)2
)⇒ min
304
![Page 306: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/306.jpg)
E((Cδ
T )2)
= E
((E(HT ) − Sδ
0 +∫ T
0(ϑ⊤
HT(s) − ϑ⊤
s )dSs + ηHT(T )
)2)
= E(E(HT ) − Sδ
0
)2
+ E(∫ T
0(ϑ⊤
HT(s) − ϑ⊤
s )2d[S]s)+ E((ηHT
(T ))2)
305
![Page 307: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/307.jpg)
=⇒
• Sδ0 = E(HT )
• ϑt = ϑHT(t), t ∈ [0, T ]
• ηHT(T ) = Cδ
T
• second moment of benchmarked P&L becomes minimal
306
![Page 308: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/308.jpg)
307
![Page 309: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/309.jpg)
Nonhedgeable Contingent Claim
• savings account S1,1t = 1
• numeraire portfolio S1,δ∗t
• Ht = Et(HT ), t ∈ [0, T ]
- independent from S1,δ∗t
- continuous
308
![Page 310: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/310.jpg)
Benchmarked risk minimization
• real world pricing formula
SδHTt = Et(HT ) = Et(HT )Et(S
1T ) = HtP (t, T )
P (t, T ) = Et(S1T )
dP (t, T ) =∂P (t, T )
∂S1dS1
t
• benchmarked NP Sδ∗t =
S1,δ∗t
S1,δ∗t
= 1
• benchmarked P&L
dCδHTt = dS
δHTt − ϑ1
HT(t)dS1
t
=
(Ht
∂P (t, T )
∂S1− ϑ1
HT(t)
)dS1
t + P (t, T )dHt
309
![Page 311: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/311.jpg)
• Assume orthogonal benchmarked primary security accounts
• product CδHT S has zero drift if
0 =d[S, CδHT ]t
dt=
(Ht
∂P (t, T )
∂S1− ϑ1
HT(t)
)d[S1]t
dt.
=⇒
ϑ1HT
(t) = Ht
∂P (t, T )
∂S1
Depends onHt !
310
![Page 312: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/312.jpg)
• Benchmarked P&L
ηHT(t) = C
δHTt =
∫ t
0
P (s, T )dHs
local martingale, orthogonal
• self-financing benchmarked hedgeable part
SδHTt − ηHT
(t) = SδHT0 +
∫ t
0
ϑ1HT
(s)dS1s
• ϑ1HT
(t) units in S1t
• remaining wealth in NP
• minimizing ddt[CδHT ]t
311
![Page 313: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/313.jpg)
• Regular benchmarked contingent claim
HT =HT
S1,δ∗T
= SδHT0 +
∫ T
0
Ht
∂P (t, T )
∂S1dS1
t +
∫ T
0
P (t, T )dHt
ϑ1HT
(t) = Ht
∂P (t, T )
∂S1
ηHT(t) =
∫ t
0
P (s, T )dHs
312
![Page 314: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/314.jpg)
Evolving Information
• consider special case when S1 true martingale
• classical risk minimization can be applied
• real world pricing formula yields
SδHTt =
SδHTt
S1t
= Et
(S1T
S1t
HT
S1,1T
)= Et
(ΛT
Λt
HT
S1,1T
)= EQ
t
(HT
S1,1T
)• equivalent minimal martingale measureQ
ΛT =dQ
dP|FT
=S1T
S1t
• same initial price as benchmarked risk minimization
SδHT0 = H0 = S
δHT0 (S1
0)−1
313
![Page 315: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/315.jpg)
• classical risk minimization invests total initial value in the savings ac-count
ϑ1CRM(t) = H0
• BRM strategy different
• P (t, T ) = S1t , ∂P (t,T )
∂S1t
= 1
=⇒
ϑ1HT
(t) = Ht = Et(HT )
• intuitive and practically appealing
• under classical risk minimization, evolving information ignored
• provides less fluctuations for benchmarked P&L
• easier diversified
314
![Page 316: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/316.jpg)
Modeling the Dynamics of Diversified Indices
Pl. & Rendek (2012)
• conjecture for normalized aggregate wealth dynamicstime transformed square root process
• Naive Diversification Theorem⇒ equity index = proxy of numeraireportfolio
• empirical stylized facts⇒ falsify models
315
![Page 317: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/317.jpg)
• ⇒ propose realistic one factor, two component long term index model
• realistic model outside classical theory⇒ benchmark approach
• exact, almost exact simulation⇒ verify empirical facts, effects of esti-mation techniques etc.
316
![Page 318: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/318.jpg)
The Affine Nature of Diversified WealthDynamics
Object: normalized units of wealth
Total wealth: Y ∆τi
Time steps: τi = i∆
Wealth unit value:√∆
317
![Page 319: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/319.jpg)
Economic activity: during [τi, τi+1) ”projects”generate each independent wealth with variance increment v2∆3/2;consume each η∆ fraction of wealth;generate together on average β
√∆ new units.
318
![Page 320: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/320.jpg)
Mean for increment of aggregate wealth:(β − ηY ∆
τi
)∆
319
![Page 321: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/321.jpg)
Variance for increment aggregate wealth: , sum of variances
=⇒ proportional to number of wealth units:Y ∆
τi√∆
=⇒ proportional to aggregate wealth=⇒ variance of increment of aggregate wealth is v2Y ∆
τi∆
320
![Page 322: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/322.jpg)
for ∆→ 0
Y ∆τi+1− Y ∆
τi=(β − ηY ∆
τi
)∆ + υ
√Y ∆τi
∆Wτi
E(∆Wτi) = 0, E((∆Wτi)2) = ∆
conjectures drift and diffusion terms
321
![Page 323: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/323.jpg)
=⇒ Conjecture viaWeak convergenceKleoden & Pl. (1992), Alfonsi (2005), Diop (2003)for parameters: β = η = υ = 1
square root process:
dYτt = (1− Yτt) dτt +√YτtdWτt
322
![Page 324: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/324.jpg)
• Quadratic variation:
[Yτ. ]t =
∫ t
0
Yτsdτs =
∫ t
0
YτsMsds
• Market activity:
Mt =dτt
dt
• integrate normalized index:can we findM =const. s.t.
M
∫ t
0
Yτsds ≈ [Yτ. ]t ?
323
![Page 325: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/325.jpg)
Quadratic variation and integrated normalized S&P500monthly data, calendar time, Shiller data
1871 1921 1971 20210
0.5
1
1.5
2
2.5
3
[ Yτ ]
t
M ∫0τ
t Yτ
s
ds
M ≈ 0.0178
average long term fit
324
![Page 326: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/326.jpg)
Market Activity: Mt =dτt
dtfrom model
1871 1921 1971 20210
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
325
![Page 327: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/327.jpg)
Quadratic variation and integrated normalized S&P500monthly data, τ -time
1871 1921 1971 20210
0.5
1
1.5
2
2.5
3
[ Yτ ]
t
∫0τ
t Yτ
s
dτs
326
![Page 328: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/328.jpg)
Eight Stylized Empirical Facts
• falsify potential models, Popper (1959)
• TOTMKWD in 26 currency denominations
about 1000 years of daily data
327
![Page 329: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/329.jpg)
(i) uncorrelated log-returns
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Average autocorrelation function for log-returns
328
![Page 330: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/330.jpg)
(ii) correlated absolute log-returns
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Average autocorrelation function for absolute log-returns
329
![Page 331: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/331.jpg)
(iii) Student-t distributed log-returns
−20 −15 −10 −5 0 5 10 15 20 2510
−6
10−5
10−4
10−3
10−2
10−1
100
Empirical log−density
Theoretical log−density
Logarithm of empirical density of normalized log-returns with Student-tdensity
330
![Page 332: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/332.jpg)
Results for log-returns of the EWI104s
Pl. & Rendek (2008)
0
0.5
1
1.5
2
−5−4
−3−2.15
−10
1 2
34
5
−2.94
−2.92
−2.9
−2.88
−2.86
−2.84
αλ
Estimated λ
Estimated LLF
× 105
SGH Student-t NIG Hyperbolic VG
σ 0.98 0.72 0.97 0.96 0.96
α 0.00 0.97 0.72
λ -2.16 1.49
ν 4.33
ln(L∗) -285796.39 -285796.39 -286448.94 -287152.08 -287499.83
Ln 0.0000004 1305.10 2711.38 3406.88
Ln = 0.0000004 < χ20.001,1 ≈ 0.000002
331
![Page 333: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/333.jpg)
(iv) volatility clustering
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 20130
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Estimated volatility
332
![Page 334: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/334.jpg)
(v) long term exponential growth
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 20133.5
4
4.5
5
5.5
6
6.5
4.178+0.048 t
Logarithm of index with trend line
333
![Page 335: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/335.jpg)
(vi) leverage effect
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 2013−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
Logarithms of normalized index and its volatility
334
![Page 336: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/336.jpg)
(vii) extreme volatility at major downturns
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 2013−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
Logarithms of normalized index and its volatility
335
![Page 337: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/337.jpg)
(viii) Integrated Normalized Index and its Quadratic Variationmonthly data, calendar time
1871 1921 1971 20210
0.5
1
1.5
2
2.5
3
[ Yτ ]
t
M ∫0τ
t Yτ
s
ds
M ≈ 0.0178
average long term fit
336
![Page 338: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/338.jpg)
⇒ Discounted Index Model generalizing conjecture
St = Aτt(Yτt)q,
Aτt = A expaτt
q > 0, a > 0,A > 0
conjecture expects q = 1
337
![Page 339: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/339.jpg)
Normalized index: (Yτt)q = St
Aτt
dYτ =
δ4−
1
2
(Γ(δ2+ q
)Γ(δ2
) ) 1q
Yτ
dτ +√Yτ dW (τ )
Long term mean: limτ→∞1τ
∫ τ
0(Ys)
qds = 1 P-a.s
338
![Page 340: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/340.jpg)
Market activity time: dτt = Mtdt
Inverse of market activity:
d
(1
Mt
)=
(ν
4γ − ϵ
1
Mt
)dt+
√γ
Mt
dWt,
where
dW (τt) =
√dτt
dtdWt =
√MtdWt
⋄ only oneWt
⋄ two component model
339
![Page 341: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/341.jpg)
Discounted index SDE:
dSt = St (µtdt+ σtdWt)
Expected rate of return:
µt =
a
Mt
−q
2
(Γ(δ2+ q
)Γ(δ2
) ) 1q
+
(δ
4q +
1
2q(q − 1)
)1
MtYτt
Mt
Volatility:
σt = q
√Mt
Yτt
Pl. & Rendek (2012c)
340
![Page 342: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/342.jpg)
Benchmark Approach
Bt – benchmark savings account
dBt = Bt
((−µt + σ2
t
)dt− σtdWt
)
σ2t ≤ µt ⇒ Bt is an (A, P )-supermartingale
⇒ no strong arbitrage; Pl. (2011)
341
![Page 343: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/343.jpg)
Assumptions:
A1. δ = 2(q + 1) A2. q2
(Γ(2q+1)Γ(q+1)
) 1q ≤ a
=⇒ σ2t ≤ µt
342
![Page 344: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/344.jpg)
Fitting the model to TOTMKWD
Step 1: Normalization of IndexAτt ≈ A exp 4aϵ
γ(ν−2)t⇒A = 65.21, 4aϵ
γ(ν−2)≈ 0.048
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 20133.5
4
4.5
5
5.5
6
6.5
4.178+0.048 t
343
![Page 345: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/345.jpg)
Normalized TOTMKWD
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 20130.4
0.6
0.8
1
1.2
1.4
1.6
1.8
344
![Page 346: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/346.jpg)
Step 2: Power q:Affine nature =⇒ conjecture: q = 1 = δ
2− 1 =⇒ δ = 4
• student-t with about four degree of freedomis considered and cannot be falsified,
=⇒ We set q = 1 and δ = 4
345
![Page 347: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/347.jpg)
Step 3: Observing Market Activity:
d[√
Y ]τt
dt= 1
4dτt
dt= Mt
4
Qti ≈[√
Y ]τti+1−[
√Y ]τti
ti+1−ti
Qti+1= α√ti+1 − tiQti + (1− α√ti+1 − ti)Qti , α = 0.92
exponential smooth
robust
346
![Page 348: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/348.jpg)
Market activity: Mt ≈ 4Qt
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 20130
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
M0 = 0.0175
347
![Page 349: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/349.jpg)
Step 4: Parameter γ:
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 20130
500
1000
1500
2000
2500
3000
66.28 t
γ = 265.12
348
![Page 350: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/350.jpg)
Step 5: Parameters ν and ϵ andStep 6: Long Term Average Net Growth Rate a:
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450
10
20
30
40
50
60
ν ≈ 4, ϵ ≈ 2.18⇒ a = 2.55⇒ no strong arbitrage
349
![Page 351: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/351.jpg)
Fitted model allows visualizingCalculated Volatility
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 20130
0.1
0.2
0.3
0.4
0.5
0.6
0.7
σt ≈√
4Qt
Yτt, average volatility: 11.9%
350
![Page 352: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/352.jpg)
Model applies also to proxies of numeraire portfolioS&P500 and VIX
1973 1977 1981 1985 1989 1993 1997 2001 2005 2009 2013−3
−2.5
−2
−1.5
−1
−0.5
0
log calculated volatility
log scaled VIX
A = 52.09, ϵ = 2.15, γ = 172.3, a = 1.5
351
![Page 353: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/353.jpg)
Simulation StudyStep 1: Market activity:
1
Mti+1
=γ(1− e−ϵ(ti+1−ti))
4ϵ
χ23,i +
√ 4ϵe−ϵ(ti+1−ti)
γ(1− e−ϵ(ti+1−ti))
1
Mti
+ Zi
2
0 5 10 15 20 25 30 35 400
0.2
0.4
0.6
0.8
1
1.2
exact simulation
352
![Page 354: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/354.jpg)
Step 2: τ -time:
τti+1− τti =
∫ ti+1
ti
Msds ≈Mti(ti+1 − ti)
0 5 10 15 20 25 30 35 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
almost exact simulation
353
![Page 355: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/355.jpg)
Step 3: Normalized index:
Yτti+1=
1− e−(τti+1−τti
)
4
χ23,i +
√√√√ 4e
−(τti+1−τti
)
1− e−(τti+1−τti
)Yτti
+ Zi
2
0 5 10 15 20 25 30 35 40
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
almost exact simulation
354
![Page 356: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/356.jpg)
Model recovers stylized empirical facts:
Model is difficult to falsify: Popper (1934)
1. Uncorrelated returns
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
1
1.2
355
![Page 357: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/357.jpg)
2. Correlated absolute returns
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6
0.8
1
1.2
356
![Page 358: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/358.jpg)
3. Student-t distributed returns
357
![Page 359: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/359.jpg)
estimated statistics
Simulation Student-t NIG Hyperbolic VG ν
1 0.008934 37.474149 102.719638 131.240780 4.012850
2 11.485226 11.175028 96.457136 132.916256 3.450916
3 0.000000 100.928524 244.190151 294.719960 3.734148
4 9.002421 35.759464 347.060676 331.014904 2.579009
5 8.767003 11.551178 121.190482 144.084964 3.170449
6 0.401429 60.570898 205.788160 252.591737 3.432435
7 12.239056 4.354888 46.411554 78.273485 3.957696
8 1.693411 23.910523 94.408789 130.623174 3.849691
9 1.232454 47.830407 202.073144 237.168411 3.236322
10 0.000000 43.037206 128.807757 162.582353 3.774957
11 0.433645 47.782681 172.736397 208.847632 3.431803
12 0.000000 56.019354 146.077121 185.624888 3.899403
13 7.137154 48.219756 579.922931 477.383441 2.293363
14 5.873948 16.515390 107.770531 135.508299 3.388307
15 0.000000 54.718046 184.112794 217.304105 3.402049
16 6.982560 3.991610 29.192198 47.105125 4.268740
17 2.966916 22.914863 108.513143 138.044416 3.553629
18 0.000000 52.066364 129.790856 160.373085 3.959605
19 0.006909 39.568695 111.398645 143.914350 3.982892
20 0.000001 56.845664 169.915512 211.260626 3.651091
21 1.674578 17.681088 61.710576 90.679738 4.265834
22 14.010840 3.279722 47.433693 73.789313 3.770825
23 11.198940 12.074044 114.888817 143.800553 3.257146
24 0.455557 27.676102 86.841704 114.452947 4.006528
358
![Page 360: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/360.jpg)
4. Volatility clustering
0 5 10 15 20 25 30 35 400
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
359
![Page 361: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/361.jpg)
5. Long term exponential growth
0 5 10 15 20 25 30 35 403.5
4
4.5
5
5.5
6
6.5
4.16+0.05 t
360
![Page 362: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/362.jpg)
6. Leverage effect and7. Extreme volatility at major market downturns
0 5 10 15 20 25 30 35 40−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
361
![Page 363: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/363.jpg)
Conclusions:
⋄ equity index model: 3 initial parameters, 3 structural parameters and 1
Wiener process (nondiversifiable uncertainty)
⋄ model recovers 7 stylized empirical facts
⋄ allows long dated derivative pricing under benchmark approach
⋄ leads outside classical theory
⋄ if benchmarked savings account local martingale=⇒ only 2 structural parameters
362
![Page 364: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/364.jpg)
Squared Bessel Processes (*)
Revuz & Yor (1999)
• squared Bessel process BESQδx
dimension δ ≥ 0
dXφ = δ dφ+ 2√|Xφ| dWφ
φ ∈ [0,∞) withX0 = x ≥ 0
• scaling property
Zφ =1
aXaφ
BESQδxa
, a > 0
363
![Page 365: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/365.jpg)
• sum of squares
Xφ =δ∑
k=1
(wk +W kφ)
2
W kφ Wiener process
=⇒
dXφ = δ dφ+ 2
δ∑k=1
(wk +W k
φ
)dW k
φ
X0 =δ∑
k=1
(wk)2 = x
364
![Page 366: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/366.jpg)
• setting
dWφ = |Xφ|−12
δ∑k=1
(wk +W k
φ
)dW k
φ
• quadratic variation
[W ]φ =
∫ φ
0
1
Xs
δ∑k=1
(wk +W k
s
)2ds = φ
Levy’s theorem
=⇒ W Wiener process in φ time
365
![Page 367: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/367.jpg)
0
100
200
300
400
500
600
700
800
0 10 20 30 40 50 60 70 80 90
phi
Squared Bessel process of dimension δ = 4 in φ-time
366
![Page 368: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/368.jpg)
• additivity property
Shiga & Watanabe (1973)
X = Xφ, φ ∈ [0,∞) BESQδx
Y = Yφ, φ ∈ [0,∞) independent BESQδ′
y
=⇒ Xφ + Yφ forms BESQδ+δ′
x+y
x, y, δ, δ′ ≥ 0
367
![Page 369: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/369.jpg)
• δ > 2, strict positivity
withX0 = x > 0
P
(inf
0≤φ<∞Xφ > 0
)= 1
• δ = 2, no strict positivity
P
(inf
0≤φ<∞Xφ > 0
)= 0
• δ ∈ [0, 2), no strict positivityX0 = x > 0
P
(inf
0≤φ≤φ′Xφ = 0
)> 0
368
![Page 370: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/370.jpg)
• transition density
δ > 0 , x > 0
pδ(ϱ, x;φ, y) =1
2(φ− ϱ)
(y
x
)ν2
exp
−
x+ y
2(φ− ϱ)
Iν
(√x y
φ− ϱ
)
Iν modified Bessel function of the first kind
index ν = δ2− 1
369
![Page 371: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/371.jpg)
80
90
100
110
120
y0.1
0.2
0.3
0.4
0.5
t
0
0.05
0.1
0.15
80
90
100
110y
Transition density of squared Bessel process for δ = 4
370
![Page 372: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/372.jpg)
• Y =Xφ
φnon-central chi-square distributed
dimension δnon-centrality parameter ℓ = x
φ
P
(Xφ
φ< u
)=
∞∑k=0
exp− ℓ
2
(ℓ2
)kk !
1−Γ(
u2; δ+2k
2
)Γ(
δ+2k2
)
Γ(·; ·) incomplete gamma function
371
![Page 373: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/373.jpg)
• Moments for α > − δ2
, φ ∈ (0,∞) and δ > 2
E(Xα
φ
∣∣A0
)=
(2φ)α exp
−X0
2φ
∑∞k=0
(X0
2φ
)k Γ(α+k+ δ2)
k ! Γ(k+ δ2)
for α > − δ2
∞ for α ≤ − δ2
=⇒E(Xφ
∣∣A0
)= X0 + δ φ
372
![Page 374: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/374.jpg)
• by monotonicity of the gamma function
E(Xα
φ
∣∣A0
)≤ (2φ)α exp
−X0
2φ
(Γ(α+ δ
2
)Γ(δ2
) + exp
X0
2φ
)
< ∞
• for δ = 4
E(X−1
φ
∣∣A0
)= X−1
0
(1− exp
−X0
2φ
)
373
![Page 375: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/375.jpg)
A Strict Local Martingale
Zφ = X−1φ inverse of a squared Bessel processXφ of dimension 4
dZφ = −2Z 32φ dWφ
=⇒ local martingale
E(Zφ
∣∣A0
)= E
(X−1
φ
∣∣A0
)= Z0
(1− exp
−12Z0 φ
)< Z0
=⇒ strict local martingale
=⇒ strict supermartingale
374
![Page 376: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/376.jpg)
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 10 20 30 40 50 60 70 80 90
phi
Inverse of a squared Bessel process of dimension δ = 4 in φ-time
375
![Page 377: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/377.jpg)
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0 10 20 30 40 50 60 70 80 90
phi
Expectation of the inverse of the squared Bessel process for δ = 4 inφ-time
376
![Page 378: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/378.jpg)
Time Transformation
• φ-time
φ(t) = φ(0) +1
4
∫ t
0
c2usudu
t ∈ [0,∞), s0 > 0
with
st = s0 exp
∫ t
0
bu du
377
![Page 379: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/379.jpg)
• φ-time
for constant b < 0, c = 0
φ(t) = φ(0) +c2
4 b s0(1− exp−b t)
• time
t(φ) = −1
bln
(1−
4 b s0
c2(φ− φ(0))
)
378
![Page 380: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/380.jpg)
0
5
10
15
20
25
30
0 10 20 30 40 50 60 70 80 90
phi
Time t(φ) against φ-time
379
![Page 381: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/381.jpg)
0
100
200
300
400
500
600
700
800
0 5 10 15 20 25 30
t
Squared Bessel process in dependence on time t
380
![Page 382: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/382.jpg)
100
150
200
250
300
350
400
450
0 5 10 15 20 25 30
t
Expectation of a squared Bessel process in dependence on time t
381
![Page 383: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/383.jpg)
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0 5 10 15 20 25 30
t
Expectation of the inverse of a squared Bessel process in dependence ontime t
382
![Page 384: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/384.jpg)
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 5 10 15 20 25 30
t
Inverse of squared Bessel process in dependence on time t
383
![Page 385: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/385.jpg)
Square Root Process
Yt = stXφ(t)
dYt = st dXφ(t) +Xφ(t) dst
= st δ dφ(t) + st 2√Xφ(t) dWφ(t) +Xφ(t) st bt dt
=
(δ
4c2t + bt Yt
)dt+ ct
√Yt
√4 st
c2tdWφ(t)
384
![Page 386: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/386.jpg)
dUt =
√4 st
c2tdWφ(t)
has quadratic variation
[U ]t =
∫ t
0
4 sz
c2zdφ(z) = t
U Wiener process
=⇒ SR process
dYt =
(δ
4c2t + bt Yt
)dt+ ct
√Yt dUt
385
![Page 387: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/387.jpg)
10
15
20
25
30
35
40
45
50
0 5 10 15 20 25 30
t
Sample path of a square root process in dependence on time t
386
![Page 388: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/388.jpg)
• moment
E(Y αt
∣∣A0
)= (2 φt st)
α exp
−Y0
2 φt
∞∑k=0
(Y0
2 φt
)kΓ(α+ k + δ2)
k! Γ(k + δ2)
δ > 2, α > − δ2
• moment estimate
E(Y αt
∣∣A0
)≤ (2 φt st)
α exp
−Y0
2 φt
(Γ(α+ δ
2)
Γ( δ2)
+ exp
Y0
2 φt
)
< ∞ for α ∈ (−δ
2, 0)
st =st
s0= exp
∫ t
0
bu du
φt = s0 (φ(t)− φ(0)) =
1
4
∫ t
0
c2usudu
387
![Page 389: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/389.jpg)
• first moment of SR process
E(Yt
∣∣A0
)= E
(Y0
∣∣A0
)exp
∫ t
0
bs ds
+
∫ t
0
δ
4c2s exp
∫ t
s
bz dz
ds
• case δ = 4
for c2t = c2 > 0 and bt = b < 0
limt→∞
E(Yt
∣∣A0
)= −
c2
b
limt→∞
E(Y −1t
∣∣A0
)= −2
b
c2
388
![Page 390: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/390.jpg)
• transition density
p(s, Ys; t, Yt) =pδ
(φ(s), Ys
ss;φ(t), Yt
st
)st
p(0, x; t, y) =1
2st φt
(y
x st
) ν2
exp
−x+ y
st
2 φt
Iν
√x y
st
φt
for 0 < t <∞, x, y ∈ (0,∞),
where ν = δ2− 1, st = expb t and φt =
c2
4 b(1− 1
st)
389
![Page 391: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/391.jpg)
• stationary density
is gamma density
pY∞(y) =
(−2 bc2
) δ2
yδ2−1
Γ( δ2)
exp
2 b
c2y
390
![Page 392: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/392.jpg)
Minimal Market ModelPl. (2001), special case of Pl.-Rendek model
Volatility Parametrization
• Diversification Theorem =⇒
well diversified stock market index approximates NP
• discounted NP
dSδ∗t = Sδ∗
t |θt| (|θt| dt+ dWt),
where
dWt =1
|θt|
d∑k=1
θkt dWkt
391
![Page 393: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/393.jpg)
0
20
40
60
80
100
120
140
160
1926 1950 1975 2000
Discounted NP
392
![Page 394: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/394.jpg)
0
1
2
3
4
5
1926 1950 1975 2000
ln(disc WSI)ln(A(t))
Logarithm of discounted NP
393
![Page 395: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/395.jpg)
Drift Parametrization
• discounted NP driftαt = Sδ∗
t |θt|2
strictly positive, predictable=⇒
• volatility
|θt| =√αt
Sδ∗t
leverage effect creates natural feedbackunder reasonably independent drift
394
![Page 396: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/396.jpg)
• discounted NP
dSδ∗t = αt dt+
√Sδ∗t αt dWt
drift has economic meaning
• transformed time
φt =1
4
∫ t
0
αs ds
395
![Page 397: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/397.jpg)
• squared Bessel process of dimension four
Xφt = Sδ∗t
dW (φt) =
√αt
4dWt
dXφ = 4 dφ+ 2√Xφ dW (φ)
Revuz & Yor (1999)
economically founded dynamics in φ-time
still no specific dynamics in t-time assumed
396
![Page 398: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/398.jpg)
Time Transformed Bessel Process
d
√Sδ∗t =
3αt
8
√Sδ∗t
dt+1
2
√αt dWt
• quadratic variation [√Sδ∗
]t=
1
4
∫ t
0
αs ds
• transformed timeφt =
[√Sδ∗
]t
observable
397
![Page 399: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/399.jpg)
0
2
4
6
8
10
12
14
1926 1950 1975 2000
Empirical quadratic variation of the square root of the discounted S&P 500
398
![Page 400: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/400.jpg)
Stylized Minimal Market Model
Pl. (2001, 2002, 2006), Pl. & Rendek (2012)
• assume discounted NP drift as
αt = α exp η t
• initial parameter α > 0
• net growth rate η
399
![Page 401: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/401.jpg)
• transformed time
φt =α
4
∫ t
0
exp η z dz
=α
4 η(exp η t − 1)
400
![Page 402: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/402.jpg)
0
2
4
6
8
10
12
14
1926 1950 1975 2000
observed transformed timefitted transformed time
Fitted and observed transformed time
401
![Page 403: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/403.jpg)
• normalized NP
Yt =Sδ∗t
αt
dYt = (1− η Yt) dt+√Yt dWt
square root process of dimension four
=⇒ parsimonious model, long term viabilityconfirmation of stylized Pl.-Rendek modelPl. & Rendek (2012)
402
![Page 404: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/404.jpg)
=⇒
• discounted NPSδ∗t = Yt αt
• NPSδ∗t = S0
t Sδ∗t = S0
t Yt αt
• scaling parameter α = 0.043
• net growth rate η = 0.0528
• reference level 1η= 18.93939
• speed of adjustment η
• half life time of major displacement ln(2)η≈ 13 years
403
![Page 405: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/405.jpg)
20
30
40
50
60
70
80
90
100
1926 1950 1975 2000
Normalized NP
404
![Page 406: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/406.jpg)
• logarithm of discounted NP
ln(Sδ∗t ) = ln(Yt) + ln(α) + η t
• no need for extra volatility process in MMM
• realistic long term dynamics
405
![Page 407: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/407.jpg)
Volatility of NP under the stylized MMM
|θt| =1√Yt
• squared volatility
d|θt|2 = d
(1
Yt
)= |θt|2 η dt−
(|θt|2
) 32 dWt
3/2 volatility model
Pl. (1997), Lewis (2000)
406
![Page 408: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/408.jpg)
0.1
0.12
0.14
0.16
0.18
0.2
0.22
1926 1950 1975 2000
Volatility of the NP under the stylized MMM
407
![Page 409: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/409.jpg)
Transition Density of Stylized MMM
• transition density of discounted NP Sδ∗
p(s, x; t, y) =1
2 (φt − φs)
(y
x
) 12
exp
−
x+ y
2 (φt − φs)
× I1( √
x y
φt − φs
)
φt =α
4 η(expη t − 1)
I1(·) modified Bessel function of the first kind
408
![Page 410: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/410.jpg)
• non-central chi-square distributed random variable:
y
φt − φs
=Sδ∗t
φt − φs
with δ = 4 degrees of freedom and non-centrality parameter:
x
φt − φs
=Sδ∗s
φt − φs
409
![Page 411: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/411.jpg)
80
90
100
110
120
y0.1
0.2
0.3
0.4
0.5
t
0
0.05
0.1
0.15
80
90
100
110y
Transition density of squared Bessel process for δ = 4410
![Page 412: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/412.jpg)
Zero Coupon Bond under the stylized MMM
• zero coupon bond
P (t, T ) = Sδ∗t Et
(1
Sδ∗T
)= P ∗
T (t)Et
(Sδ∗t
Sδ∗T
)
with savings bond
P ∗T (t) = exp
−∫ T
t
rs ds
411
![Page 413: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/413.jpg)
δ = 4 =⇒
P (t, T ) = P ∗T (t)
(1− exp
−
Sδ∗t
2 (φT − φt)
)< P ∗
T (t)
for t ∈ [0, T ), Pl. (2002)
with time transform
φt =α
4 η(expη t − 1)
P (0, T ) = 0.00423
P ∗T (0) = 0.04947
P (0,T )P ∗
T (0)≈ 0.0855
412
![Page 414: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/414.jpg)
0
0.2
0.4
0.6
0.8
1
1.2
1926 1950 1975 2000
savings bondP(t,T)
Zero coupon bond and savings bond
413
![Page 415: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/415.jpg)
-6
-5
-4
-3
-2
-1
0
1
1926 1950 1975 2000
ln(savings bond)ln(hedge portfolio)
ln(P(t,T))
ln from Zero coupon bond, hedge portfolio and savings bond
414
![Page 416: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/416.jpg)
Forward Rates under the MMM
• forward rate
f(t, T ) = −∂
∂Tln(P (t, T ))
= rT + n(t, T )
415
![Page 417: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/417.jpg)
• market price of risk contribution
n(t, T ) = −∂
∂Tln
(1− exp
−
Sδ∗t
2(φT − φt)
)
=1(
exp
Sδ∗t
2(φT −φt)
− 1
) Sδ∗t
(φT − φt)2
αT
8
limT→∞
n(t, T ) = η
Pl. (2005), Miller & Pl. (2005)
416
![Page 418: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/418.jpg)
0.020.04
0.06
0.08
0.1
eta
20
40
60
80
T
0
0.025
0.05
0.075
0.1
0.020.04
0.06
0.08eta
Market price of risk contribution in dependence on η and T417
![Page 419: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/419.jpg)
Free Snack from Savings Bond
• potential existence of weak form of arbitrage?
borrow amount P (0, T ) from savings account
Sδt = P (t, T )− P (0, T ) expr t
such that Sδ0 = 0
SδT = 1− P (0, T ) expr T > 0
lower bondSδt ≥ −P (0, T ) expr t
418
![Page 420: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/420.jpg)
• since Sδt may become negative
not strong arbitrage
Fundamental Theorem of Asset Pricing
Delbaen & Schachermayer (1998) =⇒ the MMM does not
admit an equivalent risk neutral probability measure
=⇒there is a free lunch with vanishing risk
Delbaen & Schachermayer (2006)
419
![Page 421: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/421.jpg)
Absence of an Equivalent Risk Neutral Probability Measure
• fair zero coupon bond has a lower price than the savings bond
P (t, T ) < P ∗T (t) = exp
−∫ T
t
rs ds
=S0t
S0T
• Radon-Nikodym derivative
Λt =S0t
S00
=Sδ∗0
Sδ∗t
strict supermartingale under MMM
420
![Page 422: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/422.jpg)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1926 1950 1975 2000
total massRadon Nikodym deriv
Radon-Nikodym derivative and total mass of candidate risk neutral measure
421
![Page 423: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/423.jpg)
• hypothetical risk neutral measure
total mass:
Pθ,T (Ω) = E (ΛT ) = 1− exp
−Sδ∗0
2φT
< Λ0 = 1
Pθ is not a probability measure
422
![Page 424: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/424.jpg)
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
1926 1950 1975 2000
P*^P^
Benchmarked zero coupon bond
423
![Page 425: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/425.jpg)
Hedge Simulation
• delta units in the NP
δ∗(t) =∂P (t, T )
∂Sδ∗t
= exp
−∫ T
0
rs ds
exp
−
Sδ∗t
2(φT − φt)
1
2(φT − φt)
424
![Page 426: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/426.jpg)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1926 1950 1975 2000
Ratio in the NP
425
![Page 427: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/427.jpg)
-8e-006
-6e-006
-4e-006
-2e-006
0
2e-006
4e-006
6e-006
8e-006
1926 1950 1975 2000
Benchmarked P&L for hedge portfolio
426
![Page 428: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/428.jpg)
European Call Options under the MMM
real world pricing formula =⇒
cT,K(t, Sδ∗t ) = Sδ∗
t Et
((Sδ∗
T −K)+
Sδ∗T
)
= Et
(Sδ∗t −
K Sδ∗t
Sδ∗T
)+
= Sδ∗t
(1− χ2(d1; 4, ℓ2)
)−K exp−r (T − t) (1− χ2(d1; 0, ℓ2))
χ2(·; ·, ·) non-central chi-square distribution
427
![Page 429: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/429.jpg)
with
d1 =4 ηK exp−r (T − t)
S0t αt (expη (T − t) − 1)
and
ℓ2 =2 η Sδ∗
t
S0t αt (expη (T − t) − 1)
Hulley, Miller & Pl. (2005)
Miller & Pl. (2008)
428
![Page 430: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/430.jpg)
0.6
0.8
1
1.2K2
4
6
8
10
T
0.2
0.225
0.25
0.275
0.6
0.8
1
1.2K
Implied volatility surface for the stylized MMM
429
![Page 431: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/431.jpg)
• implied volatility
in BS-formula adjust short rate to
r = −1
T − tln(P (t, T ))
otherwise put and call implied volatilities do not match
430
![Page 432: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/432.jpg)
European Put Options under the MMM
• fair put-call parity relation
pT,K(t, Sδ∗t ) = cT,K(t, Sδ∗
t )− Sδ∗t +K P (t, T )
• European put option formula
pT,K(t, Sδ∗t ) = −Sδ∗
t
(χ2(d1; 4, ℓ2)
)+K exp−r (T − t)
(χ2(d1; 0, ℓ2)− exp−ℓ2
)
Hulley, Miller & Pl. (2005)
431
![Page 433: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/433.jpg)
0
5000
10000
15000
20000
25000
30000
35000
Dec-2
5
Dec-2
8
Dec-3
1
Dec-3
4
Dec-3
7
Dec-4
0
Dec-4
3
Dec-4
6
Dec-4
9
Dec-5
2
Dec-5
5
Dec-5
8
Dec-6
1
Dec-6
4
Dec-6
7
Dec-7
0
Dec-7
3
Dec-7
6
Dec-7
9
Dec-8
2
Dec-8
5
Dec-8
8
Dec-9
1
Dec-9
4
Dec-9
7
Dec-0
0
Dec-0
3
index
K*savings bond
"risk neutral" put
fair put
Risk neutral and fair put on index
432
![Page 434: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/434.jpg)
0
50
100
150
200
250
300
30
/03
/19
28
31
/07
/19
30
30
/11
/19
32
29
/03
/19
35
30
/07
/19
37
30
/11
/19
39
31
/03
/19
42
31
/07
/19
44
29
/11
/19
46
31
/03
/19
49
31
/07
/19
51
30
/11
/19
53
30
/03
/19
56
31
/07
/19
58
30
/11
/19
60
29
/03
/19
63
30
/07
/19
65
30
/11
/19
67
31
/03
/19
70
31
/07
/19
72
29
/11
/19
74
31
/03
/19
77
31
/07
/19
79
30
/11
/19
81
30
/03
/19
84
31
/07
/19
86
30
/11
/19
88
29
/03
/19
91
30
/07
/19
93
30
/11
/19
95
31
/03
/19
98
31
/07
/20
00
29
/11
/20
02
38
44
2
index
K*savings bond
"risk neutral" put
fair put
Benchmarked “risk neutral” and fair put on index
433
![Page 435: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/435.jpg)
• put-call parity breaks down if one uses the savings bond P ∗T (t)
pT,K(t, Sδ∗t ) < cT,K(t, Sδ∗
t )− Sδ∗t +K exp−r (T − t)
for t ∈ [0, T )
434
![Page 436: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/436.jpg)
Comparison to Hypothetical Risk Neutral Prices
• hypothetical risk neutral price c∗T,K(t, Sδ∗t )
of a European call option on the GOP
• benchmarked hypothetical risk neutral call price
c∗T,K(t, Sδ∗t ) =
c∗T,K(t, Sδ∗t )
Sδ∗t
local martingale
c∗T,K(·, ·) uniformly bounded
435
![Page 437: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/437.jpg)
=⇒ c∗T,K - martingale =⇒
c∗T,K(t, Sδ∗t ) = cT,K(t, Sδ∗
t )
=⇒c∗T,K(t, Sδ∗
t ) = cT,K(t, Sδ∗t )
436
![Page 438: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/438.jpg)
• hypothetical risk neutral put-call parity
p∗T,K(t, Sδ∗t ) = c∗T,K(t, Sδ∗
t )− Sδ∗t +K P ∗
T (t)
since P ∗T (t) > P (t, T ) =⇒
pT,K(t, Sδ∗t ) < p∗T,K(t, Sδ∗
t )
t ∈ [0, T )
437
![Page 439: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/439.jpg)
Difference in Asymptotic Put Prices
• hypothetical risk neutral prices can become extreme if NP value tendstowards zero
• asymptotic fair zero coupon bond
limSδ∗
t →0PT (t, T )
a.s.= 0
for t ∈ [0, T )
• asymptotic fair European call
limSδ∗
t →0cT,K(t, Sδ∗
t )a.s.= lim
Sδ∗t →0
Sδ∗t cT,K(t, Sδ∗
t ) = 0
438
![Page 440: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/440.jpg)
• asymptotic fair put
fair put-call parity =⇒
limSδ∗
t →0pT,K(t, Sδ∗
t )a.s.= 0
439
![Page 441: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/441.jpg)
• asymptotic hypothetical risk neutral put
hypothetical risk neutral put-call parity =⇒
limSδ∗
t →0p∗T,K(t, Sδ∗
t )
a.s.= lim
Sδ∗t →0
(pT,K(t, Sδ∗
t ) +KS0t
S0T
exp
−
Sδ∗t
2(φT − φt)
)
a.s.= K
S0t
S0T
> 0
dramatic differences can arise
440
![Page 442: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/442.jpg)
0
5000
10000
15000
20000
25000
30000
35000
Dec-2
5
Dec-2
8
Dec-3
1
Dec-3
4
Dec-3
7
Dec-4
0
Dec-4
3
Dec-4
6
Dec-4
9
Dec-5
2
Dec-5
5
Dec-5
8
Dec-6
1
Dec-6
4
Dec-6
7
Dec-7
0
Dec-7
3
Dec-7
6
Dec-7
9
Dec-8
2
Dec-8
5
Dec-8
8
Dec-9
1
Dec-9
4
Dec-9
7
Dec-0
0
Dec-0
3
index
K*savings bond
"risk neutral" put
fair put
“Risk neutral” and fair put on index
441
![Page 443: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/443.jpg)
Pricing Annuities
Baldeaux and Pl. (2012)
Interest Indexed Payouts
Q(t, T ) = S∗t Et
(BT
S∗T
)S∗t =
S∗t
Bt
Bt - savings account
442
![Page 444: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/444.jpg)
e.g.: Minimal Market Model (MMM), Pl. (2001)=⇒
Q(t, T ) = 1− exp
−
2 η S∗t
α (expη T − expη t)
443
![Page 445: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/445.jpg)
Interest Indexed Life Annuity
θ-entitlement levelT -retirement dateT0 < T1 < T2 < · · ·ξx - remaining life time of individual aged x at time 0
Uθx,T (t) = S∗
t Et
∑Ti≥T
Iξx>Tiθ BTi
S∗Ti
= θ
∑Ti≥T
Pt (ξx > Ti) Q(t, Ti)
444
![Page 446: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/446.jpg)
Threshold Life Table
P (ξx > T ) =
exp− b
ln(C)
(Cx+T − 1
)for T ≤ N − x
q(1 + λ
(x+T−N
ω
))− 1λ
for T > N − x
?)
445
![Page 447: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/447.jpg)
60 70 80 90 100 110 1200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ung
radu
ated
mor
talit
y ra
te
age
Figure 1: Mortality rates fitted to mortality data of the US population in 2007with threshold atN = 93.
446
![Page 448: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/448.jpg)
2
4
6
8
10
12
14
16
18
10 20 30 40
Figure 2: Discounted fair interest indexed life annuity as function of time toretirement.
447
![Page 449: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/449.jpg)
0
2
4
6
8
10
12
14
16
18
20
1920 1940 1960
discounted hedge portfoliodiscounted fair annuity
Figure 3: Discounted hedging portfolio and discounted fair interest indexedlife annuity evolving over the years.
448
![Page 450: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/450.jpg)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1920 1940 1960
Figure 4: Fraction of the hedging portfolio invested in the benchmark port-folio.
449
![Page 451: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/451.jpg)
Targeted Pension
Γl(t)-account of l th member in units of interest indexed life annuity
• total discounted valueMti∑l=1
Γl(ti)Uθti
xl,Tl(ti) = δ0total(ti) + δ∗total(ti) S
∗ti
=⇒ entitlement level
θti =δ0total(ti) + δ∗total S
∗ti∑Mi
l=1 Γl(ti)∑
Tk≥TlPti(ξxl > Tk)Q(ti, Tk)
450
![Page 452: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/452.jpg)
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1920 1940 1960
Figure 5: Entitlement level over the years.
451
![Page 453: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/453.jpg)
0.99
0.995
1
1.005
1.01
1.015
1.02
1973 1980 1990 2000 2010
Figure 6: Entitlement level using fast growing portfolio and daily hedging.
452
![Page 454: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/454.jpg)
Variable Annuity
Guaranteed Minimum Death Benefits
Marquardt, Pl. & Jaschke (2008)
• payout to the policyholder
max(egτV0, Vτ )
τ time of death
g ≥ 0 is the guaranteed instantaneous growth rate
V0 is the initial account value
Vτ is the unit value of the policyholder’s account at time of death τembedded put option
453
![Page 455: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/455.jpg)
0 5 10 15 20 25 300.8
0.85
0.9
0.95
1
1.05
1.1
MMM
T
GM
DB
0
0 5 10 15 20 25 300.8
0.85
0.9
0.95
1
1.05
1.1
risk neutral
T
GM
DB
0
0 5 10 15 20 25 300.8
0.85
0.9
0.95
1
1.05
1.1
1.15Black Scholes
T
GM
DB
0
0 5 10 15 20 25 300.9
0.95
1
1.05
1.1
1.15MMM
T
GM
DB
0
0 5 10 15 20 25 300.9
0.95
1
1.05
1.1
1.15risk neutral
T
GM
DB
0
0 5 10 15 20 25 300.9
0.95
1
1.05
1.1
1.15Black Scholes
T
GM
DB
0g=0.025
g=0
g=0.025g=0.025
g=0g=0
Present value of the GMDB under the real world pricing formula (left), the risk
neutral pricing formula (middle) and the Black Scholes formula (right) for
η = 0.05, α0 = 0.05, r = 0.05, ξ = 0.01 and Y0 = 20.454
![Page 456: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/456.jpg)
ReferencesArrow, K. J. (1965). Aspects of the Theory of Risk-Bearing. Helsinki: Yrjo Hahnsson Foun-
dation.
Best, M. J. & R. R. Grauer (1991). On the sensitivity of mean-variance-efficient portfoliosto changes in asset means: Some analytical and computational results. Rev. FinancialStudies 4(2), 315–342.
Black, F. & M. Scholes (1973). The pricing of options and corporate liabilities. J. PoliticalEconomy 81, 637–654.
Breiman, L. (1960). Investment policies for expanding business optimal in a long run sense.Naval Research Logistics Quarterly 7(4), 647–651.
Breiman, L. (1961). Optimal gambling systems for favorable games. In Proceedings of theFourth Berkeley Symposium on Mathematical Statistics and Probability, Volume I, pp.65–78.
Cochrane, J. H. (2001). Asset Pricing. Princeton University Press.
Constantinides, G. M. (1992). A theory of the nominal structure of interest rates. Rev. Finan-cial Studies 5, 531–552.
Davis, M. H. A. (1997). Option pricing in incomplete markets. In M. A. H. Dempster and
455
![Page 457: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/457.jpg)
S. R. Pliska (Eds.), Mathematics of Derivative Securities, pp. 227–254. Cambridge Uni-versity Press.
Debreu, G. (1959). Theory of Value. Wiley, New York.
Delbaen, F. & W. Schachermayer (1998). The fundamental theorem of asset pricing for un-bounded stochastic processes. Math. Ann. 312, 215–250.
Delbaen, F. & W. Schachermayer (2006). The Mathematics of Arbitrage. Springer Finance.Springer.
DeMiguel, V., L. Garlappi, & R. Uppal (2009). Optimal versus naive diversification: Howinefficient is the 1/n portfolio strategy? Rev. Financial Studies 22(5), 1915–1953.
Doob, J. L. (1953). Stochastic Processes. Wiley, New York.
Duffie, D. (2001). Dynamic Asset Pricing Theory (3rd ed.). Princeton, University Press.
Fernholz, E. R. & I. Karatzas (2005). Relative arbitrage in volatility-stabilized markets. An-nals of Finance 1(2), 149–177.
Geman, S., N. El Karoui, & J. C. Rochet (1995). Changes of numeraire, changes of proba-bility measures and pricing of options. J. Appl. Probab. 32, 443–458.
Hakansson, N. H. (1971a). Capital growth and the mean-variance approach to portfolio se-lection. J. Financial and Quantitative Analysis 6(1), 517–557.
Hakansson, N. H. (1971b). Multi-period mean-variance analysis: towards a general theory
456
![Page 458: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/458.jpg)
of portfolio choice. J. Finance 26, 857–884.
Hakansson, N. H. & W. T. Ziemba (1995). Capital growth theory. In R. Jarrow, V. Maksi-movic, and W. T. Ziemba (Eds.), Handbooks in Operations Research and ManagementScience: Finance, Volume 9, pp. 65–86. Elsevier Science.
Harrison, J. M. & D. M. Kreps (1979). Martingale and arbitrage in multiperiod securitiesmarkets. J. Economic Theory 20, 381–408.
Hulley, H., S. Miller, & E. Platen (2005). Benchmarking and fair pricing applied to twomarket models. Kyoto Economic Review 74(1), 85–118.
Ingersoll, J. E. (1987). Theory of Financial Decision Making. Studies in Financial Eco-nomics. Rowman and Littlefield.
Jamshidian, F. (1997). LIBOR and swap market models and measures. Finance Stoch. 1(4),293–330.
Kardaras, C. & E. Platen (2010). Minimizing the expected market time to reach a certainwealth level. SIAM J. Financial Mathematics. 1(1), 16–29.
Kelly, J. R. (1956). A new interpretation of information rate. Bell Syst. Techn. J. 35, 917–926.
Khanna, A. & M. Kulldorff (1999). A generalization of the mutual fund theorem. FinanceStoch. 3(2), 167–185.
Latane, H. (1959). Criteria for choice among risky ventures. J. Political Economy 38, 145–
457
![Page 459: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/459.jpg)
155.
Lewis, A. L. (2000). Option Valuation Under Stochastic Volatility. Finance Press, NewportBeach.
Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stockportfolios and capital budgets. Rev. Econom. Statist. 47, 13–37.
Loewenstein, M. & G. A. Willard (2000). Local martingales, arbitrage, and viability: Freesnacks and cheap thrills. Econometric Theory 16(1), 135–161.
Long, J. B. (1990). The numeraire portfolio. J. Financial Economics 26, 29–69.
Markowitz, H. (1952). Portfolio selection. J. Finance VII(1), 77–91.
Markowitz, H. (1959). Portfolio Selection: Efficient Diversification of Investment. Wiley,New York.
Markowitz, H. (1976). Investment for the long run: New evidence for an old rule. J. Fi-nance XXXI(5), 1273–1286.
Marquardt, T., E. Pl., & J. Jaschke (2008). Valuing guaranteed minimum death benefit op-tions in variable annuities under a benchmark approach. Technical report, University ofTechnology, Sydney. QFRC Research Paper 221.
Merton, R. C. (1973a). An intertemporal capital asset pricing model. Econometrica 41, 867–888.
458
![Page 460: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/460.jpg)
Merton, R. C. (1973b). Theory of rational option pricing. Bell J. Econ. Management Sci. 4,141–183.
Miller, S. & E. Platen (2005). A two-factor model for low interest rate regimes. Asia-PacificFinancial Markets 11(1), 107–133.
Miller, S. & E. Platen (2008). Analytic pricing of contingent claims under the real worldmeasure. Int. J. Theor. Appl. Finance 11(8), 841–867.
Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica 3, 768–783.
Novikov, A. A. (1972). On an identity for stochastic integrals. Theory Probab. Appl. 17,717–720.
Platen, E. (1997). A non-linear stochastic volatility model. Technical report, Australian Na-tional University, Canberra, Financial Mathematics Research Reports. FMRR 005-97.
Platen, E. (2001). A minimal financial market model. In Trends in Mathematics, pp. 293–301. Birkhauser.
Platen, E. (2002). Arbitrage in continuous complete markets. Adv. in Appl. Probab. 34(3),540–558.
Platen, E. (2005). An alternative interest rate term structure model. Int. J. Theor. Appl. Fi-nance 8(6), 717–735.
Platen, E. (2006). A benchmark approach to finance. Math. Finance 16(1), 131–151.
459
![Page 461: A Benchmark Approach to Investing, Pricing and Hedging · A Benchmark Approach to Investing, Pricing and Hedging Eckhard Platen University of Technology Sydney Pl. & Heath (2006,](https://reader034.fdocuments.us/reader034/viewer/2022050122/5f52437aecf7155f90329880/html5/thumbnails/461.jpg)
Platen, E. (2008). Law of the minimal price. Technical report, University of Technology,Sydney. QFRC Research Paper 215.
Pratt, J. W. (1964). Risk aversion in the small and in the large. Econometrica 32, 122–136.
Revuz, D. & M. Yor (1999). Continuous Martingales and Brownian Motion (3rd ed.).Springer.
Roll, R. (1973). Evidence on the ”Growth-Optimum” model. J. Finance 28(3), 551–566.
Ross, S. A. (1976). The arbitrage theory of capital asset pricing. J. Economic Theory 13,341–360.
Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditionsof risk. J. Finance 19, 425–442.
Sharpe, W. F. (1966). Mutual fund performance. J. Business 39, 119–138.
Shiga, T. & S. Watanabe (1973). Bessel diffusion as a one parameter family of diffsuionprocesses. Z. Wahrsch. Verw. Gebiete 27, 37–46.
Thorp, E. O. (1972). Portfolio choice and the Kelly criterion. In Proceedings of the 1971Business and Economics Section of the American Statistical Association, Volume 21,pp. 5–224.
Tobin, J. (1958). Liquidity preference as behaviour towards risk. Rev. of Economic Stud-ies 25(2), 65–86.
460