A Benchmark Approach to Quantitative Finance · A Benchmark Approach to Quantitative Finance...
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A Benchmark Approach to Quantitative Finance
Eckhard PlatenSchool of Finance and Economics and Department of Mathematical Sciences
University of Technology, Sydney
Lit: Platen, E. & Heath, D.: A Benchmark Approach to Quantitative Finance
Springer Finance, 700 pp., 199 illus., Hardcover, ISBN-10 3-540-26212-1 (2006).
Platen, E.: A benchmark approach to finance. Mathematical Finance16(1), 131-151 (2006).
Buhlmann, H. & Platen, E.: A discrete time benchmark approach for insurance and finance
ASTIN Bulletin33(2), 153-172 (2003).
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Contents1 Asset Pricing 1
2 Continuous Financial Markets 55
3 Portfolio Optimization 91
4 Modeling Stochastic Volatility 107
5 Minimal Market Model 154
6 Stylized Multi-Currency MMM 214
7 Valuing Guaranteed Minimum Death Benefits 219
8 Markets with Event Risk 273
References 337
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1 Asset Pricing
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.
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-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)
Figure 1.1: Logarithms of savings bond, fair zero coupon bondand savings
account.c© Copyright Eckhard Platen 10 BA - Toronto C 1 2
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Financial Market
• jth primary security account
Sjt
j ∈ 0, 1, . . . , d
• savings account
S0t
t ≥ 0
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• strategy
δ = δt = (δ0t , δ1t , . . . , δ
dt )⊤, t ≥ 0
predictable
• portfolio
Sδt =
d∑
j=0
δjt S
jt
• self-financing
dSδt =
d∑
j=0
δjt dS
jt
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• strictly positive portfolio
Sδ0 = x > 0
Sδt ∈ (0,∞) for all t ∈ [0,∞)
=⇒ Sδ ∈ V+x
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Numeraire Portfolio
Definition 1.1 Sδ∗ ∈ V+x numeraire portfolioif
Et
Sδt+h
Sδ∗
t+h
Sδt
Sδ∗
t
− 1
≤ 0
for all nonnegativeSδ and t, h ∈ [0,∞).
• Sδ∗ “best” performing portfolio
Long (1990), Becherer (2001), Pl. (2002, 2006),
Buhlmann& Pl. (2003), Goll & Kallsen (2003), Karatzas & Kardaras (2007)
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Main Assumption of the Benchmark Approach
Assumption 1.2 There exists a numeraire portfolioSδ∗ ∈ V+x .
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EWI
MCI
DWI
WSI
0
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
1/1/73 28/9/76 25/6/80 22/3/84 18/12/87 15/9/91 12/6/95 9/3/99 4/12/02 31/8/06
$US
Figure 1.2: Constructed MCI, DWI, EWI and WSI. In the long term the WSI
and EWI outperform all other indices, Le& Platen (2006).
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• benchmarked value
Sδt =
Sδt
Sδ∗
t
Supermartingale Property
Corollary 1.3 For nonnegativeSδ
Sδt ≥ Et
(
Sδs
)
0 ≤ t ≤ s < ∞
Sδ supermartingale
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• long term growth rate
gδ = lim supt→∞
1
tln
(
Sδt
Sδ0
)
Theorem 1.4 For Sδ ∈ V+x
gδ ≤ gδ∗ .
• “pathwise best” in the long run
Karatzas & Shreve (1998), Pl. (2004a),
Karatzas & Kardaras (2007)
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Pl. (2004a)
Definition 1.5 Nonnegative portfolioSδ outperforms systematicallySδ if
(i) Sδ0 = Sδ
0 ;
(ii) P(
Sδt ≥ Sδ
t
)
= 1
(iii) P(
Sδt > Sδ
t
)
> 0.
• “relative arbitrage” Fernholz & Karatzas (2005)
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-5
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-3
-2
-1
0
1
1930 1940 1950 1960 1970 1980 1990 2000
time
ln(fair zero coupon bond)ln(savings account)
Figure 1.3: Logarithms of fair zero coupon bond and savings account.
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Theorem 1.6 Numeraire portfolio cannot be outperformed systematically.
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• portfolio ratio
Aδt,h =
Sδt+h
Sδt
• expected growth
gδt,h = Et
(
ln(
Aδt,h
))
t, h ≥ 0
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Definition 1.7 Sδ growth optimal if
gδt,h ≤ g
δt,h
for all t, h ≥ 0 and nonnegativeSδ.
• expected log-utility
Kelly (1956)
Hakansson (1971a)
Merton (1973a)
Roll (1973)
Markowitz (1976)
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Theorem 1.8 The numeraire portfolio is growth optimal.
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Strong Arbitrage
• market participants can only exploit arbitrage
• limited liability
=⇒ nonnegative total wealth of each market participant
Definition 1.9 A nonnegativeSδ is a strong arbitrageif Sδ0 = 0 and
P(
Sδt > 0
)
> 0.
Pl. (2002)-mathematical arguments (super-martingale property)
Loewenstein & Willard (2000)-economic arguments
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Theorem 1.10 There isno strong arbitrage.
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• Delbaen & Schachermayer (1998)
free lunches with vanishing risk (FLVR)
• Loewenstein & Willard (2000)
free snacks& cheap thrills
=⇒ no-arbitrage pricing makes no sense under BA
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-0.2
0
0.2
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1
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 1.4:P (t, T ) minus savings account.
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-0.0004
-0.0002
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
0.0018
1920 1922 1924 1926 1928
time
Figure 1.5:P (t, T ) minus savings account.
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• Existence of equivalent risk neutral probability measure is
more amathematical conveniencethan an economic necessity.
• Basing quantitative methods on classical risk neutral pricing is
restrictive.
• Trends are ignored under
Fundamental Theorem of Asset Pricing
Delbaen & Schachermayer (1998)
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• 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)
All ignore trends !
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• One needs consistent pricing concept that exploits trends
• All benchmarked nonnegative securities are supermartingales
(no upward trend)
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Definition 1.11 Price is fair if, when benchmarked, forms martingale
(no trend)
Sδt = Et
(
Sδs
)
0 ≤ t ≤ s < ∞.
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Lemma 1.12 Theminimal nonnegative supermartingale that reaches a
given benchmarked contingent claimis a martingale.
see Pl.& Heath (2006)
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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
Figure 1.6: Benchmarked savings bond and benchmarked fair zero coupon
bond.c© Copyright Eckhard Platen 10 BA - Toronto C 1 27
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Law of the Minimal Price
Pl. (2008)
Theorem 1.13 If a fair portfolio replicates a nonnegative payoff, then this
represents the minimal replicating portfolio.
• least expensive
• minimal hedge
• economically correct price in a competitive market
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• contingent claim
HT
E0
(
HT
Sδ∗
T
)
< ∞
• SδH
t fair if
SδH
t =SδH
t
Sδ∗
t
= Et
(
HT
Sδ∗
T
)
Link with real economy !
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Law of the Minimal Price =⇒
Corollary 1.14
Minimal price for replicableHT is fair and given by
real world pricing formula
SδH
t = Sδ∗
t Et
(
HT
Sδ∗
T
)
.
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• most pricing concepts are unified and generalized by real world pricing
=⇒
actuarial pricing
risk neutral pricing
pricing with stochastic discount factor
pricing with numeraire change
pricing with deflator
pricing with state pricing density
pricing with pricing kernel
pricing with numeraire portfolio
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WhenHT independentof Sδ∗
T
=⇒ rigorous derivation of
• actuarial pricing formula
SδH
t = P (t, T )Et(HT )
with zero coupon bond
P (t, T ) = Sδ∗
t Et
(
1
Sδ∗
T
)
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• real world pricing formula =⇒
SδH
0 = E0
(
ΛT
S00
S0T
HT
)
Λt =S0
t
S00
supermartingale
1 = Λ0 ≥ E0(ΛT )
=⇒
SδH
0 ≤E0
(
ΛTS0
0
S0T
HT
)
E0(ΛT )
similar for any numeraire (hereS0 savings account)
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0
10
20
30
40
50
60
70
80
90
100
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 1.7: Discounted S&P500 total return index.
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1.2
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time
Radon-Nikodym derivativeTotal RN-Mass
Figure 1.8: Radon-Nikodym derivative and total mass of putative risk neutral
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• special case whensavings account is fair:
=⇒ ΛT = dQ
dPforms martingale; E0(ΛT ) = 1;
equivalent risk neutral probability measureQ exists;
Bayes’ formula =⇒risk neutral pricing formula
SδH
0 = EQ0
(
S00
S0T
HT
)
Harrison & Kreps (1979), Ingersoll (1987),
Constantinides (1992), Duffie (2001), Cochrane (2001),. . .
• otherwise “risk neutral price”≥ real world price
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-1
0
1
2
3
4
5
6
7
8
9
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time
ln(index)ln(savings account)
Figure 1.9: ln(S&P500 accumulation index) and ln(savings account).
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Extreme Maturity Bond
• short rate deterministic
• savings bond
P ∗(t, T ) =S0
t
S0T
• index S1 as numeraire portfolioSδ∗
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• fair zero coupon bond
Law of the Minimal Price
=⇒ real world pricing formula
P (t, T ) = Sδ∗
t Et
(
1
Sδ∗
T
)
• benchmarked fair zero coupon bond
P (t, T ) =P (t, T )
Sδ∗
t
martingale
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• discounted numeraire portfolio
Sδ∗
t =Sδ∗
t
S0t
numeraire portfolio for continuous market:
dSδ∗
t = αt dt+
√
Sδ∗
t αt dWt
time transformed squared Bessel process
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• model the drift of discounted index as
αt = α expη t=⇒
Sδ∗
t time transformed squared Bessel process of dimension four
=⇒ Minimal Market Model
MMM, see Pl.& Heath (2006)
• net growth rate
η ≈ 0.054 with R2 of 0.89
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-2
-1
0
1
2
3
4
5
1930 1940 1950 1960 1970 1980 1990 2000
time
ln(discounted index)trendline
Figure 1.10: Logarithm of discounted S&P500 and trendline.
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• normalized index
Yt =Sδ∗
t
αt
dYt = (1 − η Yt) dt+√
Yt dWt
• quadratic variation of√Yt
d√
Yt = . . .+1
2dWt
it∑
ℓ=1
(
√
Ytℓ−√
Ytℓ−1
)2
≈[√Y]
t=t
4
• scaling parameter α ≈ 0.01468 with R2 of 0.995
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0
5
10
15
20
25
30
1930 1940 1950 1960 1970 1980 1990 2000
time
[sqrt Y]trendline
Figure 1.11: Quadratic Variation of√Yt.
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• fair zero coupon bond
P (t, T ) = P ∗(t, T )
(
1 − exp
− 2 η Sδ∗
t
α (expη T − expη t)
)
• initial prices
P ∗(0, T ) = 0.0335
P (0, T ) = 0.00077
P (0,T )P ∗(0,T )
< 0.023 =⇒ 2.3%
no strong arbitrage
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0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
1930 1940 1950 1960 1970 1980 1990 2000
time
benchmarked savings bondbenchmarked fair zero coupon bond
Figure 1.12: Benchmarked savings bond and benchmarked fair zero coupon
bond.c© Copyright Eckhard Platen 10 BA - Toronto C 1 46
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1930 1940 1950 1960 1970 1980 1990 2000
time
savings bondfair zero coupon bond
savings account
Figure 1.13: Savings bond, fair zero coupon bond and savings account.
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-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)
Figure 1.14: Logarithms of fair zero coupon bond and savings account.
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• savings account isoutperformed systematically
by fair zero coupon bond
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-8
-7
-6
-5
-4
-3
-2
-1
0
1
1930 1940 1950 1960 1970 1980 1990 2000
time
ln(fair zero coupon bond)ln(savings account)
Figure 1.15: Logarithms of fair zero coupon bond and savings account.
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• self-financing hedge portfoliohedge ratio
δ∗t =
∂P (t, T )
∂Sδ∗
t
= P ∗(0, T ) exp
−2 η Sδ∗
t
α (expη T − expη t)
× 2 η
α (expη T − expη t)
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-8
-7
-6
-5
-4
-3
-2
-1
0
1
1930 1940 1950 1960 1970 1980 1990 2000
time
ln(zero coupon bond)ln(self financing hedge portfolio)
Figure 1.16: Logarithm of zero coupon bond and self-financinghedge port-
folio.c© Copyright Eckhard Platen 10 BA - Toronto C 1 52
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-2e-005
-1e-005
0
1e-005
2e-005
3e-005
4e-005
5e-005
6e-005
7e-005
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 1.17: Benchmarked profit and loss.
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0
0.2
0.4
0.6
0.8
1
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time
Figure 1.18: Fraction invested in the index.
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2 Continuous Financial Markets
• d sources of continuoustraded uncertainty
independent standard Wiener processes
W 1,W 2, . . . ,W d, d ∈ 1, 2, . . .
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Primary Security Accounts
Sjt - jth primary security account at timet,
j ∈ 0, 1, . . . , d
cum-dividend share value or savings account value,
dividends or interest are accumulated, reinvested
• vector process of primary security accounts
S = St = (S0t , . . . , S
dt )⊤, t ∈ [0, T ]
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• assumeunique strong solutionof SDE
dSjt = Sj
t
(
ajt dt+
d∑
k=1
bj,kt dW k
t
)
t ∈ [0,∞), Sj0 > 0, j ∈ 1, 2, . . . , d
• predictable, integrable appreciation rate processesaj
• predictable, square integrable volatility processesbj,k
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• savings account
S0t = exp
∫ t
0
rs ds
predictable, integrable short rate process
r = rt, t ∈ [0,∞)
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• Market price of risk
The unique invariant of the market
θt = (θ1t , θ
2t , . . . , θ
dt )⊤
solution of relation
bt θt = at − rt 1
where
at = (a1t , a
2t , . . . , a
dt )
⊤
1 = (1, 1, . . . , 1)⊤
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Assumption 2.1
Volatility matrixbt = [bj,kt ]dj,k=1 is invertible
=⇒• market price of risk equals
θt = b−1t (at − rt 1)
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• rewritejth primary security account SDE
dSjt = Sj
t
rt dt+d∑
k=1
bj,kt (θk
t dt+ dW kt )
t ∈ [0,∞), j ∈ 1, 2, . . . , d
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• strategy
δ = δt = (δ0t , . . . , δdt )⊤, t ∈ [0, T ]
predictable andS-integrable,
δjt number of units ofjth primary security account
• portfolio
Sδt =
d∑
j=0
δjt S
jt
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• Sδ self-financing⇐⇒
dSδt =
d∑
j=0
δjt dS
jt
changes in the portfolio value only due to gains from trading,
self-financing in each denomination
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• jth fraction
πjδ,t = δj
t
Sjt
Sδt
j ∈ 0, 1, . . . , d
d∑
j=0
πjδ,t = 1
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• 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=1
πjδ,t b
j,kt
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• logarithm of strictly positive portfolio
d ln(Sδt ) = gδ
t dt+
d∑
k=1
bkδ,t dW
kt
with growth rate
gδt = rt +
d∑
k=1
bkδ,t
(
θkt − 1
2bk
δ,t
)
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Numeraire Portfolio Sδ∗ = Growth Optimal Portfolio
=⇒ maximize the growth rate
gδt = rt +
d∑
k=1
d∑
j=1
πjδ,t b
j,kt
(
θkt − 1
2
d∑
ℓ=1
πℓδ,t b
ℓ,kt
)
=⇒ for eachj ∈ 1, 2, . . . , d
0 =
d∑
k=1
bj,kt
(
θkt −
d∑
ℓ=1
πℓδ∗,t b
ℓ,kt
)
=⇒θ⊤
t = π⊤δ∗,t bt
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Sincebt invertible=⇒
πδ∗,t = (π1δ∗,t, . . . , π
dδ∗,t)
⊤
= (b−1t )⊤ θt
=⇒ NP
dSδ∗
t = Sδ∗
t
([
rt +
d∑
k=1
(θkt )2
]
dt+
d∑
k=1
θkt dW
kt
)
finite NP exists
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Benchmarked Portfolios
Sδt =
Sδt
Sδ∗
t
dSδt = Sδ
t
rt dt+
d∑
k=1
d∑
j=0
πjδ,t b
j,kt
(
θkt dt+ dW k
t
)
dSδ∗
t = Sδ∗
t
(
rt dt+d∑
k=1
θkt
(
θkt dt+ dW k
t
)
)
dSδt = −Sδ
t
d∑
k=1
d∑
j=0
πjδ,t s
j,kt dW k
t
driftless =⇒ local martingale =⇒ supermartingalewhen nonnegative
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Theorem 2.2 Any nonnegative benchmarked portfolio is an
(A, P )-supermartingale(no upward trend).
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=⇒ NP is numeraire portfolio
use NP asbenchmark for investment and asnumeraire for pricing
• markets have thousands of primary security accounts
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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
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• 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 overT = 32 yearsd = 50 risky primary security accounts
σ = 0.15, r = 0.05
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0
5
10
15
20
25
0 5 10 15 20 25 30 35
time
Figure 2.1: Simulated primary security accounts.
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0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30 35
time
GOPB
Figure 2.2: Simulated NP and savings account.
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Diversified Portfolios
Fundamental phenomenon ofdiversification leads naturally toNP:
diversified =⇒ fractions “small”
Definition 2.3 A strictly positive portfolioSδ is adiversified portfolio if
∣
∣
∣πj
δ,t
∣
∣
∣≤ K2
d12+K1
a.s. for allj ∈ 0, 1, . . . , d andt ∈ [0, T ] for K1,K2 ∈ (0,∞) and
K3 ∈ 1, 2, . . ., independent ofd ∈ K3,K3 + 1, . . ..
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• jth benchmarked primary security account
Sjt =
Sjt
Sδ∗
t
dSjt = −Sj
t
d∑
k=1
sj,kt dW k
t
• (j, k)th specific volatility
sj,kt = θk
t − bj,kt
measuresspecific market risk of Sj with respect toW k
• θkt measuresgeneral market risk with respect toW k
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• kth total specific absolute volatility
skt =
d∑
j=0
|sj,kt |
regular =⇒ variance of specific returns “bounded”
Definition 2.4 A market is calledregular if
E(
(
skt
)2)
≤ K5.
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Tracking Rate
Variance of benchmarked portfolio returns
Rdδ(t) =
d∑
k=1
d∑
j=0
πjδ,t s
j,kt
2
Rdδ(t) - quantifies distance betweenSδ and Sδ∗
benchmarked NP constant=⇒
Rdδ∗
(t) = 0
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Approximate NP
tracking rate “small”
Definition 2.5 A strictly positive portfolioSδ is an approximate NP if
for all t ∈ [0, T ] and ε > 0
limd→∞
P(
Rdδ(t) > ε
)
= 0.
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Diversification Theorem
Theorem 2.6 In a regular market
any diversified portfolio is an approximate NP.
• model independent
• similarity to CLT
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Equal Weighted Index (EWI)
SδEWI holds equal fractions
πjδEWI,t
=1
d+ 1
j ∈ 0, 1, . . . , d
• tracking rate for multi-asset BS example:
RdδEWI
(t) =
d∑
k=1
(
1
d+ 1
(
θkt − bk,k
t
)
)2
= d
(
1
d+ 1
(
σ√d
− σ
))2
≤ σ2
(d+ 1)→ 0
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0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30 35
time
GOPEWI
Figure 2.3: Simulated NP and EWI.
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0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30 35
time
GOPindex
Figure 2.4: Simulated diversified accumulation index and NP.
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0
20,000
40,000
60,000
80,000
100,000
120,000
1/1/73 28/9/76 25/6/80 22/3/84 18/12/87 15/9/91 12/6/95 9/3/99 4/12/02 31/8/06
Figure 2.5: World industry sector stock market indices as primary security
accounts.
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EWI
MCI
DWI
WSI
0
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
1/1/73 28/9/76 25/6/80 22/3/84 18/12/87 15/9/91 12/6/95 9/3/99 4/12/02 31/8/06
$US
Figure 2.6: Constructed MCI, DWI, EWI and WSI. In the long term the WSI
and EWI outperform all other indices, Le& Platen (2006).
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0
50
100
150
200
250
300
350
400
450
0 5 9 14 18 23 27 32
Figure 2.7: Primary security accounts under the MMM.
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0
10
20
30
40
50
60
0 5 9 14 18 23 27 32
Figure 2.8: Benchmarked primary security accounts.
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0
10
20
30
40
50
60
70
80
90
100
0 5 9 14 18 23 27 32
EWI
GOP
Figure 2.9: NP and EWI.
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0
10
20
30
40
50
60
70
80
90
100
0 5 9 14 18 23 27 32
Market index
GOP
Figure 2.10: NP and market index.
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3 Portfolio Optimization
• NP best long term investment
Kelly (1956), Latane (1959), Breiman (1961),
Hakansson (1971b), Thorp (1972)
• optimal portfolio separated into two funds
Tobin (1958), Sharpe (1964)
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• mean-variance efficient portfolio
Markowitz (1959)
• intertemporal capital asset pricing model
Merton (1973a)
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Expected Utility Maximization
• utility function U(·), U ′(·) > 0 U ′′(·) < 0
• examples
power utility
U(x) =1
γxγ
for γ 6= 0 andγ < 1
log-utility
U(x) = ln(x)
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5 10 15 20x
-15
-10
-5
5
U
Figure 3.1: Examples for power utility (upper graph) and log-utility (lower
graph).
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• assumeS0 is scalar diffusion
• maximize expected utility
vδ = maxSδ∈V+
S0
E0
(
U(
SδT
))
V+S0
strictly positive, discounted, fair portfolios
Sδ0 = S0 > 0
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Theorem 3.1
Benchmarked, expected utility maximizing portfolio:
Sδt = u(t, S0
t ) = Et
(
U ′−1(
λ S0T
)
S0T
)
,
λ s.t. S0 = Sδ0 S
δ∗
0 = u(0, S0)Sδ∗
0
two fund separation with risk aversion coefficient
J δt =
1
1 − S0t
u(t,S0t )
∂u(t,S0t )
∂S0t
=⇒ invest 1
J δt
of wealth in NP and remainder in savings account
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• 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 ) = Et
(
U ′−1(
λ S0T
)
S0T
)
= Et
(
1
λ S0T
S0T
)
=1
λ
Lagrange multiplier
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λ =Sδ∗
0
S0
risk aversion coefficient
J δt = 1
• expected log-utility
vδ = E0
(
ln(
Sδ∗
T
))
= ln(λ) + ln(S0) +1
2
∫ T
0
E0
(
(
θ(s, Sδ∗
s ))2)
ds
if local martingale part forms a martingale
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• power utility
U(x) =1
γxγ
for γ < 1, γ 6= 0
U ′(x) = xγ−1
U ′−1(y) = y1
γ−1
U ′′(x) = (γ − 1)xγ−2
concave
u(t, S0t ) = Et
(
λ
Sδ∗
T
)1
γ−11
Sδ∗
T
= λ1
γ−1 Et
(
(
Sδ∗
T
)γ
1−γ
)
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Sδ∗ geometric Brownian motion
u(t, S0t ) = λ
1γ−1
(
Sδ∗
t
)
γ1−γ
×Et
(
exp
γ
1 − γ
(
θ2
2(T − t) + θ (WT −Wt)
))
= λ1
γ−1(
Sδ∗
t
)
γ1−γ exp
θ2
2
γ
(1 − γ)2(T − t)
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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 coefficient
J δt = 1 − γ
expected utility
vδ = E0
(
1
γ
(
SδT
)γ)
= exp
−θ2
2
γ
1 − γT
(S0)γ
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Utility Indifference Pricing
discounted payoff H = HS0
T
(not perfectly hedgable)
vδε,V = E0
(
U
(
(S0 − ε V )Sδ
T
S0
+ ε H
))
• assumeS0 is scalar diffusion
Definition 3.2 utility indifference price V s.t.
limε→0
vδε,V − vδ
0,V
ε= 0
Davis (1997)
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• Taylor expansion
vδε,V ≈ E0
(
U(
SδT
)
+ εU ′(
SδT
)
(
H − V SδT
S0
))
= vδ0,V + εE0
(
U ′(
SδT
)
H)
− V εE0
(
U ′(
SδT
) SδT
S0
)
SδT = Sδ
T /S0T = U ′−1
(
λ S0T
)
=⇒
vδε,V − vδ
0,V = ε
(
E0
(
λ S0T H
)
− V E0
(
λ S0T
SδT
S0
))
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=⇒• utility indifference price
V =E0
(
λ
Sδ∗
T
H)
E0
(
λ
Sδ∗
T
SδT
S0
) =E0
(
H
Sδ∗
T
)
1S0
S0
Sδ∗
0
independent ofU and of given model =⇒
• real world pricing formula
V = Sδ∗
0 E0
(
H
Sδ∗
T
)
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Hedging
• benchmarked derivative price
UH(t) = Et
(
H
Sδ∗
T
)
• real world martingale representation
H
Sδ∗
T
= UH(t) +
d∑
k=1
∫ T
t
xkH(s) dW k
s +MH(t)
MH -orthogonal martingale
E([
MH ,Wk]
t
)
= 0
Follmer-Schweizer decomposition, Follmer & Schweizer (1991)
least square projection
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Theorem 3.3 Derivative price
SδH
t = Sδ∗
t Et
(
H
Sδ∗
T
)
fractions
πδH(t) =
(
bδH(t)⊤ b−1
t
)⊤
with portfolio volatilities
bkδH
(t) =1
UH(t)
d∑
j=0
δjH(t) Sj
t bj,kt =
xkH(t)
UH(t)+ θk
t .
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4 Modeling Stochastic Volatility
• consider well diversified accumulation index≈ NP
dSδ∗
t = Sδ∗
t
(
rt dt+ |θt|(
|θt| dt+ dWt
))
d ln(
Sδ∗
t
)
=
(
rt dt+1
2|θt|2
)
dt+ |θt| dWt
• volatility
|θt| =
√
d
dt[ln (Sδ∗)]t
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0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
1982 1992 2002
time
Figure 4.1: Estimated volatility of WSI from 1973–2004.
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Log-Returns of a Stock Index
-10 -5 0 5 10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Figure 4.2: Histogram for S&P500 hourly deseasonalized log-returns.
fitted Gaussian density, much fatter tails
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Implied Volatilities
• European call for Black-Scholes model
cτ,K(0, S, σ)
maturity τ ∈ [0, T ]
strike priceK > 0
underlying indexS > 0
volatility σ
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• observed European call option price
Vc,τ,K(0, S0)
• implied volatility
σcallBS (0, S0, τ,K)
obtained such that
Vc,τ,K(0, S0) = cτ,K
(
0, S0, σcallBS (0, S0, τ,K)
)
by some root finding method
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100
200
300
time 0.6
0.8
1
1.2
1.4
moneyness
20
40
60
100
200
300
time
Figure 4.3: Implied volatilities for S&P500 three month options.
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implied volatility surface
0.5
0.75
1
1.25
1.5
KS
0.20.4
0.60.8
11.2
1.41.6T
0.22
0.24
0.26
0.28
0.5
0.75
1
1.25KS
0.20.4
0.60.8
11.2
1.4
Figure 4.4: Average S&P500 implied volatility surface.
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A Modified Constant Elasticity of Variance Model
• CEV Model
Cox (1975), Cox & Ross (1976)
Andersen & Andreasen (2000)
Lewis (2000)
Lo, Yuen & Hui (2000)
Brigo & Mercurio (2005)
Heath& Pl. (2005b)
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• classical risk neutral CEV
Beckers (1980)
Schroder (1989)
Delbaen & Shirakawa (2002)
certain problems in risk neutral pricing of derivatives
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Modified CEV Model
Heath& Pl. (2005b)
• savings account
dS0t = r S0
t dt
total market price for risk process
θt ≥ 0, t ∈ [0, T ]
• drifted Wiener process
dWθ(t) = θt dt+ dWt
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• NP
dSδ∗
t = Sδ∗
t (rt dt+ θt dWθ(t))
• NP volatility
θt = (Sδ∗
t )a−1 ψ
exponenta ∈ (−∞,∞)
scaling parameterψ > 0
stochastic fora 6= 1
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• hypothetical risk neutral CEV dynamics
dSδ∗
t = Sδ∗
t rt dt+ (Sδ∗
t )a ψ dWθ(t)
Wθ - Wiener process under a hypothetical risk neutral measurePθ
• If a = 1 =⇒ BS model
=⇒ equivalent risk neutral probability measurePθ
with market price of riskθt = ψ
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• real world CEV dynamics
dSδ∗
t =(
Sδ∗
t rt +(
Sδ∗
t
)2a−1ψ2)
dt+(
Sδ∗
t
)aψ dWt
modified CEV model remains fora < 1 strictly positive
this isnot the case for its “risk neutral” dynamics
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• squared Bessel process
Xt =(
Sδ∗
t
)2 (1−a)
for a 6= 1 with SDE
dXt =(
2(1 − a) rtXt + ψ2 (1 − a) (3 − 2a))
dt
+2ψ (1 − a)√
Xt dWt
anddimension
δ =3 − 2 a
1 − a
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• Radon-Nikodym derivative process
Λθ = Λθ(t), t ∈ [0, T ](state price density)
• hypothetical risk neutral measurePθ
dPθ
dP= Λθ(T )
Λθ(t) =Sδ∗
0
Sδ∗
t
S0t =
(
X0
Xt
)q
S0t
q =1
2 (1 − a)
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• squared Bessel process
dXt =(
2 (1 − a) rtXt + ψ2 (1 − a)(1 − 2 a))
dt
+2ψ (1 − a)√
Xt dWθ(t)
dimension
δθ =1 − 2 a
1 − a
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• dimension
risk neutral
δθ =1 − 2 a
1 − a
real world
δ =3 − 2 a
1 − a
• if a < 1 =⇒ δ > 2 andδθ < 2
if a > 1 =⇒ δ < 2 andδθ > 2
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-1 0 1 2 3
-4
-2
0
2
4
6
8
Risk Neutral
Real World
Figure 4.5: Dimensionsδ andδθ as a function of the exponenta.
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• hypothetical risk neutral probability
Pθ(A) =
∫
A
dPθ(ω) =
∫
A
dPθ(ω)
dP (ω)dP (w)
=
∫
A
Λθ(T ) dP (ω)
=⇒
Pθ(Ω) =
∫
Ω
Λθ(T ) dP (ω) = E0 (Λθ(T ))
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• for a < 1 =⇒ δθ < 2
=⇒ Λθ is strict supermartingale =⇒
Pθ(Ω) < Λθ(0) = 1
=⇒ Pθ is not a probability measure
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• for a 6= 1 P andPθ are not equivalent
=⇒ modified CEV model doesnot haveequivalent risk neutral
probability measure
• consider onlya < 1 since fora > 1 NP explodes
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Real World Pricing
• contingent claim
Hτ = Hτ (Sδ∗
τ ), E
(
Hτ
Sδ∗
τ
)
< ∞
• fair price
uHτ(t, Sδ∗
t ) = Sδ∗
t uHτ(t, Sδ∗
t )
benchmarked price
uHτ(t, Sδ∗
t ) = Et
(
Hτ (Sδ∗
τ )
Sδ∗
τ
)
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Martingale Representation
Hτ
Sδ∗
τ
= uHτ(τ, Sδ∗
τ )
= uHτ(t, Sδ∗
t ) +
∫ τ
t
(
Sδ∗
s
)aψ∂uHτ
(s, Sδ∗
s )
∂Sδ∗
dWs
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Stochastic Volatility Models
• volatility function of underlying diffusion=⇒ LV model
• volatility as separate, possibly correlated process
Hull & White (1987, 1988)Johnson & Shanno (1987)Scott (1987)Wiggins (1987)Chesney & Scott (1989)Melino & Turnbull (1990)Stein & Stein (1991)Hofmann, Pl.& Schweizer (1992)Heston (1993),. . .
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Empirical Evidence on Log-Returns
• Markowitz & Usmen (1996)
daily S&P500 → Studentt(4.5)
• Hurst & Platen (1997)
daily stock market indices→ Studentt(4)
• Fergusson& Pl. (2006), Pl.& Rendek (2008)
daily WSI, EWI104s with high accuracy→ Studentt(4)
• Breymann, Dias & Embrechts (2003)
copula of joint distribution of log-returns of exchange rates →Studentt copula with roughly four degrees of freedom
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−10 −5 0 5 10
10−4
10−3
10−2
10−1
Figure 4.6: Log-histogram of the EWI104s log-returns and Studentt density
with four degrees of freedom.c© Copyright Eckhard Platen 10 BA - Toronto C 4 132
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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
Figure 4.7: Log-likelihood function based on the EWI104s.
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Country Student-t NIG Hyperbolic VG ν
Australia 0.000000 76.770817 150.202282 181.632971 4.281222
Austria 0.000000 39.289103 77.505683 102.979330 4.725907
Belgium 0.000000 31.581622 60.867570 83.648470 4.989912
Brazil 2.617693 5.687078 63.800349 60.078395 2.713036
Canada 0.000000 47.506215 79.917741 104.297607 5.316154
Denmark 0.000000 41.509921 87.199686 114.853658 4.512101
Finland 0.000000 28.852844 68.677271 88.553080 4.305638
France 0.000000 26.303544 57.639325 80.567283 4.722787
Germany 0.000000 27.290205 52.667918 71.120798 5.005856
Greece 0.000000 60.432172 104.789463 125.601499 4.674626
Hong.Kong 0.000000 42.066531 100.834255 122.965326 3.930473
India 0.000000 74.773701 163.594078 198.002956 3.998713
Ireland 0.000000 77.727856 136.505582 170.013644 4.761519
Italy 0.000000 25.196598 55.185625 75.481897 4.668983
Japan 0.000000 37.630363 77.163656 102.967380 4.649745
Korea.S. 0.000000 120.904983 304.829431 329.854620 3.289204
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Malaysia 0.000000 79.714054 186.013963 221.061290 3.785195
Netherlands 0.000000 26.832761 51.625813 71.541627 5.084056
Norway 0.000000 42.243851 89.012090 115.059003 4.472349
Portugal 0.000000 61.177624 137.681039 165.689683 3.984860
Singapore 0.000000 36.379685 77.600590 98.124375 4.251472
Spain 0.000000 56.694545 109.533768 138.259224 4.517153
Sweden 0.000000 77.618384 143.420049 178.983373 4.546640
Taiwan 0.000000 41.162560 96.283628 115.186585 3.914719
Thailand 0.000000 78.250621 254.590254 267.508143 3.032038
UK 0.000000 26.693076 55.937248 80.678494 4.952843
USA 0.000000 40.678242 79.617362 100.901197 4.636661
Table 1:Ln test statistic of the EWI104s for different currency denominations
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Index Log-returns
• Student t distributed with about 4 degrees of freedom
• clustering
Find corresponding stationary volatility processes !
Kessler (1997), Prakasa Rao (1999)
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• normal mixing
log-return conditionally Gaussian distributed
with stochastic variance process
for small time step sizeh > 0
∆ ln(Sδ∗
t ) = ln
(
Sδ∗
t+h
Sδ∗
t
)
∼ N(
θ2t
2h, θ2
t h
)
whenθ2t has inverse gamma stationary density
=⇒ estimated log-return appears to be Studentt distributed
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A Class of Continuous Stochastic Volatility Models
• discounted NP
Sδ∗
t =Sδ∗
t
S0t
dSδ∗
t = Sδ∗
t (θ2t dt+ θt dWt)
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Particular Stochastic Volatility Models
• Hull and White’87 model
Hull & White (1987)
dθ2t = µ θ2
t dt+ ξ θ2t dWt
squared volatility does not have a stationary dynamics
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• Heston model
Hull & White (1988), Heston (1993)
mean reverting dynamics
k, θ andγ are positive constants
=⇒
dθ2t = k (θ2 − θ2
t ) dt+ γ |θt| dWt
θ20 > 0, κ
γ2 ≥ 12
=⇒ has as stationary density gamma density
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• Scott model
Scott (1987)
Stein & Stein (1991)
Ornstein-Uhlenbeck process
k, θ, γ positive constants
dθt = k (θ − θt) dt+ γ dWt
θ0 ∈ ℜ
=⇒ θ2t has as stationary density a gamma density
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• Wiggins model
Wiggins (1987)
Chesney & Scott (1989)
Melino & Turnbull (1990)
d ln(θt) = k(
ln(θ) − ln(θt))
dt+ γ dWt
θ0 > 0
θ2t has as stationary density a log-normal density
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• inverted gamma distribution
pθ2(y) =(12ν)
12 ν
ε2 Γ(12ν)
(
y
ε2
)− 12 ν−1
exp
−12ν ε2
y
y > 0, degrees of freedomν > 0, scaling parameterε
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• SDE for squared volatility
dθ2t = k θ
4(ξ−1)t (θ2 − θ2
t ) dt+ γ θ2 ξt dWt
degree of freedom
ν =2 (2 ξ − 1)
1 − ε2
θ2
=⇒ stationary density inverted gamma
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•32
volatility model
ξ = 32
Pl. (1997), Lewis (2000), Pl. (2001)
dθ2t = k θ2
t (θ2 − θ2t ) dt+ γ θ3
t dWt
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• ARCH diffusion
ξ = 1
Engle (1982), Nelson (1990) and Frey (1997)
dθ2t = k (θ2 − θ2
t ) dt+ γ θ2t dWt
continuous time limit of innovation process
GARCH(1,1) and NGARCH(1,1)
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Implied Volatilities for the ARCH Diffusion Model
Hurst (1997), Lewis (2000), Heath, Hurst& Pl. (2001)
GARCH(1,1) of Bollerslev (1986) whenW andW are independent
continuous time limit of NGARCH(1, 1) model of Engle & Ng (1993)
two-factor model
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80
90
100
110
120K 0.2
0.4
0.6
0.8
1
T
0.2
0.21
0.22
80
90
100
110
120K
Figure 4.8: Implied volatility surface for zero correlation.
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80
90
100
110
120K -0.4
-0.2
0
0.2
0.4
rho
0.18
0.2
0.22
80
90
100
110
120K
Figure 4.9: Effect of changing correlation on implied volatilities.
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8090
100
110
120K
5
10
15
20
k
0.1950.1975
0.2
0.2025
0.205
8090
100
110
120K
Figure 4.10: Effect of changing speed of adjustment on implied volatilities.
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8090
100
110
120K
0.5
1
1.5
2
gamma
0.19
0.2
0.21
8090
100
110
120K
Figure 4.11: Effect of changing volatility of the squared volatility on implied
volatilities.
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Moneyness
Opt
ion
Pric
e D
iffer
ence
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
-0.1
5-0
.10
-0.0
50.
00.
05
GLMRHestonBlack-Scholes
Figure 4.12: Option price differences between the ARCH diffusion (GLMR),
Heston and Black-Scholes models, with time to maturity of three months.
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Moneyness
Impl
ied
Vol
atili
ty
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
0.14
0.16
0.18
0.20
GLMRHestonBlack-Scholes
Figure 4.13: Implied volatilities of the ARCH diffusion (GLMR), Heston
and Black-Scholes models, with time to maturity of three months.
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5 Minimal Market Model
• search for parsimonious one-factor model
• 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 dW
kt
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0
10
20
30
40
50
60
70
80
90
100
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 5.1: Discounted S&P500 total return index.
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-2
-1
0
1
2
3
4
5
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 5.2: Logarithm of discounted S&P500.
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• logarithm of discounted NP
• is not a transformed Wiener process
• is not a process with independent increments
• has somefeedback effect
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Drift Parametrization
• discounted NP drift
αt = Sδ∗
t |θt|2
strictly positive, predictable
=⇒• volatility
|θt| =
√
αt
Sδ∗
t
leverage effect creates naturalfeedbackunder independent drift
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• discounted NP
dSδ∗
t = αt dt+
√
Sδ∗
t αt dWt
drift has economic meaning,
fundamental value
• transformed time
ϕt =1
4
∫ t
0
αs ds
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• squared Bessel process of dimension four
Xϕt= Sδ∗
t
dW (ϕt) =√αt dWt
dXϕ = 4 dϕ+ 2√
Xϕ dW (ϕ)
Revuz & Yor (1999)
economically founded dynamics inϕ-time
still no specific dynamics int-time
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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 time
ϕt =[√
Sδ∗
]
t
observable
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0
1
2
3
4
5
6
7
8
9
10
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 5.3: Empirical quadratic variation of the square rootof the discounted
NP.c© Copyright Eckhard Platen 10 BA - Toronto C 5 162
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Stylized Minimal Market Model
Pl. (2001, 2002, 2006)
• assumediscounted NP drift as
αt = α exp η t
• initial parameter α > 0
• net growth rate η
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• transformed time
ϕt =α
4
∫ t
0
exp η z dz
=α
4 η(exp η t − 1)
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0
1
2
3
4
5
6
7
8
9
10
1930 1940 1950 1960 1970 1980 1990 2000
time
[sqrt(discounted index)]theoretical variation
Figure 5.4: Fitted and observed transformed time.
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• normalized NP
Yt =Sδ∗
t
αt
dYt = (1 − η Yt) dt+√
Yt dWt
square root process of dimension four
=⇒ parsimonious model, long term viability
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=⇒• discounted NP
Sδ∗
t = Yt αt
• NP
Sδ∗
t = S0t S
δ∗
t = S0t Yt αt
• scaling parameterα0 = 0.01468
• net growth rateη = 0.054
• reference level1η
= 18.5
• speed of adjustmentη
• half life time of major displacement ln(2)η
≈ 13 years
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0
10
20
30
40
50
60
70
80
90
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 5.5: Normalized NP.
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• logarithm of discounted NP
ln(Sδ∗
t ) = ln(Yt) + ln(α0) + η t
• no need for extra volatility process in MMM
• realistic long term dynamics
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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
Platen (1997), Lewis (2000)
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0.1
0.15
0.2
0.25
0.3
0.35
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 5.6: Volatility of the NP under the stylized MMM.
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Transition Density of Stylized MMM
• transition density ofdiscounted 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
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• non-central chi-squaredistributed random variable:
y
ϕt − ϕs
=Sδ∗
t
ϕt − ϕs
with δ = 4 degrees of freedom andnon-centrality parameter:
x
ϕt − ϕs
=Sδ∗
s
ϕt − ϕs
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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
Figure 5.7: Transition density of squared Bessel process forδ = 4.
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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
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δ = 4 =⇒
P (t, T ) = P ∗T (t)
(
1 − exp
− Sδ∗
t
2 (ϕT − ϕt)
)
< P ∗T (t)
for t ∈ [0, T ), Platen (2002)
with time transform
ϕt =α
4 η(expη t − 1)
P (0, T ) = 0.00077
P ∗T (0) = 0.0335
P (0,T )P ∗
T (0)≈ 0.023
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1930 1940 1950 1960 1970 1980 1990 2000
time
savings bondfair zero coupon bond
savings account
Figure 5.8: Savings bond, fair zero coupon bond and savings account.
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-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)
Figure 5.9: Logarithms of savings bond and fair zero coupon bond.
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Forward Rates under the MMM
• forward rate
f(t, T ) = − ∂
∂Tln(P (t, T ))
= rT + n(t, T )
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• 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. (2005a), Miller& Pl. (2005)
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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
Figure 5.10: Market price of risk contribution in dependence on η andT .
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Free Snack from Savings Bond
• potential existence of weak form of arbitrage?
borrow amountP (0, T ) from savings account
Sδt = P (t, T ) − P (0, T ) expr t
such thatSδ0 = 0
SδT = 1 − P (0, T ) expr T > 0
lower bound
Sδt ≥ −P (0, T ) expr t
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• sinceSδt may become negative
allowed by strong arbitrage concept
• cheap thrills may exist
Loewenstein & Willard (2000)
MMM doesnot admit an equivalent risk neutral probability measure
=⇒there is afree lunch with vanishing risk
Delbaen & Schachermayer (2006)
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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
=S0
t
S0T
• Radon-Nikodym derivative
Λt =S0
t
S00
=Sδ∗
0
Sδ∗
t
strict (A, P )-supermartingaleunder MMM
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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
Figure 5.11: Radon-Nikodym derivative and total mass of putative risk neu-
tral measure.c© Copyright Eckhard Platen 10 BA - Toronto C 5 185
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• hypothetical risk neutral measure
total mass:
Pθ,T (Ω) = E0 (ΛT ) = 1 − exp
− Sδ∗
0
2ϕT
< Λ0 = 1
Pθ is not a probability measure
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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
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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)
Hulley & Pl. (2008)
Miller & Pl. (2008)
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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
Figure 5.12: Implied volatility surface for the stylized MMM.
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• 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
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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)
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• put-call paritybreaks down if one uses the savings bondP ∗T (t)
pT,K(t, Sδ∗
t ) < cT,K(t, Sδ∗
t ) − Sδ∗
t +K exp−r (T − t)
for t ∈ [0, T )
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• excellent hedge performance for extreme maturities
Hulley & Pl. (2008)
European calls
European puts
barrier options
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Comparison to Hypothetical Risk Neutral Prices
• hypothetical risk neutral price c∗T,K(t, Sδ∗
t )
of a European call option on the NP
• 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
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=⇒ 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 )
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• hypothetical risk neutral put-call parity
p∗T,K(t, Sδ∗
t ) = c∗T,K(t, Sδ∗
t ) − Sδ∗
t +K P ∗T (t)
sinceP ∗T (t) > P (t, T ) =⇒
pT,K(t, Sδ∗
t ) < p∗T,K(t, Sδ∗
t )
t ∈ [0, T )
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Difference in Asymptotic Put Prices
• hypothetical risk neutral prices can become extreme if NP value tends
towards zero
• asymptotic fair zero coupon bond
limS
δ∗
t →0PT (t, T )
a.s.= 0
for t ∈ [0, T )
• asymptotic fair European call
limS
δ∗
t →0cT,K(t, Sδ∗
t )a.s.= lim
Sδ∗
t →0Sδ∗
t cT,K(t, Sδ∗
t ) = 0
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• asymptotic fair putfair put-call parity =⇒
limS
δ∗
t →0pT,K(t, Sδ∗
t )a.s.= 0
• asymptotic hypothetical risk neutral puthypothetical risk neutral put-call parity =⇒
limS
δ∗
t →0p∗
T,K(t, Sδ∗
t )
a.s.= lim
Sδ∗
t →0
(
pT,K(t, Sδ∗
t ) +KS0
t
S0T
exp
− Sδ∗
t
2(ϕT − ϕt)
)
a.s.= K
S0t
S0T
> 0
dramatic differences can arise
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0
2
4
6
8
1930 1940 1950 1960 1970 1980 1990 2000
time
ln(K*savings bond)ln(risk neutral put)
ln(put)
Figure 5.13: Logarithms of savings bond timesK, risk neutral put and fair
put.c© Copyright Eckhard Platen 10 BA - Toronto C 5 199
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MMM with Random Market Activity
• squared Bessel process with dimensionδ = 4
Sδ∗ = Sδ∗
t , t ∈ [0,∞)
dSδ∗
t = αt dt+
√
αt Sδ∗
t dWt
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• NP
Sδ∗
t = S0t S
δ∗
t
• volatility
|θt| =
√
αt
Sδ∗
t
• discounted NP drift
αt to be modeled
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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
Figure 5.14: Implied volatility surface for the stylized MMM.
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0.5
0.75
1
1.25
1.5
KS
0.20.4
0.60.8
11.2
1.41.6T
0.22
0.24
0.26
0.28
0.5
0.75
1
1.25KS
0.20.4
0.60.8
11.2
1.4
Figure 5.15: Average S&P500 implied volatility surface.
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100
200
300
time 0.6
0.8
1
1.2
1.4
moneyness
20
40
60
100
200
300
time
Figure 5.16: Implied volatilities for S&P500 three month options.
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100
200
300
time 0.6
0.8
1
1.2
1.4
moneyness
1020304050
100
200
300
time
Figure 5.17: Implied volatilities for S&P500 one year options.
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• market activity mt
Breymann, Kelly& Pl. (2006)
Pl. & Rendek (2008)
αt = ξtmt
with
ξt = ξ0 exp η t
dmt = k(mt)β2 dt+ βmt
(
dWt +√
1 − 2 dWt
)
- correlation parameter
activity volatility β > 0
W - independent Wiener process
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Example:
• drift
k(mt) = (p− gmt)mt
2
• stationary density
pm(y) =gp−1
Γ(p− 1)yp−2 exp−g y
gamma density with meanp−1g
and variance1g
for p > 1 andg > 0
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0.5
1
1.5y
10
20
30
40
50
g
0
1
2
0.5
1
1.5y
Figure 5.18: Stationary density of market activitymt = y as function ofy
and speed of adjustment parameterg.c© Copyright Eckhard Platen 10 BA - Toronto C 5 208
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Zero Coupon Bond
• benchmarked zero coupon bond
PT (t, Sδ∗
t ,mt) = Et
(
1
Sδ∗
T
)
= Et
(
1
S0T S
δ∗
T
)
• zero coupon bond
PT (t, Sδ∗
t ,mt) = Sδ∗
t PT (t, Sδ∗
t ,mt) = S0t S
δ∗
t PT (t, Sδ∗
t ,mt)
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European Options on a Market Index
• put option price
pT,K(t, Sδ∗
t ,mt) = S0t S
δ∗
t Et
(
K
S0T S
δ∗
T
− 1
)+
effect of random market time on implied volatilities
curvature for the short dated implied volatility surface
Heath& Pl. (2005a)
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80
90
100
110
120
K 0.2
0.4
0.6
0.8
1
T
0.30.32
0.34
0.36
80
90
100
110K
Figure 5.19: Implied volatilities for put options on index asa function of
strikeK and maturityT .c© Copyright Eckhard Platen 10 BA - Toronto C 5 211
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80
90
100
110
120
K
2
4
6
8
10
g
0.3
0.32
0.34
80
90
100
110K
Figure 5.20: Implied volatilities for put options as a function of strikeK and
speed of adjustmentg.
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80
100
120K 2
4
6
8
10
T
0.30.3250.35
0.375
0.4
80
100
120K
Figure 5.21: Implied volatilities for long dated put optionsas a function of
strikeK and maturityT .
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6 Stylized Multi-Currency MMM
Pl. (2001)
Heath& Pl. (2005a)
d+ 1 currencies
Sδ∗
i (t) NP in ith currency
rit - ith short rate
θki (t) - market price of risk
Sδ∗
i (t) = αit Y
it S
ii(t)
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αit = αi
0 expηi t
Sii(t) = expri t
• ith normalized NP
dY it =
(
1 − ηi Y it
)
dt+
√
Y it
d+1∑
k=1
qi,k dW kt
scaling levelsqi,k
d+1∑
k=1
(qi,k)2 = 1
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• (i, j)th exchange rate
Xi,jt =
Sδ∗
i (t)
Sδ∗
j (t)=Y i
t αit S
ii(t)
Y jt α
jt S
jj (t)
dXi,jt = Xi,j
t
(ri − rj) dt+
d+1∑
k=1
qi,k
√
Y it
− qj,k
√
Y jt
(
qi,k
√
Y it
dt+ dW kt
)
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• jth savings account inith currency
Sji (t) = Xi,j
t Sjj (t)
dSji (t) = Sj
i (t)
ri dt+
d+1∑
k=1
qi,k
√
Y it
− qj,k
√
Y jt
(
qi,k
√
Y it
dt+ dW kt
)
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• stochastic market price of risk
θki (t) =
qi,k
√
Y it
• (j, k)th volatility
bj,ki (t) = θk
i (t) − θkj (t)
equity prices
commodity prices
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7 Valuing Guaranteed Minimum Death Benefits
• variable annuitiesfund-linked
tax-deferred
guarantees
• guaranteed minimum death benefits(GMDBs)
roll-ups:
original investment accrued at a pre-defined interest rate
ratchets:
death benefit based upon the highest anniversary account value
embedded options:
hedging against market downturn occurrence of death
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• GMDB put, floating put and/or look-back put option
long maturities of contracts
standard option pricing theory ?
no obvious best choice for price:
IFRS Phase II
CFO-Forum (2008)
CRO-Forum (2008)
Solvency II
Verheugen & Hines (2008)
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• pricing and hedging of GMDBs
benchmark approachin
Pl. (2002, 2004b), Pl.& Heath (2006)
Buhlmann& Pl. (2003)
best performing portfolio isbenchmark
doesnot require
existence of an equivalent risk neutral probability measure
uniquefair price is theminimal price
may provide significantly lower prices
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Financial Model
• underlying risky security (unit)
dSt = (µt − γ)Stdt+ σtSt dWt
γ ≥ 0 management fee rates
Wt - Brownian motion
• savings account
dBt = rtBt dt
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• market price of risk
θt =µt − γ − rt
σt
• risky asset
dSt
St
= rt dt+ σtθt dt+ σt dWt
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• self-financing portfolio
Vt = δ0tBt + δ1tSt
dVt = δ0t dBt + δ1t dSt
• fractions
π0t = δ0t
Bt
Vt
, π1t = δ1t
St
Vt
π0t + π1
t = 1
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• instantaneous portfolio return
dVt
Vt
= π0t
dBt
Bt
+ π1t
dSt
St
= rt dt+ π1tσt(θt dt+ dWt)
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• numeraire portfolio (NP) as benchmark
Long (1990)
V ∗ is best performing in several ways
• is growth optimal portfolio
Kelly (1956)
V ∗ = maxE(log VT )
π1∗t =
µt − γ − rt
σ2t
=θt
σt
dV ∗t
V ∗t
= rt dt+ θt(θt dt+ dWt)
Merton (1992)
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• NP best performing portfolio
lim supT →∞
1
Tlog
(
V ∗T
V ∗0
)
≥ lim supT →∞
1
Tlog
(
VT
V0
)
global well diversified index≈ NP
Pl. (2005b)
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0
10
20
30
40
50
60
70
80
90
100
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 7.1: Discounted S&P500 total return index.
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-2
-1
0
1
2
3
4
5
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 7.2: Logarithm of discounted S&P500.
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• benchmarked price
Ut = Ut
V ∗
t
U nonnegative =⇒
Ut ≥ Et
(
Us
)
t ≤ s
supermartingales (no upward trend)
Pl. (2002)
no strong arbitrage
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• benchmarked securities that form martingales are calledfair.
Ut = Et
(
Us
)
t ≤ s
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• Law of the Minimal Price
Pl. (2008)
Fair prices are minimal prices
V nonnegative fair portfolio
V ′ nonnegative portfolio
VT = V ′T
supermartingale property
=⇒Vt ≤ V ′
t
t ∈ [0, T ],
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0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
1930 1940 1950 1960 1970 1980 1990 2000
time
benchmarked savings bondbenchmarked fair zero coupon bond
Figure 7.3: Benchmarked savings bond and benchmarked fair zero coupon
bond.c© Copyright Eckhard Platen 10 BA - Toronto C 7 233
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• real world pricing formulafor
Et
(
HT
V ∗T
)
< ∞
UH(t) = V ∗t Et
(
HT
V ∗T
)
t ∈ [0, T ]
no risk neutral probability needs to exist
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• actuarial pricing formula
whenHT is independent ofV ∗T
=⇒UH(t) = P (t, T )Et(HT )
zero coupon bond
P (t, T ) = V ∗t Et
(
1
V ∗T
)
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• standard risk neutral pricing
Ross (1976), Harrison & Pliska (1983)
complete market
candidate Radon-Nikodym derivative
Λt =dQ
dP
∣
∣
∣
∣
At
=BtV
∗0
B0V∗
t
supermartingale =⇒
1 = Λ0 ≥ E0 (ΛT )
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=⇒
UH(0) = E
(
V ∗0
V ∗T
HT
)
= E
(
ΛT
B0
BT
HT
)
≤E(
ΛTB0
BTHT
)
E(ΛT )
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Only in the special case whenΛT martingale
=⇒ risk neutral pricing formula:
UH(0) = EQ
(
B0
BT
HT
)
Ignores any real trend !
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• Trends ignored also when using:
stochastic discount factor, Cochrane (2001);
deflator, Duffie (2001);
pricing kernel , Constantinides (1992);
state price density, Ingersoll (1987);
NP as Long (1990)
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• Example zero coupon bond
benchmarked fair zero coupon
P (t, T ) =P (t, T )
V ∗t
= Et
(
1
V ∗T
)
martingale (no trend)
for rt - deterministic
P (t, T ) = V ∗t Et
(
1
V ∗T
)
= exp
−T∫
t
rs ds
Et
(
V ∗t
V ∗T
)
,
V ∗t =
V ∗
t
Bt- discounted NP
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0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
1930 1940 1950 1960 1970 1980 1990 2000
time
benchmarked savings bondbenchmarked fair zero coupon bond
Figure 7.4: Benchmarked savings bond and benchmarked fair zero coupon
bond.c© Copyright Eckhard Platen 10 BA - Toronto C 7 241
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=⇒ downward trend reflects equity premium
=⇒ Λ strict supermartingale (downward trend), then
P (t, T ) < exp
−T∫
t
rs ds
=Bt
BT
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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
Figure 7.5: Radon-Nikodym derivative and total mass of putative risk neutral
measure.c© Copyright Eckhard Platen 10 BA - Toronto C 7 243
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1930 1940 1950 1960 1970 1980 1990 2000
time
savings bondfair zero coupon bond
savings account
Figure 7.6: Savings bond, fair zero coupon bond and savings account.
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-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)
Figure 7.7: Logarithms of savings bond and fair zero coupon bond.
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-2
-1
0
1
2
3
4
5
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 7.8: Logarithm of discounted S&P500.
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• discounted NP
V ∗t =
V ∗t
Bt
dV ∗t = V ∗
t θt (θt dt+ dWt)
= αt dt+
√
αt V∗
t dWt
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• discounted NP drift
αt := V ∗t θ
2t
=⇒ volatility
θt =
√
αt
V ∗t
reflects leverage effect
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• minimal market model
Pl. (2001, 2002)
MMM
discounted NP drift
assume
αt = α0 expηt
α0 > 0
net growth rateη > 0
=⇒ MMM
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-2
-1
0
1
2
3
4
5
1930 1940 1950 1960 1970 1980 1990 2000
time
ln(discounted index)trendline
Figure 7.9: Logarithm of discounted S&P500 and trendline.
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• normalized NP
Yt =V ∗
t
αt
dYt = (1 − ηYt) dt+√
Yt dWt
square root process of dimension four
Only one parameter !
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• volatility of NP
θt =1√Yt
=
√
αt
V ∗t
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• Zero coupon bond under the MMM
rt - deterministic
P (t, T ) = exp
−T∫
t
rs ds
(
1 − exp
− V ∗t
2(ϕ(T ) − ϕ(t))
)
Explicit formula !
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-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)
Figure 7.10: Logarithms of savings bond and fair zero coupon bond.
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0
0.2
0.4
0.6
0.8
1
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 7.11: Fraction invested in the savings account.
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-2e-005
-1e-005
0
1e-005
2e-005
3e-005
4e-005
5e-005
6e-005
7e-005
1930 1940 1950 1960 1970 1980 1990 2000
time
Figure 7.12: Benchmarked profit and loss.
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since
Et
(
ΛT
Λt
)
= Et
(
V ∗t
V ∗T
)
=
(
1 − exp
− V ∗t
2(ϕ(T ) − ϕ(t))
)
=⇒ P (t, T ) < Bt
BT
Overpricing by savings bond
no equivalent risk neutral probability measure
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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
Figure 7.13: Radon-Nikodym derivative and total mass of putative risk neu-
tral measure.c© Copyright Eckhard Platen 10 BA - Toronto C 7 258
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• European put option under the MMM
Hulley, Miller & Pl. (2005)
p(t, V ∗t , T,K, r) = V ∗
t Et
(
(K − V ∗τ )+
V ∗τ
)
p(t, V ∗t , T,K, r) = −V ∗
t χ2(d1; 4, l2)
+Ke−r(T −t)(
χ2(d1; 0, l2) − exp −l2/2)
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with
d1 =4ηK exp−r(T − t)
Btαt(expη(T − t) − 1)
and
l2 =4ηV ∗
t
Btαt(expη(T − t) − 1)
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χ2(x;n, l) non-central chi-square distribution function
n ≥ 0 degrees of freedom
non-centrality parameterl > 0
χ2(x;n, l) =
∞∑
k=0
exp− l
2
(
l2
)k
k!
1 −Γ(
x2; n+2k
2
)
Γ(
n+2k2
)
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• putative risk neutral price
p(t, V ∗t , T,K, r) = p(t, V ∗
t , T,K, r)+Ke−r(T −t) exp
− V ∗t
2(ϕ(T ) − ϕ(t))
risk neutral is overpricing
for V ∗t → 0 =⇒ p(t, V ∗
t , T,K, r) → 0
and
p(t, V ∗t , T,K, r) → K e−r(T −t) > 0
risk neutral ignores trends
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0
2
4
6
8
1930 1940 1950 1960 1970 1980 1990 2000
time
ln(K*savings bond)ln(risk neutral put)
ln(put)
Figure 7.14: Logarithms of savings bond timesK, risk neutral put and fair
put.c© Copyright Eckhard Platen 10 BA - Toronto C 7 263
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Guaranteed Minimum Death Benefit (GMDB)
• payout to the policyholder
max(egτV0, Vτ )
time of deathτ
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 τ
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• insurance chargesξ ≥ 0
=⇒ policyholder’s unit value
Vt = e−ξtV ∗t
=⇒
max(egτV0, Vτ ) = max(egτV0, e−ξτV ∗
τ )
= e−ξτ max(e(g+ξ)τV0, V∗
τ )
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insurance company invests the entire fund valueV in the NP V ∗
=⇒ payoff
HT = GMDBT = e−ξT[
(e(g+ξ)TV ∗0 − V ∗
T )+ + V ∗T
]
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fair valueGMDB0 of the total claim
real world pricing formula
GMDB0 = V ∗0 E
(
GMDBT
V ∗T
)
= e−ξT(
p(0, V ∗0 , T, e
(g+ξ)TV ∗0 , r) + V ∗
0
)
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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
Figure 7.15: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 andY0 = 20.
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• lifetime τ is stochastic
GMDB0 = V ∗0 E
(
E
(
GMDBτ
V ∗τ
∣
∣
∣
∣
Fτ
))
GMDB0 =
∫ T
0
(
p(0, V ∗0 , t, e
(g+ξ)tV ∗0 , r) + V ∗
0
)
e−ξtfτ (t) dt
fτ (·) - mortality density
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0 20 40 60 80 100 12010
−5
10−4
10−3
10−2
10−1
100
Age x
log(
q x)
malefemale
Figure 7.16: German mortality data
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rational investors will lapse when embedded put-option outof the moneyGMDBs have stochastic maturityTitanic options, Milevsky & Posner (2001)
• roll-up GMDB payoff
Ht =
(1 − βt)V∗
t , if lapsed at timet,
max(egτV0, V∗
τ ), if death occurs at timet = τ
surrender chargeβt
βt =
(8 − ⌈t⌉)%, t ≤ 7,
0, t > 7
no credit riskno accumulation phase
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20 30 40 50 60 70 801
1.05
1.1
1.15
Age x
GM
DB
0
malefemale
Figure 7.17: Value of the GMDB under the MMM for male and female pol-
icyholders agedx, assuming an irrational lapsation ofl = 1%.
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8 Markets with Event Risk
Jump Diffusion Markets
• filtered probability space(Ω,A,A, P )
• continuous trading uncertainty
m ∈ 1, 2, . . . , d
Wiener processesW k = W kt , t ∈ [0,∞), k ∈ 1, 2, . . . ,m
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• events
counting processpk = pkt , t ∈ [0,∞)
intensity hk = hkt , t ∈ [0,∞)
hkt > 0
∫ t
0
hks ds < ∞
• jump martingale
dqkt =
(
dpkt − hk
t dt) (
hkt
)− 12
k ∈ 1, 2, . . . , d−m
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E
(
(
qkt+∆ − qk
t
)2 ∣∣
∣At
)
= ∆
• trading uncertainty
W = W t = (W 1t , . . . , W
mt , q1
t , . . . , qd−mt )⊤, t ∈ [0,∞)
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Primary Security Accounts
dSjt = Sj
t−
(
ajt dt+
d∑
k=1
bj,kt dW k
t
)
j ∈ 1, 2, . . . , d
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Assumption 8.1
bj,kt ≥ −
√
hk−mt
for all t ∈ [0,∞), j ∈ 1, 2, . . . , d andk ∈ m+ 1, . . . , d.
Assumption 8.2 bt is invertible
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• market price of risk
θt = (θ1t , . . . , θ
dt )⊤ = b−1
t [at − rt 1]
at = (a1t , . . . , a
dt )
⊤
1 = (1, . . . , 1)⊤
=⇒
dSjt = Sj
t−
(
rt dt+
d∑
k=1
bj,kt (θk
t dt+ dW kt )
)
j ∈ 0, 1, . . . , d
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• strategy
δ = δt = (δ0t , . . . , δdt )⊤, t ∈ [0,∞)
• portfolio
Sδt =
d∑
j=0
δjt S
jt
• self-financing if
dSδt =
d∑
j=0
δjt dS
jt
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Growth Optimal Portfolio
• fraction
πjδ,t = δj
t
Sjt
Sδt
πδ,t = (π1δ,t, . . . , π
dδ,t)
⊤
dSδt = Sδ
t−
rt dt+ π⊤δ,t− bt (θt dt+ dW t)
strictly positive ⇐⇒
d∑
j=1
πjδ,t b
j,kt > −
√
hk−mt
k ∈ m+ 1, . . . , d
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d ln(Sδt ) = gδ
t dt+
m∑
k=1
d∑
j=1
πjδ,t b
j,kt dW k
t
+d∑
k=m+1
ln
1 +d∑
j=1
πjδ,t−
bj,kt
√
hk−mt
√
hk−mt dW k
t
• growth rate
gδt = rt +
m∑
k=1
d∑
j=1
πjδ,t b
j,kt θk
t − 1
2
d∑
j=1
πjδ,t b
j,kt
2
+
d∑
k=m+1
d∑
j=1
πjδ,tb
j,kt
(
θkt −
√
hk−mt
)
+ln
1+
d∑
j=1
πjδ,t
bj,kt
√
hk−mt
hk−mt
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Assumption 8.3√
hk−mt > θk
t
k ∈ m+ 1, . . . , d
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• fractions of candidate benchmark
πδ∗,t = (π1δ∗,t, . . . , π
dδ∗,t)
⊤ =(
c⊤t b
−1t
)⊤
ckt =
θkt for k ∈ 1, 2, . . . ,m
θkt
1−θkt (h
k−mt )−
12
for k ∈ m+ 1, . . . , d
if θkt
√
hk−mt
≪ 1
=⇒ ckt ≈ θk
t
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• candidate benchmark
dSδ∗
t = Sδ∗
t−(
rt dt+ c⊤t (θt dt+ dW t)
)
= Sδ∗
t−
(
rt dt+
m∑
k=1
θkt (θk
t dt+ dW kt )
+
d∑
k=m+1
θkt
1 − θkt (hk−m
t )− 12
(θkt dt+ dW k
t )
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Definition 8.4 Sδ ∈ V+ NP if gδt ≤ g
δt for all Sδ ∈ V+.
=⇒
Sδ∗ is a NP.
=⇒
gδ∗
t = rt+1
2
m∑
k=1
(θkt )2−
d∑
k=m+1
hk−mt
ln
1 +θk
t√
hk−mt − θk
t
+θk
t√
hk−mt
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for θkt
√
hk−mt
≪ 1 asymptotically
gδ∗
t ≈ rt +1
2
d∑
k=1
(θkt )2 = rt +
|θt|22
d ln(Sδ∗
t ) ≈ gδ∗
t dt+
d∑
k=1
θkt dW
kt
dSδ∗
t ≈ Sδ∗
t
(
rt +
d∑
k=1
θkt
(
θkt dt+ dW k
t
)
)
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• jump diffusion market (JDM) SJD(d) = S, a, b,~r, h,A, P
• benchmarked portfolio
Sδt =
Sδt
Sδ∗
t
dSδt =
m∑
k=1
d∑
j=1
δjt S
jt b
j,kt − Sδ
t θkt
dW kt
+
d∑
k=m+1
d∑
j=1
δjt S
jt− b
j,kt
1 − θkt
√
hk−mt
− Sδt− θ
kt
dW kt
driftless
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• benchmarked savings account
dS0t = −S0
t−
d∑
k=1
θkt dW
kt
driftless
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benchmarked portfolioSδ driftless =⇒ (A, P )-local martingale
Ansel & Stricker (1994)
=⇒
Theorem 8.5 In a JDM a nonnegativeSδ is an (A, P )-supermartin-
gale
Sδt ≥ E
(
Sδτ
∣
∣
∣At
)
τ ∈ [0,∞) andt ∈ [0, τ ].
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A nonnegative portfolio is a strongarbitrage if it can get out of zero.
Corollary 8.6
A JDM does not allow nonnegative portfolios that permit strong arbitrage.
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Law of the Minimal Price
Corollary 8.7 Aτ -measurable payoffH withE( H
Sδ∗
τ
|A0) < ∞. If
there exists a fair nonnegative portfolioSδ that replicates the payoff
Sδτ = H,
then this is the minimal nonnegative replicating portfolio.
=⇒• fair portfolios provide the cheapest hedge
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H Aτ -measurable payoff, withE( H
Sδ∗
τ
) < ∞
=⇒
real world pricing formula
UH(t) = Sδ∗
t E
(
H
Sδ∗
τ
∣
∣
∣
∣
At
)
• equivalent to riskneutral pricing as long as candidate Radon-Nikodym
derivative S0t
S00
forms(A, P )-martingale
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• zero coupon bond
P (t, T ) = E
(
Sδ∗
t
Sδ∗
T
∣
∣
∣
∣
At
)
• benchmarked zero coupon bond
P (t, T ) = P (t,T )
Sδ∗
t
dP (t, T ) = −P (t−, T )
d∑
k=1
σk(t, T ) dW kt
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=⇒
ln(P (t, T )) = ln(P (0, T )) −m∑
k=1
(∫ t
0
σk(s, T )dW ks +
1
2
∫ t
0
(σk(s, T ))2 ds
)
+
d∑
k=m+1
(
∫ t
0
σk(s, T )√
hk−ms ds+
∫ t
0
ln
(
1 − σk(s, T )√
hk−ms
)
dpks
)
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• forward rate
f(t, T ) = − ∂
∂Tln(P (t, T )) = − ∂
∂Tln(P (t, T ))
=⇒• forward rate equation
f(t, T ) = f(0, T ) +
m∑
k=1
∫ t
0
(
∂
∂Tσk(s, T )
)
(
σk(s, T ) ds+ dW ks
)
+
d∑
k=m+1
∫ t
0
1
1 − σk(s,T )√h
k−ms
∂
∂Tσk(s, T )
(
σk(s, T ) ds+ dW ks
)
similar to HJM equation
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Sequences of Diversified Portfolios
Sequence(Sδ(d))d∈N is a sequence of DPs if
∣
∣
∣πj
δ,t
∣
∣
∣≤ K2
d12+K1
for all j ∈ 0, 1, . . . , d, t ∈ [0,∞) and d ∈ d0, d0 + 1, . . ..
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• benchmarked primary security account
dSj
(d)(t) = −Sj
(d)(t−)
d∑
k=1
σj,k
(d)(t) dWkt
• kth total specific volatility
σk(d)(t) =
d∑
j=0
|σj,k
(d)(t)|
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Sequence of JDMsregular if
E
(
(
σk(d)(t)
)2)
≤ K5
for all t ∈ [0,∞), d ∈ N and k ∈ 1, 2, . . . , d.
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Sequence of Approximate NPs
• tracking rate
Rδ(d)(t) =
d∑
k=1
d∑
j=0
πjδ,t σ
j,k
(d)(t)
2
Sδ(d) is NP ⇐⇒ Sδ
(d) ≡ constant
⇐⇒Rδ
(d)(t) = 0
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(Sδ(d))d∈N is a sequence ofapproximate NPsif
limd→∞
Rδ(d)(t)
P= 0
• expected tracking rate
eδ(d)(t) = E
(
Rδ(d)(t)
)
(Sδ(d))d∈N has vanishing expected tracking rate, if
limd→∞
eδ(d)(t) = 0
=⇒
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limd→∞
P(
Rδ(d)(t) > ε
)
≤ limd→∞
1
εeδ(d)(t) = 0
Lemma 8.8 For a sequence of JDMs, any sequence of strictly positive
portfolios with vanishing expected tracking rate is a sequence of approximate
NPs.
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Diversification Theorem
Platen (2005a)
For a regular sequence of JDMs(SJD(d))d∈N , each sequence(Sδ
(d))d∈N
of DPs is a sequence of approximate NPs. Moreover
eδ(d)(t) ≤ (K2)
2K5
d2K1.
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Diversification in an MMM Setting
• savings account
S0(d)(t) = expr t
• discounted NP drift
αδ∗
t = α0 expη t
• jth benchmarked primary security account
Sj
(d)(t) =1
Y jt α
δ∗
t
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• SR process
dY jt =
(
1 − η Y jt
)
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)
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0
50
100
150
200
250
300
350
400
450
0 5 9 14 18 23 27 32
Figure 8.1: Primary security accounts under the MMM.
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0
10
20
30
40
50
60
0 5 9 14 18 23 27 32
Figure 8.2: Benchmarked primary security accounts.
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0
10
20
30
40
50
60
70
80
90
100
0 5 9 14 18 23 27 32
EWI
GOP
Figure 8.3: NP and EWI.
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0
10
20
30
40
50
60
70
80
90
100
0 5 9 14 18 23 27 32
Market index
GOP
Figure 8.4: NP and market index.
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Real World Pricing for Two Market Models
Specifying a Continuous NP
market prices of event risks are zero
θkt = 0
NP
dSδ∗
t = Sδ∗
t
(
rt dt+
m∑
k=1
θkt
(
θkt dt+ dW k
t
)
)
Sδ∗
0 = 1
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Benchmarked Primary Security Accounts
dSjt = −Sj
t−
d∑
k=1
σj,kt dW k
t
Sjt = Sj
0 exp
−1
2
∫ t
0
m∑
k=1
(
σj,ks
)2ds−
m∑
k=1
∫ t
0
σj,ks dW k
s
× exp
∫ t
0
d∑
k=m+1
σj,ks
√
hk−ms ds
d∏
k=m+1
pk−mt∏
l=1
1 −σj,k
τkl −
√
hk−m
τkl −
pivotal objects of study
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• simplifying notation:
|σjt | =
√
√
√
√
m∑
k=1
(
σj,kt
)2
• aggregate continuous noise processes
W jt =
m∑
k=1
∫ t
0
σj,ks
|σjs|dW k
s
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• generalized volatility matrix bt = [bj,kt ]dj,k=1 invertible
bj,kt = θk
t − σj,kt
for k ∈ 1, 2, . . . ,m
bj,kt = −σj,k
t
k ∈ m+ 1, . . . , d
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• constant intensities
hkt = hk > 0
σj,kt = σj,k ≤
√hk−m
• benchmarkedjth primary security account:
Sjt = Sj,c
t Sj,dt
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• continuous part
Sj,ct = Sj
0 exp
−1
2
∫ t
0
|σjs|2 ds−
∫ t
0
|σjs| dW j
s
• compensated jump part
Sj,dt = exp
d∑
k=m+1
σj,k√hk−m t
d∏
k=m+1
(
1 − σj,k
√hk−m
)pk−mt
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Merton Model
rt = r, σj,kt = σj,k
Sj,ct = Sj
0 exp
−1
2|σj|2 t− |σj| W j
t
Girsanov’s theorem, see Section 9.5
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A Minimal Market Model with Jumps
Hulley, Miller & Pl. (2005)
• discounted NP drift
αj(t) = αj0 expηjt
ηj - net growth rate
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• jth square root process
dY jt =
(
1 − ηjY jt
)
dt+
√
Y jt dW
jt
• continuous part
Sj,ct =
1
αj(t)Y jt
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• time transformation
ϕj(t) = ϕj0 +
1
4
∫ t
0
αj(s) ds
• squared Bessel process of dimension four
Xj
ϕj(t)= αj(t)Y j
t =1
Sj,ct
S0 is a strict local martingale
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Zero Coupon Bonds
P (t, T ) = Sδ∗
t E
(
1
Sδ∗
T
∣
∣
∣
∣
At
)
=1
S0t
E
(
exp
−∫ T
t
rs ds
S0T
∣
∣
∣
∣
At
)
• MM case
P (t, T ) = exp−r(T−t) 1
S0t
E(
S0T
∣
∣
∣At
)
= exp−r(T−t)
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• MMM case
λjt =
1
Sjt (ϕ
j(t) − ϕj(T ))
S0T = S0,c
T
P (t, T ) = E
(
exp
−∫ T
t
rs ds
∣
∣
∣
∣
At
)
1
S0t
E(
S0T
∣
∣
∣At
)
= E
(
exp
−∫ T
t
rs ds
∣
∣
∣
∣
At
)
(
1 − exp
−1
2λ0
t
)
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Forward Contracts
Sδ∗
t E
(
F j(t, T ) − SjT
Sδ∗
T
∣
∣
∣
∣
At
)
= 0
forward price
F j(t, T ) =Sδ∗
t E(
SjT
∣
∣
∣At
)
Sδ∗
t E(
1
Sδ∗
T
∣
∣
∣At
) =
Sjt
P (t,T )1
Sjt
E(
SjT
∣
∣At
)
if Sjt > 0
0 if Sjt = 0
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• MM caseF j(t, T ) = Sj
t expr(T − t)standard expression
• MMM case
1
Sjt
E(
SjT
∣
∣
∣At
)
=1
Sj,ct
E(
Sj,cT
∣
∣
∣At
) 1
Sj,dt
E(
Sj,dT
∣
∣
∣At
)
= 1 − exp
−1
2λj
t
forward price
F j(t, T ) = Sjt
1 − exp
−12λj
t
1 − exp−1
2λ0
t
(
E
(
exp
−∫ T
t
rs ds
∣
∣
∣
∣
At
))−1
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Asset-or-Nothing Binaries
Ingersoll (2000), Buchen (2004) and Buchen & Konstandatos (2005)
rt = r
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Aj,k(t, T,K) = Sδ∗
t E
(
1SjT ≥K
SjT
Sδ∗
T
∣
∣
∣
∣
At
)
=Sj
t
Sjt
E(
1SjT ≥K(S0
T )−1S0T S
jT
∣
∣
∣At
)
=Sj
t
Sj,ct
E
(
1Sj,cT ≥g(p
k−m
T −pk−mt )S0
T
× exp
σj,k√hk−m (T − t)
(
1 − σj,k
√hk−m
)pk−m
T −pk−mt
Sj,cT
∣
∣
∣
∣
At
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Aj,k(t, T,K) =
=
∞∑
n=0
exp−hk−m(T − t)
(hk(T − t))n
n!exp
σj,k√hk−m(T − t)
×(
1 − σj,k
√hk−m
)nSj
t
Sj,ct
E(
1Sj,cT ≥g(n)S0
T Sj,cT
∣
∣
∣At
)
for all t ∈ [0, T ], where
g(n) =K
S0t S
j,dt
exp
−(
r + σj,k√hk−m
)
(T − t)
(
1 − σj,k
√hk−m
)−n
for all n ∈ N
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MM case
Aj,k(t, T,K) =∞∑
n=0
exp−hk−m (T − t)
(hk−m (T − t))n
n!
× exp
σj,k√hk−m (T − t)
(
1 − σj,k
√hk−m
)n
Sjt N(d1(n))
for all t ∈ [0, T ], where
d1(n) =
ln(
Sjt
K
)
+
r + σj,k√hk−m + n
ln
(
1− σj,k√hk−m
)
T −t+ 1
2
(
σ0,j)2
(T − t)
σ0,j√T − t
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σi,j =√
|σi|2 − 2 i,j |σi| |σj| + |σj|2
i,j - correlation
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MMM case
S0 and Sj,c are independent
Aj,k(t, T,K) =
∞∑
n=0
exp−hk−m(T − t)
(hk−m(T − t))n
n!
× exp
σj,k√hk−m(T − t)
(
1 − σj,k
√hk−m
)n
×Sjt
(
G′′0,4
(ϕ0(T ) − ϕ0(t)
g(n);λj
t , λ0t
)
− exp
−1
2λj
t
)
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G′′0,4(x;λ, λ
′) equals probabilityP ( ZZ′
≤ x)
non-central chi-square distributed random variableZ ∼ χ2(0, λ)
non-central chi-square distributed random variableZ′ ∼ χ2(4, λ′)
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Bond-or-Nothing Binaries
MM case
Bj,k(t, T,K) =
∞∑
n=0
exp−hk−m (T − t)(hk−m (T − t))n
n!
×K exp−r(T − t)N(d2(n))
for all t ∈ [0, T ], where
d2(n) =
ln(
Sjt
K
)
+
(
r + σj,k√hk−m + n
ln(1− σj,k√hk−m
)
T −t− 1
2
(
σ0,j)2
)
(T − t)
σ0,j√T − t
= d1(n) − σ0,j√
T − t
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MMM case
Bj,k(t, T,K)
=
∞∑
n=0
exp−hk−m(T − t)(hk−m (T − t))n
n!
×K exp−r(T − t)(
1 −G′′0,4
(
(ϕj(T ) − ϕj(t))g(n);λ0t , λ
jt
))
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European Call Options
cj,kT,K(t) = Sδ∗
t E
(
SjT −K
)+
Sδ∗
T
∣
∣
∣
∣
At
= Sδ∗
t E
(
1SjT ≥K
SjT −K
Sδ∗
T
∣
∣
∣
∣
At
)
= Aj,k(t, T,K) −Bj,k(t, T,K)
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MM case
cj,kT,K(t) =
∞∑
n=0
exp−hk−m(T − t)(hk−m(T − t))n
n!
×(
exp
σj,k√hk−m (T − t)
(
1 − σj,k
√hk−m
)n
Sjt N(d1(n))
−K exp−r(T − t)N(d2(n))
)
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MMM case
cj,kT,K(t) =
∞∑
n=0
exp−hk−m(T − t)(hk−m(T − t))n
n!
×[
exp
σj,k√hk−m (T − t)
(
1 − σj,k
√hk−m
)n
×Sjt
(
G′′0,4
((
ϕ0(T ) − ϕ0(t)
g(n)
)
;λjt , λ
0t
)
− exp
−1
2λj
t
)
−K exp−r(T − t)(
1 −G′′0,4
(
(ϕj(T ) − ϕj(t))g(n);λ0t , λ
jt
))
]
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Defaultable Zero Coupon Bonds
maturityT
defaults at the first jump timeτk−m1 of pk−m
zero recovery
P k−m(t, T ) = Sδ∗
t E
(
1τk−m1 >T Sδ∗
T
∣
∣
∣
∣
At
)
= Sδ∗
t E
(
1
Sδ∗
T
∣
∣
∣
∣
At
)
E(
1τk−m1 >T
∣
∣
∣At
)
= P (t, T )P (pk−mT = 0
∣
∣At)
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• conditional probability of survival
P(
pk−mT = 0
∣
∣At
)
= E(
1pk−mt =0 1p
k−m
T −pk−mt =0
∣
∣
∣At
)
= 1pk−mt =0 P
(
pk−mT − pk−m
t = 0∣
∣
∣At
)
= 1pk−mt =0E
(
exp
−∫ T
t
hk−ms ds
∣
∣
∣
∣
∣
At
)
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