Stochastic Volatility Modelling
Bruno Dupire
Nice 14/02/03
Bruno Dupire
Structure of the talk1.Model review
2.Forward equations
3.Forward volatility
4.Arbitrageable & admissible smile dynamics
Bruno Dupire
Model RequirementsHas to fit static/current data:
• Spot Price• Interest Rate Structure• Implied Volatility Surface
Should fit dynamics of:• Spot Price (Realistic Dynamics)• Volatility surface when prices move• Interest Rates (possibly)
Has to be• Understandable• In line with the actual hedge• Easy to implement
Bruno Dupire
• : Bachelier 1900
• : Black-Scholes 1973
• : Merton 1973
• : Merton 1976
• : Hull&White 1987
A Brief History of Volatility (1)
Qt
t
t dWtdttrS
dS )( )(
Qtt dWdS
tLt
Qtt
t
t
dZdtVVad
dWdtrS
dS
)(
2
Qt
t
t dWdtrS
dS
dqdWdtkrS
dS Qt
t
t )(
Bruno Dupire
• Dupire 1992, arbitrage modelwhich fits term structure of volatility given by log contracts.
• Dupire 1993, minimal model to fit current volatility surface.
A Brief History of Volatility (2)
Qt
Tt
Qtt
t
t
dZdtT
tLd
dWS
dS
2
2
22
2
,2
2
2 2,
),( )(
K
CK
KC
rKTC
TK
dWtSdttrS
dS
TK
Qt
t
t
Bruno Dupire
• Heston 1993, semi-analytical
formulae.
• Dupire 1996 (UTV), Derman 1997,
stochastic volatility model
which fits current volatility
surface HJM treatment.
A Brief History of Volatility (3)
ttt
t
ttt
t
dZdtbd
dWdtrS
dS
)(
222
2
K
V
dZbdtdV
TK
QtTKTKTK
T
,
,,,
S
tolconditiona
varianceforward ousinstantane :
Bruno Dupire
• Bates 1996, Heston + Jumps:
• Local volatility + stochastic volatility:– Markov specification of UTV– Reech Capital Model: f is quadratic– SABR: f is a power function
A Brief History of Volatility (4)
ttt
t
ttt
t
dWdtbd
dqdZdtrS
dS
)(
222
2
Qtt
t
t dZtSfdtrS
dS ,
Bruno Dupire
• Lévy Processes• Stochastic clock:
– VG (Variance Gamma) Model: BM taken at random time (t)
– CGMY model: same, with integrated square root process
• Jumps in volatility• Path dependent volatility• Implied volatility modelling• Incorporate stochastic interest rates• n dimensional dynamics of • n assets stochastic correlation
A Brief History of Volatility (5)
Bruno Dupire
BWD Equation: price of one option CK0,T0
for different (S,t)
FWD Equation: price of all options CK,T for current (S0,t0)
Advantage of FWD equation:• If local volatilities known, fast computation
of implied volatility surface,• If current implied volatility surface known,
extraction of local volatilities,• Understanding of forward volatilities
and how to lock them.
Forward Equations (1)
Bruno Dupire
Several ways to obtain them:
• Fokker-Planck equation:– Integrate twice Kolmogorov Forward Equation
• Tanaka formula:– Expectation of local time
• Replication– Replication portfolio gives a much
more financial insight
Forward Equations (2)
Bruno Dupire
dT
dT/2
FWD Equation: dS/S = (S,t) dW
T
CCCS TKTTKT
TK ,,
, Define
2
22
2
2
,
K
CK
TK
T
C
Equating prices at t0:
TTKC ,TKC , ST
Tat ,TTKCS
STST
0T
K
KK
TKK
TK,
22
2
,
Bruno Dupire
FWD Equation: dS/S = r dt + (S,t) dW
Tat ,TTKCS
K
CrK
K
CK
TK
T
C
2
22
2
2
,
Time Value + Intrinsic Value(Strike Convexity) (Interest on Strike)
TTKC ,
TrKe K ST
TKC ,
TV IV
Equating prices at t0:
TV IV
TrKe K
rK
ST
0T
TKrKDig ,
K ST
TrT KeS
KST
TKK
TK,
22
2
,
Bruno Dupire
TV + Interests on K – Dividends on S
Tat ,TTKCS
CdK
CKdr
K
CK
TK
T
C
2
22
2
2
,
TTKC ,
TrdKe KSTTKC ,
TV IV
Equating prices at t0:
TrdKe )( K
Kdr ST
0T
TKDigKdr , )(
TKK
TK,
22
2
,
K ST
TrTdT KeeS
TKCd , TKCd ,
TrT KeS
FWD Equation: dS/S = (r-d) dt + (S,t) dW
Bruno Dupire
• If known, quick computation of all today,
• If all known:
Local volatilities extracted from vanilla prices and used to price exotics.
Stripping Formula
TK ,
CdK
CKdr
K
CKTK
T
C
2
222
2
,
00, , tSC TK
2
22
2,
KC
K
dCKC
KdrTC
TK
00, , tSC TK
Bruno Dupire
FWD Equation for Americans
• Local Volatility model in the continuation region
if ,
if ,
• In Black Scholes :
in the continuation region (Carr 2002)
Same FWD Eq. than in the European case (+Boundary
Condition)
02
22
PSPrPS
P SSSt
StS , Tt PP
ttS ,
KKSS
KS
PKPS
KPSPP22
tS ,
KKKT rKPPKP 22
2
Bruno Dupire
FWD Equation for Americans in LVM
• If do we have
for Americans?
BS LVM
BWD Eq. Same for Eur/Am
Same for Eur/Am
FWD Eq. Same for Eur/Am
?
dWtSrdtS
dS,
KKKT rKPPK
TKP 2
2
2
,
Bruno Dupire
Counterexample
• Assume
for , FWD Eq. for Europeans:
When indeed
21 ,, tttotftS
21, ttT
0 KT rKPP
0AmTP
0
01t 2t
Bruno Dupire
Risk
NeutralDynamics
SmileCalibrated
1DDiffusion
Local Volatility Model
dWtSdtdrS
dS, : Simplest model which fits the smile
Models of interest
Local VolatilityModel
Bruno Dupire
When , gives at T:
Equating Prices at t0:
from local vol model
Any stochastic volatility model which matches the initial smiles has to satisfy:
0T TKCS , KTK 22
2
1
2
222 |
2
1
K
CKKSE
T
CTT
TK
KC
K
TC
KSE TT ,2| 2
2
22
2
TKKSE TT ,| 22
FWD Equation when dS/S = t dW
Bruno Dupire
Convexity Bias
0,
?| 20
22
ZW
KSEdZd
dWdS
ttt
t
20
2only NO! tE
tlikely to be high if 00 or SSSS tt
KSE tt | 2
0S
20
K
Bruno Dupire
Impact on Models
dWtSfS
dSt, tS ,
Risk Neutral drift for instantaneous forward variance
Markov Model:
fits initial smile with local vols
]|[
,,
2
2
SSE
tStSf
tt
Bruno Dupire
Locking FWD Variance | ST=K
T
CCC
dKCC
TKTTKTTK
K
K TK
TK
,,,,
,'
,
and
2
' Define
2
K
ST
Tat ,TTKCS
ST
0T
1
K
4
222 K
KK
Constantcurvature
K KK
2
1
K KK STST
1
2
4
221 K
Tat ,,TTKC
Bruno Dupire
Portfolio
Initial Cost = 0
at T: if
0 if
Replicates the pay-off of a conditional variance swap
Conditional Variance Swap
2,
2 ,,2,2,
TKTKTKTK
DigDigTKCS
KPF
2
,22 TKT KKST ,
Local Vol strippedfrom initial smile
KKST ,
TKKSE TT ,22
It is possible to lock this value
TKPF ,
Bruno Dupire
It is not possible to prescribe just any future smile
If deterministic, one must have
Not satisfied in general
Deterministic future smiles
S0
K
dSTSCTStStSC TKTK 1,10000, ,,,,,22
T1 T2t0
Bruno Dupire
If
stripped from SmileS.t
Then, there exists a 2 step arbitrage:Define
At t0 : Sell
At t:
gives a premium = PLt at t, no loss at T
Conclusion: independent of from initial smile
Det. Fut. smiles & no jumps => = FWD smile
TTKKTKTKtSVTKtS imp
TK
TK
,,,lim,,/,,, 2
00
2,
TKtSK
CtSVTKPL TKt ,,,,,
2
2
,2
tStSt DigDigPL ,, t0 t T
S0
S
K
TKt TKK
SSS ,2
TK,2, sell , CS
2buy , if
TKtSVtS TK ,,, 200, tSV TK ,,
Bruno Dupire
• Sticky Strike assumption: Each (K,T) has a fixed impl(K,T) independent of (S,t)• Sticky Delta assumption: impl(K,T) depends only on moneyness and residual maturity
• In the absence of jumps,
- Sticky Strike is arbitrageable- Sticky is (even more) arbitrageable
Consequence of det. future smiles
Bruno Dupire
Each CK,T lives in its Black-Scholes (impl(K,T))world
P&L of Delta hedge position over t:
If no jump
21C
12C
1tS
ttS
Example of arbitrage with Sticky Strike
21,2,1 assume2211
TKTK CCCC
!
free , no
02
22
21
2211221
22
22
21
2
12
12
21
1
tSCCPL
tSSCPL
tSSCPL
Bruno Dupire
average of 2 weighted by S20
: model with local vols , 0 with 0
Where 0= of C under 0, and =RN density under
Implied vol solution of
From Local Vols to implied Vols
T
dtdStStStStSCC0 0 0
20
20 , , ,,
2
1
2implied
dSdtS
dSdtS
02
022
20
(equivalent to density of Brownian Bridge from (S,t0) to (K,T))
Bruno Dupire
Market Model of Implied Volatility
• Implied volatilities are directly observable• Can we model directly their dynamics?
where is the implied volatility of a given• Condition on dynamics?
0r
2211
1
ˆ
ˆdWudWudt
d
dWS
dS
TKC ,
Bruno Dupire
Drift Condition
• Apply Ito’s lemma to
• Cancel the drift term
• Rewrite derivatives of
gives the condition that the drift of must
satisfy.
For short T,
(Short Skew Condition :SSC)
where
tSC ,ˆ,
tSC ,ˆ,
Tt
tSC ,ˆ,
ˆ
ˆd
222
12ˆ XuXu
SKX lnln
Bruno Dupire
Local Volatility Model Case
det. function of det. function of
and : ,
SSC:
solved by
ˆ
ˆ1
Su S 02 u
SXSXXu
S
S
ˆˆ
1
ˆ
ˆ
ˆ1ˆ 1
K
S uuduX
0,
ˆ
, tS tS ,
1ˆ
ˆ
ˆ
ˆdW
Sdt
d S
Bruno Dupire
“Dual” Equation
The stripping formula
can be expressed in terms of
When
solved by
KK
T
CK
C2
2 2
:
KXK
ˆˆ
1
ˆ
0T
K
S uuduX
0,
ˆ
Bruno Dupire
Large Deviation Interpretation
The important quantity is
If then satisfies:
and
K
S uu
du
0,
dWxadx x
x ua
duxy
0
dWdtdy
K
S
K
S
K
S
uudu
SK
uu
du
u
du
0,
lnlnˆ
0,ˆˆ
KyWKx tt
Bruno Dupire
Consider, for one maturity, the smiles associated to 3 initial spot values
Skew case
-ATM short term implied follows the local vols-Similar skews
Smile dynamics: Local Vol Model 1
S0SS
Local vols
S Smile
0 Smile S
S Smile
K
Bruno Dupire
Pure Smile case
-ATM short term implied follows the local vols-Skew can change sign
Smile dynamics: Local Vol Model 2
KS 0S S
Local vols
S Smile
0 Smile S
S Smile
Bruno Dupire
Skew case (<0)
- ATM short term implied still follows the local vols
- Similar skews as local vol model for short horizons- Common mistake when computing the smile for anotherspot: just change S0 forgetting the conditioning on :if S : S0 S+ where is the new ?
Smile dynamics: Stoch Vol Model 1
TKKSE TT ,22
Local vols
S0SS
S Smile
0 Smile S
S Smile
K
Bruno Dupire
Pure smile case
- ATM short term implied follows the local vols- Future skews quite flat, different from local vol model-Again, do not forget conditioning of vol by S
Smile dynamics: Stoch Vol Model 2
Local volsS Smile
0 Smile S
S Smile
S 0S S K
Bruno Dupire
Skew case
- ATM short term implied constant (does not follows the local vols)- Constant skew- Sticky Delta model
Smile dynamics: Jump Model
S SmileS Smile 0 Smile S
Local vols
S 0SS K
Bruno Dupire
Pure smile case
- ATM short term implied constant (does not follows the local vols)- Constant skew- Sticky Delta model
Smile dynamics: Jump Model
S SmileS Smile
Local vols
0 Smile SS 0S S K
Bruno Dupire
2 ways to generate skew in a stochastic vol model
-Mostly equivalent: similar (St,t) patterns, similar future evolutions-1) more flexible (and arbitrary!) than 2)-For short horizons: stoch vol model local vol model + independent noise on vol.
Spot dependency
0,)2
0,,,)1
ZW
ZWtSfxtt
0S ST
ST
0S
Bruno Dupire
Conclusion• Local vols are conditional forward
values that can be locked
• All stochastic volatility models have to respect the local vols in expectation
• Without jumps, the only non arbitrageable deterministic future smiles are the forward smile from the local vol model
• In particular, without jump, sticky strikes and sticky delta are arbitrageable
• Jumps needed to mimic market smile move
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