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Stochastic Volatility Modelling Bruno Dupire Nice 14/02/03

Stochastic Volatility Modelling

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Stochastic Volatility Modelling. Bruno Dupire Nice 14/02/03. Structure of the talk. Model review Forward equations Forward volatility Arbitrageable & admissible smile dynamics. Model Requirements. Has to fit static/current data: Spot Price Interest Rate Structure - PowerPoint PPT Presentation

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Page 1: Stochastic Volatility Modelling

Stochastic Volatility Modelling

Bruno Dupire

Nice 14/02/03

Page 2: Stochastic Volatility Modelling

Bruno Dupire

Structure of the talk1.Model review

2.Forward equations

3.Forward volatility

4.Arbitrageable & admissible smile dynamics

Page 3: Stochastic Volatility Modelling

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

Page 4: Stochastic Volatility Modelling

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 )(

Page 5: Stochastic Volatility Modelling

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

Page 6: Stochastic Volatility Modelling

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 :

Page 7: Stochastic Volatility Modelling

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 ,

Page 8: Stochastic Volatility Modelling

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)

Page 9: Stochastic Volatility Modelling

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)

Page 10: Stochastic Volatility Modelling

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)

Page 11: Stochastic Volatility Modelling

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

,

Page 12: Stochastic Volatility Modelling

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

,

Page 13: Stochastic Volatility Modelling

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

Page 14: Stochastic Volatility Modelling

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

Page 15: Stochastic Volatility Modelling

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

Page 16: Stochastic Volatility Modelling

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

,

Page 17: Stochastic Volatility Modelling

Bruno Dupire

Counterexample

• Assume

for , FWD Eq. for Europeans:

When indeed

21 ,, tttotftS

21, ttT

0 KT rKPP

0AmTP

0

01t 2t

Page 18: Stochastic Volatility Modelling

Bruno Dupire

Risk

NeutralDynamics

SmileCalibrated

1DDiffusion

Local Volatility Model

dWtSdtdrS

dS, : Simplest model which fits the smile

Models of interest

Local VolatilityModel

Page 19: Stochastic Volatility Modelling

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

Page 20: Stochastic Volatility Modelling

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

Page 21: Stochastic Volatility Modelling

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

Page 22: Stochastic Volatility Modelling

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

Page 23: Stochastic Volatility Modelling

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 ,

Page 24: Stochastic Volatility Modelling

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

Page 25: Stochastic Volatility Modelling

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 ,,

Page 26: Stochastic Volatility Modelling

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

Page 27: Stochastic Volatility Modelling

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

Page 28: Stochastic Volatility Modelling

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))

Page 29: Stochastic Volatility Modelling

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 ,

Page 30: Stochastic Volatility Modelling

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

Page 31: Stochastic Volatility Modelling

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 ,

ˆ

ˆ

ˆdW

Sdt

d S

Page 32: Stochastic Volatility Modelling

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

Page 33: Stochastic Volatility Modelling

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

Page 34: Stochastic Volatility Modelling

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

Page 35: Stochastic Volatility Modelling

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

Page 36: Stochastic Volatility Modelling

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

Page 37: Stochastic Volatility Modelling

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

Page 38: Stochastic Volatility Modelling

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

Page 39: Stochastic Volatility Modelling

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

Page 40: Stochastic Volatility Modelling

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

Page 41: Stochastic Volatility Modelling

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