1

Gaussian Approximations for Option Prices in Stochastic

Volatility Models

Chuanshu Ji

(joint work with Ai-ru Cheng, Ron Gallant, Beom Lee)

UNC-Chapel Hill

2

• Calibration of SV models using both return and

option data

• Gaussian approximations in numerical integration

for computing option prices

• Numerical results

• Conclusion

Outline

3

Several approaches in volatility modelling--- important in ``return vs risk’’ studies

• Constant: Black-Scholes model• Function of returns: ARCH / GARCH models• Realized volatility with high frequency returns• With latent random factors: SV models

4

Simple historical SV model

• Discretization via Euler approximation with

Goal : estimate (parameter)

(latent variables)

2log( )t th

/ 2 (1)

(2)1

(1)

(2)

tht t

t t t

y e

h h

(1) (2)

:

: (0,1)

t

t t

y

N

where return process (observed)

and independent

1

( , , )

( , , )Th h

h

5

Inference for SV models (return data only)

• Frequentist: efficient method of moments (EMM), e.g. Gallant, Hsu & Tauchen (1999)

• Bayesian: MCMC, particle filter, SIS, … e.g. Jacquier, Polson & Rossi (1994), Chib, Nardari & Shephard (2002)

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MCMC Algorithm

• Want to sample

/ 2 (1)

(2)1

(1)

(2)

tht t

t t t

y e

h h

1( , , ) ( , , ) ( , | )Th h h p h y and from

(Step 1) Initialize 1( , , ) ( , , )Th h h and

(Step 2) Sample 1( , , ) ( | , )Th h h p h y from

(Step 3) Sample ( , , ) ( | , )p h y from

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SIS-based MCMC

iteration (i -1) SIS

iteration (i) SIS

iteration (i+1) SIS

( 1) ( 1) ( 1) ( 1)1( , , , , )i i i i

t Th h h h

( 1) ( 1) ( 1) ( 1)1( , , , , )i i i i

t Th h h h

( ) ( ) ( ) ( )1( , , , , )i i i i

t Th h h h

Keepupdating

hby MCMC

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Implementation

• Sample from

hproposal vs hcurrent

Consider

i.e.,

where

accept h′ with probability

1( , , )Th h h ( | , )p h y

)

)

h

h

pro

curren

a

t

pos limportance weight (

importance weight (

1 2

11 2

1

2

2

( ) ( ) ( )( | , ) ( )

( | , ) ( ) ( ) ( ) ( )TT

T T

u u

h h

u

u

p y g

p y g

h h hh

h u hu h

h

1( | , ) ( | )( )

( )t t t t

t tt

p h h p y hu h

g h

1

( )min ,1

( )

T

tt

t

t

tu

u

h

h

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Some simulation result

• 100,000 iterations (after discarding 10,000 iterations)

  Posterior Mean Stand. Dev.

(-0.8) -0.7572 0.2409

(0.9) 0.9062 0.0296

(0.6) 0.5902 0.0816

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Some plots of simulation results

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A challenging problem in empirical finance

• Hybrid SV model = historical volatility + ``implied’’ volatility

• Historical volatility: (stock) return data under real world

probability measure

• ``implied’’ volatility: option data under risk-neutral probability

measure

Stock Data Option Data

Hybrid SV Model

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Why need option data to fit a SV model?

• To price various derivatives, we must fit risk-neutral probability models

• To understand the discrepancy between risk-neutral measure estimated from option data and physical measure estimated from return data (different preferences towards risk ?)

• See discussions in several papers, e.g. Garcia, Luger and Renault (2003, JE)

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Some references

• EMM: Chernov & Ghysels (2000), Pan (2002)

• MCMC: Jones (2001), Eraker (2004)

Almost all follow the affine model in Heston (1993) (maybe add jumps), why?

--- a closed-form solution reduces computational intensity …

--- any alternatives ?

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Hybrid SV model(under a risk-neutral measure Q)

• Discretized version

• Additional Setting

– Simple version of European call option pricing formula

where

– Assume

where Ct : observed call option price

/ 2 (1)

(2)1

(3)

(4)( )

tht t

t t t

y e

h h

,

,logE ( ) ( )t t

t tt

xrKt t tS

V e S e K p x dx

,

( ) (0, )

( )2

u

u

t h

th

t

t t t

p x N e du

er du

(3)log tt

t

C

V

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Idea of Hybrid Model

historical volatility (real world measure P)

future volatility (risk-neutral measure Q)

• No arbitrage ⇐ Existence of an equivalent martingale measure Q (risk-neutral

measure)

defined by its Radon-Nikodým derivative w.r.t. P

[Girsanov transformation, see Øksendal (1995)]

1, , , ,t Th h h

1 1 1 1 1( , , ) ( , , ) ( , , )t t T Th h h h h h

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Algorithm

• Want to sample 1( , , , , ) ( , , ) ( , | , )Th h h p h y C and from

(Step 1) Initialize 1( , , , , ) ( , , )Th h h and

(Step 2) Sample 1( , , ) ( | , , )Th h h p h y C from

(Step 3) Sample ( , , , , ) ( | , , )p h y C from

/ 2 (1)

(2)1

(3)

(4)( )

tht t

t t t

y e

h h

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More details in (Step 2)

• Sample from

hproposal vs hcurrent

Consider

i.e.,

where

Accept h′ with probability

1( , , )Th h h ( | , , )p h y C

)

)

h

h

proposal

current

importance weight (

importance weight (

1 1

1 1

( ) ( )( | , , ) ( )

( | , , ) ( ) ( ) ( )T T

T T

u h u hp h y C g h

p h y C g h u h u h

1 ( |( | , ) ( | )( )

(

, ,

)

)t t t tt t

t

tt p C hp h h yp y hu h

g h

1

( )min ,1

( )

Tt t

t t t

u h

u h

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Sample in (Step 3)

Consider vs

through

where

( | , , , )p h y C from

1

( | , , , , , , ) ( | , , , ) ( | , , , , , )T

tt

p h y C p h p C h y

1

1

( | ) ( | ) ( )

( | ) ( | ) ( )

T

ttT

tt

p p C g

p p C g

2

2

log( | ) exp

2t t

t

C Vp C

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Modified Algorithm

Sample 1( , , , , ) ( , , ) ( , | , )Th h h p h y C and from

(Step 1) Retrieve estimates of from historical volatility model

Then, initialize

( , , ) h and

(Step 2) Compute option prices Vt by approximation

(Step 3) Sample and

/ 2 (1)

(2)1

(3)

(4)( )

tht t

t t t

y e

h h

and

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Computing option price Vt (uncorrelated)

•

depends on the 1D statistic

• Theorem 1 (Conditional CLT)

where enjoy explicit expressions in terms

updated at each iteration

• No need to generate the future volatility under risk-neutral measure

➩ Simply sample

,

,logE ( ) ( )t t

t tt

xrKt t tS

V e S e K p x dx

1

u u

nt h hnt

j

e du e U

(0,1) as( )

n n

n

U EUΝ n

Var U

from ( , ( ))n n nU Ν EU Var U

and ( )n nEU Var U

, , , & th

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Some simulation result (uncorrelated)

• 20,000 iterations (after discarding 5,000 iterations)

• 3 hours (Gaussian approximation)

vs

27 hours (“brute force” numerical integration)

maturity of option = 30 days

# of sequences of future volatility = 100

  Posterior Mean Stand. Dev.

(0.01) 0.0122 0.0003

(-0.02) -0.0161 0.0054

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Correlated case (leverage effect)

• Historical SV model

• Hybrid SV model with option data

• Sample

• To use Gaussian approximations in computing option prices,

we need asymptotic distribution of the 2D stat

/ 2 2 (1) (2)

(2)1

1tht t t

t t t

y e

h h

(2)1( )t t th h

2

11 1

andj j

n nh h

n n jj j

U e V e

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Computing option price Vt (correlated)

• Theorem 2 (an extension of Theorem 1)

where enjoy explicit expressions

in terms of updated at each iteration

see Cheng / Gallant / Ji / Lee (2005) for details

• Significant dimension reduction: from generating future volatility paths to simulating bivariate normal samples of ,

11 12

21 22

01, as

0n n

n n

U EU a aΝ n

V EV a an

nU

, , , , 1, 2,n n ijEU EV a i j

, , , , & th

nV

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Some simulation result (correlated)

• 100,000 iterations (after discarding 30,000 iterations) (7 hours)

• 5,000 iterations (after discarding 2,000 iterations)

by Gaussian approximations (1 hour and 20 minutes)

  Posterior Mean Stand. Dev.

(0.01) 0.0125 0.0003

(-0.05) -0.0515 0.0048

  Posterior Mean Stand. Dev.

(-0.8) -0.7293 0.2156

(0.9) 0.9109 0.0260

(0.6) 0.5748 0.0731

(-0.3) -0.2874 0.1044

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Diagnostics of convergence

• Brooks and Gelman (1998)

based on Gelman and Rubin (1992)

• Consider independent multiple MCMC chains

• Consider the ratio

against # of iterations

between-chain variance

within-chain variance

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Historical SV model, correlated

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Hybrid SV model, correlated

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Summary

• Why the proposed Gaussian approximations are useful?

The method reduces high dimensional numerical integrals (brutal force Monte Carlo) to low dimensional ones; it applies to many different SV models (frequentist and Bayesian).

• Other development

- real data (option data, not easy), see Cheng / Gallant / Ji / Lee (2005)- more realistic and complicated SV models: Chernov, Gallant, Ghysels & Tauchen (2006, JE), two-factor SV model [one AR(1), one GARCH diffusion]; see Cheng & Ji (2006);- more elegant probability approximations

More references: Ghysels, Harvey & Renault (1996), Fouque, Papanicolaou & Sircar (2000)