gaussian approximations for option prices in stochastic volatility models
DESCRIPTION
Gaussian Approximations for Option Prices in Stochastic Volatility Models. Chuanshu Ji (joint work with Ai-ru Cheng, Ron Gallant, Beom Lee) UNC-Chapel Hill. Outline. Calibration of SV models using both return and option data - PowerPoint PPT PresentationTRANSCRIPT
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)
6
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
7
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
8
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
9
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
10
Some plots of simulation results
11
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
12
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)
13
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 ?
14
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
15
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
16
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
17
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
18
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
19
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
20
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
21
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
22
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
23
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
24
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
25
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
26
Historical SV model, correlated
27
Hybrid SV model, correlated
28
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)