Gaussian Approximations for
Option Prices in Stochastic
Volatility Models
Chuanshu Ji
(joint work with Ai-ru Cheng, Ron Gallant, Beom Lee)
UNC-Chapel Hill
1
Outline
• Calibration of SV models using both return and
option data
• Gaussian approximations in numerical integration
for computing option prices
• Numerical results
• Conclusion
2
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
3
Simple historical SV model
• Discretization via Euler approximation with ht  log( t2 )
yt  eht / 2 t(1)
(1)
ht     ht 1   t(2)
(2)
where yt : return process (observed)
 t(1) and  t(2) : independent N (0,1)
 Goal : estimate
  (  ,  , )
h  ( h1 ,
(parameter)
, hT ) (latent variables)
4
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)
5
MCMC Algorithm
yt  eht / 2 t(1)
(1)
ht     ht 1   t(2)
(2)
• Want to sample   ( ,  , ) and h  (h1 , , hT ) from p( , h | y)
(Step 1) Initialize   ( ,  , ) and h  (h1 , , hT )
, hT ) from p(h |  , y)
(Step 2) Sample
h  (h1 ,
(Step 3) Sample
  (  ,  , ) from p( | h, y )
6
SIS-based MCMC



 (h1( i 1) ,
, ht( i 1) ,
(i )
(i )
SIS  h  ( h1 ,
, ht(i ) ,
iteration (i+1) SIS  h(i 1)  (h1(i 1) ,



, ht(i 1) ,
iteration (i -1) SIS  h( i 1)
iteration
(i)
, hT( i 1) )
Keep
updating
, hT(i ) )
h
by MCMC
, hT(i 1) )
7
Implementation
• Sample h  (h1 , , hT )
 hproposal vs hcurrent
Consider
from
p(h |  , y )
importance weight (hproposal )
importance weight (hcurrent )
p(h |  , y ) g ( h) u1 (h1 )u2 (h2 )

i.e.,
p ( h |  , y ) g ( h)
u1 ( h1 ) u2 ( h2 )
where
ut (ht ) 
 accept h′ with probability
uT (hT  )
uT ( hT )
p (ht | ht 1 ,  ) p ( yt | ht )
g (ht )
 T u (h  ) 
min   t t ,1
 t 1 ut (ht ) 


8
Some simulation result
• 100,000 iterations (after discarding 10,000 iterations)
(-0.8)
 (0.9)
 (0.6)
Posterior Mean
Stand. Dev.
-0.7572
0.2409
0.9062
0.0296
0.5902
0.0816
9
Some plots of simulation results
10
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
Option Data
Stock Data
Hybrid SV Model
11
Why need option data to fit a SV model?
• To price various derivatives, we must fit riskneutral probability models
• To understand the discrepancy between riskneutral 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)
12
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 ?
13
Hybrid SV model
(under a risk-neutral measure Q)
•
•
Discretized version
yt  eht / 2 t(1)
(3)
ht  (   )   ht 1   t(2)
(4)
Additional Setting
– Simple version of European call option pricing formula
Vt  e
 r
 

x t ,t 
Et   K
( St e
 K ) p ( x)dx 
log t ,t 


St
where
p( x)
t ,t 
– Assume
 Ct 
(3)
log 



t
V
 t 
where
N (0, 
t 
e hu du )
t
t 
e hu
  (r 
) du
t
2
Ct : observed call option price
14
Idea of Hybrid Model
historical volatility
(real world measure P)
future volatility
(h11 ,
(risk-neutral measure Q)
h1 ,
, h1 )
, ht ,
(ht 1 ,
, ht  )
, hT
(hT 1 ,
, hT  )
• 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)]
15
Algorithm
yt  eht / 2 t(1)
(3)
ht  (   )   ht 1   t(2)
(4)
• Want to sample   ( ,  , ,  , ) and h  (h1, , hT ) from p( , h | y, C)
(Step 1) Initialize   ( ,  , ,  , ) and h  (h1 ,
, hT )
, hT ) from p(h |  , y, C )
(Step 2) Sample
h  (h1 ,
(Step 3) Sample
  (  ,  , ,  , ) from p( | h, y, C )
16
More details in (Step 2)
• Sample h  (h1 , , hT )
 hproposal vs hcurrent
from p(h |  , y, C )
Consider
i.e.,
importance weight (hproposal )
importance weight (hcurrent )
p(h |  , y, C ) g (h) u1 (h1 )

p ( h |  , y , C ) g ( h)
u1 (h1 )
where ut (ht ) 
uT (hT  )
uT (hT )
p(ht | ht 1 ,  ) p( yt | ht ) p (Ct | ht ,  , y )
g (ht )
 Accept h′ with probability
 T u (h  ) 
min   t t ,1
 t 1 ut (ht ) 


17
Sample  from p( |  , h, y, C ) in (Step 3)
T
p( | h, y, C ,  ,  , ,  )  p( | h,  ,  , ) p(Ct | h, y,  ,  , ,  )
 Consider

vs
t 1

T
p (  | ) p (Ct | ) g ( )
through
t 1
T
p( | ) p(Ct | ) g ( )
t 1
where
 log  C V 2 
t
t

p(Ct | )  exp  
2


2


18
Modified Algorithm
Sample
yt  eht / 2 t(1)
(3)
ht  (   )   ht 1   t(2)
(4)
  ( ,  , ,  , ) and h  (h1, , hT ) from p( , h | y, C)
(Step 1) Retrieve estimates of (  ,  , ) and h
from historical volatility model
Then, initialize  and 
(Step 2) Compute option prices Vt by approximation
(Step 3) Sample  and 
19
Computing option price Vt (uncorrelated)
•
Vt  e
 r
 

x t ,t 
Et   K
( St e
 K ) p ( x)dx 
log t ,t 
 St

depends on the 1D statistic
•
Theorem 1 (Conditional CLT)

t 
t
n
e du   ehu  U n
hu
U n  EU n
Var (U n )
j 1
 Ν (0,1)
as n  
EU n and Var (U n ) enjoy explicit expressions in terms
 ,  ,  ,  & ht updated at each iteration
where
•
No need to generate the future volatility under risk-neutral measure
➩ Simply sample
U n from Ν ( EU n ,Var (U n ))
20
Some simulation result (uncorrelated)
• 20,000 iterations (after discarding 5,000 iterations)
 (0.01)
 (-0.02)
•
Posterior Mean
Stand. Dev.
0.0122
0.0003
-0.0161
0.0054
3 hours (Gaussian approximation)
vs
27 hours (“brute force” numerical integration)
maturity of option = 30 days
# of sequences of future volatility = 100
21
Correlated case (leverage effect)
• Historical SV model
yt  e ht / 2

1   2  t(1)   t( 2)

ht     ht 1   t( 2)
• Hybrid SV model with option data
ht  (   )   ht 1   t(2)
• Sample 
• To use Gaussian approximations in computing option prices,
we need asymptotic distribution of the 2D stat
n
Un   e
j 1
hj
n
and Vn   e
j 1
 j 1
hj 2
22
Computing option price Vt (correlated)
•
Theorem 2 (an extension of Theorem 1)
  0   a11
1  U n  EU n 

  Ν  , 
n  Vn  EVn 
  0   a21
where
EU n , EVn , aij , i, j  1, 2,
in terms of
 ,  ,  , ,  & ht
a12  

a22  
as n  
enjoy explicit expressions
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
U, V
n
n
23
Some simulation result (correlated)
• 100,000 iterations (after discarding 30,000 iterations) (7 hours)
(-0.8)
 (0.9)
 (0.6)
(-0.3)
•
Posterior Mean
Stand. Dev.
-0.7293
0.2156
0.9109
0.0260
0.5748
0.0731
-0.2874
0.1044
5,000 iterations (after discarding 2,000 iterations)
by Gaussian approximations (1 hour and 20 minutes)
 (0.01)
 (-0.05)
Posterior Mean
Stand. Dev.
0.0125
0.0003
-0.0515
0.0048
24
Diagnostics of convergence
• Brooks and Gelman (1998)
based on Gelman and Rubin (1992)
• Consider independent multiple MCMC chains
• Consider the ratio
between-chain variance
within-chain variance
against # of iterations
25
Historical SV model, correlated
26
Hybrid SV model, correlated
27
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)
28