1. (joint work with Ai-ru Cheng, Ron Gallant, Beom Lee)
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. Outline Calibration of SV models using both return and option data
Gaussian approximations in numerical integration for computing option prices
Numerical results
Conclusion

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)

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 (Step 1) Initialize (Step 2) Sample

7. SIS-based MCMC  iteration (i -1) SIS  iteration (i) SIS  Keep
updating h
by MCMC

8. Implementation Sample from  hproposal vs hcurrent Consider i.e.,
where
 accept h′ with probability

9. Some simulation result
100,000 iterations (after discarding 10,000 iterations)
Posterior Mean
Stand. Dev.
(-0.8)
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
Option Data
Stock 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

15. Idea of Hybrid Model historical volatility future 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)]

16. Algorithm Want to sample (Step 1) Initialize (Step 2) Sample

17. More details in (Step 2) Sample from  hproposal vs hcurrent Consider
i.e.,
where
 Accept h′ with probability

18. Sample in (Step 3)
 Consider vs
through
where

19. Modified Algorithm Sample
(Step 1) Retrieve estimates of from historical volatility model
Then, initialize
(Step 2) Compute option prices Vt by approximation
(Step 3) Sample

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

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)
0.0054

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

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 ,

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.
(-0.8)
0.2156
(0.9)
0.9109
0.0260
(0.6)
0.5748
0.0731
(-0.3)
0.1044
Posterior Mean
Stand. Dev.
(0.01)
0.0125
0.0003
(-0.05)
0.0048

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

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)