Gaussian Approximations for Option Prices in Stochastic Volatility Models

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. 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 (Step 3) Sample

7. SIS-based MCMC    iteration (i -1) SIS iteration (i) SIS iteration (i+1) 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)

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 (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 (Step 3) 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

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)

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)