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Published byLisa Parrish Modified over 6 years ago
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(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
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Outline Calibration of SV models using both return and option data
Gaussian approximations in numerical integration for computing option prices Numerical results Conclusion
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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
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Simple historical SV model
Discretization via Euler approximation with Goal : estimate (parameter) (latent variables)
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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 (Step 1) Initialize (Step 2) Sample
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SIS-based MCMC iteration (i -1) SIS iteration (i) SIS Keep
updating h by MCMC
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Implementation Sample from hproposal vs hcurrent Consider i.e.,
where accept h′ with probability
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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
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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 Option Data Stock 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
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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)]
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Algorithm Want to sample (Step 1) Initialize (Step 2) Sample
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More details in (Step 2) Sample from hproposal vs hcurrent Consider
i.e., where Accept h′ with probability
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Sample in (Step 3) Consider vs through where
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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
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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
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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) 0.0054
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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
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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 ,
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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. (-0.8) 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
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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
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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)
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