基于非仿射随机波动率模型的期权定价研究

Abstract

Abstract: By applying the fast Fourier transform (FFT) method, the problem of option pricing when the underlying asset follows the non-affine stochastic volatility models is considered in this paper. Firstly, by utilizing a perturbation method to the partial differential equation of the characteristic function for the underlying log-asset price, an approximate solution for the characteristic function is derived. Then, a quasi-analytical approximate formula for European options is attained by means of Fourier transform and its inverse. This formula is easy to implement and can be accurately and quickly computed by the FFT algorithm. Numerical examples show that the FFT-based option pricing method is very accurate and efficient. Finally, an empirical study of call warrants on Hang Seng index is presented. Empirical results demonstrate that the non-affine stochastic volatility option pricing model is more accurate than the classical Black-Scholes model.

Key words: option pricing, non-affine stochastic volatility, fast Fourier transform, perturbation method

CLC Number:

F830.9

WU Xin-yu, YANG Wen-yu, MA Chao-qun, WANG Shou-yang. Option Pricing under Non-Affine Stochastic Volatility Model[J]. Chinese Journal of Management Science, 2013, (1): 1-7.

References

[1] Black F, Scholes M. The pricing of options and corporate liabilities [J]. Journal of Political Economy, 1973, 81(3): 637-654.
[2] Black F. Studies of stock price volatility changes [C]. Proceedings of the 1976 Meeting of Business and Economic Statistics Section, American Statistical Association, 1976.
[3] Christie A A. The stochastic behavior of common stock variances: value, leverage, and interest rate effects[J]. Journal of Financial Economics, 1982, 10(4): 407-432.
[4] Cox J C, Ross S A. The valuation of options for alternative stochastic processes[J]. Journal of Financial Economics, 1976, 3(1-2): 145-166.
[5] Merton R C. Option pricing when underlying stock returns are discontinuous[J]. Journal of Financial Economics, 1976, 3(1-2): 125-144.
[6] Hull J C, White A D. The pricing of options on asset with stochastic volatilities[J]. Journal of Finance, 1987, 42(2): 281-300.
[7] Stein E M, Stein J C. Stock price distributions with stochastic volatility: an analytic approach[J]. Review of Financial Studies, 1991, 4: 727-752.
[8] Heston S L. A closed-form solution for options with stochastic volatility with applications to bond and currency options [J]. Review of Financial Studies, 1993, 6(2): 327-343.
[9] Andersen T G, Benzoni L, Lund J. Estimating jump-diffusions for equity returns[J]. Journal of Finance, 2002, 57: 1239-1284.
[10] Chacko G, Viceira L. Spectral GMM estimation of continuous-time processes[J]. Journal of Econometrics, 2003, 116: 259-292.
[11] Chernov M, Gallant A R, Ghysels E, Tauchen G. Alternative models for stock price dynamics [J]. Journal of Econometrics, 2003, 116: 225-257.
[12] Jones C. The dynamics of stochastic volatility: evidence from underlying and options markets [J]. Journal of Econometrics, 2003, 116: 181-224.
[13] Ait-Sahalia Y, Kimmel R. Maximum likelihood estimation of stochastic volatility models[J]. Journal of Financial Economics, 2007, 83: 413-452.
[14] Chourdakis K. Non-affine option pricing [J]. Journal of Derivatives, 2004, 11(3): 10-25.
[15] Christoffersen P, Jacobs K, Mimouni K. An empirical comparison of affine and non-affine models for equity index options [R]. Working Paper, 2006.
[16] Christoffersen P, Jacobs K, Mimouni K. Volatility dynamics for the S&P 500: evidence from realized volatility, daily returns, and option prices[J]. Review of Financial Studies, 2010, 23(8): 3141-3189.
[17] Durham G B. Risk-neutral modelling with affine and non-affine models [R]. Working Paper, 2010.
[18] Hansis A. Affine versus non-affine stochastic volatility and the impact on asset allocation [R]. Working Paper, 2010.
[19] Ignatieva K, Rodrigues P J M, Seeger N. Stochastic volatility and jumps: exponentially affine yes or no? an empirical analysis of S&P 500 dynamics[R]. Working Paper, 2009.
[20] Chourdakis K, Dotsis G. Maximum likelihood estimation of non-affine volatility processes[J]. Journal of Empirical Finance, 2011, 18(3): 533-545.
[21] Drimus G G. Options on realized variance by transform methods: a non-affine stochastic volatility model[J]. Quantitative Finance, forthcoming, 2012.
[22] Carr P, Madan D. Option valuation using the fast Fourier transform[J]. Journal of Computational Finance, 1999, 2: 61-73.
[23] Yu Jun. On leverage in a stochastic volatility model[J]. Journal of Econometrics, 2005, 127: 165-178.
[24] Chan K C, Karolyi G A, Longstaff F A, et al. An empirical comparison of alternative models of the short-term interest rate[J]. Journal of Finace, 1992, 47(3): 1209-1227.
[25] Nelson D B. ARCH models as diffusion approximations[J]. Journal of Econometrics, 1990, 45: 7-38.
[26] Kevorkian J, Cole J D. Perturbation methods in applied mathematics[M]. New York: Springer-Verlag, 1981.
[27] Mikhailov S, Nogel U. Heston's stochastic volatility model implementation, calibration and some extensions[M]//Wilmott P.The best of Wilmott.Chichester:John Wiley & Sons,2005.

Option Pricing under Non-Affine Stochastic Volatility Model

PDF (PC)

Knowledge

Cited

Abstract

Cite this article

share this article

References

Related Articles 13

Metrics

Comments

Recommended 10

[1]	WANG Xian-dong, HE Jian-min. Pricing Asian Options under Uncertain Environment with Fuzziness and Randomness Considering Decision Maker's Subjective Judgment [J]. Chinese Journal of Management Science, 2020, 28(9): 33-44.
[2]	YANG Chang-hui, SHAO Zhen, LIU Chen, FU Chao. A Hybrid Modeling Framework and Its Application for Exchange Traded Fund Options Pricing [J]. Chinese Journal of Management Science, 2020, 28(12): 44-53.
[3]	LI Qing, ZHANG Hu. Single-index Nonparametric Option Pricing Model——A Modified Nonparametric Pricing Approach [J]. Chinese Journal of Management Science, 2020, 28(10): 43-53.
[4]	WU Xin-yu, ZHAO Kai, LI Xin-dan, MA Chao-qun. Option Pricing Under Time-Varying Risk Aversion: An Empirical Study Based on SSE 50ETF Options [J]. Chinese Journal of Management Science, 2019, 27(11): 11-22.
[5]	HUANG Shou-jun, YANG Jun, CHEN Qi-an. Coordination Mechanism of V2G Reserve Contract Based on B-S Option Pricing Model [J]. Chinese Journal of Management Science, 2016, 24(10): 10-21.
[6]	ZHAO Pan, XIAO Qing-xian. Pricing of European Options Based on Tsallis Distribution and Jump-diffusion Process [J]. Chinese Journal of Management Science, 2015, 23(6): 41-48.
[7]	HAN Li-yan, YE Hao, LI Wei. Nonparametric Methods Based Stock Index Option Pricing [J]. Chinese Journal of Management Science, 2012, (1): 23-29.
[8]	ZHENG Hong, YOU Chun. Barrier Option Pricing Method of Supplemental Medical Expense Insurance and its Application [J]. Chinese Journal of Management Science, 2011, 19(6): 169-176.
[9]	ZHANG Hong-yan. Study on the Sensitivity of Implied volatility Based on Artificial Intelligence [J]. Chinese Journal of Management Science, 2008, 16(3): 125-130.
[10]	LIU Xin-bao, YE Qiang, LIU Lin, YANG Shan-lin. Branching Ant Colony with Dynamic Perturbation Algorithm for Solving TSPs [J]. Chinese Journal of Management Science, 2005, (6): 57-63.
[11]	WU Yun, HE Jian-min. Stvdy on the Solutions of Up-Knock-out Option Pricing Model [J]. Chinese Journal of Management Science, 2002, (6): 23-26.
[12]	CHEN Zhan-feng, ZHANG Ke. Approaching Mathematical Logic of the Option Pricing Principles [J]. Chinese Journal of Management Science, 2001, (2): 10-15.
[13]	Huang Kai. Application of OPT in Corporation Strategic Investment——A Case Study [J]. Chinese Journal of Management Science, 1998, (2): 14-21.