TEL  Vol.2 No.1 , February 2012
Efficient Pricing of European-Style Options under Heston’s Stochastic Volatility Model
Abstract: Heston’s stochastic volatility model is frequently employed by finance researchers and practitioners. Fast pricing of European-style options in this setting has considerable practical significance. This paper derives a computationally efficient formula for the value of a European-style put under Heston’s dynamics, by utilizing a transform approach based on inverting the characteristic function of the underlying stock’s log-price and by exploiting the characteristic function’s symmetry. The value of a European-style call is computed using a parity relationship. The required characteristic function is obtained as a special case of a more general solution derived in prior research. Computational advantage of the option value formula is illustrated numerically. The method can help to mitigate the time cost of algorithms that require repeated evaluation of European-style options under Heston’s dynamics.
Cite this paper: O. Zhylyevskyy, "Efficient Pricing of European-Style Options under Heston’s Stochastic Volatility Model," Theoretical Economics Letters, Vol. 2 No. 1, 2012, pp. 16-20. doi: 10.4236/tel.2012.21003.

[1]   S. L. Heston, “A Closed-Form Solution for Options with Sto-chastic Volatility with Applications to Bond and Currency Options,” Review of Financial Studies, Vol. 6, No. 2, 1993, pp. 327-343. doi:10.1093/rfs/6.2.327

[2]   Y. A?t-Sahalia and R. Kimmel, “Maximum Likelihood Estimation of Stochastic Volatility Models,” Journal of Financial Economics, Vol. 83, No. 2, 2007, pp. 413-452. doi:10.1016/j.jfineco.2005.10.006

[3]   F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-654. doi:10.1086/260062

[4]   Y. A?t-Sahalia and A. W. Lo, “Non-parametric Estimation of State-Price Densities Implicit in Financial Asset Prices,” Journal of Finance, Vol. 53, No. 2, 1998, pp. 499-547. doi:10.1111/0022-1082.215228

[5]   D. Duffie, J. Pan and K. Singleton, “Transform Analysis and Asset Pricing for Affine Jump-Diffusions,” Econometrica, Vol. 68, No. 6, 2000, pp. 1343-1376. doi:10.1111/1468-0262.00164

[6]   G. Bakshi and D. Madan, “Spanning and Derivative- Security Valuation,” Journal of Financial Economics, Vol. 55, No. 3, 2000, pp. 205-238. doi:10.1016/S0304-405X(99)00050-1

[7]   O. Zhylyevskyy, “A Fast Fourier Transform Technique for Pricing American Options under Stochastic Volatility,” Review of Derivatives Research, Vol. 13, No. 1, 2010, pp. 1-24. doi:10.1007/s11147-009-9041-6

[8]   R. Geske and H. E. Johnson, “The American Put Option Valued Analytically,” Journal of Finance, Vol. 39, No. 5, 1984, pp. 1511-1524. doi:10.2307/2327741

[9]   R. Zvan, P. A. Forsyth and K. R. Vetzal, “Penalty Methods for American Options with Stochas-tic Volatility,” Journal of Computational and Applied Mathematics, Vol. 91, No. 4, 1998, pp. 199-218. doi:10.1016/S0377-0427(98)00037-5

[10]   N. Clarke and K. Parrott, “Multigrid for American Option Pricing with Stochastic Volatility,” Applied Mathematical Finance, Vol. 6, No. 3, 1999, pp. 177-195. doi:10.1080/135048699334528

[11]   C. W. Oosterlee, “On Multigrid for Linear Complementarity Problems with Application to American-Style Options,” Electronic Transactions on Numerical Analysis, Vol. 15, 2003, pp. 165-185.

[12]   S. Ikonen and J. Toivanen, “Componentwise Splitting Methods for Pricing American Options under Stochastic Vola-tility,” International Journal of Theoretical and Applied Finance, Vol. 10, No. 2, 2007, pp. 331-361. doi:10.1142/S0219024907004202

[13]   J. M. Harrison and D. M. Kreps, “Martingales and Arbitrage in Multiperiod Securities Markets,” Journal of Economic Theory, Vol. 20, 1979, pp. 381-408. doi:10.1016/0022-0531(79)90043-7

[14]   M. Chernov and E. Ghysels, “A Study towards a Unified Approach to the Joint Estimation of Objective and Risk Neutral Measures for the Purpose of Options Valuation,” Journal of Financial Economics, Vol. 56, No. 3, 2000, pp. 407-458. doi:10.1016/S0304-405X(00)00046-5

[15]   T. W. Epps, “Option Pricing under Stochastic Volatility with Jumps,” Unpub-lished manuscript, University of Virginia, 2004.

[16]   J. Gil-Pelaez, “Note on the Inversion Theorem,” Biometrika, Vol. 38, No. 3-4, 1951, pp. 481-482.

[17]   R. C. Merton, “The-ory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183. doi:10.2307/3003143

[18]   W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, “Numerical Recipes in Fortran 77: The Art of Scientific Computing,” 2nd Edition (Reprint with Corrections), Cambridge University Press, Cambridge, MA, 2001.