Joint Characteristic Function of Stock Log-Price and Squared Volatility in the Bates Model and Its Asset Pricing Applications


The model of Bates specifies a rich, flexible structure of stock dynamics suitable for applications in finance and economics, including valuation of derivative securities. This paper analytically derives a closed-form expression for the joint conditional characteristic function of a stock’s log-price and squared volatility under the model dynamics. The use of the function, based on inverting it, is illustrated on examples of pricing European-, Bermudan-, and American-style options. The discussed approach for European-style derivatives improves on the option formula of Bates. The suggested approach for American-style derivatives, based on a compound-option technique, offers an alternative solution to existing finite-difference methods.

Share and Cite:

O. Zhylyevskyy, "Joint Characteristic Function of Stock Log-Price and Squared Volatility in the Bates Model and Its Asset Pricing Applications," Theoretical Economics Letters, Vol. 2 No. 4, 2012, pp. 400-407. doi: 10.4236/tel.2012.24074.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Y. Ait-Sahalia and J. Jacod, “Analyzing the Spectrum of Asset Returns: Jump and Volatility Components in High Frequency Data,” Journal of Economic Literature, 2011.
[2] S. L. Heston, “A Closed-Form Solution for Options with Stochastic 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
[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] D. S. Bates, “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options,” Review of Financial Studies, Vol. 9, No. 1, 1996, pp. 69-107. doi:10.1093/rfs/9.1.69
[5] Y. Ait-Sahalia and J. Jacod, “Testing for Jumps in a Discretely Observed Process,” Annals of Statistics, Vol. 37, No. 1, 2009, pp. 184-222. doi:10.1214/07-AOS568
[6] 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
[7] G. Bakshi and D. Madan, “Spanning and Derivative-Security Valuation,” Journal of Financial Economics, Vol. 55, No. 2, 2000, pp. 205-238. doi:10.1016/S0304-405X(99)00050-1
[8] R. Geske and H. E. Johnson, “The American Put Option Valued Analytically,” Journal of Finance, Vol. 39, No. 5, 1984, pp. 1511-1524.
[9] C. Chiarella, B. Kang, G. H. Meyer and A. Ziogas, “The Evaluation of American Option Prices under Stochastic Volatility and Jump-Diffusion Dynamics Using the Method of Lines,” Quantitative Finance Research Centre, University of Technology, Sydney, 2008.
[10] 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
[11] J. M. Harrison and D. M. Kreps, “Martingales and Arbitrage in Multiperiod Securities Markets,” Journal of Economic Theory, Vol. 20, No. 3, 1979, pp. 381-408. doi:10.1016/0022-0531(79)90043-7
[12] 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
[13] P. Protter, “Stochastic Integration and Differential Equations: A New Approach,” Springer-Verlag, New York, 1990.
[14] K. L. Chung, “A Course in Probability Theory,” 3rd Edition, Academic Press, San Diego, 2001.
[15] 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
[16] R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183. doi:10.2307/3003143
[17] 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, Cambridge University Press, Cambridge, 2001.
[18] N. G. Shephard, “From Characteristic Function to Distribution Function: A Simple Framework for the Theory,” Econometric Theory, Vol. 7, No. 4, 1991, pp. 519-529. doi:10.1017/S0266466600004746
[19] T. W. Epps, “Pricing Derivative Securities,” World Scientific, River Edge, 2000.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.