Abstract
We introduce a hybrid stochastic volatility model where the asset price process follows the Heston model and interest rates are governed by a two-factor stochastic model. Two cases are considered. First, it is assumed that interest rates and asset prices are uncorrelated. The characteristic function method is used to derive semi-analytical pricing formulae for plain vanilla options. In the second case we introduce a correlation between the asset price process and the short rate process and use Monte Carlo simulations for pricing options. To reduce the stochastic error, we implement the control variate method where an estimator of the option value for the uncorrelated case is used as a control variate. The options are priced with a varying correlation coefficient. We observe that the control variate method allows us to speed up Monte Carlo computations by a factor with the magnitude of several hundreds. The efficiency of the method is higher for smaller values of the correlation coefficient. We then study the impact a correlation between the two processes has on option prices. It has been noticed that the call option price is an increasing function of the correlation coefficient.
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References
Ahlip, R., Rutkowski, M.: Pricing of foreign exchange options under the Heston stochastic volatility model and CIR interest rates. Quant. Financ. 13 (6), 955–966 (2013)
Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Pol. Econ. 81 (3), 637–654 (1973)
Blanchard, A.: The two-factor Hull-White model: pricing and calibration of interest rates derivatives. Accessed 28 Mar 2014
Brigo, D., Mercurio, F.: Interest Rate Models-Theory and Practice: With Smile, Inflation and Credit. Springer, Berlin/New York (2007)
Cox, J.C., Ingersoll, J.E., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53 (2), 385–407 (1985)
Duffie, D., Pan, J.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68 (6), 1343–1376 (2000)
Glynn, P.W., Szechtman, R.: Some new perspectives on the method of control variates. In: Monte Carlo and Quasi-Monte Carlo methods, 2000 (Hong Kong), pp. 27–49. Springer, Berlin (2002)
Grzelak, L.A., Oosterlee, C.W.: On the Heston model with stochastic interest rates. SIAM J. Financ. Math. 2 (1), 255–286 (2011)
Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6 (2), 327–343 (1993)
Hull, J., White, A.: Pricing interest-rate-derivative securities. Rev. Financ. Stud. 3 (4), 573–592 (1990)
Kienitz, J., Kammeyer, H.: An implementation of the Hybrid-Heston-Hull-White model. Available at SSRN 1399389 (2009)
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Volume 23 of Applications of Mathematics (New York). Springer, Berlin (1992)
Rouah, F.D.: The Heston Model and Its Extensions in Matlab and C. Wiley, Hoboken (2013)
Van Haastrecht, A., Lord, R., Pelsser, A., Schrager, D.: Pricing long-maturity equity and FX derivatives with stochastic interest rates and stochastic volatility. Insur.: Math. Econ. 45 (3), 436–448 (2009)
Vasicek, O.: An equilibrium characterization of the term structure [reprint of J. Financ. Econ. 5 (2), 177–188 (1977)]. In: Financial Risk Measurement and Management. Volume 267 of International Library of Critical Writings in Economics, pp. 724–735. Edward Elgar, Cheltenham (2012)
Zhu, J.: Applications of Fourier Transform to Smile Modeling, 2nd edn. Springer Finance. Springer, Berlin (2010) Theory and implementation
Acknowledgements
R. Makarov wishes to acknowledge the generous support of the NSERC Discovery Grant program.
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Jones, G., Makarov, R. (2016). Pricing Options with Hybrid Stochastic Volatility Models. In: Bélair, J., Frigaard, I., Kunze, H., Makarov, R., Melnik, R., Spiteri, R. (eds) Mathematical and Computational Approaches in Advancing Modern Science and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-30379-6_50
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DOI: https://doi.org/10.1007/978-3-319-30379-6_50
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