Semi-Analytical Option Pricing Under Double Heston Jump-Diffusion Hybrid Model

Year 2018, Volume: 1 Issue: 3, 138 - 152, 30.12.2018

Rehez Ahlip Laurence A. F. Park Ante Prodan

https://doi.org/10.33187/jmsm.432019

Cited By: 2

Abstract

We examine European call options in the jump-diffusion version of the Double Heston stochastic volatility model for the underlying price process to provide a more flexible model for the term structure of volatility. We assume, in addition, that the stochastic interest rate is governed by the Cox-- Ross -- Ingersoll (CIR) dynamics. The instantaneous volatilities are correlated with the dynamics of the stock price process, whereas the short-term rate is assumed to be independent of the dynamics of the price process and its volatility. The main result furnishes a semi-analytical formula for the price of the European call option in the hybrid call option/interest rates model. Numerical results show that the model implied volatilities are comparable for in-sample but outperform out-of-sample implied volatilities compared to the benchmark Heston model[1], and Double Heston volatility model put forward by Christoffersen et al., [2] for calls on the S&P 500 index.

Keywords

Finance, Double Heston Jump Diffusion model, L\'evy process, Affine processes, Calibration of stochastic volatility

References

• [1] S.L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Fin. Stud., 6 (1993), 327–343.
• [2] P. Christoffersen, S. Heston, C. Jacobs, The shape and term structure of the index option smirk: Why multifactor stochastic volatility models work so well, Management Sci., 55 (2009), 1914–1932.
• [3] J. C. Cox, J. E. Ingersoll, S. A Ross, A theory of term structure of interest rates, Econometrica, 53 (1985), 385–408.
• [4] J. Gatheral, The Volatility Surface, John Wiley, 2006.
• [5] M. C. Recchioni, Y. Sun, An explicitly solvable Heston model with stochastic interest rates, European J. Oper. Res., 249 (2016), 359–377.
• [6] G. Bakshi, C. Cao, Z. Chen, Empirical performance of alternative option pricing models, J. Finance, 5 (1997), 2003–2049.
• [7] D. Bates, Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options, Rev. Fin. Stud., 9 (1996), 69–107.
• [8] D. Duffie, J. Pan, K. Singleton, Transform analysis and asset pricing for affine jump-diffusions, Econometrica, 68 (2000), 1343–1376.
• [9] T. Andersen, J. Andreasen, Jump-diffusion processes: Volatility smile fitting and numerical methods, J. Fin. Econ., 4 (2000), 231–262.
• [10] B. Eraker, M. Johannes, N. Polson, The impacts of jumps in volatility and returns, J. Finance, 58 (2003), 1269–1300.
• [11] A. L. Lewis, Option Valuation under Stochastic Volatility II, Finance Press, New Port Beach, California, USA 2016.
• [12] R. Cont, T. Kokholm, A consistent pricing model for index options and volatility derivatives, Math. Finance, 23 (2013), 248–274.
• [13] P. Carr, L. Wu, Stochastic skewness in currency options, J. Fin. Econ., 86 (2007), 213–244.
• [14] A. Van Haastrecht, R. Lord, A. Pelsser, D. Schrager, Pricing long-maturity equity and FX derivatives with stochastic interest rates and stochastic volatility, Insurance Math. Econom., 45 (2009), 436–448.
• [15] R. Schöbel, J. Zhu, Stochastic volatility with an Ornstein-Uhlenbeck process: An extension, Europ. Finance Rev., 3 (1999), 23–46.
• [16] O. Vasicek, An equilibrium characterisation of the term structure, J. Fin. Econ., 5 (1977), 177–188.
• [17] A. Van Haastrecht, A. Pelsser, Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility, Quant. Fin., 11 (2011), 665–691.
• [18] L. A. Grzelak, C. W. Oosterlee, On the Heston model with stochastic interest rates, SIAM J. Fin. Math., 2 (2011), 255–286.
• [19] L. A. Grzelak, C. W. Oosterlee, On the Heston model with stochastic interest rates, Appl. Math. Finance, 19 (2012), 1–35.
• [20] L. A. Grzelak, C. W. Oosterlee, S. Van Weeren, Extension of stochastic volatility models with Hull-White interest rate process, Quant. Finance, 12 (2012), 89–105.
• [21] A. Cozma, C. Reisinger, Convergence of an Euler discretisation scheme for the Heston stochastic-local volatility model with CIR interest rates, available at http://ideas.repec.org/p/arx/papers/1501.06084.html
• [22] R. Ahlip, M. Rutkowski, Semi-analytical pricing of currency options in the Heston/CIR jump-diffusion Hybrid model, Appl. Math. Finance, 22 (2015), 1–27.
• [23] B. Wong, C. C. Heyde, On the martingale property of stochastic exponentials, J. Appl. Probab., 41 (2004), 654–664.
• [24] M. Musiela, M. Rutkowski, Martingale Methods in Financial Modelling, Springer, Berlin, 2005.
• [25] M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial Markets, Springer, Berlin, 2009.
• [26] R. Ahlip, M. Rutkowski, Pricing of foreign exchange options under the Heston stochastic volatility model and the CIR interest rates, Quant. Finance, 13 (2013), 955–966.
• [27] A. S. Trolle, Unspanned stochastic volatility and the pricing of commodity derivatives, Rev. Financ. Stud., 22 (2009), 4423–4461.
• [28] C. Bernard, Z. Cui, C. McLeish, Nearly exact option price simulation using characteristic functions, Int. J. Theor. Appl. Finance, 2012, 1250047, 29 pages.
• [29] D. Brigo, A. Alfonsi, Credit default swaps calibration and option pricing with the SSRD stochastic intensity and interest-rate model, Finance Stochast., 9 (2005), 29–42.
• [30] P. Carr, D. Madan, Option valuation using the fast Fourier transform, J. Comput. Finance, 2 (1999), 61–73.
• [31] P. Carr, D. Madan, Saddlepoint methods for option pricing, J. Comput. Finance, 13 (2009), 49–61.
• [32] P. Carr, L. Wu, Time-changed Levy Processes and option pricing, J. Financ. Econ., 17 (2004), 113–141.
• [33] S. Levendorski˘i, Efficient pricing and reliable calibration in the Heston model, Int. J. Theor. Appl. Finance, 15 (2012), 1250050, 44 pages.
• [34] R. Lord, C. Kahl, Optimal Fourier inversion in semi-analytical option pricing, J. Comput. Finance, 10 (2007), 1–30.
• [35] R. Lord, C. Kahl, Complex logarithms in Heston-like models, Math. Finance, 20 (2010), 671–694.
• [36] F.D. Rouah, The Heston model: And it’s extensions in Matlab and C, J. Wiley & Sons, 2013.