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PRICING HOLDER-EXTENDABLE CALL OPTIONS WITH MEAN-REVERTING STOCHASTIC VOLATILITY

  • S. N. I. IBRAHIM (a1), A. DÍAZ-HERNÁNDEZ (a2), J. G. O’HARA (a3) and N. CONSTANTINOU (a1) (a2) (a3)

Abstract

Options with extendable features have many applications in finance and these provide the motivation for this study. The pricing of extendable options when the underlying asset follows a geometric Brownian motion with constant volatility has appeared in the literature. In this paper, we consider holder-extendable call options when the underlying asset follows a mean-reverting stochastic volatility. The option price is expressed in integral forms which have known closed-form characteristic functions. We price these options using a fast Fourier transform, a finite difference method and Monte Carlo simulation, and we determine the efficiency and accuracy of the Fourier method in pricing holder-extendable call options for Heston parameters calibrated from the subprime crisis. We show that the fast Fourier transform reduces the computational time required to produce a range of holder-extendable call option prices by at least an order of magnitude. Numerical results also demonstrate that when the Heston correlation is negative, the Black–Scholes model under-prices in-the-money and over-prices out-of-the-money holder-extendable call options compared with the Heston model, which is analogous to the behaviour for vanilla calls.

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[1] Ananthanarayanan, A. L. and Schwartz, E. S., “Retractable and extendible bond: the Canadian experience”, J. Finance 35 (1980) 3147; doi:10.1111/j.1540-6261.1980.tb03469.x.
[2] Black, F. and Scholes, M., “The pricing of options and corporate liabilities”, J. Political Economy 81 (1973) 637654; doi:10.1086/260062.
[3] Brennan, M. J. and Schwartz, E. S., “Savings bonds, retractable bonds and callable bonds”, J. Financ. Econ. 5 (1977) 6788; doi:10.1016/0304-405X(77)90030-7.
[4] Buchen, P. W., “The pricing of dual-expiry exotics”, Quant. Finance 4 (2004) 101108; doi:10.1088/1469-7688/4/1/009.
[5] Carr, P. and Madan, D., “Option valuation using the fast Fourier transform”, J. Comput. Finance 2 (1999) 6173; doi:10.21314/JCF.1999.043.
[6] Chung, Y. P. and Johnson, H., “Extendible options: the general case”, Finance Res. Lett. 8 (2011) 1520; doi:10.1016/j.frl.2010.09.003.
[7] Cont, R. and Voltchkova, E., “A finite difference scheme for option pricing in jump diffusion and exponential Lévy models”, SIAM J. Numer. Anal. 43 (2005) 15961626; doi:10.1137/S0036142903436186.
[8] Cox, J. C., Ingersoll, J. E. and Ross, S. A., “A theory of the term structure of interest rate”, Econometrica 53 (1985) 385407; doi:10.2307/1911242.
[9] Dias, M. A. G. and Rocha, K. M. C., “Petroleum concessions with extendible options using mean reversion with jumps to model oil prices”. Working paper, IPEA, Brazil (1999) 1–27; http://realoptions.org/papers1999/MarcoKatia.pdf.
[10] Glasserman, P., Monte Carlo methods in financial engineering (Springer, New York, 2004).
[11] Griebsch, S. A. and Wystup, U., “On the valuation of fader and discrete barrier options in Heston’s stochastic volatility model”, Quant. Finance 11 (2011) 693709; doi:10.1080/14697688.2010.503375.
[12] Gukhal, C. R., “The compound option approach to American options on jump-diffusion”, J. Econom. Dynam. Control 28 (2004) 20552074; doi:10.1016/j.jedc.2003.06.002.
[13] Hauser, S. and Lauterbach, B., “Empirical tests of the Longstaff extendible warrant model”, J. Empir. Finance 3 (1996) 114; doi:10.1016/0927-5398(95)00019-4.
[14] Heston, S. L., “A closed-form solution for options with stochastic volatility with applications to bond and currency options”, Rev. Financial Stud. 6 (1993) 327343; doi:10.1093/rfs/6.2.327.
[15] Hirsa, A., Computational methods in finance (Chapman and Hall/CRC Press, London, 2013.).
[16] Howe, J. S. and Wei, P., “The valuation effects of warrant extensions”, J. Finance 48 (1993) 305314; doi:10.1111/j.1540-6261.1993.tb04711.x.
[17] Huang, J., Zhu, W. and Ruan, X., “Fast Fourier transform based power option pricing with stochastic interest rate, volatility, and jump intensity”, J. Appl. Math. 2013 (2013) 17; doi:10.1155/2013/875606.
[18] Hurd, T. R. and Zhou, Z., “A Fourier transform method for spread option pricing”, SIAM J. Financial Math. 1 (2010) 142157; doi:10.1137/090750421.
[19] Ibrahim, S. N. I., Ng, T. W., O’Hara, J. G. and Nawawi, A., “Pricing holder-extendable options in a stochastic volatility model with an Ornstein–Uhlenbeck process”, Malays. J. Math. Sci. 11 (2017) 18; http://einspem.upm.edu.my/journal/fullpaper/vol11/1.pdf.
[20] Ibrahim, S. N. I., O’Hara, J. G. and Constantinou, N., “Pricing extendible options using the fast Fourier transform”, Math. Prob. Eng. 2014 (2014) 17; doi:10.1155/2014/831470.
[21] Kiusalaas, J., Numerical methods in engineering with Python, 2nd edn (Cambridge University Press, Cambridge, 2012).
[22] Koussis, N., Martzoukos, S. H. and Trigeorgis, L., “Multi-stage product development with exploration, value-enhancing, preemptive and innovation options”, J. Banking Finance 37 (2013) 174190; doi:10.1016/j.jbankfin.2012.08.020.
[23] Longstaff, F. A., “Pricing options with extendible maturities: analysis and applications”, J. Finance 45 (1990) 935957; doi:10.1111/j.1540-6261.1990.tb05113.x.
[24] Merton, R. C., “Option pricing when underlying stock returns are discontinuous”, J. Financ. Econ. 3 (1976) 125144; doi:10.1016/0304-405X(76)90022-2.
[25] Moyaert, T. and Petitjean, M., “The performance of popular stochastic volatility option pricing models during the subprime crisis”, Appl. Financ. Econ. 21 (2011) 10591068; doi:10.1080/09603107.2011.562161.
[26] Neftci, S. N. and Santos, A. O., “Puttable and extendible bonds: developing interest rate derivatives for emerging markets”. IMF Working paper, WP/03/201 (International Monetary Fund, 2003) http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.585.3038&rep=rep1&type=pdf.
[27] Peng, B. and Peng, F., “Pricing extendible option under jump-fraction process”, J. East China Norm. Univ. Natur. Sci. 2012 (2012) 3040; doi:10.3969/j.issn.1000-5641.2012.03.006.
[28] Pillay, E. and O’Hara, J. G., “FFT based option pricing under a mean reverting process with stochastic volatility and jumps”, J. Comput. Appl. Math. 235 (2011) 33783384; doi:10.1016/j.cam.2010.10.024.
[29] Rouah, F. D., The Heston model and its extensions in Matlab and C# (John Wiley & Sons, New Jersey, 2013).
[30] Santa-Clara, P. and Yan, S., “Crashes, volatility, and the equity premium: lessons from S&P 500 options”, Rev. Econ. Stat. 92 (2010) 435451; doi:10.1162/rest.2010.11549.
[31] Shevchenko, P. V., “Holder-extendible European option: corrections and extensions”, ANZIAM J. 56 (2015) 359372; doi:10.1017/S1446181115000097.
[32] Sophocleous, C., O’Hara, J. G. and Leach, P. G. L., “Algebraic solution of the Stein–Stein model for stochastic volatility”, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 17521759; doi:10.1016/j.cnsns.2010.08.008.
[33] Zhang, S. and Wang, L., “A fast Fourier transform technique for pricing European options with stochastic volatility and jump risk”, Math. Probl. Eng. 2012 (2012) 117; doi:10.1155/2012/761637.
[34] Zhang, S. and Wang, L., “Fast Fourier transform option pricing with stochastic interest rate, stochastic volatility and double jumps”, Appl. Math. Comput. 219 (2013) 1092810933; doi:10.1016/j.amc.2013.05.008.
[35] Zhang, S. and Wang, L., “A fast numerical approach to option pricing with stochastic interest rate, stochastic volatility and double jumps”, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 18321839; doi:10.1016/j.cnsns.2012.11.010.
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PRICING HOLDER-EXTENDABLE CALL OPTIONS WITH MEAN-REVERTING STOCHASTIC VOLATILITY

  • S. N. I. IBRAHIM (a1), A. DÍAZ-HERNÁNDEZ (a2), J. G. O’HARA (a3) and N. CONSTANTINOU (a1) (a2) (a3)

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