Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T05:09:58.987Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  14 October 2019

S. N. I. IBRAHIM*
Affiliation:
Department of Mathematics, Faculty of Science and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia email [email protected]
A. DÍAZ-HERNÁNDEZ
Affiliation:
Faculty of Economics and Business, Universidad Anahuac Mexico-Norte, Huixquilucan 52786, Mexico email [email protected]
J. G. O’HARA
Affiliation:
School of Computer Science and Electronic Engineering, University of Essex, Colchester CO4 3SQ, UK email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Copyright
© 2019 Australian Mathematical Society

Footnotes

*

Deceased.

References

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.Google Scholar
Black, F. and Scholes, M., “The pricing of options and corporate liabilities”, J. Political Economy 81 (1973) 637654; doi:10.1086/260062.Google Scholar
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.Google Scholar
Buchen, P. W., “The pricing of dual-expiry exotics”, Quant. Finance 4 (2004) 101108; doi:10.1088/1469-7688/4/1/009.Google Scholar
Carr, P. and Madan, D., “Option valuation using the fast Fourier transform”, J. Comput. Finance 2 (1999) 6173; doi:10.21314/JCF.1999.043.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Glasserman, P., Monte Carlo methods in financial engineering (Springer, New York, 2004).Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Hirsa, A., Computational methods in finance (Chapman and Hall/CRC Press, London, 2013.).Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Kiusalaas, J., Numerical methods in engineering with Python, 2nd edn (Cambridge University Press, Cambridge, 2012).Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Rouah, F. D., The Heston model and its extensions in Matlab and C# (John Wiley & Sons, New Jersey, 2013).Google Scholar
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.Google Scholar
Shevchenko, P. V., “Holder-extendible European option: corrections and extensions”, ANZIAM J. 56 (2015) 359372; doi:10.1017/S1446181115000097.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar