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AN ANALYTICAL OPTION PRICING FORMULA FOR MEAN-REVERTING ASSET WITH TIME-DEPENDENT PARAMETER

Part of: Equations of mathematical physics and other areas of application Mathematical finance

Published online by Cambridge University Press:  23 August 2021

P. NONSOONG Open the ORCID record for P. NONSOONG [Opens in a new window] ,
K. MEKCHAY Open the ORCID record for K. MEKCHAY [Opens in a new window]  and
S. RUJIVAN Open the ORCID record for S. RUJIVAN [Opens in a new window]
P. NONSOONG
Affiliation:
Department of Mathematics and Computer Science, Chulalongkorn University, Bangkok, Thailand; e-mail: piyapoom.n@hotmail.com
K. MEKCHAY*
Affiliation:
Department of Mathematics and Computer Science, Chulalongkorn University, Bangkok, Thailand; e-mail: piyapoom.n@hotmail.com
S. RUJIVAN
Affiliation:
Center of Excellence in Data Science for Health Study, Division of Mathematics and Statistics, School of Science, Walailak University, Nakhon Si Thammarat, Thailand; e-mail: rsanae@wu.ac.th
*
e-mail: khamron.m@chula.ac.th
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Abstract

We present an analytical option pricing formula for the European options, in which the price dynamics of a risky asset follows a mean-reverting process with a time-dependent parameter. The process can be adapted to describe a seasonal variation in price such as in agricultural commodity markets. An analytical solution is derived based on the solution of a partial differential equation, which shows that a European option price can be decomposed into two terms: the payoff of the option at the initial time and the time-integral over the lifetime of the option driven by a time-dependent parameter. Finally, results obtained from the formula have been compared with Monte Carlo simulations and a Black–Scholes-type formula under various kinds of long-run mean functions, and some examples of option price behaviours have been provided.

Keywords

option pricingmean-reverting processFeynman–Kac formula

MSC classification

Primary: 91G20: Derivative securities
Secondary: 35Q91: PDEs in connection with game theory, economics, social and behavioral sciences
Type
Research Article
Information
The ANZIAM Journal , Volume 63 , Issue 2 , April 2021 , pp. 178 - 202
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Copyright
© Australian Mathematical Society 2021

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References

Black, F., “The pricing of commodity contracts”, J. Financ. Econ. 3 (1976) 167–179; doi:10.1016/0304-405X(76)90024-6.CrossRefGoogle Scholar
Black, F. and Karasinski, P., “Bond and option pricing when short rates are lognormal”, Financial Anal. J. 47 (1991) 52–59; doi:10.2469/faj.v47.n4.52.CrossRefGoogle Scholar
Black, F. and Scholes, M., “The pricing of options and corporate liabilities”, J. Pol. Econ. 81 (1973) 637–654; doi:10.1086/260062.CrossRefGoogle Scholar
Brigo, D. and Mercurio, F., Interest rate models theory and practice, 1st edn (Springer, Berlin–Heidelberg, 2001); doi:10.1007/978-3-662-04553-4.CrossRefGoogle Scholar
Carr, P., Jarrow, R. and Mynen, R., “Alternative characterizations of American put options”, Math. Finance 2 (1992) 87–106; doi:10.1111/j.1467-9965.1992.tb00040.x.CrossRefGoogle Scholar
Chiu, M. C., Lo, Y. W. and Wong, H. Y., “Asymptotic expansion for pricing options for a mean-reverting asset with multiscale stochastic volatility”, Oper. Res. Lett. 39 (2011) 289–295; doi:10.1016/j.orl.2011.06.002.CrossRefGoogle Scholar
Chunhawiksit, C. and Rujivan, S., “Pricing discretely-sampled variance swaps on commodities”, Thai J. Math. 14 (2016) 711–724, available at http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/view/1638.Google Scholar
Clewlow, L. and Strickland, C., “Valuing energy options in a one factor model fitted to forward prices”, SSRN 160608 (1999); doi:10.2139/ssrn.160608.Google Scholar
Folland, G. B., Real analysis: modern techniques and their applications, 2nd edn (John Wiley & Sons, New York, 1999), available at https://www.wiley.com/en-us/Real+Analysis%3A+Modern+Techniques+and+Their+Applications%2C+2nd+Edition-p-9781118626399.Google Scholar
Hull, J. and White, A., “Pricing interest-rate-derivative securities”, Rev. Financ. Stud. 3 (1990) 573–592; doi:10.1093/rfs/3.4.573.CrossRefGoogle Scholar
Hull, J. C., Options, futures, and other derivatives, 8th edn (Prentice Hall, Boston, MA, 2012), available at www.pearson.com/us/higher-education/product/Hull-Options-Futures-and-Other- Derivatives-8th-Edition/9780132164948.html.Google Scholar
Jacka, S. D., “Optimal stopping and the American put”, Math. Finance 1 (1991) 1–14; doi:10.1111/j.1467-9965.1991.tb00007.x.CrossRefGoogle Scholar
Kim, I. J., “The analytic valuation of American options”, Rev. Financ. Stud. 3 (1990) 547–572; doi:10.1093/rfs/3.4.547.CrossRefGoogle Scholar
Merton, R. C., “Theory of rational option pricing”, Bell J. Econ. Manag. Sci. 4 (1973) 141–183; doi:10.2307/3003143.CrossRefGoogle Scholar
Øksendal, B., Stochastic differential equations: an introduction with applications, 6th edn (Springer, Berlin, 2003), available at www.springer.com/gp/book/9783540047582.CrossRefGoogle Scholar
Rujivan, S., “A novel analytical approach for pricing discretely sampled gamma swaps in the Heston model”, ANZIAM J. 57 (2016) 244–268; doi:10.1017/S1446181115000309.Google Scholar
Rujivan, S., “Analytically pricing variance swaps in commodity derivative markets under stochastic convenience yields”, Commun. Math. Sci. 19 (2021) 111–146; doi:10.4310/CMS.2021.v19.n1.a5.CrossRefGoogle Scholar
Rujivan, S. and Rakwongwan, U., “Analytically pricing volatility swaps and volatility options with discrete sampling: nonlinear payoff volatility derivatives”, Commun. Nonlinear Sci. Numer. Simul. 100 (2021) 1–29; doi:10.1016/j.cnsns.2021.105849.CrossRefGoogle Scholar
Rujivan, S. and Zhu, S.-P., “A simplified analytical approach for pricing discretely-sampled variance swaps with stochastic volatility”, Appl. Math. Lett. 25 (2012) 1644–1650; doi:10.1016/j.aml.2012.01.029.CrossRefGoogle Scholar
Rujivan, S. and Zhu, S.-P., “A simple closed-form formula for pricing discretely-sampled variance swaps under the Heston model”, ANZIAM J. 56 (2014) 1–27; doi:10.1017/S1446181114000236.Google Scholar
Schwartz, E. S., “The stochastic behavior of commodity prices: implications for valuation and hedging”, J. Finance 52 (1997) 923–973; doi:10.1111/j.1540-6261.1997.tb02721.x.CrossRefGoogle Scholar
Swishchuk, A. V., “Explicit option pricing formula for a mean-reverting asset in energy market”, J. Numer. Appl. Math. 1 (2008) 216–233; https://ssrn.com/abstract=1492843.Google Scholar
Underwood, R. and Wang, J., “An integral representation and computation for the solution of American options”, Nonlinear Anal. Real World Appl. 3 (2002) 259–274; doi:10.1016/S1468-1218(01)00028-1.CrossRefGoogle Scholar
Weraprasertsakun, A. and Rujivan, S., “A closed-form formula for pricing variance swaps on commodities”, Vietnam J. Math. 45 (2017) 255–264; doi:10.1007/s10013-016-0224-9.CrossRefGoogle Scholar
Wong, H. Y. and Lo, Y. W., “Option pricing with mean reversion and stochastic volatility”, European J. Oper. Res. 197 (2009) 179–187; doi:10.1016/j.ejor.2008.05.014.CrossRefGoogle Scholar