Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T07:02:59.108Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© Australian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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