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Nonsoong, P.; Mekchay, K.; Rujivan, S.

An analytical option pricing formula for mean-reverting asset with time-dependent parameter. (English) Zbl 1471.91582

ANZIAM J. 63, No. 2, 178-202 (2021).
Summary: 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.

Cited in 1 Document

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences

Keywords:

option pricing; mean-reverting process; Feynman-Kac formula
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References:

[1] Black, F., “The pricing of commodity contracts”, J. Financ. Econ.3 (1976) 167-179; doi: doi:10.1016/0304-405X(76)90024-6.
[2] Black, F. and Karasinski, P., “Bond and option pricing when short rates are lognormal”, Financial Anal. J.47 (1991) 52-59; doi: doi:10.2469/faj.v47.n4.52.
[3] Black, F. and Scholes, M., “The pricing of options and corporate liabilities”, J. Pol. Econ.81 (1973) 637-654; doi: doi:10.1086/260062. · Zbl 1092.91524
[4] Brigo, D. and Mercurio, F., Interest rate models theory and practice, 1st edn (Springer, Berlin-Heidelberg, 2001); doi: doi:10.1007/978-3-662-04553-4. · Zbl 1038.91040
[5] Carr, P., Jarrow, R. and Mynen, R., “Alternative characterizations of American put options”, Math. Finance2 (1992) 87-106; doi: doi:10.1111/j.1467-9965.1992.tb00040.x. · Zbl 0900.90004
[6] 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: doi:10.1016/j.orl.2011.06.002. · Zbl 1242.91185
[7] 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. · Zbl 1414.91368
[8] Clewlow, L. and Strickland, C., “Valuing energy options in a one factor model fitted to forward prices”, SSRN160608 (1999); doi: doi:10.2139/ssrn.160608.
[9] 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 · Zbl 0924.28001
[10] Hull, J. and White, A., “Pricing interest-rate-derivative securities”, Rev. Financ. Stud.3 (1990) 573-592; doi: doi:10.1093/rfs/3.4.573. · Zbl 1386.91152
[11] 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. · Zbl 1087.91025
[12] Jacka, S. D., “Optimal stopping and the American put”, Math. Finance1 (1991) 1-14; doi: doi:10.1111/j.1467-9965.1991.tb00007.x. · Zbl 0900.90109
[13] Kim, I. J., “The analytic valuation of American options”, Rev. Financ. Stud.3 (1990) 547-572; doi: doi:10.1093/rfs/3.4.547.
[14] Merton, R. C., “Theory of rational option pricing”, Bell J. Econ. Manag. Sci.4 (1973) 141-183; doi: doi:10.2307/3003143. · Zbl 1257.91043
[15] Øksendal, B., Stochastic differential equations: an introduction with applications, 6th edn (Springer, Berlin, 2003), available at www.springer.com/gp/book/9783540047582. · Zbl 1025.60026
[16] Rujivan, S., “A novel analytical approach for pricing discretely sampled gamma swaps in the Heston model”, ANZIAM J.57 (2016) 244-268; doi: doi:10.1017/S1446181115000309. · Zbl 1415.91289
[17] Rujivan, S., “Analytically pricing variance swaps in commodity derivative markets under stochastic convenience yields”, Commun. Math. Sci.19 (2021) 111-146; doi: doi:10.4310/CMS.2021.v19.n1.a5. · Zbl 1479.91413
[18] 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: doi:10.1016/j.cnsns.2021.105849. · Zbl 1464.91073
[19] 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: doi:10.1016/j.aml.2012.01.029. · Zbl 1260.91102
[20] 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: doi:10.1017/S1446181114000236. · Zbl 1298.91169
[21] Schwartz, E. S., “The stochastic behavior of commodity prices: implications for valuation and hedging”, J. Finance52 (1997) 923-973; doi: doi:10.1111/j.1540-6261.1997.tb02721.x.
[22] 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. · Zbl 1164.60047
[23] 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: doi:10.1016/S1468-1218(01)00028-1. · Zbl 1011.91049
[24] Weraprasertsakun, A. and Rujivan, S., “A closed-form formula for pricing variance swaps on commodities”, Vietnam J. Math.45 (2017) 255-264; doi: doi:10.1007/s10013-016-0224-9. · Zbl 1366.91156
[25] Wong, H. Y. and Lo, Y. W., “Option pricing with mean reversion and stochastic volatility”, European J. Oper. Res.197 (2009) 179-187; doi: doi:10.1016/j.ejor.2008.05.014. · Zbl 1157.91375
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