Abstract
In this paper, we introduce a dynamic model for the spot foreign exchange rate which is driven by a standard Brownian motion and a stationary compound Poisson process under the domestic real measure. In order to price the derivatives on the foreign exchange rate, we need to find an equivalent probability measure under which the discounted process of the foreign exchange rate by the domestic free interest rate minus the foreign free interest rate is a martingale. The Esscher transform is an efficient technique to find an equivalent martingale measure. Applying the tool of the characteristic function, we derive some Esscher transform parameters with respect to the spot foreign exchange rate. At the same time, we get the corresponding Esscher martingale measure which is the domestic risk-neutral measure Q equivalent to the domestic real measure. Moreover, we reconsider the dynamic process of the spot foreign exchange rate under the measure Q. Furthermore, we hope that the exchange rate fluctuates within a certain range, since too large fluctuation will bring a series of serious problems. In fact, the foreign exchange rate is usually stable in a certain range. Thus, studying the pricing of foreign exchange rate derivatives, we often assume that the foreign exchange rate fluctuates within a certain range. Based on the above work, we combine European option with the power option to propose a new type of the foreign exchange power option whose payoff function is controlled by multiplying an indicative function on the interval of the foreign exchange rate and further obtain the pricing formulas under this model. At last, we utilize the actual market data of the foreign exchange rate of USD/CNY to obtain the value of the foreign exchange power option and investigate the implied volatility.
Similar content being viewed by others
References
Ahlip, R., & King, R. (2010). Computational aspects of pricing forergn exchange options with stochastic volatility and stochastic interest rates. Journal of Statistical Planning and Inference, 140, 1256–1268.
Andrew, C. N., & Johnny, S. L. (2011). Valuing variable annuity guarantees with the multivariate Esscher transform. Insurance: Mathematics and Economics, 49, 393–400.
Bo, L. J., Wang, Y. J., & Yang, X. W. (2010). Markov-modulated jump-diffusions for currency option pricing. Insurance: Mathematics and Economics, 46, 461–469.
Fard, F. A. (2015). Analytical pricing of vulnerable options under a generalized jump-diffusion model. Insurance: Mathematics and Economics, 60, 19–28.
Gerber, H. U., & Shiu, E. S. W. (1994). Option pricing by Esscher transforms. Transactions of the Society of Actuaries, 46, 91–99.
Gerber, H. U., & Shiu, E. S. W. (1996). Actuarial bridges to dynamic hedging and option pricing. Insurance: Mathematics and Economics, 18, 183–218.
Harrison, J. M., & Kreps, D. M. (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20(3), 381–408.
Harrison, J. M., & Pliska, S. R. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and Their Applications, 11(3), 215–260.
Klebaner, F. C. (2005). Introduction to stochastic calculus with applications (2nd ed.). London: Imperial College Press.
Lau, J. W., & Siu, T. K. (2008). On option pricing under a completely random measure via a generalized Esscher transform. Insurance: Mathematics and Economics, 43, 99–107.
Li, W. H., Liu, L. X., Lv, G. W., & Li, C. X. (2018a). Exchange option pricing in jump-diffusion models based on Esscher transform. Communications in Statistics - Theory and Methods, 42(19), 4661–4672.
Li, Z., Zhang, W. G., & Liu, Y. J. (2018b). European Guanto option pricing in presence of liquidity risk. The North American Journal of Economics and Finance, 45, 230–244.
Miao, D. W., Lin, X. C., & Yu, S. H. (2016). A note on the never-early-exercise region of American power exchange options. Operations Research Letters, 44, 129–135.
Rao, B. P. (2016). Pricing geometric Asian power options under mixed fractional Brownian motion environment. Physica A, 446, 92–99.
Shreve, S. (2004). Stochastic calculus for finance II: Continuous-time models. New York: Springer.
Sun, Q., & Xu, W. (2015). Pricing foreign equity option with stochastic volatility. Physica A, 437, 89–100.
Su, X. N., Wang, W., & Wang, W. S. (2013). Valuing power options under a regime-switching model. Journal of East China Normal University, 6, 32–39.
Swishchuk, A., Tertychnyi, M., & Elliott, R. (2014). Pricing currency derivatives with Markov-modulated Levy dynamics. Insurance: Mathematics and Economics, 57, 67–76.
Funding
This funding was provided by The Social Science Foundation Project of Hebei Province, Grant No. HB19YJ055.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, W., Li, C., Liu, L. et al. Foreign Currency Power Option Pricing Based on Esscher Transform. Comput Econ 58, 535–548 (2021). https://doi.org/10.1007/s10614-020-10046-w
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10614-020-10046-w