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Quasi-closed-form solution and numerical method for currency option with uncertain volatility model

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Abstract

There exist some non-stochastic factors in the financial market, so the dynamics of the exchange rate highly depends on human uncertainty. This paper investigates the pricing problems of foreign currency options under the uncertain environment. First, we propose an currency model under the assumption that exchange rate, volatility, domestic interest rate and foreign interest rate are all driven by uncertain differential equations; especially, the exchange rate exhibits mean reversion. Since the analytical solutions of nested uncertain differential equations cannot always be obtained, we design a new numerical method, Runge–Kutta-99 hybrid method, for solving nested uncertain differential equations. The accuracy of the designed numerical method is investigated by comparison with the analytical solution. Subsequently, the quasi-closed-form solutions are derived for the prices of both European and American foreign currency options. Finally, in order to illustrate the rationality and the practicability of the proposed currency model, we design several numerical algorithms to calculate the option prices and analyze the price behaviors of foreign currency options across strike price and maturity.

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Notes

  1. Our paper completed the writing of original manuscript in 2016, and we submitted it to journal in January 2017. As far as we know, there was no uncertain currency model considering both mean-reversion and floating interest rates at that time. It was not until we recently received comments from the editorial board that we noticed that the work of Wang and Chen (2019) had studied the mean-reversion uncertain currency model with floating rates. Nevertheless, our proposed model can still be regarded as an extended version of them.

  2. In general, it is difficult to obtain the analytic solution of uncertain differential equation, let alone the nested uncertain differential equation. So far, many studies have proposed different types of numerical methods to solve this problem, see, for example, Yang and Ralescu (2015), Yang and Shen (2015), Wang et al. (2015), Gao (2016), Zhang et al. (2017), Ji and Zhou (2018) and among others. However, the widely used Runge–Kutta method is very simple and has high accuracy for solving uncertain differential equations. Additionally, this paper mainly focuses on proposing a new dynamic model for exchange rate and giving the quasi-closed-form solutions of the prices of European and American foreign currency options. Thus, this paper only presents a numerical method for calculating the inverse uncertainty distribution of the solution of nested uncertain differential equations based on Runge–Kutta method and does not consider the applicability of other numerical methods. One can explore the capabilities of other numerical methods, such as Adams method, Milne method and Hamming method.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 71901124, U1901223 and 71720107002), the Natural Science Foundation of Jiangsu Province (Grant No. BK20190695) and the Fundamental Research Funds for the Central Universities (Grant No. 2019ZD13). The authors would also like to thank the Editor and anonymous reviewers for their constructive comments and suggestions.

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Authors and Affiliations

  1. Business School, Nanjing Normal University, Nanjing, 210023, China

    Zhe Li

  2. School of Business Administration, South China University of Technology, Guangzhou, 510641, China

    Yong-Jun Liu & Wei-Guo Zhang

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  1. Zhe Li
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  2. Yong-Jun Liu
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  3. Wei-Guo Zhang
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Correspondence to Wei-Guo Zhang.

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Li, Z., Liu, YJ. & Zhang, WG. Quasi-closed-form solution and numerical method for currency option with uncertain volatility model. Soft Comput 24, 15041–15057 (2020). https://doi.org/10.1007/s00500-020-04854-3

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