Quasi-closed-form solution and numerical method for currency option with uncertain volatility model
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METHODOLOGIES AND APPLICATION
Quasi-closed-form solution and numerical method for currency option with uncertain volatility model Zhe Li1
· Yong-Jun Liu2 · Wei-Guo Zhang2
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Currency option pricing · Uncertain volatility model · Runge–Kutta-99 hybrid method · Uncertain differential equation · Uncertainty theory
1 Introduction The globalization of world economy has made a greater progress in recent decades. The different economies can exchange goods, services and currencies. Consequently, the foreign exchange market is one of the largest financial markets in the world, yet the pricing of foreign exchange derivatives poses challenges for modeling. Under the premise of stock price followed a geometric Brownian motion, Black and Scholes (1973) originally developed European option pricing formula. After that, a great deal of researchers Communicated by V. Loia.
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Wei-Guo Zhang [email protected] Zhe Li [email protected] Yong-Jun Liu [email protected]
1
Business School, Nanjing Normal University, Nanjing 210023, China
2
School of Business Administration, South China University of Technology, Guangzhou 510641, China
devoted to developing foreign currency option pricing models. Garman and Kohlhagen (1983) supposed that both the domestic and foreign interest rates were constants and the exchange rate followed a stochastic process and then derived the pricing formula for foreign currency option. Grabbe (1983) and Hilliard et al. (1991) developed foreign currency option pricing models with stochastic interest rates. Heston (1993) assumed that the volatility of underlying asset price was driven by a stochastic process and then derived the closed-form solutions for bond and foreign currency options. Ekvalla et al. (1997) considered a model for the pric
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