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

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.