Abstract
Suppose that the interest rates obey stochastic differential equations, while the exchange rate follows an uncertain differential equation; this paper proposes a new currency model. Under the proposed currency model, the pricing formula of European currency options is then derived. Some numerical examples recorded illustrate the quality of pricing formulas. Meanwhile, this paper analyzes the relationship between the pricing formula and some parameters.
Highlights
Nowadays, the currency option is one of the best investment tools for companies and individuals to hedge against adverse movements in exchange rates
Suppose that the interest rates obey stochastic differential equations, while the exchange rate follows an uncertain differential equation; this paper proposes a new currency model
Assume that the exchange rate follows an uncertain differential equation; a new currency model is proposed as follows: drt = a1 (b1 − rt) dt + σ1dW1t dft = a2 (b2 − ft) dt + σ2dW2t dZt = μZtdt + σZtdCt where σ1 is the diffusion of rt; σ2 is the diffusion of ft; a1, a2, b1, and b2 are the constant parameters; W1t and W2t are independent Wiener processes and Wit and Ct are independent for i = 1, 2
Summary
The currency option is one of the best investment tools for companies and individuals to hedge against adverse movements in exchange rates. In the above-mentioned literature, the exchange rate follows a stochastic differential equation under the framework of probability theory. Some emergencies coming from wars, political policies, or natural disasters may affect the exchange rate In this case, it is difficult to obtain available statistical data about exchange rate, and the assumption that the exchange rate follows a stochastic differential equation may be out of work. Sheng and Shi [14] proposed the mean-reverting currency model under Asian currency option In these currency models listed above, the exchange rate is governed by an uncertain process instead of stochastic process, and the interest rates are taken as constant. Taking into account two factors, randomness and uncertainty, we propose a new currency model in this paper In detail, both domestic and foreign interest rates follow stochastic differential equations, while the exchange rate follows an uncertain differential equation.
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