CGMY模型下 的欧式外汇期权定价

修正传统有效市场假说,重新假设外汇汇率存在扩散和跳跃,并结合CGMY模型,采用傅里叶变换方法,推导出了CGMY模型下欧式外汇期权价格满足的分数阶偏微分方程(FPDE).尽管因分数阶偏导数引发的“全局性”很难处理,仍然推导出CGMY模型下欧式外汇期权的定价公式及其满足的平价公式.同时,引入一个新的缩放参数m来控制指数函数的增长率以克服被积函数衰减引起的计算困难,使其与Lévy密度函数的衰减在速度上达到一个平衡.最后,从数学与金融意义上分析了关键参数变化对欧式外汇期权价格的影响.

Analytically Pricing Foreign Exchange Options under the CGMY Model

Modifying the traditional effective market hypothesis(EMH)and assuming that the foreign exchange rate exists jumps and spreads,the fractional partial differential equation(FPDE)of the European foreign exchange option price under the CGMY model is derived by using Fourier transform method.Although the“globalness”caused by fractional partial derivatives is difficult to handle,the explicit closed-form analytical solution and the put-call parity of European foreign exchange options under the CGMY model are still derived.In addition,to effectively overcome the computational difficulties caused by the integrand decay,a new scaling parameter m is introduced to control the growth rate of the exponential function,so that it can achieve a balance with the attenuation of the Lévy density function.Finally,the influence of key parameter changes on the price of European foreign exchange options is analyzed from the perspective of mathematics and finance.