Generalized hyperbolic distributions and their applications to finance: parameter estimation, vanilla European option pricing and algorithms

在回顾了广义双曲分布，尤其是其具有良好卷积性质的正态逆高斯分布的特征后，使用期望折现方法对普通欧式看涨期权定价问题进行了研究，并利用调和分析中的积分逼近理论提出了普通欧式看涨期权的渐进解析表达式.为了解决参数估计和解非线性方程（组）的问题，笔者提出将模拟退火算法和遗传算法混合，对于解决像包含Bessel函数这样复杂、具有大量局部最优解的优化问题，具有很现实的意义.对于解非线性方程（组），可以通过把问题转化为最小化误差函数，直接使用这种算法.分析证明，这个算法可以在合理的时间内以任意精度逼近解.After a brief literature review of the current study on the application of the Generalized Hyperbolic Distribution to finance, we use the well-known discounted expectation method to price the vanilla European calls. Then we derive an analytical asymptotic approximation of the complex integral containing Bessel functions. To solve the problem of parameter estimation and non-linear equation system of the Esscher Transformation, we devise a hybrid algorithm of the genetic algorithm embedded with the annealing algorithm with backfire. The algorithm proves to be robust and fast convergent in the optimization of complex functions with Bessel functions. The same algorithm also lends itself to the solution of nonlinear equation system after reformulating the original problem to minimizing the sum of the absolute errors of the two sides of the equation system.中国科学引文数据库(CSCD)03202-2091