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

ZOU Jian (Applied Economics Department, Guanghua School of Management, Peking University, Beijing 100871, China)  
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.
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