基于跳-分形模型的美式看涨期权定价

Pricing American Call Option Under the Jump-Fraction Model

假设股票变化过程服从跳一分形布朗运动,根据风险中性定价原理对股票发生跳跃次数的收益求条件期望现值推导出M次离散支付红利的美式看涨期权解析定价方程,并使用外推加速法求出当M趋于无穷时方程的二重、三重正态积分多项式表达,依此计算连续支付红利美式看涨期权价值.数值模拟表明通常仅需二重正态积分多项式能产生精确价值,而在极实值状态下则需三重正态积分多项式才能满足,结合两种多项式可以编出有效数字程序评价支付红利的美式看涨期权.

With the assumption that stock follows the jump fractional Brownian motion,an analytic pricing equation for the American call option on stocks paying M dividends was derived using the risk neutral pricing principle which can solve the expected present value of cashflow by conditioning the number of jumps for the stock.Polynomial expressions requiring bivariate normal integrals and trivariate normal integrals respectively were deduced for evaluating the equation when M tend to infinite with help of the extrapolation acceleration approach,thus American call option with continuous dividends is calculated.Numerical simulations show that Polynomial expression yields in most cases accurate values using nothing more than bivariate normals.In the more difficult（deep- in- the- money） cases,trivariate normals suffice.A combination of these Polynomial expressions yields a numerically efficient procedure for valuation of American call option that is useful even for personal computers.