线性与非线性扩散和鞅期权定价
Linear vs. Nonlinear Diffusion and Martingale Option Pricing
摘要：
首先，类马尔可夫过程的规模正好与赫斯特指数H导出的封闭形式。一类的特殊情况是Tsallis密度，在其他地方称为非线性扩散或非线性反馈扩散。但是，Tsallis模型只是一个非常大的类的线性扩散与学生t一样的密度之一。其次，我们证明了随机微积分，我们的推广的Black-Scholes偏微分方程（PDE）的可变扩散系数是等价于风险中性的股票价格的鞅贴现。以前，这是证明了有限的情况下，高斯对数回报率的哈里森和Kreps，但我们证明它在这里的大类经验有用的和理论上有趣的回报模型，其中扩散系数D（x，t）取决于对数回报率x和时间t。最后，我们证明了如果收益率x的厚尾包含在市场分布中，期权价格会爆炸。
英文摘要：
First, classes of Markov processes that scale exactly with a Hurst exponent H are derived in closed form. A special case of one class is the Tsallis density, advertised elsewhere as nonlinear diffusion or diffusion with nonlinear feedback. But the Tsallis model is only one of a very large class of linear diffusion with a student-t like density. Second, we show by stochastic calculus that our generalization of the Black-Scholes partial differential equation (pde) for variable diffusion coefficients is equivalent to a Martingale in the risk neutral discounted stock price. Previously, this was proven for the restricted case of Gaussian logarithmic returns by Harrison and Kreps, but we prove it here for large classes of empirically useful and theoretically interesting returns models where the diffusion coefficient D(x,t) depends on both logarithmic returns x and time t. Finally, we prove that option prices blow up if fat tails in returns x are included in the market distribution.
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