A theory of non‐Gaussian option pricing
基于非高斯股票收益模型推导期权定价公式，使用Tsallis熵最大化得到广义Black-Scholes方程，并给出欧式看涨期权的闭式解。该模型能复现波动率微笑，对日元期货期权的实证表明仅用一个波动率参数即可接近市场价格。
Abstract Option pricing formulae are derived from a non‐Gaussian model of stock returns. Fluctuations are assumed to evolve according to a nonlinear Fokker‐Planck equation which maximizes the Tsallis nonextensive entropy of index q. A generalized form of the Black‐Scholes differential equation is found, and we derive a martingale measure which leads to closed‐form solutions for European call options. The standard Black‐Scholes pricing equations are recovered as a special case (q = 1). The distribution of stock returns is well modelled with q circa 1.5. Using that value of q in the option pricing model we reproduce the volatility smile. The partial derivatives (or Greeks) of the model are also calculated. Empirical results are demonstrated for options on Japanese Yen futures. Using just one value of σ across strikes we closely reproduce market prices, for expiration times ranging from weeks to several months.