波动率微笑、相对偏差和交易策略——基于非线性生灭过程的股价波动一般扩散模型

Volatility Smile,Relative Deviation and Trading Strategies:A General Diffusion Model for Stock Price Movements Based on Nonlinear Birth-death Process

对股票收益的波动率微笑和股票指数相对偏差稳定的观察，引入期权定价理论的新方向。描写群体行为的非线性随机过程，比几何布朗运动为主的代表者模型能更好地描写股价波动。行为金融学中，我们将投资者简化为两类不同交易策略的投资群体——反转投资者和动量投资者，引入生灭过程来直观刻画他们产生的股价涨落，并取极限得到一般扩散过程。我们建立的一般框架可以统一理解目前已经熟知的几何布朗运动、广义方差常弹性、残余波动率、利率的期限结构等理论模型，也能描写观察到的波动率微笑和稳定的相对偏差。用S＆P 500数据对这些模型进行了参数估计和检验的结果很好；有可能在期权定价中取代流行的Black—Scholes模型。

Empirical observation of volatility smile in stock returns and stable relative deviations in stock indices introduce new direction in modeling option pricing. Nonlinear stochastic model of population dynamics is a better alternative for the representative agent models in stock price movements. In behavioral finance, investors can be simplified into two cate- gories with different trading strategies： the contrarian investors and the momentum investors The birth-death process is introduced to describe stock price fluctuations. The general diffusion process can be obtained as a limiting case, so that we have a unified framework in understanding existing models in option pricing theory,including the geometric Brownian Motion, the model of Constant Elasticity of Varianc, the Residual Volatility and term structure of interest rates. Our model is capable of demonstrating volatility＇smile and stable relative devia tion from real data. Parameter estimations and empirical tests are conducted in fitting S＆P500 indices and price forecast in index option. It is a better alternative to the Black- Scholes model in option pricing.