Abstract: The existing stochastic volatility models have such a problem: A single-factor volatility model can generate steep curves or flat curves at a given volatility, but it cannot generate both for given parameters, which is inconsistent with the actual observed data. To precisely describe the market implied volatility curve, this paper studied a two-factor 4/2 stochastic volatility model that includes, as special instances, the Heston model and the 3/2 model. Besides, it applied Lewis’s fundamental transform approach to deduce the partial differential equation(PDE). In addition, by adopting the data on S&P 500, it estimated the parameters of the 4/2 model. Furthermore, it investigated the 4/2 model along with the Heston model and the 3/2 model, and compare their different performances. The results indicate that the option price fitting error of the 4/2 model is smaller than that of other two models.

Key words: stochastic volatility, fundamental transform, 4/2 model, Lie's symmetries, Laplace transform

摘要： 现有的随机波动率模型存在这样一个问题：给定一组参数，在一定的标的资产波动率水平下，单因子模型只能产生陡峭或平滑的期权隐含波动率曲线，而不能同时存在两种形态，这与实际观察的数据不符。为了更准确地刻画市场隐含波动率曲面，研究一种双因子4/2随机波动率模型，该模型结合了Heston模型和3/2 模型。采用Lewis的基础变换法将期权定价问题转化为求解偏微分方程（PDE）的问题；利用标普500指数期权数据估计模型的参数，比较了不同模型在期权定价上的差异。结果表明，4/2模型的期权价格拟合误差小于另外两种模型，弥补了原模型在这方面的不足。

关键词: 随机波动率, 基础变负, 4/2模型, 李对称, 拉普拉斯变换