均值回归乘性噪声：行为与对期权的影响

What sort of behaviour

专题: Finance / 金融
难度: L4
来源: QuantQuestion

题目详情

Assume the stock price process $S_{t}$ follows the stochastic differential equation

dS_{t} = \alpha (\mu - S_{t}) dt + S_{t} \sigma dW_{t}

with $\alpha , \sigma$ and $\mu > 0$ . What sort of qualitative behaviour does the stock price exhibit? What is the impact of the parameters $\alpha$ and $\mu$ ? What effects do they have on the price of a call option on this stock?

解析

SDE：

dS_t=\alpha(\mu-S_t)dt+\sigma S_t dW_t.

当 $S_t>\mu$ 时漂移为负，向下拉回；当 $S_t<\mu$ 时漂移为正，向上拉回，因此呈均值回归，均值水平为 $\mu$ 。
$\alpha$ 越大，回归速度越快； $\mu$ 决定长期均值水平。

对期权：在无套利定价下，真实漂移会被换成风险中性漂移；均值回归结构通常会降低远离均值的扩散，从而倾向降低远期期权的时间价值（具体取决于风险中性下的方差结构）。