确定性时变波动率：定价与对冲

Time dependent volatility

专题: Finance / 金融
难度: L4
来源: QuantQuestion

题目详情

Suppose an asset has a deterministic time dependent volatility. How would I price an option on it using the Black- Scholes theory? How would I hedge it?

解析

若 $\sigma(t)$ 为确定性函数，风险中性下

\frac{dS_t}{S_t}=(r-q)dt+\sigma(t)dW_t.

对数收益仍为正态，累计方差为

\Sigma^2(t,T)=\int_t^T \sigma(u)^2\,du.

欧式期权仍可用 BS 形式，只需把常数波动率项 $\sigma\sqrt{\tau}$ 替换为 $\sqrt{\Sigma^2(t,T)}$ 。

对冲仍用

\boxed{\Delta=\partial_S V}.