0.1 Structure
We replace the constant volatility in the Black-Scholes-Merton Model \(\sigma\) with a deterministic function in time \(\sigma_{t}\) then we can show that Black-Scholes still holds if \(\sigma^2\) is replaced with \(\frac{1}{T}\int _{0}^T\sigma^2_{s} \, ds\)
\begin{proof} To see this consider the stock price dynamics under \(\mathbb{Q}\) \[
dS_{t}=rS_{t}dt+\sigma_{t}S_{t}dW_{t}^\mathbb{Q}.
\] Applying the log transformation and Itô Formula we obtain \[
S_{t}=S_{0}\exp\left\{ \int _{0}^t\left( r- \frac{\sigma^2_{s}}{2} \right) \, ds+\int _{0}^t\sigma_{s} \, dW_{s}^\mathbb{Q} \right\}.
\] As a result, we simply replace \(\sigma^2t\) with \(\int _{0}^t\sigma_{s}^2 \, ds\) and replace \(\sigma dW_{t}^\mathbb{Q}\) with \(\int _{0}^t \sigma_{s} \, dW_{s}^\mathbb{Q}\). \end{proof}
The implied volatility surface \(I(T,K)\) is not flat in \(T\) but still flat in \(K\). This is a subtle improvement on BS but is still far from ideal for practical purposes.