1 Importance
Goal: \(\mathbb{E}[g(X)]\) Think \(X_{T}\sim N(0,T)\) and \(g(x)=(L-S_{0}e^{(r- \frac{1}{2}\sigma^2)T+\sigma x})_{+}\) \[ \begin{align} \int_{}^{}{g(x) \frac{f_{X_{t}}(x)}{\tilde{f}(x)}}\tilde{f}(x)~d{x}= \underbrace{\mathbb{E}\left[ \frac{g(Y)f(Y)}{\tilde{f}(Y)} \right] }_{\frac{1}{N}\sum_{n=1}^{N} \frac{g(Y^{(n)})f(Y^{(n)})}{\tilde{f}(Y^{(n)})}}~~\text{where }Y\sim \tilde{f} \end{align} \]