Importance sampling for Monte Carlo with local volatility in practice

I am given a diffusion with a local volatility to price barrier options:

$$dX(t)=X(t)\mu dt+X(t)\sigma(t,X)dW_t$$

I want to use Importance Sampling to price barrier options "far" out of the money. I did some research and found https://pdfs.semanticscholar.org/4fe5/94e3c7667c762cf1f7d841fcd0a4bf30f255.pdf

This is the simplest I found about the subject as I am looking to simply implement the method.

However, I am struggling to use this in practice. From my understand one can use (with the same notation as in the paper)

$$V \approx V_g = E_g[G(X)\frac{f(X)}{g(X)}]$$

$$V \approx V_f = E_f[G(X)]$$

with $$f$$ the original density and $$g$$ the newly define one that minimises the variance of the monte carlo estimator.

If $$f$$ is given by : $$f(x)=(2 \pi)^{-\frac{n}{2}} \mathrm{e}^{-\frac{1}{2} x^{T} x}$$

then $$g$$ can be defined by: $$g_{\mu}(x)=(2 \pi)^{-\frac{n}{2}} \mathrm{e}^{-\frac{1}{2}(x-\mu)^{T}(x-\mu)}$$

with $$\mu$$: $$\min _{\mu} E_{f}\left[G^{2}(X) \frac{f(X)}{g_{\mu}(X)}\right]$$

How can I practicaly use this in my Monte Carlo simulation, for the $$X(t)$$-process given above?

• That’s an interesting question. I am wondering whether and how it could be (economically) feasible to calculate the density of each path under the two measures $f$ and $g$, or at least the densities of the discretely sampled points along each path. Aug 6 '21 at 19:32

As far as I know, there is not an analytical formula or approximation telling you what value of $$\mu$$ is a solution for the minimizing-equation
$$\min _{\mu} E_{f}\left[G^{2}(X) \frac{f(X)}{g_{\mu}(X)}\right].$$
However, usually a good initial guess is to take $$\mu$$ such that the new distribution is centered around the strike of your option (or closer to the barrier). Then, using that value of $$\mu$$ as a starting guess, you can try and fine-tune the parameter a bit more. You can do that by looking at how the price you get from your Monte Carlo evolves as you increment the number of samples.
The naive idea there is that you have a known ratio $$f(X)//g_\mu (X)$$ weighting the values of your sample, that is now centered around the strike, therefore giving you a higher sensitivity to the underlying process.