Importance Sampling for Barrier Options with Local Volatility
Summary
The document discusses using importance sampling to estimate prices for far out-of-the-money barrier options under local volatility. It describes changing the distribution of simulated inputs from the original density to a shifted normal density, then weighting each payoff by the likelihood ratio so the estimator still targets the original price. In principle, the shift can be chosen to reduce estimator variance, but the document gives no closed-form solution for the minimizing parameter.
The suggested practical starting point is to shift the sampling distribution toward the option’s strike or barrier, where payoffs are more likely and sensitivity may be greater. The responses recommend first studying simpler fixed-volatility calls and barrier options, and comparing sample variance as the shift changes. With local volatility, a full optimization may add instability without enough benefit; a simple shift rule learned from preliminary analysis can be preferable. No implementation details or quantitative results are provided.
Key ideas
- Importance sampling changes the simulation distribution and applies likelihood-ratio weights to estimate the original expectation.
- A shift toward the strike or barrier can make rare payoffs more frequent in the simulation.
- The variance-minimizing shift generally has no analytical solution in the discussion.
- Explore the shift by examining estimator variance across candidate centers.
- For local-volatility barriers, simple empirically chosen shifts may be more stable than full optimization.
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# Importance sampling for Monte Carlo with local volatility in practice
# 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)
$$V \approx V_g = E_g[G(X)\frac{f(X)}{g(X)}]$$
instead of using:
$$V \approx V_f = E_f[G(X)]$$
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?
## Answer by KT8 (score 1)
https://quant.stackexchange.com/a/68792
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.
Hope that helps.
## Answer by Andrea (score 1)
https://quant.stackexchange.com/a/81305
Just to extend the answer from KT8.
Depending on your level of expertise, starting with a barrier + local vol might be too complicated.
Are you familiar with the behaviour of importance sampling for a call with fixed vol? And for a barrier option?
As KT8 says, one normally wants to centre the distribution where there is more convexity and for a call it is at the strike. For a barrier, there is a lot of convexity between strike next to the barrier too, so you could plot the sample variance as you move the centre around.
Once you add local vol, you might find that running a full optimisation is an overkill and makes the whole calculation less stable. So maybe you want to select some simple "rule" you have learnt in the analysis above and use it even it is not the "bestest".Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.