Estimating the Optimal Importance Sampling Drift with Robbins–Monro
Summary
The document discusses finding an importance sampling drift that minimizes the variance of a least squares Monte Carlo estimate. It defines an objective as the second moment of a weighted payoff and gives an expression for its gradient. The proposed Robbins–Monro update uses a noisy observation of that gradient at each iteration, so the unknown expectation does not need to be calculated exactly before running the algorithm.
The answer explains how to form that observation: simulate paths, evaluate the gradient’s integrand for each sampled path, and average the values to estimate the gradient. The document also points to a paper for further reading. It provides no numerical experiment, convergence demonstration, or guidance on choosing step sizes or sample counts, so it gives the basic estimator rather than a complete implementation recipe. Performance will depend on the payoff, sampling design, and tuning of the stochastic approximation.
Key ideas
- Importance sampling can reduce the variance of a Monte Carlo payoff estimate by changing the sampling drift.
- The drift objective is the weighted payoff’s second moment, and its gradient characterizes candidate optima.
- Robbins–Monro can use a sampled gradient contribution instead of an exact expectation.
- Estimate the gradient by averaging its integrand across simulated paths.
- The document gives no convergence results or step size selection guidance.
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Full text
# Finding optimal drift, importance sampling, least square monte carlo
# Finding optimal drift, importance sampling, least square monte carlo
I am working with Importance sampling for Least Squared monte carlo and have now problems understanding the implementation of the Robbins-Monro algorithm for finding the optimal drift for finding minimum variance of my estimate. The original problem formulation that is now answered is given here.
The article I am following for Robbins-Monro algorithm is this link
The problem i want to solve is to find a optimal drift $\theta^*$ by solving:
$H(\theta^*)=\min_{\theta}H(\theta)$
Where $H(\theta)=\mathbb{E}\left[ G^2(Z)e^{-\theta Z+\frac{1}{2}\theta^2}\right]$, the second moment of the payoff function $G(Z)=\max(K-S(t),0)$. Indeed, we have: $\nabla H(\theta)=0$
Now following the Morris monro algorithm in the link, the general formulation of the stochastic algorithm is given in equation (10) and is given by:
$X_{n+1}=X_n-\gamma_{n+1}F(X_n,Z_{n+1})$
and going further to equation (15) we have the second moment (the gradient of $H(\theta)$) given by:
$h(\theta)=\nabla H(\theta)=\mathbb{E}\left[(\theta-Z)G^2(Z)e^{-\theta Z+\frac{1}{2}\theta^2}\right]$.
Now I wonder, since I don't know the second moment, how should I approximate it numerically in order to evaluate the algorithm? Given in the article, they don't really explain how the second moment is found?
Appreciate for help. Thank you!
## Answer by Mark Joshi (score 1)
https://quant.stackexchange.com/a/29861
$h(\theta)=\nabla H(\theta)=\mathbb{E}\left[(\theta-Z)G^2(Z)e^{-\theta Z+\frac{1}{2}\theta^2}\right]$
so just take a bunch of paths and evaluate $$ (\theta-Z)G^2(Z)e^{-\theta Z+\frac{1}{2}\theta^2} $$on them and take the average.
## Answer by M. Jeunesse (score 0)
https://quant.stackexchange.com/a/27831
Here is a good paper which can help you.
https://www.rocq.inria.fr/mathfi/Premia/free-version/doc/premia-doc/pdf_html/mc_jourdainlelong_doc.pdfShown 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.