Group for Research in Decision Analysis

# Monte Carlo efficiency improvement via adaptive Importance Sampling and applications to pricing high-dimensional options

## Thanos Avramidis

We consider the problem of integration via Monte Carlo simulation. Importance sampling (IS) is a well-known technique for improving efficiency, and the main difficulty in applying IS is to identify a 'good' probability density for sampling random variates. We define 'good' to mean 'large efficiency improvement with high probability'. To automate this task, we propose a Markov Chain Adaptive Estimation (MCAE) Procedure, whose key features are sampling from the optimal importance sampling density via Markov Chain Monte Carlo and estimation of an IS density as a mixture of multivariate Normal (or $$t$$) densities with modes at certain local maxima of the importance function (defined as the product of integrand times the original density). When all simulation inputs are Normal, dimension reduction via principal components combines well with MCAE, allowing fast identification of a good IS density, even in high dimensions. We present Monte Carlo experimental results on randomly generated European option-pricing problems (including path-dependent options), demonstrating consistent, considerable efficiency improvement, even in high dimension (over 100).