Abstract
We propose an approach to a twofold optimal parameter search for a combined variance reduction technique of the control variates and the important sampling in a suitable pure-jump Levy process framework. The parameter search procedure is based on the two-time-scale stochastic approximation algorithm with equilibrated control variates component and with quasi-static importance sampling one. We prove the almost sure convergence of the algorithm to a unique optimum. The parameter search algorithm is further embedded in adaptive Monte Carlo simulations in the case of the gamma distribution and process. Numerical examples of the CDO tranche pricing with the Gamma copula model and the intensity Gamma model are provided to illustrate the effectiveness of our method.
Highlights
The class of Levy processes has drawn a great amount of attention, for example, in the field of mathematical finance, where the Brownian motion has long been considered to be insufficient to describe a variety of asset price dynamics
We have developed and analyzed an application of the two-time-scale stochastic approximation algorithm in the optimal parameter search for the combined control variates and importance sampling in the Levy process framework
We have proved that the algorithm converges almost surely to a unique root of the gradient of the variance in the sense of “equilibrated control variates and quasi-static importance sampling,” and the almost sure convergence guarantees the incorporation of the algorithms into the adaptive Monte Carlo variance reduction procedure
Summary
The class of Levy processes has drawn a great amount of attention, for example, in the field of mathematical finance, where the Brownian motion has long been considered to be insufficient to describe a variety of asset price dynamics. We take the same route as in [17]: (i) we combine two variance reduction techniques, the control variates (CV) and the importance sampling (IS), and investigate the two-fold optimality of the combination, (ii) we apply the two-time-scale stochastic approximation algorithm in parameter search for the combination and prove almost sure convergence of the algorithm to a unique optimum, and (iii) we incorporate the parameter search procedure into adaptive Monte Carlo simulation. The idea of incorporating the parameter search with a stochastic approximation into adaptive Monte Carlo simulation is studied in Arouna [3] and is applied to the two-time-scale version of [17] Those methods make full use of a property of the Gaussian distribution that after the measure change by the Girsanov theorem, it again follows a Gaussian distribution with its mean shifted and with the same shape (variance).
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