This paper presents a new control variate method for general multi-dimensional stochastic differential equations (SDEs) including jumps in order to reduce the variance of Monte Carlo method. Our control variate method is based on an asymptotic expansion technique, and does not require an explicit characteristic function nor a closed form probability density function of SDEs. This is the first one which derives the control variate method for such general models. Moreover, in our control variate method, the regression estimators can be chosen for each number of jump times, and improve the efficiency of the variance reduction. This paper also provides a variance estimate of our method in terms of its terminal time and a small noise parameter used in an asymptotic expansion method. For an application of our method, we evaluate multi-asset options under general local stochastic volatility with jumps models in finance, and show calculation scheme of control variates for Greeks.
In numerical experiments, we apply the new control variate method to pricing basket
options for ZABR type local stochastic volatility model with jumps, and confirm that our
method works very well.