Better Numerical Approximation for Multi-dimensional SDEs

Today, better numerical approximations are required for multidimensional SDEs to improve on the poor performance of the standard Monte Carlo integration. Usually in finance, it is the weak convergence property of numerical discretizations, which is most important, because with financial applications, one is mostly concerned with the accurate estimation of expected payoffs. However, recent studies for hedging, portfolio optimization, and the valuation of exotic options show that the strong convergence property plays a crucial role. When one prices an exotic option or wants to approximate a portfolio, the SDEs used are not important. What really matters is that the SDEs approximate correctly the real distribution of the process. Using this principle, this research suggests that, instead of considering a given no-commutative multi-dimensional SDE that represents our process, we consider another SDE that has the same distribution but with a different strong convergence order. Manipulating the new SDE, which has an extra process it becomes commutative and we avoid the simulation of the Lévy Area (extremely expensive with respect to the computational time). The new SDE obtains solutions that in a weak sense, which is in a distributional sense, coincide with those of the original SDE. If certain conditions are satisfied, scheme gives a first order strong convergence without the simulation of the Lévy Area. Conversely, for the original nocommutative SDE, the Milstein scheme, neglecting the Lévy Area, has 0.5 order strong convergence. If the conditions are not satisfied, this study confirms experimentally that scheme has a better strong approximation than using the standard Milstein scheme in the original SDEs (both schemes neglecting the simulation of the Lévy Area). AMS subject classifications: 60G20, 65CXX, 65C20, 37H10, 41A25. KeywordsDiscrete time approximation, stochastic schemes, stochastic volatility models, Milstein Scheme, Lévy Area, scheme, Orthogonal Milstein Scheme, orthogonal transformation, strong convergence.