On weak and strong convergence rate for the Heston stochastic volatility model
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The Heston stochastic volatility model is one of the most fundamental models in mathematical ﬁnance. Recently, many numerical schemes have been developed for the Heston model. However, in the literature, there is no weak or strong convergence rate obtained for the full parameter regime. In this PhD thesis, we shall focus on the numerical scheme that simulates the variance process exactly and applies the stochastic trapezoidal rule to approximate the time integral of the variance process in the SDE of the logarithmic asset process. Our goal is to obtain the weak and strong convergence rates of such a numerical scheme for the Heston model. The weak convergence rate is of traditional interest, because it is an important measure on how fast the bias of a numerical scheme decays. We prove that the numerical scheme we consider converges at rate two for the whole parameter regime, and the test function can be any polynomial of the logarithmic asset process. The rate is consistent with the standard rate of the stochastic trapezoidal rule, although the Lipschitz assumption is not satisﬁed. The strong convergence analysis is meaningful in the framework of Multi-level Monte Carlo (MLMC). The MLMC can be regarded as a variance reduction technique for numerical schemes on SDEs, as long as there is a MLMC estimator with a good strong convergence rate. We establish efﬁcient MLMC estimators, separately for the path-independent and path-dependent simulations. We are able to provide the strong convergence rates in both situations.