This work deals with the simulation of Wishart processes and affine diffusions on positive semidefinite matrices. To do so, we focus on the splitting of the infinitesimal generator in order to use composition techniques as did Ninomiya and Victoir [Appl. Math. Finance 15 (2008) 107-121] or Alfonsi [Math. Comp. 79 (2010) 209-237]. Doing so, we have found a remarkable splitting for Wishart processes that enables us to sample exactly Wishart distributions without any restriction on the parameters. It is related but extends existing exact simulation methods based on Bartlett's decomposition. Moreover, we can construct high-order discretization schemes for Wishart processes and second-order schemes for general affine diffusions. These schemes are, in practice, faster than the exact simulation to sample entire paths. Numerical results on their convergence are given. . This reprint differs from the original in pagination and typographic detail. 1 2 A. AHDIDA AND A. ALFONSI kind. Such processes solve the following SDE:A. AHDIDA AND A. ALFONSI Of course, the generators L M and L S are equivalent; one can be deduced from the other. However, L S already embeds the fact that the process lies in S d (R), which reduces the dimension from d 2 to d(d + 1)/2 and gives, in practice, shorter formulas. This is why we will mostly work in the sequel with infinitesimal generators on S d (R). Unless it is necessary to make the distinction with L M , we will simply denote L = L S .1.2. The characteristic function of Wishart processes. As for other affine processes, the characteristic function of affine processes on positive semidefinite matrices can be obtained by solving two ODEs. In the case of Wishart processes, it is possible to solve explicitly these ODEs by solving a matrix Riccati equation (see Levin [20]). Here, we give the closed formula for the Laplace transform and a precise description of its set of convergence.ds and m t = exp(tb). We introduce the set of convergence of the Laplace transform of X x t , D b,a;t = {v ∈ S d (R), E[exp(Tr(vX x t ))] < ∞}. This is a convex open set that is given explicitly by
We introduce a mean-reverting SDE whose solution is naturally defined on the space of correlation matrices. This SDE can be seen as an extension of the well-known Wright-Fisher diffusion. We provide conditions that ensure weak and strong uniqueness of the SDE, and describe its ergodic limit. We also shed light on a useful connection with Wishart processes that makes understand how we get the full SDE. Then, we focus on the simulation of this diffusion and present discretization schemes that achieve a second-order weak convergence. Last, we explain how these correlation processes could be used to model the dependence between financial assets.We first calculate the quadratic covariation of M RC d (x, κ, c, a). By Lemma 27, we get:We remark in particular that d (X t ) i,j , d(X t ) k,l = 0 when i, j, k, l are distinct.We are now in position to calculate the infinitesimal generator of M RC d (x, κ, c, a). The infinitesimal generator on M d (R) is defined by:
We propose an affine extension of the Linear Gaussian term structure Model (LGM) such that the instantaneous covariation of the factors is given by an affine process on semidefinite positive matrices. First, we set up the model and present some important properties concerning the Laplace transform of the factors and the ergodicity of the model. Then, we present two main numerical tools to implement the model in practice. First, we obtain an expansion of caplets and swaptions prices around the LGM. Such a fast and accurate approximation is useful for assessing the model behavior on the implied volatility smile. Second, we provide a second order scheme for the weak error, which enables to calculate exotic options by a Monte-Carlo algorithm. These two pricing methods are compared with the standard one based on Fourier inversion.
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