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Quantitative Methods in Financial and Insurance Mathematics
“Numerical solution of the Hamilton-Jacobi-Bellman formulation for continuous time mean variance asset allocation”
We solve the optimal asset allocation problem using a mean variance approach. There is no direct dynamic programming principle for continuous time mean variance asset allocation. However, we can transform the original mean variance optimization problem into a problem having the same optimal control as the original mean variance problem. We then formulate this transformed problem as a Hamilton Jacobi Bellman (HJB) Partial Differential Equation (PDE). We use a finite difference method with fully implicit timestepping to solve the resulting non-linear HJB PDE, and present the solutions in terms of an efficient frontier and an optimal asset allocation strategy. The numerical scheme satisfies sufficient conditions to ensure convergence to the viscosity solution of the HJB PDE. We handle various constraints on the optimal policy. Numerical tests indicate that realistic constraints can have a dramatic effect on the optimal policy compared to the unconstrained solution. If time permits, we will include numerical results comparing time inconsistent and time consistent mean variance strategies, as well as mean quadratic variation policies. Peter Forsyth, Cheriton School of Computer Science, University of Waterloo
“Numerical approach of solving jump-diffusion PIDE via pseudo-parabolic equations”
“New dual methods for single and multiple exercise options”
Part I: We recap the dual method of Rogers (2002) / Haugh & Kogan (2002) and present some new results concerning optimal dual martingales. Based on these results we build a regression based algorithm which computes both an upper bound and a lower bound for an American option at the same time. Unlike the Andersen & Broadie algorithm, our new algorithm doesn't require nested simulation and is therefore fast. Click here for pdf document
Part II) In the second part of the talk we extend the dual method to the multiple stopping problem and we outline an algorithm for the evaluation of multiple exercise options such as swing options. Click here for pdf document
Part I) J. Schoenmakers and J. Huang: Optimal dual martingales and their stability; fast evaluation of Bermudan products via dual backward regression (new manuscript WIAS preprint 1574).
Part II) J. Schoenmakers: A pure martingale dual for multiple stopping (Finance and Stoch., forthcoming (2010)).