Numerical Methods for Pricing Multi - Asset Options

Numerical Methods for Pricing Multi-Asset Options Yuwei Chen Master of Science Graduate Department of Computer Science University of Toronto 2017 We consider the pricing of two-asset European and American options by numerical Partial Differential Equation (PDE) methods, and compare the results with certain analytical formulae. Two cases of options are tested: exchange option and spread option. For exchange options, the analytical formula considered is the (exact) Margrabe formula. For spread options, we consider Kirk’s formula and the formula by Li, Deng and Zhou. In pricing European two-asset options, the basic numerical PDE model is the two-dimensional Black-Scholes PDE. Different boundary conditions are considered, and the effect of them on the solution at various points of the grid is studied. Furthermore, various types of non-uniform grids are considered, aiming at reducing the error at certain areas of the grid. Moreover, the effect of the truncated domain on the PDE approximation is studied. We also discuss the effect of certain problem parameters, such as the length of maturity time, and the values of volatility and correlation, to the accuracy and convergence of the PDE approximations. The experiments indicate that the numerical PDE computed price and Greeks are second-order, for appropriately chosen grid discretizations. In the American pricing problem, the discrete penalty method is applied to the linear complementarity problem involving the two-dimensional Black-Scholes PDE and additional constraints. The convergence of the American Spread put option approximation computed with penalty iteration remains second-order, with the number of penalty iterations per timestep remaining small (2-3). We also consider an iterative method with preconditioning techniques for solving the arising large sparse linear system at each timestep, and show that this solution technique is asymptotically optimal.