Convergence of Lattice and Pde Methods for Pricing Asian Options

In a recent article, Barraquand and Pudet (1996) state that the lattice based Forward Shooting Grid (FSG) method is convergent for Asian options if either nearest lattice point or linear interpolation is used. Moreover, this result is claimed to be independent of any relationship between the grid quantization parameter (for the spacing of the nodal averages) and the timestep size. However, a more detailed analysis of the propagation of interpolation error reveals a problem. A worst case error analysis shows that the error may be large as the number of timesteps becomes large if nearest lattice point interpolation is used. We demonstrate that the worst case error is indeed attained in some numerical examples. Moreover, if a linear interpolation scheme is employed the FSG algorithm does not converge to the correct price, being o by a constant error which does not vanish in the limit as t! 0, unless the limit is carried out in a certain way. Similarly, the method proposed by Hull and White (1993) is convergent provided that the node spacing in the average direction is selected appropriately as t ! 0. It is also a straightforward matter to show that partial di erential equation (PDE) based methods are convergent. Numerical examples comparing convergence for all three techniques are presented.