Computational Methods for Maximum Drawdown Options Under Jump-Diffusion

Abstract

Recently, the maximum drawdown (MD) has been proposed as an alternative risk measure ideal for capturing downside risk. Furthermore, the maximum drawdown is associated with a Pain ratio and therefore may be a desirable insurance product. This thesis focuses on the pricing of the discrete maximum drawdown option under jump-diffusion by solving the associated partial integro differential equation (PIDE). To achieve this, a finite difference method is used to solve a set of one-dimensional PIDEs and appropriate observation conditions are applied at a set of observation dates. We handle arbitrary strikes on the option for both the absolute and relative maximum drawdown and then show that a similarity reduction is possible for the absolute maximum drawdown with zero strike, and for the relative maximum drawdown with arbitrary strike. We present numerical tests of validation and convergence for various grid types and interpolation methods. These results are in agreement with previous results for the maximum drawdown and indicate that scaled grids using a tri-linear interpolation achieves the best rate of convergence. A comparison with mutual fund fees is performed to illustrate a possible rationalization for why investors continue to purchase such funds, with high management fees.