Pricing American Drawdown Options under Markov Models

The drawdown in the price of an asset shows how much the price falls relative to its historical maximum. This paper considers the pricing problem of American style drawdown call options, which allow the holder to optimally choose the time to receive a call payoff written on the drawdown. Our pricing framework includes classical Russian options and American lookback puts as special cases after a suitable equivalent measure change. We approximate the original asset price model by a continuous time Markov chain and develop two types of algorithms to solve the optimal stopping problem for the drawdown process. The first one is a transform based algorithm which is applicable to general exponential Levy models. The second approach solves the linear complementarity problem (LCP) associated with the variational inequalities for the value function and it applies to general Markov models. We propose an efficient Block-LCP method that reduces an LCP with big size to a sequence of sub-LCPs with mild size which can be solved by a variety of LCP solvers and we identify the best solver through numerical experiments. Convergence of Markov chain approximation is proved and various numerical examples are given to demonstrate their computational efficiency and convergence properties