Carr Memorial: Maximum Drawdown Derivatives to a Hitting Time

Abstract

Drawdown is defined as the amount a portfolio has decreased from its running maximum. Drawdown has become ensconced in finance practice with some hedge funds shutting down portfolio managers who reach a certain drawdown limit. In this article, we show that, for every continuous local martingale that hits a given point <i>m</i> with probability 1, the running maximum of drawdown at the time of hitting <i>m</i> has the same inverse exponential distribution. We then derive prices and hedge ratios for binary calls on maximum absolute and relative drawdown maturing at the hitting time for <i>m</i>. We also derive prices for call spreads on maximum drawdown at the hitting time for <i>m</i>. These prices and hedge ratios are model independent across all continuous arbitrage-free stochastic processes that, with probability 1, either hit <i>m</i> or reach a drawdown equal to the strike price. This includes stochastic volatility models whose volatility is bounded away from 0 before hitting <i>m</i> or the strike. These results are both simpler and more general than prior work, which, while allowing for a fixed maturity, require infinite series representations, the use of complex derivatives to hedge and greater restrictions on the stochastic process. The key fact that facilitates our form of model independence is that the values of the derivatives at maturity are invariant to time changes.