Active Management of the Maximum Drawdown

Risk management of drawdowns and portfolio optimization with drawdown constraints is becoming increasingly important among practitioners. In this paper, we introduce new techniques how to control the maximum drawdown (MDD). One can view the maximum drawdown as a contingent claim, and price and hedge it accordingly as a derivative contract. Trading drawdown contracts or replicating them by hedging would directly address the concerns of portfolio managers who would like to control them. Similar contracts can be written on the maximum drawup (MDU). We show that buying a contract on MDD or MDU is equivalent to adopting a momentum trading strategy, while selling it corresponds to contrarian trading. We also discuss more complex products, such as plain vanilla options on the maximum drawdown or drawup, or related barrier options on MDD and MDU which we call crash and rally options, respectively. 1 Drawdowns and Drawups Suppose that we have an underlying asset whose price process at time t is given by St. For example, the price process could be a stock price, index, interest rate or exchange rate. Denote by Mt its running maximum up to time t: Mt = max u∈[0,t] Su. Drawdown Dt is defined as the drop of the asset price from its running maximum: (1) Dt = Mt − St. Maximum drawdown MDDt is defined as the maximal drop of the asset price from its running maximum over a given period of time: (2) MDDt = max u∈[0,t] Du. Assume that you have an investor who enters the market at a certain point and leave it at some following point within a given time period. Maximum drawdown measures the worst loss of such an investor, meaning that he buys the asset at a local maximum and sells it at the subsequent lowest point, and this drop is the largest in the given time period. This represents the worst period for holding this asset and could be written mathematically as: (3) MDDt = sup qu∈{0,1} one switch (0 → 1) and (1 → 0) ∫ t