Contracts on Maximum and Average Drawdowns or Drawups

Risk management of drawdowns and portfolio optimization with drawdown constraints is becoming increasingly important among practitioners. In this paper, we introduce new types of contracts which depend on the maximum drawdown or on the average drawdown. Trading drawdown contracts would address directly the concerns of portfolio managers who would like to manage them. The maximum or the average drawdown can be viewed as a derivative contract which could be priced and hedged accordingly as an option. Similar contracts can be written on the maximum drawup or on the average drawup. We also discuss more complex products, such as barrier knock in option on the maximum drawdown or drawup, 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, it 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. Average drawdown ADDt is given by