Measuring Severity of the Market Crashes by Using Contracts on Maximum Relative Drawdown

Maximum Relative Drawdown measures the largest percentage drop of the price process on a given time interval. More recently, Maximum Relative Drawdown has become more popular as an alternative measure of risk. In contrast to the Value at Risk measure, it captures the path property of the price process. In this article, we propose a partial differential equation approach to determine the theoretical distribution of the Maximum Relative Drawdown. We also discuss the possibility of constructing an option contract that would insure the event that the Maximum Relative Drawdown exceeds a certain fixed percentage. We call these contracts Crash Options. We compute the theoretical prices and hedging strategies for the Crash Option. This gives us a new method how to measure the severity of the market crash by comparing the actual size of the market drop to the theoretical values of the crash option contract. Although the crash option is not currently traded contract, we can use our methods to find its market value by constructing the hedging strategy which replicates this option.