Drawdowns and the Speed of Market Crashes

Apart from the finite time-horizon properties of drawdowns and drawups, the issue of how fast a market crash occurred is also of vital importance to investors and portfolio or hedge fund managers. This motivates us to study the joint distribution of the drawdown and the speed at which it is realized in Chapter 3. In particular, we measure the speed of market crash by the time elapsed between the first passage time of the drawdown to level a > 0, τ+D(a), and the last reset time of the running maximum prior to that drawdown ga := sup{t < τ+ D(a) : Xt = Xt}. The analysis involves studying of the last reset time of maximum ga, for which we make use of the technique of progressive enlargement of filtration and path decomposition at the random time ga. By deriving the joint Laplace transform of ga and τ+ D(a), we provide the analytical basis for the pricing of financial claims based on the drawdown and its speed, which can be used to hedge against dramatic market crashes.