Expected maximum drawdowns under constant and stochastic volatility

by Nouri, Suhila Lynn.

The maximum drawdown on a time interval [0, T] of a random process can be defined as the largest drop from a high water mark to a low water mark. In this project, expected maximum drawdowns are analyzed in two cases: maximum drawdowns under constant volatility and stochastic volatility. We consider maximum drawdowns of both generalized and geometric Brownian motions. Their paths are numerically simulated and their expected maximum drawdowns are computed using Monte Carlo approximation and plotted as a function of time. Only numerical representation is given for stochastic volatility since there are no analytical results for this case. In the constant volatility case, the asymptotic behavior is described by our simulations which are supported by theoretical findings. The asymptotic behavior can be logarithmic for positive mean return, square root for zero mean return, or linear for negative mean return. When the volatility is stochastic, we assume it is driven by a mean-reverting process, in which case we discovered that if one uses the effective volatility in the formulas obtained for the constant volatility case, the numerical results suggest that similar asymptotic behavior holds in the stochastic case. 3 Introduction Quantifying risk is a primary concern of any investor. If the standard deviation of returns for a manager is large enough to produce a loss during some time period, that manager will experience drawdowns. Many consider a manager’s drawdowns to be a better measure of risk than simply considering the volatility of returns or a return/risk measure such as the Shape ratio [1]. Also, taking drawdowns as a description of a manager’s historical performance has the distinct quality of referring to a physical reality. It is known that the Commodity Futures Trading Commission (CFTC) has a mandatory disclosure regime that requires futures traders to disclose as a part of their performance their “worst peak-to-valley drawdown” [6]. Particularly in hedge funds, estimating drawdown and maximum drawdown is imperative for estimating the probability of reaching a stop-loss that may set off large liquidations and of reaching the high water mark prior to the end of the year that will result in a performance fee [7]. A drawdown is defined as change in value of a portfolio from any established peak (high water mark) to the subsequent trough (low water mark). A high water mark has occurred if it is higher than any previous net asset value and is followed by a loss. A low water mark has occurred if it is the lowest net asset value between two high water marks. A maximum drawdown of a portfolio is the largest drop from a high water mark to a low water mark. Even though a manager can only have one maximum drawdown, it is informative to look at the distribution from which the maximum drawdown was drawn. If one considers several managers, all with the same or similar track records and return characteristics, it is feasible to see what their distributions of worst drawdowns look like 4 [1]. Simulated drawdowns (DD) and corresponding maximum drawdowns (MDD) can be seen below, with mean return .2, 0, and -.2, sigma .3, and time interval 1. This project is structured as follows: Chapter one will cover the theoretical and empirical results of drawdowns and expected maximum drawdowns under the assumption of constant volatility. Chapter two will reveal what happens to expected maximum drawdowns when the volatility function is taken to be stochastic. Chapter three will show numerical examples of expected maximum drawdowns under the cases of constant and stochastic volatility. Finally, Chapter four concludes the theoretical and empirical findings reported in this project and contains recommendations for future work. 5