# Expected maximum drawdowns under constant and stochastic volatility

Abstract (Summary)

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.
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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
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[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.
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Bibliographical Information:

Advisor:

School:Worcester Polytechnic Institute

School Location:USA - Massachusetts

Source Type:Master's Thesis

Keywords:stochastic processes portfolio management brownian motion

ISBN:

Date of Publication: