Title page for ETD etd-050406-151319

Document Type thesis

Author Name Nouri, Suhila Lynn

Email Address suhilanouri at hotmail.com

URN etd-050406-151319

Title Expected Maximum Drawdowns Under Constant and Stochastic Volatility

Degree MS

Department Mathematical Sciences

Advisors
Luis Roman, Advisor

Keywords
drawdowns
maximum drawdowns

Date of Presentation/Defense 2006-05-04

Availability unrestricted

Abstract
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.

Files
snouri.pdf

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Document Type	thesis
Author Name	Nouri, Suhila Lynn
Email Address	suhilanouri at hotmail.com
URN	etd-050406-151319
Title	Expected Maximum Drawdowns Under Constant and Stochastic Volatility
Degree	MS
Department	Mathematical Sciences
Advisors	Luis Roman, Advisor
Keywords	drawdowns maximum drawdowns
Date of Presentation/Defense	2006-05-04
Availability	unrestricted
Abstract 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.
Files	snouri.pdf