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# On the Correlation of Maximum Loss and Maximum Gain of Stock Price Processes

Abstract (Summary)
One of the primary issues in mathematical finance is the ability to construct portfolios that are optimal with respect to the risk. The stock price is subject to stochastic variability so the risk an investor encounters is due to the stock prices. A commonly used measure of risk is the expected maximum loss of a stock, in other words, how much one can lose. It can be defined informally as the largest drop from a stock peak to a stock nadir. Over a certain fixed length of time, a reasonably low expected maximum loss is as crucial to the success of any fund as a high maximum gain or maximum profit. The correlation coefficient of the maximum loss and the maximum gain indicates the relation between the gain and the risk using measures which are functions of the Sharpe ratio. The price of one share of the risky asset, the stock, is modeled by geometric Brownian motion. By taking the log of geometric Brownian motion, Brownian motion can be used as basis of the calculations related to the geometric Brownian motion. In this dissertation work, we present analytical results related to the joint distribution of the maximum loss and maximum gain of a Brownian motion and the correlation of them, and detailed explanation of this theoretical result which requires a review of standard but difficult literature. We have given an analytical expression for the correlation of the supremum and the infimum of standard Brownian motion up to an independent exponential time, we have shown convexity of the maximum gain and the maximum loss, and we have calculated some bounds for the expected values of maximum gain and maximum loss. We also search for a relation between the Sharpe ratio and the correlation coefficient for Brownian motion with drift and geometric Brownian motion with drift. Using the scaling property, we have shown that the correlation coefficient does not depend on the diffusion coefficient for Brownian motion. And finally, using real-life data, we have presented the correlation of maximum gain and maximum loss and the correlation of the supremum and the infimum of stock prices.
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