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

Vardar, Ceren
http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1224274306

Abstract Details

2008, Doctor of Philosophy (Ph.D.), Bowling Green State University, Mathematics/Probability and Statistics.
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 asa 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.
Gabor Szekely (Advisor)
Craig Zirbel (Advisor)
Bullerjahn George (Committee Member)
Rizzo Maria (Committee Member)
Chen John (Committee Member)
162 p.
Mathematics
Brownian Motion; Geometric Brownian Motion; Sharpe Ratio; Strong Markov Property; Scaling Property; Bessel Process; Doob's h-transform; Path Decomposition

Recommended Citations

Citations

  • Vardar, C. (2008). On the Correlation of Maximum Loss and Maximum Gain of Stock Price Processes [Doctoral dissertation, Bowling Green State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1224274306

    APA Style (7th edition)

  • Vardar, Ceren. On the Correlation of Maximum Loss and Maximum Gain of Stock Price Processes. 2008. Bowling Green State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1224274306.

    MLA Style (8th edition)

  • Vardar, Ceren. "On the Correlation of Maximum Loss and Maximum Gain of Stock Price Processes." Doctoral dissertation, Bowling Green State University, 2008. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1224274306

    Chicago Manual of Style (17th edition)

bgsu1224274306
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© 2008, all rights reserved.
This open access ETD is published by Bowling Green State University and OhioLINK.