Advances in Financial Mathematics
Session Organizers:
Gunduz Caginalp, University of Pittsburgh
Ahmet Duran, University of Michigan-Ann Arbor6 Speakers (Meyer, Goodman, Stojanovic, Zhang, Sturm, Duran)
Gunter Meyer ( meyer@math.gatech.edu ), Department of Mathematics, Georgia Institute of Technology
Option Pricing for Jump Diffusion with Uncertain or Stochastic Volatility
Abstract: Using elementary maximum principle arguments we shall derive a Black Scholes Barenblatt (BSB) type partial-integro-differential equation which can be used to bound option prices when the underlying asset follows a jump diffusion process with uncertain but bounded volatility. A straightforward iterative algorithm for the numerical solution of the BSB equation will be outlined and analyzed. We shall close with comments on solving the jump diffusion problem when the volatility is assumed to be stochastic. (Received May 7, 2008)
Victor Goodman (goodmanv@indiana.edu), Department of Mathematics, Indiana University-Bloomington
Principal Component Analysis of Forward Interest Rates
Abstract: Observations of forward interest rates determine a high-dimensional covariance matrix that summarizes the volatility of yields within a central bank's bond market. A principal component analysis of these rates, using data from different decades and from different countries, displays a common pattern under the assumption of a Gaussian trading noise. I will describe this (well-known) pattern and will also describe a mathematical obstacle which has prevented any viable model from expressing the PCA results. Then I show that, by conditioning on certain favorable market events, we can produce three factor models which have the volatility characteristics of PC market-data analyses. (Received January 28, 2008)
Srdjan D. Stojanovic (srdjan.stojanovic@uc.edu), Department of Mathematics, University of Cincinnati
Quantitative Equity: Dividend Policy, Risk Premium, and Market Share Dynamics
Abstract: We employ the recent theory of pricing of liquid financial contracts in multi-factor incomplete markets with risk premium determination to value equity. In particular the so called "Dividend Puzzle" of F. Black in regard to the Miller-Modigliani theorem of dividend policy irrelevancy, is resolved quantitatively. We further study the firm's market share dynamics effect on the equity value. Finally, some empirical results are presented. (Received April 9, 2008)
Qiang Zhang (mazq@cityu.edu.hk), Department of Mathematics, and Department of Economics and Finance, City University of Hong Kong
The Risk of Bankruptcy in Long-term Investment
Abstract: In recent years, various continuous-time strategies in portfolio management have been developed with different objectives. However the risks associated with these strategies are not well understood. We focus on one particular measure of risk in this talk, namely the probability of bankruptcy occurring while applying these strategies. We demonstrate that applications of untamed strategies in long-term investments always lead to sure bankruptcy. This is true even when the target-return rate is only slightly above the risk-free interest rate. For tamed strategies, if the target-return rate is set above certain critical value, then the probability of being in bankruptcy will be one hundred percent for a long-term investor. Empirical study based on the market data confirms these findings. (Received April 10, 2008)
Ray R. Sturm (rsturm@bus.ucf.edu), Department of Finance, College of Business Administration, University of Central Florida
The 52-week high strategy: Momentum and Overreaction in Large Firm Stocks
Abstract: Prior studies have documented momentum profits to stock portfolios formed from 52-week highs in prices. In this study, I primarily examine the pattern of returns to portfolios formed from other highs besides the 52-week high and from the time interval between current prices and the prior high. I find evidence suggesting that investors attach value to prior highs and lows besides the 52-week high/low, but the 52-week high/low appears to have more value than the others. My results imply that prior price extremes contain information about future returns, and present a challenge to market efficiency. (Received May 7, 2008)
Ahmet Duran (durana@umich.edu), Department of Mathematics, University of Michigan-Ann Arbor
Quantitative Behavioral Finance and Out-of-sample Prediction via Asset Flow Differential Equations
Quantitative behavioral finance is a new discipline that uses mathematical and statistical methodology to understand behavioral (cognitive and emotional) biases in conjunction with valuation. A system of nonlinear asset flow differential equations (AFDE) incorporates behavioral concepts with the finiteness of assets and microeconomic principles. They have been developed and analyzed asymptotically by Caginalp and collaborators since 1989. I will focus on how the elimination of "noise" or changes in valuation and an inverse problem involving parameter optimization for AFDE can be used in order to forecast near term market returns by following out-of-sample procedure.