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MSRI-UP 2011: Mathematical Finance

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During the first two weeks of MSRI-UP, in preparation for their research, students will be introduced to several topics in mathematical finance, including special topics in probability and stochastic processes, arbitrage-free derivative pricing, the Black-Scholes-Merton partial differential equation, and liquidity models.During the remainder of the program, the students will work in teams on research projects.Below, we give examples of two research areas.

Project 1: Liquidity Modeling

In the fields of mathematical finance and financial engineering, a standard assumption is one of infinite liquidity of securities. Under this assumption all market agents are price takers, meaning that one can buy or sell any number of shares of a security instantaneously at the market price without affecting that market price. In reality, this assumption does not hold.One popular model which relaxes the assumption of infinite liquidity is presented by Cetin, Jarrow, and Protter . This model postulates the existence of a supply curve S_t(x) which gives the price of an asset as a function of trade size and is an extension of the standard Black-Scholes-Merton model. For highly liquid stocks, this curve has been found to be a linear function of trade size with a slope that changes randomly in time. For assets which are illiquid, the supply curve lacks this linear property but seems to have a piecewise-linear structure in trade size.Using financial data, students will investigate the supply curve for moderately liquid and illiquid assets and research methods for modeling the supply curve in these cases. Models will be tested statistically for goodness of fit using spline techniques, and the distribution of model parameters will be examined using tools from probability and stochastic processes.

Project 2: Cointegration and the Capital Asset Pricing Model

The ability to predict excess returns has been a goal of financial economists for decades. The Capital Asset Pricing Model (CAPM) is a commonly used factor model for predicting expected returns, but there are several unrealistic assumptions associated with this model that make the results unreliable, such as the assumption that supply equals demand for all assets and the assumption that investors act rationally when investing their money. This model also relies solely on how risky the asset in question is compared to the overall risk of the market portfolio when predicting returns. Fama and French improve the CAPM by incorporating additional risk factors into the model, but its predictive ability is still in question.Students will use the statistical technique of cointegration to to investigate long term relationships between macroeconomic factors, such as dividend yields and interest rates, and use these results to build factor models that expand on the CAPM and the Fama and French model. Financial data will be used to test the predictive power of the proposed factor models.