Market Expectations, Long Term Risk, and Stochastic Spectral Theory

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This project focuses on learning expectations of market participants about probability distributions of future asset returns from the current prices of options on those assets combined with assumptions and historical data about the underlying risk-return trade-offs in the economy. Risk assessments are based on historical data. The limitation of the historical approach is in potentially underestimating the probability of events that did not occur in the historical data under consideration. The goal of this research is to improve probability models of markets by developing theory and methods for calibrating them to additional sources of information in addition to historical data. This will help put risk management and investment decision making on a more solid foundation and will aid the financial services industry and market regulators in extracting implied probability distributions from market prices of options to improve risk management. The project is interdisciplinary, drawing on the fields of operations research, economics, probability theory and mathematical analysis and will have a positive impact on education and human resources development.The methodology is based on far-reaching extensions of the recent Recovery Theorem of Ross that shows that when all uncertainty (risk) in the economy is modeled as a discrete-time irreducible finite-state Markov chain and the stochastic discount factor is transition independent, then there exists a unique recovery of the Markov chain's transition probability matrix from options prices. We aim to extend the recovery methodology to general classes of continuous-time Markov processes, including diffusions and jump-diffusions. On the other hand, we aim to relax the transition independence assumption by building on the fundamental work of Hansen and Scheinkman on long term risk. Our approach aims to combine structural assumptions on the stochastic discount factor drawn from the macro-finance literature with the joint calibration of the resulting models to currently observed market options prices together with historical time series data on the underlying asset returns. This will involve analytical development of the spectral theory for Markov processes and computational implementations of recoveries in specific classes of models.