Market Expectations, Long Term Risk, and Stochastic Spectral Theory

This proposal 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. The recent global financial crisis showcased the inadequacy of current models in financial risk management. 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 the proposed research is to improve probability models of financial markets by developing theory and methods for calibrating them to additional sources of information in addition to historical data. This has the potential to put financial risk management on a more solid foundation. It is also of great value in making investment decisions. Specifically, we propose to develop methods for using the current market prices of options in addition to historical data on the underlying primary assets (equities and bonds), to learn the current market's expectations of the future, i.e., the market-implied probability distributions that represent the expectations of market participants in the aggregate. We propose to apply it to improve portfolio optimization and risk assessment. The methodology is based on far-reaching extensions of the recent Recovery Theorem due to Ross (2013) 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. On one hand, 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 (2009) 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.