Financial decisions about complicated securities or investments must be made subject to considerable uncertainty about the returns they will ultimately produce. Moreover, as in the case of the recent crisis, there is often a limit to what can be deduced from history about what to expect in the future. Significant nonstationarities weaken the connection between past and future and thus can render useless standard methods of statistical inference. It has been argued that the (perceived) failures of the theory of finance in the context of the recent crisis are due to inadequate attention paid to nonstationarities and related complexities of financial markets. Indeed, the dominant theory of portfolio choice and asset pricing models investors as completely confident in their understanding of the environment and in their ability to predict the future. This project is concerned with modeling the behavior and interaction of individuals who are aware of the complexity of their environment and who are therefore suitably cautious when making decisions: they recognize that the past is an imperfect guide to the future and they seek decisions that are robust to possible errors in their theory or model of the environment.
More specifically, the project's goal is to generalize formal continuous-time asset pricing theory to incorporate the preceding features (limited confidence, robust decision-making, nonstationarities and less ambitious learning). Continuous time modeling is prominent in financial economics because of its analytical advantages and because we live in continuous time. A technical challenge that ensues is that the familiar tools of stochastic analysis employ and build on probability theory. However, as indicated further below, the above features cannot be accommodated within a probability space framework. Thus the project will have a sizable technical component that will adapt and extend very recent developments in stochastic analysis that eliminate the need for a probability space framework in continuous time modeling. The resulting expansion of the tool box of financial economics is a contribution in its own right.
A major focus of the project is on the modeling of volatility. Stochastic time varying volatility is important for understanding features of asset returns, particularly empirical regularities in derivative markets, and also the dynamics of real macroeconomic variables; and evidence suggests that volatilities follow complicated dynamics. The common modeling response, in the so-called stochastic volatility literature, is to postulate correspondingly complicated parametric laws of motion, including specification of the dynamics of the volatility of volatility. However, one might question whether investors can learn these laws of motion precisely, and more generally, whether it is plausible to assume that investors become completely confident in any particular law of motion. It has been argued that such confidence is particularly problematic because the quantity being modeled is not directly observable, and because noise in the data makes it difficult to draw sharp inferences about volatility from observables, such as the market prices of derivative securities. Furthermore, it is often the case that alternative models of volatility have differing implications for the application at hand but cannot be distinguished empirically from past data. This motivates modeling decision-making that is robust to possible misspecifications of the dynamics of volatility, providing thereby a way to robustify stochastic volatility modeling. It is known that robustness to uncertainty about the true model of volatility cannot be accommodated within a probability space framework, thus motivating the technical component described above.
Broader impacts include the possible impact of this work on the methods used by regulators and others to understand financial markets.
