Dr. Wenjian Liu (UCSB)

Title: Ross Recovery in Continuous Time

Abstract: In this talk, we will be concerned with a theoretically interesting and practically important question: under what assumptions can one uniquely recover physical probabilities from observed option prices? Ross has contributed an important breakthrough against the conventional wisdom by showing that along with a restriction on preferences, option prices forecast not only the average return, but also the entire return distribution. Specifically, Ross shows that one can recover the real-world transition probability matrix from given Arrow-Debreu state prices, if all uncertainty in an arbitrage-free economy follows a finite state, discrete time irreducible Markov chain and the pricing kernel satisfies a structural assumption of transition independence. By placing the structure on the dynamics of the numeraire portfolio rather than Ross's restriction on the preferences of the representative agent, Carr and Yu have extended Ross’s recovery result to one-dimensional diffusions on bounded intervals with regular boundaries at both ends. By means of the refined spectrum analysis of the elliptic operator, we are able to extend this single driver Ross Recovery problem to multivariate driving state variables. Also let us see some extensions to unbounded cases.