Alec Kercheval, Florida State University
Self-excited Black-Scholes models for option pricing
Beginners first learn to price stock options with a simple binomial tree model for random price changes. It is well known that this classical one-dimensional random walk converges weakly to Brownian motion in the proper space-time scaling limit. Actual stock prices changes occur not at regular times but at random times according to the order flow in an electronic limit order book (LOB), and these are observed to have heteroscedastic and self-exciting characteristics.
In this talk, we consider random walks in which jumps occur at random times described by an independent general point process, which could be a self-exciting process such as a Hawkes process. We show that in the correct scaling limit, this converges to a time-changed Brownian motion, where the time change is the compensator of the original point process. The resulting stock price process can exhibit many of the stylized properties of observed stock prices. We establish a familiar formula for the price of an option for this model, forming a connection between models of LOB dynamics and financial derivative pricing. (This paper is joint work with Navid Salehy and Nima Salehy.)
Alec Kercheval is professor and associate chair of mathematics at Florida State University, and currently a visiting scholar at CDAR and the mathematics department at UC Berkeley. Until recently he was director of the FSU financial mathematics graduate program. His research has been funded by NSF and the Simons Foundation, and he has been on the faculties of the University of Texas at Austin, Indiana University, and Boston University. From 1999-2001 he was a member of the research group of MSCI Barra, Inc., in Berkeley.