Speaker: Alec Kercheval, Florida State University
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 heterscedastic 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.)