Pricing and hedging options conditional on market activity

Alec Kercheval, Navid Salehy, Nima Salehy

We replicate and price European options on stocks modeled by time-changed ge- ometric Brownian motion. The time change is obtained as the integrated intensity of random arrival times of price changes of the underlier over the life of the op- tion. For European call options we obtain explicit hedging and pricing formulas. This approach is motivated by the need to connect option prices directly to the microstructure properties of the limit order book that determines tick-by-tick stock price changes. The continuous time model is obtained as an appropriate limit of discrete time random walks with random jump times, in the limit of infinitely many independent representative agents.