Continuous limits of random walks over point processes and self-excited Black-Scholes models.
Alec Kercheval, Navid Salehy, Nima Salehy
We modify the classical one-dimensional random walk by letting the time of each step be determined by a random point process, such as a self-exciting Hawkes process. We do not require the random walk have independent increments or satisfy the Markov property. In a suitable rescaled limit as space and time step sizes tend to zero, we show, as a generalization of Donsker's theorem, that the random walk weakly converges to a time-changed Brownian motion, where the time change is the compensator of the original counting process.
As an application, we view stock price changes as determined by random arrival times in a limit order book. For a stock price process driven by the limiting time-changed Brownian motion, we establish conditions under which European option payoffs are attainable as the terminal value of a self-financing strategy in the stock and a bond, and establish a unique no-arbitrage pricing formula. For a European call option we obtain an explicit formula parametrized by the integrated intensity of arrival times over the life of the option.