On the running maximum of Brownian motion and associated lookback options

• Main Author: Ho, Tak Yui
• Published: London School of Economics and Political Science (University of London) 2018
• Subjects: HA Statistics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.762914

The running maximum of Brownian motion appears often in mathematical finance. In derivatives pricing, it is used in modelling derivatives with lookback or barrier hitting features. For path dependent derivatives, valuation and risk management rely on Monte Carlo simulation. However, discretization schemes are often biased in estimating the running maximum and barrier hitting time. For example, it is hard to know if the underlying asset has crossed the barrier between two discrete time points when the simulated asset prices are on one side of the barrier but very close. We apply several martingale methods, such as optional stopping and change of measure, also known as importance sampling including exponential tilting, on simulating the stopping times, and positions in some case, of the running maximum of Brownian motion. This results in more accurate and computationally cheap Monte Carlo simulations. In the linear deterministic barrier case, close-form distribution functions are obtained from integral transforms. The stopping time and position can hence be simulated exactly and efficiently by acceptancerejection method. Examples in derivative pricing are constructed by using the stopping time as a trigger event. A differential equation method is developed in parallel to solve for the Laplace transform and has the potential to be extended to other barriers. In the compound Poisson barrier case, we can reduce the variance and bias of the crossing probabilities simulated by different importance sampling methods. We have also addressed the problem of heavy skewness when applying importance sampling.