Pricing methods for barrier options

Abstract

The main goal of this project is to better understand barrier options and try to propose some methods to price them. We start with a review of the financial concepts necessary to develop our methods, where we explain what are derivates, options and the core of our work, barrier options. We also review the classical methods to price options which are the base for the methodologies that we propose. The two methods that we suggest to price barrier options and observe some of their properties are: a quantum mechanical approach using the path integral technique and a modification of the Monte Carlo simulation. For the first one we have considered the stock price as a free particle moving in a space bounded by two barriers and performing trajectories starting at the initial stock price and finishing at maturity time. We have developed the path integral formulation to find the probability of a certain trajectory and we have found that it depends on the inverse of the exponential of a quantity that we call action, in analogy to the path integral, which depends on the followed path by the stock price. The procedure proposed is to start from a completely random path and evolve it in order to reduce the action, thus increasing the probability of such path. We have founded that starting from a random path arising from a Gaussian distribution gives better results than starting from a uniform distribution when we compare with the classical methods. The reason is that the classical methods make the hypothesis that the dynamics of the stock price is based on the Brownian motion, which is Gaussian distributed. We call probabilistic Monte Carlo the other method because we let the stock price penetrate the barriers with some probability, with this methodology we try to observe the role of the barriers in options and the role of the drift in the Brownian motion governing the stock price. We have found that relaxing a bit the barriers is enough to recover an option without them. The reason is that, how far the stock price arrive is limited by the variance of the Brownian motion, then, if the barriers are far enough, is the same as not having them. Finally, we observe that the drift makes the stock price prone to rise, this makes the upper barrier more sensitive to changes than the lower one