Pricing options with the SABR Model

Abstract

This thesis discusses the pricing of stock options using the recently developed SABR model. The SABR model assumes the volatility to be stochastic, and its main advantage compared to other stochastic volatility models is that there exists a direct formula which approximates prices of European options. The main question of this thesis is: • Is the SABR model a good model to use when pricing European and American options? To answer this question, three different subquestions are dealt with, namely: • In which situations is the approximating direct formula good enough to price European options? • What is a good numerical method to price European options under the SABR model? • What is good method to price American options under the SABR model?

To price European options under the SABR model, we design two numerical methods. The first one uses Monte Carlo simulations, and the second one is based on the Vellekoop and Nieuwenhuis tree method. By comparing the results of these two methods and the approximating direct formula, for different chosen values of the parameters, it is shown that the direct formula approximates prices of European options well for most values of the parameters, but not for all. Furthermore, it is demonstrated that the two numerical methods give similar results for the price of European options, but that the tree method is much faster. This is why we use this tree method, when applying the SABR model to real market data of European options. To price American options, once more two numerical methods are designed. The first one is based on the Least-Squares Monte Carlo method, and the second one is the same tree method used to price European options under the SABR model, with some slight modifications. By comparing the results of these two methods for the different chosen values of the parameters, we see that the two numerical methods give similar results for the prices of American options, but that the tree method is again much faster. This is why we use this tree method again, when applying the SABR model to real market data of American options. From the results of the fitting of the SABR model to real market data, it follows that the approximating direct formula can be used to price real European options. Furthermore, the results show that the SABR model is indeed a good model to use when pricing European and American options. It is even demonstrated that the SABR model prices American options with short maturity times more accurately than the commonly used Heston model.