Numerisk prissättning av exotiska optioner

Abstract

This paper examines Asian, lookback and barrier options of European style on the time interval [0; T], where T is the time of maturity. The purpose is to investigate numerical methods to compute their price within the Black-Scholes model. This is carried out in both C++ and Matlab with the objective of comparing the computational performance of the two programming languages. Moreover, various sensitivity quantities, the so called Greeks, are investigated. The numerical pricing methods selected are the Monte Carlo method and the Crank-Nicolson finite difference method. Implementation of the Monte Carlo method consists of iterating an unbounded random sample N times in order to generate an approximate price. This is achieved through partitioning the interval [0; T] into n subintervals and simulating a discrete path for the value S(t) of the underlying asset. The Crank-Nicolson method is applied through inverting the time variable of the Black-Scholes partial differential equation, where the space derivatives are centered and the time derivatives are estimated in a forwardbackward manner. Implementation of the method implies partitioning both the time interval [0; T] and the space interval [0; xmax] into n and m subintervals, respectively. Examination of the selected Greeks show that Asian options are less sensitive to volatility than lookback and barrier options are. Furthermore, it is concluded that the Crank-Nicolson method is superior to the Monte Carlo method for the pricing of all of the examined options. One of the reasons for this is a convergence problem that arises for the lookback and barrier options, which causes the Monte Carlo method to be very time-consuming. Lastly, C++ is shown to be the language of choice for a fast and relatively accurate approximation.