PRICING BARRIER OPTIONS - USE OF NUMERICAL SIMULATION METHODS

The objective of this research was to develop a general model for the pricing of Exotic options (in particular, barrier options). Exotic options are increasing in popularity because of the amount of flexibility they offer. They can be tailor made to suit the risk - return profile of any investor and hence they are an important tool in Modem Finance. Barrier options are the most popular amongst the various exotic options in the market and it is essential to develop a general model for pricing them. Both theoretical and simulation methods have been used in the past to price barrier options. The various theoretical models that have been developed in the past are first illustrated in this thesis. However, as the complexity of the options increases it becomes too difficult to arrive at a closed form theoretical solution. Various simplifying assumptions have to be made and this reduces the flexibility of these models. To overcome this difficulty simulation methods are used to price barrier options. The two categories of simulation methods that have been used are the Lattice method and the modified Monte Carlo simulation method. The various models used in past have focused on developing a solution for a particular case but no attempt has been made to develop a model for a general case. So, a very general case was formulated (a portfolio of 'k' number of securities) and the two simulation methods were used to price 'outside' barrier options on them. (If the value of the barrier depends on the price of an instrument other than the underlying instrument of an option the barrier is called an outside barrier) For the case of Lattice method the basic unit was taken to be a trinomial lattice as this shows a faster rate of convergence than the traditional binomial lattice. The lattice structure was developed for a portfolio of 'k' securities. The barrier was an outside one and call and put options for various cases were calculated. For the modified Monte Carlo methods simulations were done for both the pseudo random numbers and the deterministic sequences of numbers. These were then used in pricing the outside barrier option. The deterministic sequences showed a faster rate of convergence than the pseudo random numbers that are generally used. Finally a comparison of the two pricing methods has been done. The Lattice method achieves convergence faster and is more intuitive. Hence it can be used where a number of different options have to be priced quickly. The Monte Carlo method on the other hand, although takes a much longer time to converge, is more flexible and can handle a lot of complexity in the contract features. So it can be used where precision is of utmost importance. The code for these methods has been written using MATLB version 4.1.2. This code can be slightly modified to price other categories of exotic options too.

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• CHIRANJEET