Title:

Pricing of options in interrupted markets using utility maximisation theory

In the financial industry, the standard method of pricing options is using the Black & Scholes model or one of its immediate extensions. One of the basic assumptions in this model is that the underlying of the option contract is available for continuous trading, thus, via hedging, one can completely offset the risk generated by trading in such derivatives. In this thesis an attempt is made to price options in cases where the Black & Scholes model cannot be applied. The particular restriction considered, is the a priori known unavailability of the underlying for continuous trading. The objectives of this thesis are the following:
a) investigate the effect such deterministic trading interruptions, under a generic utility maximisation framework,
b) exploit the concepts developed by [11] , in order to obtain a more tractable model, for options on one underlying asset, and
c) extend the model developed using the [11] concepts and investigate the effects of deterministic interrupts for options with two underlyings.
By investigating a parallel branch of pricing options to Black & Scholes, namely the utility maximization theory (which is popular in portfolio pricing), it is possible to treat many of the shortcomings of the Black & Scholes model, including the problem dealt with in this thesis. We will consider the Black & Scholes price, as the base case for the options price and compare the prices obtained from our models to this base case price. The advantage of using utility maximization theory is that, the results yielded from such a model converge to the Black & Scholes base case price when we assume that no restrictions
apply to our underlyings.
Three models are developed, all based on an initial portfolio composed of a bond and risky assets. Initially, we consider a general form of the utility function from the HARA class of utility functions and provide prices for both base and restricted cases under three different utility functions. Then, a negative exponential utility function is used to develop a second model. By using this utility function, we can simplify the form of our problem since we can ignore the initial level of the bond in the portfolio, which is an optimisation parameter that, this utility function is insensitive to.
The first two models developed can price options based on one underlying only. An improvement is achieved by increasing to two the number of underlyings of the option, hence widening the class of options that can be priced. Options such as the spread option belong to this class, an option very popular in the energy industry. In the end of this thesis, it is shown how the twoasset model can be applied to a real problem from the energy sector. The case of a petroleum refinery is investigated, and results are drawn on how the refinery value can be increased by rescheduling its maintenance periods.
