The thesis is based around the path integral framework offered by Chiarella et al. (1999). In their method, the underlying is expanded into a Fourier-Hermite series. At each time step, the coefficients of the series are determined in a backward recur- sive manner, using recurrence relations. These relationships are formed utilising the orthogonal properties of Hermite orthogonal polynomials. In Chapter 3, an anal- ysis of the method described by Chiarella et al. is presented. This will assist in understanding the remaining chapters and comparison of techniques used to solve the same problem.
The first approach is similar to that offered by Chiarella et al. The main difference being the use of normalised Hermite orthogonal polynomials. A set of recurrence relations are formed, as with the previous method. The benefits of using the nor- malised polynomials are the form of the recurrence relations as well as the speed to find accurate results (especially for the European option). Some relations have one less exponential term. Given this fact, the speed of computation should be improved for a large number of basis functions.
The next approach, using the same path integral framework, also converts the un- derlying price at each time step. The price is represented by a series of interpolation polynomials. In this method, integration is performed only once, at the beginning of the process. Using the result of the integration and the interpolation polynomial coefficients found, the option price is evaluated. This process is repeated at each time step. The method requires no transformations and is quite straight forward to implement. The path integral framework is converted from an infinite interval to a finite interval.
The major issues arising from this method include, the determination of the interval of integration and the node point allocation. The problem of the interval of integra- tion is solved via the properties of the Gaussian in the integral. Node allocation or
distribution will vary depending on the derivative security being priced. Similar to Chiarella et al., the resultant derivative security price is continuous and infinitely differentiable allowing for fast and accurate evaluation of the hedge ratios (if re- quired). The major advantage of this method is the very high accuracy obtained and the easy adaptation for American and Barrier type options.
The final approach uses traditional quadrature rules such as the trapezoid and Simp- son’s rule. Using a similar set up to that of the previous technique, a quadrature scheme is formed to represent the derivative security price at each time step. The rules used show that accurate results can be found in relatively quick time. Issues as those that have arisen in the previous approach such as node allocation also exists in this approach. The quadrature rules can also be easily applied to American and Barrier type options.
The thesis is a numerical investigation of the path integral framework. The thesis will emphasise the performance and accuracy of each of the methods for the framework and particular parameters. Trade offs between accuracy and computational effort are addressed. The ease of implementation (in the case of the European options) allows an insight into the behaviour and performance of the method for the path integral framework and more complex options like, American put and down and out call options.
The Black and Scholes Paradigm
This chapter shows the evolution of the Black & Scholes (1973) paradigm. It begins with the major assumptions in which a derivative security like an option is modeled and priced. We present the formulation of the Black and Scholes equation (a partial differential equation) using a replicating portfolio. In deriving the Black and Scholes equation, a formula is presented for both a European Call and a Put option. Finally, the development of the Chiarella et al. (1999) path integral is presented, which is constructed based on the Black and Scholes paradigm.2.1
Introduction
Prior to presenting the path integral framework used in this thesis for option pric- ing, an understanding of the Black & Scholes (1973) paradigm is required. Since many option pricing models are based on this paradigm, the chapter will describe the fundamentals of the assumptions, equations and the derivation of the formula. We initially present the major assumptions on which a model using the paradigm must satisfy. There are many assumptions which exist and continue to be used since the creation of the Black and Scholes formula well over three decades ago.
Following the assumptions, we present a summary version of the creation of the Black and Scholes equation (a partial differential equation) using a replicating portfolio. The presented method is based on that in Wilmott (1999). The partial differential equation (pde), is derived using a portfolio containing a long position in the option and a short position in a quantity of the underlying. The portfolio is replicating because it changes continuously with respect to time and a change in value of the underlying. The pde is also derived using common financial principles of delta hedg- ing and no arbitrage.
We finally present the formulation of the path integral framework based on Chiarella et al. (1999). This is the framework which is central to this thesis. The framework developed uses the assumptions and ideas described in this paradigm. The frame- work is built based on the technique of path integrals in statistical physics.