Research On The Path Integral For Option Pricing

There are many mathematical methodologies'and techniques in modern financial theory applying to preventing and controlling financial risks. Financial derivatives have become more and more main stream instruments for risk management and controlling due to their powerful function to avoid risk and maintain value of financial position. Option is one of the most important instrument of financial derivatives. Along with development of financial innovation,many kinds of securitisation assets come to be exist, such as stocks?bonds?commodities?foreign exchange?stocks index and volatility all of these can be treated as underling asset of securitisation assets. Given a certain underling asset, it is a more complicate problem for pricing corresponding options. After Black?Scholes and Merton constructed theory of modern option pricing, option pricing has become a core field in modern financial research. Based on the BSM framework, there are many other models and techniques which have been broadly applied in engineering and natural science , developed to deal with the option pricing. Paht integral which act as an important methodology in modern physics is drawn into the option pricing in the same way.This dissertation will concentrate on option pricing with path integral methodology and how to compute the path integral of option pricing. The use of path integrals has been commonplace in science for many years since the creation of the path integral in Feynman (1942). Its application to finance, in particular the pricing of derivative securities, has been less common. The thesis will offer various alternative techniques to construct and solve some particular path integral models. This dissertation can be diveded into two sections . In the first section , based on a path integral framework of option pricing which was presented by Chiarella?El-Hassan and Kucera, we deal with how to compute price of european option?america option and barrier option . Their method involves the use of a Fourier-Hermite series expansion which represents the option value at each time step. The Hermite orthogonal polynomials and their associated properties are employed to create a set of recurrence relations so that a final option pricing polynomial is formed. A similar approach using normalised Hermite orthogonal polynomials is also presented. Similar methods and techniques are utilised to form a new set of recurrence relations. The accuracy obtained for both types of orthogonal polynomials are of the same magnitude. Compared to Fourier-Hermite series expansion method, normalized Fourier-Hermite method ease the declination in computing accuracy when difference between underling asset price and expiration price is big with Fourier-Hermite method, and alleviate the fluctuation of Fourier-Hermite series method.The core idea of Fouier-Hermite series method to compute the option pricing path integral is to decompose the path integral in simplify form with Fourier-Hermite series. In the other approaches, the path integral is transformed from an infinite interval integral into one of a finite interval with a bound on the resulting error. This is achieved by using the weight (in the form of a Gaussian) within the integrand of the path integral. Using an a-priori value, the tails of the Gaussian are eliminated to form the finite interval. Two numerical methods are used to approximate the option price namely, mathematical interpolation and various quadrature (Newton-Cotes) rules. The interpolation approach takes a series of Hermite interpolation polynomials (of order 2) to represent the option price at each time step. Since there is no closed form solution of the path integral, converting the option price function to a series of polynomials allows an approximation of the option price to be found. By discretizing the underlying, a series of integrations are evaluated for each time step. Various discretization schemes are implemented including a fixed number of partitions (equally spaced over each time step), equally spaced partitions (over each time step) and an adaptive node distribution. In this final discretization scheme, the partitions are formed so that the errors caused by interpolation are controlled. The option price approximations are highly accurate with some discretization schemes working better than others. The final approach takes the finite interval path integral and uses various quadrature (Newton-Cotes) rules. Endpoint, Midpoint, Trapezoidal and Simpson's rules are employed to approximate the option price. The underlying is discretized using a fixed number of partitions, equally spaced over all time steps for each of the rules implemented. The results obtained using the various rules are highly accurate for the European option and the down and out call option but require a large number of partitions to obtain the same accuracy as the other methods for the American put option.After investigating the computing method for the option pricing path integral which has been represent in the first section, we step further to investigate the deeper application of path integral in option pricing. The idea of the option pricing path integral presented by Chiarella et.l is based on the geometry intuition of path integral applied to describe history of a stochastic motion . Except for this superfluous application of path integral in financial field, Path integral has been an powerful technique in dealing with complex stochastic system in physics and this means path integral has more deeper idea which can be applied to deal with option pricing. The key point of view in this notion of path integral is to construct a Lagrangian functional for a stochastic dynamic system, and the least action functional of every path can exactly describe the dynamics of the stochastic dynamic system. Because option pricing can be described as a similar stochastic dynamic system, we can use this ideas of path integral to construct a mathematic model for option pricing. And this idea of modeling make a natural bridge between neutral-risk and no-arbitrage. In the same way, we use this modeling idea to investigate a non-gaussian option pricing path integral drived by a Tsallis distribution, since a stochastic process drived by Tsallis distribution is more properly to describe fat-tail and peak which can be found in the real financial market. The formulation of the Lagrangian functional is not unique and really depends on the structure of the SDE, we therefore cannot apply the same Lagrangian functional for all SDE. While we had carried out a general formulation of the Lagrangian, this formulation is also not unique and must take into account the dependence of drift and diffusion termon time and process variable. These Lagrangian formulations enabled us to write down the Lagrangian functional for the non-Gaussian processes drived by Tsallis distribution. Unfortunately these formulations lead to intractable path integrals and we therefore cannot evaluate the path integral with the techniques introduced in Chiarella et.l model. An alternative approach is to use the method of least action—this finds the path that contributes the most in the path integral and the path which minimizes the action functional. We referred to this approach as the instanton method. We can obtain such a path by directly solving the Euler–Lagrange equation. The Euler– Lagrange equation often leads to highly non–trivial, and non–linear differential equations. Nevertheless it was possible to solve such differential equations and obtain a solution. The instanton method is the most promising way in evaluating the path integral when using the Lagrangian method. Of course, except the instanton method, there are other alternative techniques to get numerical solutions of the non-Gaussian path integral. The one is try to transform the non-Gaussian path integral into a Gaussian one. The other is perturbation method which can decompose the non-Gaussian path integral into more simple form and easy to deal with. These will be the subject of future research and propects.