The use of path integrals has developed into a viable option pricing model represen- tation in the past decade or so. Since the creation of the Black-Scholes PDE and the various techniques to solve (1.1), authors have attempted to model vanilla and non vanilla options in alternative forms. Path integrals has been one of the alternative methods.
Path integrals have been used in various areas of science over the years, especially in quantum physics. One of the advantages of using path integrals is the variety of techniques used to solve them. From Monte Carlo simulation to various quadrature methods, the techniques have been developed and applied to finance.
The following review will present the use of path integrals to model and the tech- niques to evaluate option prices. One of the early uses of a path integral in derivative security pricing was from Makivic (1994). The author presents a Monte Carlo ap- proach (using the Metropolis algorithm) to price a security.
Makivic also states that the main advantages of a path integral approach are:
(1) partial derivatives of the price with respect to all of the input parameters can be computed in a single simulation,
(2) results for multiple setsof parameters can be computed in a single simulation, and
(3) suitability for implementation on a parallel or distributed computing environ- ment.
It must be said that his assertions are correct for a path integral approach using Monte Carlo simulation to evaluate the price. The best results show errors of order 10−4.
Baaquie (1997) presents a path integral approach to option pricing with stochastic volatility. Baaquie generalises the results of Hull & White (1987) for cases when the stock price and volatility have non-zero correlation. Ingber (2000) also presents a path integral approach to options with stochastic volatilities. The author uses an Adaptive Simulated Annealing approach to determine the behaviour of diffusion. This behaviour is determined by daily Eurodollar future prices and implied volatil- ities. An algorithm called PATHINT is used to evaluate prices.
Linetsky (1998) offers a path integral approach to financial modeling and option pricing. The author states that ”the path integral formalism constitutes a conve- nient and intuitive language for stochastic modeling in finance”. Linetsky presents various path integrals, including a framework for the Black-Scholes paradigm path dependent options and multi-asset derivatives. The author finally develops evalu- ations for various options using analytical approximations and numerical methods (Monte Carlo simulation and/or discretization schemes).
Some authors have investigated the use of path integrals to model path dependent options. Matacz (2000) uses a partial averaging method to price path dependent
options (Asian options and occupation time derivatives). The method of partial averaging reduces the dimension of the integral. The evaluation can be performed by Monte Carlo simulation methods. Baaquie, Corian`o & Srikant (2003) also offer a path integral approach to solve for path dependent options. They build their model using the Black-Scholes paradigm and then extend it to create more complex secu- rities such as exotic and path dependent options. Baaquie et al. (2003) evaluate the option prices by Monte-Carlo simulation. Bormetti, Montagna, Moreni & Nicrosini (2006) also present a path integral framework to evaluate (via Monte Carlo simula- tion) prices for various path dependent options.
An interesting application using a path integral approach is offered by Otto (1999). The author presents a model to price interest rate derivatives. Path integrals for the short term and bond option are developed. Otto suggests two techniques to solve these derivatives, they are a lattice method or the use of Monte Carlo simulation.
Bennati, Rosa-Clot & Taddei (1999) develop a path integral approach for various stochastic equations that best represent financial markets. The path integrals are designed to cater for one and multi dimensional cases. The authors then present some analytic results for various models such as Black-Scholes, Cox-Ingersoll-Ross and others. Rosa-Clot & Taddei (2002) offer numerical methods to price some of the derivative securities presented in Bennati et al. (1999). Rosa-Clot and Tad- dei use two methods to evaluate prices, Monte Carlo simulation and deterministic evaluations (quadrature rules). The deterministic evaluations has its advantages in low dimensional problems but in high dimensions the technique has issues with large matrix dimensions. Various options (European options , Zero-coupon bonds, Caplets, American options and Bermudan swaptions) are priced.
Some authors have investigated the use and evaluation of path integrals to price op- tions using unique and less common techniques. Kleinert (2002) presents a Natural Martingale for non-Gaussian fluctuations of the underlying. Decamps, De Schepper & Goovaerts (2006) develop a path integral approach to asset-liability manage-
ment. Chiarella, El-Hassan & Kucera (1999) present an evaluation of a European and American option in a path integral framework. The novel approach to the eval- uation is the use of a Fourier-Hermite series. The technique takes into consideration the form of the integrand of the path integral (1.3),
fk−1(ξk−1) = e −rΔt √ π ∞ −∞ e−(ξk−μ(ξk−1))2fk(√2Δt ξ k)dξk. (1.3)
The Gaussian in the integrand is in the form of the weight of a Hermite orthogonal polynomial. The price function, fk(ξ
k), is expanded into a Fourier-Hermite series. This series is continuous and is a differentiable representation of the underlying. Given the form of the Fourier-Hermite series, the Deltas are easily found as well as the option price.
In Chapter 2 we present the development of the path integral (1.3). Chapter 3, in this thesis, gives a thorough overview of the technique used to find the option price. In this overview of the technique, errors were found in the formulation and in the results presented. The path integral is formed using an application of Ito’s Lemma. Chapter 4 offers a modification to the technique used to evaluate the option price. The alternative method uses normalised Hermite orthogonal polynomials. The use of the normalised polynomials has its advantages, especially when a large number of basis functions are used.
An extension of the previous approach is offered by Chiarella, El-Hassan & Kucera (2004) to incorporate the evaluation of point barrier option prices. The path integral is very similar with the only difference being the integral domain. The path integral (1.3) with a finite domain, namely,
fk−1(ξk−1) = e −rΔt √ π zk,u zk,l e−(ξk−μ(ξk−1))2fk(√2Δt ξ k)dξk, (1.4) where
zk,l = ln (bk,l)
σk2Δtk, and zk,u =
ln (bk,u)
σk2Δtk, (1.5) fork =K−1, . . . ,1 withbk,l andbk,ubeing the lower and upper barriers respectively, at time step k.
Chapters 5 and 6 offer alternative techniques to evaluate the same path integral framework (1.3) and (1.4). Prices are approximated for European, American and Barrier options. The techniques take into account the form of the integrand such that interpolation polynomials and various quadrature rules can be used. The tech- niques employed are highly accurate and very fast to compute.
Given the literature review presented in this thesis, it is clear that the methods and techniques used in evaluating the option price are vast. From the early days of Black, Scholes and Merton to the introduction of many scientific approaches, option pricing is a growing area in both finance and mathematics. Path integrals in finance is relatively new in comparison, with the last decade seeing an increase in activity. Path integrals have been used in areas such as quantum physics for many years since the initial work by Feynman (1942).