Spread Option Pricing on Single-Core and Parallel Computing Architectures

This paper introduces parallel computation for spread options using two-dimensional Fourier transform. Spread options are multi-asset options whose payoffs depend on the difference of two underlying financial securities. Pricing these securities, however, cannot be done using closed-form methods; as such, we propose an algorithm which employs the fast Fourier Transform (FFT) method to numerically solve spread option prices in a reasonable amount of short time while preserving the pricing accuracy. Our results indicate a significant increase in computational performance when the algorithm is performed on multiple CPU cores and GPU. Moreover, the literature on spread option pricing using FFT methods documents that the pricing accuracy increases with FFT grid size while the computational speed has opposite effect. By using the multi-core/GPU implementation, the trade-off between pricing accuracy and speed is taken into account effectively

Fourier pricing of two-asset options: a comparison of methods

Fourier methods form an integral part in the universe of option pricing due to their speed, accuracy and diversity of use. Two types of methods that are extensively used are fast Fourier transform (FFT) methods and the Fourier-cosine series expansion (COS) method. Since its introduction the COS method has been seen to be more efficient in terms of rate of convergence than its FFT counterparts when pricing vanilla options; however limited comparison has been performed for more exotic options and under varying model assumptions. This paper will expand on this research by considering the efficiency of the two methods when applied to spread and worst-of rainbow options under two different models - namely the Black-Scholes model and the Variance Gamma model. In order to conduct this comparison, this paper considers each option under each model and determines the number of terms until the price estimate converges to a certain level of accuracy. Furthermore, it tests the robustness of the pricing methodologies to changes in certain discretionary parameters. It is found that although under the Black-Scholes model the COS method converges in fewer terms than the FFT method for both spread options (32 versus 128 terms) and the rainbow options (64 versus 512 terms), this is not the case under the more complex Variance Gamma model where the terms to convergence of both methods are similar. Both the methodologies are generally robust against changes in the discretionary variables; however, a notable issue appears under the implementation of the FFT methodology to worst-of rainbow options where the choice of the truncated integration region becomes highly influential on the ability of the method to price accurately. In sum, this paper finds that the improved speed of the COS method against the FFT method diminishes with a more complex model - although the extent of this can only be determined by testing for increasingly complex characteristic functions. Overall the COS method can be seen to be preferable from a practical point of view due to its higher level of robustness

Analysis of Fourier transform valuation formulas and applications

The aim of this article is to provide a systematic analysis of the conditions such that Fourier transform valuation formulas are valid in a general framework; i.e. when the option has an arbitrary payoff function and depends on the path of the asset price process. An interplay between the conditions on the payoff function and the process arises naturally. We also extend these results to the multi-dimensional case, and discuss the calculation of Greeks by Fourier transform methods. As an application, we price options on the minimum of two assets in L\'evy and stochastic volatility models.Comment: 26 pages, 3 figures, to appear in Appl. Math. Financ

The evaluation of early exercise exotic options

University of Technology, Sydney. Faculty of Business.Research on the pricing of multifactor American options has been growing at a slow pace due to the curse of dimensionality. If we start to consider the pricing of American option contracts written on more than one underlying asset or relax the constant volatility assumption of the Black and Scholes (1973) model, the computational burden increases as more computing power is required to handle the increasing number of dimensions. This thesis deals with the problem of pricing multifactor American options under both constant and stochastic volatility. The main focus of the thesis is to extend the representation results of Kim (1990) and Carr, Jarrow and Myneni (1992) and to devise higher dimensional numerical techniques for pricing multifactor American options. We present numerical examples for two and three factor models. The pricing problems are formulated using the well known hedging arguments. We adopt two main approaches; the first involves deriving integral expressions for the American option prices with the aid of Jamshidian’s (1992) transformation of the associated partial differential equation from a homogeneous problem on a restricted domain to an inhomogeneous problem on an unrestricted domain, Duhamel’s principle and integral transform methods. The second technique involves implementing the method of lines algorithm for American exotic options, with the spread call option under stochastic volatility being the main example – this approach tackles directly the pricing partial differential equation. Chapter 1 contains an overview of the American option pricing problem from the viewpoint of the applications in this thesis. The chapter concludes with some technical results used in the rest of the thesis. The main contributions of the thesis are contained in the subsequent chapters. Chapter 2 extends the integral transform approach of McKean (1965) and Chiarella and Ziogas (2005) to the pricing of American options written on two underlying assets under Geometric Brownian motion. A bivariate transition density function of the two underlying stochastic processes is derived by solving the associated backward Kolmogorov partial differential equation. Fourier transform techniques are used to transform the partial differential equation to a corresponding ordinary differential equation whose solution can be readily found by using the integrating factor method. An integral expression of the American option written on any two assets is then obtained by applying Duhamel’s principle. A numerical algorithm for calculating American spread call option prices is given as an example, with the corresponding early exercise boundaries approximated by linear functions. Numerical results are presented and comparisons made with other alternative approaches. Chapter 3 considers the pricing of an American call option whose underlying asset evolves under the influence of two independent stochastic variance processes of the Heston (1993) type. We derive the associated partial differential equation (PDE) for the option price using standard hedging arguments. An integral expression for the general solution of the PDE is derived using Duhamel’s principle, which is expressed in terms of the yet to be determined trivariate transition density function for the driving stochastic processes. We solve the backward Kolmogorov PDE satisfied by the transition density function by first transforming it to the corresponding characteristic PDE using a combination of Fourier and Laplace transforms. The characteristic PDE is solved by the method of characteristics. Having determined the density function, we provide a full representation of the American call option price. By approximating the early exercise surface with a bivariate log-linear function, we develop a numerical algorithm to calculate the pricing function. Numerical results are compared with those from the method of lines algorithm. The approach is generalised in Chapter 4 to the case when the underlying asset evolves under the influence of more than two stochastic variance processes by using a combination of induction proofs and some lengthy derivations

Option Pricing in Multivariate Stochastic Volatility Models of OU Type

We present a multivariate stochastic volatility model with leverage, which is flexible enough to recapture the individual dynamics as well as the interdependencies between several assets while still being highly analytically tractable. First we derive the characteristic function and give conditions that ensure its analyticity and absolute integrability in some open complex strip around zero. Therefore we can use Fourier methods to compute the prices of multi-asset options efficiently. To show the applicability of our results, we propose a concrete specification, the OU-Wishart model, where the dynamics of each individual asset coincide with the popular Gamma-OU BNS model. This model can be well calibrated to market prices, which we illustrate with an example using options on the exchange rates of some major currencies. Finally, we show that covariance swaps can also be priced in closed form.Comment: 28 pages, 5 figures, to appear in SIAM Journal on Financial Mathematic