The COS method: An efficient Fourier method for pricing financial derivatives

When valuing and risk-managing financial derivatives, practitioners demand fast and accurate prices and sensitivities. Aside from the pricing of non-standard exotic financial derivatives, so-called plain vanilla European options form the basis for the calibration of financial models. As any pricing and risk management system has to be able to calibrate to these plain vanilla options, it is important to be able to value these options quickly and accurately. By means of the risk-neutral valuation formula the price of any option, without early exercise features, can be written as an expectation of the discounted payoff of this option. Starting from this representation one can apply three types of numerical methods to calculate the price itself: Monte Carlo simulation, numerical solution of the corresponding partial-(integro) differential equation (P(I DE) and numerical integration. An important aspect of research in Computational Finance is to continuously improve the performance of the pricing methods. In this dissertation we present a novel and efficient option pricing method based on the integration representation for various financial derivatives. The method is called the COS method, because the key idea is to replace the probability density function, appearing in the risk neutral valuation formula, by its Fourier-cosine series expansion. Fourier-cosine series coefficients have an elegant closed-form relation with the characteristic function (i.e. the Fourier transform of the underlying density function), which is readily known, for example for exponential Lévy processes, and more general for the broader class of regular affine processes. For European options, presented in Chapter 2 of this thesis, the risk-neutral valuation formula appears as an inner product of the probability density function and the payoff function. Approximating the density function by its Fourier-cosine series expansion transforms the inner product into combinations of products of cosine basis functions and the (payoff) function which is known analytically. The method's convergence is therefore directly connected to the convergence of the Fourier-cosine series approximation of the density function. For smooth density functions, we show that the convergence is exponential. Since the computational complexity grows only linearly with the number of terms in the expansion, the COS method is optimal in error convergence and in computational complexity for European options. In other chapters of this thesis, the applicability of the COS method has been generalized to pricing options with early-exercise features and discretely-monitored barrier options, as well as to the calibration of Credit Default Swaps, under exponential Lévy processes. Furthermore, an efficient pricing method based on the COS method has been developed for pricing early-exercise options under the (two-dimensional) Heston stochastic volatility dynamics. The main insight for generalizing to the options with early-exercise features under Lévy processes, as presented in Chapter 3, is that between any two adjacent early-exercise dates the COS formula for European options can be applied, and that the series coefficients of the option values at each time lattice can be recovered recursively from those of the payoff function. This recursion can be written as a matrix-vector product with the matrix being the sum of two special matrices: a Hankel matrix and a Toeplitz matrix. Multiplication of a vector to these special matrices can be written as circular convolution of two vectors and can therefore be computed rapidly by means of the Fast Fourier Transform (FFT) algorithm. The overall computational complexity for early-exercise (and discretely monitored barrier) options with M exercise dates is O((M-1)Nlog2N), with N being the number of terms in the cosine expansion. It is then shown in Chapter 4 that the COS method can be used to calibrate Credit Default Swaps (CDSs), that are the basic building blocks of the credit risk market. The (bid-ask) spread for a CDS depends on the default probability of the underlying reference entity. In the approach adopted here, the credit default spreads are related to a series of survival/default probabilities with different maturities. These survival probabilities can be viewed as prices of binary down-and-out barrier options (without discounting). As such, the COS method for discrete barrier options can be used to recover the probabilities. A special scheme has been developed to reduce the computational time by computing several survival probabilities simultaneously. The method's capabilities have been demonstrated by the calibration to quotes of the constituents of the iTraxx Series. Finally, in Chapter 5, the COS method has been extended by quadrature rules to efficiently deal with two-dimensional pricing problems, originating for early-exercise options under the Heston stochastic volatility model. We focus especially on parameter values for which the volatility can reach zero. This is a nontrivial situation for which many other numerical schemes fail, including the finite difference PDE schemes. The problem is related to parameter values for which the Feller condition is not satisfied, and for which the pricing problem is close to singular. The variance of the stock process follows a noncentral chi-square distribution, and, for some combinations of the relevant parameters, the density function values tend to infinitely large numbers when the variance approaches zero. We handle this problem by a transformation from the variance domain to the log-variance domain. The two-dimensional discrete pricing formula is then obtained by applying the Fourier-cosine series expansion approximation in the log-stock dimension and a high-order quadrature rule in the log-variance dimension. The overall computational complexity is almost-linear in the log-stock dimension and quadratic in the log-variance dimension with very satisfactory error convergence. In this thesis we show that the COS method can price various financial derivatives under exponential Lévy and Heston stochastic volatility models. Highly efficient pricing results are presented, due to a fast error convergence and a lean computational complexity. We further determine the stability of the pricing method by a rigorous error analysis and by many numerical experiments under extreme parameter values.