A NOVEL PRICING METHOD FOR EUROPEAN OPTIONS BASED ON FOURIER-COSINE SERIES EXPANSIONS

Here we develop an option pricing method for European options based on the Fourier-cosine series, and call it the COS method. The key insight is in the close relation of the characteristic function with the series coefficients of the Fourier-cosine expansion of the density function. In most cases, the convergence rate of the COS method is exponential and the computational complexity is linear. Its range of application covers different underlying dynamics, including L\'evy processes and Heston stochastic volatility model, and various types of option contracts. We will present the method and its applications in two separate parts. The first one is this paper, where we deal with European options in particular. In a follow-up paper we will present its application to options with early-exercise features

Pricing Early-Exercise and Discrete Barrier Options by Fourier-Cosine Series Expansions

We present a pricing method based on Fourier-cosine expansions for early-exercise and discretely-monitored barrier options. The method works well for exponential Levy asset price models. The error convergence is exponential for processes characterized by very smooth transitional probability density functions. The computational complexity is $O((M-1) N \log{N})$ with $N$ a (small) number of terms from the series expansion, and $M$, the number of early-exercise/monitoring dates.

Pricing Bermudan options under local L\'evy models with default

We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential L\'evy-type martingale. This class of models allows for a local volatility, local default intensity and a locally dependent L\'evy measure. We present a pricing method for Bermudan options based on an analytical approximation of the characteristic function combined with the COS method. Due to a special form of the obtained characteristic function the price can be computed using a Fast Fourier Transform-based algorithm resulting in a fast and accurate calculation. The Greeks can be computed at almost no additional computational cost. Error bounds for the approximation of the characteristic function as well as for the total option price are given

Highly efficient pricing of exotic derivatives under mean-reversion, jumps and stochastic volatility

The pricing of exotic derivatives continues to attract much attention from academics and practitioners alike. Despite the overwhelming interest, the task of finding a robust methodology that could derive closed-form solutions for exotic derivatives remains a difficult challenge. In addition, the level of sophistication is greatly enhanced when options are priced in a more realistic framework. This includes, but not limited to, utilising jump-diffusion models with mean-reversion, stochastic volatility, and/or stochastic jump intensity. More pertinently, these inclusions allow the resulting asset price process to capture the various empirical features, such as heavy tails and asymmetry, commonly observed in financial data. However, under such a framework, the density function governing the underlying asset price process is generally not available. This leads to a breakdown of the classical risk-neutral option valuation method via the discounted expectation of the final payoff. Furthermore, when an analytical expression for the option pricing formula becomes available, the solution is often complex and in semi closed-form. Hence, a substantial amount of computational time is required to obtain the value of the option, which may not satisfy the efficiency demanded in practice. Such drawbacks may be remedied by utilising numerical integration techniques to price options more efficiently in the Fourier domain instead, since the associated characteristic functions are more readily available. This thesis is concerned primarily with the efficient and accurate pricing of exotic derivatives under the aforementioned framework. We address the research opportunity by exploring the valuation of exotic options with numerical integration techniques once the associated characteristic functions are developed. In particular, we advocate the use of the novel Fourier-cosine (COS) expansions, and the more recent Shannon wavelet inverse Fourier technique (SWIFT). Once the option prices are obtained, the efficiency of the two techniques are benchmarked against the widely-acclaimed fast Fourier transform (FFT) method. More importantly, we perform extensive numerical experiments and error analyses to show that, under our proposed framework, not only is the COS and SWIFT methods more efficient, but are also highly accurate with exponential rate of error convergence. Finally, we conduct a set of sensitivity analyses to evaluate the models’ consistency and robustness under different market condition

Efficient pricing of discrete arithmetic Asian options under mean reversion and jumps based on Fourier-cosine expansions

We propose an efficient pricing method for arithmetic Asian options based on Fourier-cosine expansions. In particular, we allow for mean reversion and jumps in the underlying price dynamics. There is an extensive body of empirical evidence in the current literature that points to the existence and prominence of such anomalies in the prices of certain asset classes, such as commodities. Our efficient pricing method is derived for the discretely monitored versions of the European-style arithmetic Asian options. The analytical solutions obtained from our Fourier-cosine expansions are compared to the benchmark fast Fourier transform based pricing for the examination of its accuracy and computational efficienc