A highly efficient pricing method for European-style options based on Shannon wavelets

In the search for robust, accurate and highly efficient financial option valuation techniques, we present here the SWIFT method (Shannon Wavelets Inverse Fourier Technique), based on Shannon wavelets. SWIFT comes with control over approximation errors made by means of sharp quantitative error bounds. The nature of the local Shannon wavelets basis enables us to adaptively determine the proper size of the computational interval. Numerical experiments on European-style options confirm the bounds, robustness and efficiency

SWIFT valuation of discretely monitored arithmetic Asian options

In this work, we propose an efficient and robust valuation of discretely monitored arithmetic Asian options based on Shannon wavelets. We employ the so-called SWIFT method, a Fourier inversion numerical technique with several important advantages with respect to the existing related methods. Particularly interesting is that SWIFT provides mechanisms to determine all the free-parameters in the method, based on a prescribed precision in the density approximation. The method is applied to two general classes of dynamics: exponential Lévy models and square-root diffusions. Through the numerical experiments, we show that SWIFT outperforms state-of-the-art methods in terms of accuracy and robustness, and shows an impressive speed in execution time

Two-dimensional Shannon wavelet inverse Fourier technique for pricing European options

The SWIFT method for pricing European-style options on one underlying asset was recently published and presented as an accurate, robust and highly efficient technique. The purpose of this paper is to extend the method to higher dimensions by pricing exotic option contracts, called rainbow options, whose payoff depends on multiple assets. The multidimensional extension inherits the properties of the one-dimensional method, being the exponential convergence one of them. Thanks to the nature of local Shannon wavelets basis, we do not need to rely on a-priori truncation of the integration range, we have an error bound estimate and we use fast Fourier transform (FFT) algorithms to speed up computations. We test the method for similar examples with state-of-the-art methods found in the literature, and we compare our results with analytical expressions when available

The CTMC-Heston model: calibration and exotic option pricing with SWIFT

This work presents an efficient computational framework for pricing a general class of exotic and vanilla options under a versatile stochastic volatility model. In particular, we propose the use of a finite state continuous time Markov chain (CTMC) model, which closely approximates the classic Heston model but enables a simplified approach for consistently pricing a wide variety of financial derivatives (...

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

Pricing early-exercise and discrete barrier options by Shannon wavelet expansions

We present a pricing method based on Shannon wavelet expansions for early-exercise and discretely-monitored barrier options under exponential Lévy asset dynamics. Shannon wavelets are smooth, and thus approximate the densities that occur in finance well, resulting in exponential convergence. Application of the Fast Fourier Transform yields an efficient implementation and since wavelets give local approximations, the domain boundary errors can be naturally resolved, which is the main improvement over existing methods

Pricing early-exercise and discrete barrier options by Shannon wavelet expansions

We present a pricing method based on Shannon wavelet expansions for early-exercise and discretely-monitored barrier options under exponential Lévy asset dynamics. Shannon wavelets are smooth, and thus approximate the densities that occur in finance well, resulting in exponential convergence. Application of the Fast Fourier Transform yields an efficient implementation and since wavelets give local approximations, the domain boundary errors can be naturally resolved, which is the main improvement over existing methods

SWIFT Calibration of the Heston model

In the present work, the SWIFT method for pricing European options is extended to Heston model calibration. The computation of the option price gradient is simplified thanks to the knowledge of the characteristic function in closed form. The proposed calibration machinery appears to be extremely fast, in particular for a single expiry and multiple strikes, outperforming the state-of-the-art method we compare it with. Further, the a priori knowledge of SWIFT parameters makes a reliable and practical implementation of the presented calibration method possible. A wide range of stress, speed and convergence numerical experiments is carried out, with deep in-the-money, at-the-money and deep out-of-the-money options for very short and very long maturitie

A Shannon wavelet method for pricing foreign exchange options under the Heston multi-factor CIR model

We present a robust and highly efficient Shannon wavelet pricing method for plain-vanilla foreign exchange European options under the jump-extended Heston model with multi-factor CIR interest rate dynamics. Under a Monte Carlo and partial differential equation hybrid computational framework, the option price can be expressed as an expectation, conditional on the variance factor, of a convolution product that involves the densities of the time-integrated domestic and foreign multi-factor CIR interest rate processes. We propose an efficient treatment to this convolution product that effectively results in a significant dimension reduction, from two multi-factor interest rate processes to only a single-factor process. By means of a state-of-the-art Shannon wavelet inverse Fourier technique, the resulting convolution product is approximated analytically and the conditional expectation can be computed very efficiently. We develop sharp approximation error bounds for the option price and hedging parameters. Numerical experiments confirm the robustness and efficiency of the method