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

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

Highly Efficient Shannon Wavelet-based Pricing of Power Options under the Double Exponential Jump Framework with Stochastic Jump Intensity and Volatility

We propose a highly efficient and accurate valuation method for exotic-style options based on the novel Shannon wavelet inverse Fourier technique (SWIFT). Specifically, we derive an efficient pricing methods for power options under a more realistic double exponential jump model with stochastic volatility and jump intensity. Inclusion of such innovations may accommodate for the various stylised facts observed in the prices of financial assets, and admits a more realistic pricing framework as a result. Following the derivation of our SWIFT pricing method for power options, we perform extensive numerical experiments to analyse both the method's accuracy and efficiency. In addition, we investigate the sensitivities in the resulting prices, as well as the inherent errors, to changes in the underlying market conditions. Our numerical results demonstrate that the SWIFT method is not only more efficient when benchmarked to its close competitors, such as the Fourier- cosine (COS) and the widely-acclaimed fast-Fourier transform (FFT) methods, but it is also robust across a range of different market conditions exhibiting exponential error convergence

Extension of the SWIFT option pricing scheme for european options calibration under Heston stochastic volatility model

A Heston model calibration technique is presented for European options under the Heston model. The novel Shannon Wavelets Inverse Fourier Technique (SWIFT) is extended for European option price calibration (previously it was used only for pricing European, Asian, barrier, and Bermudan options). This method has different expressions and speed-up techniques, adequate to different set-ups. These are discussed and new expressions and properties are presented for the gradient computation and option calibration. The Heston characteristic function expression recently proposed by \cite{cui17} is used in the SWIFT implementation due to its analytic gradient and its continuity properties. The time performance, robustness, and convergence under set-ups representative of real markets is studied for different implementations of the SWIFT technique and compared with the option calibration scheme presented by \cite{cui17} The SWIFT implementations are coded in C++ and uploaded to a public GitHub repository. The libray implements several of the different SWIFT expressions for GBM and Heston European options

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

Polynomial models in finance

This thesis presents new flexible dynamic stochastic models for the evolution of market prices and new methods for the valuation of derivatives. These models and methods build on the recently characterized class of polynomial jump-diffusion processes for which the conditional moments are analytic. The first half of this thesis is concerned with modelling the fluctuations in the volatility of stock prices, and with the valuation of options on the stock. A new stochastic volatility model for which the squared volatility follows a Jacobi process is presented in the first chapter. The stock price volatility is allowed to continuously fluctuate between a lower and an upper bound, and option prices have closed-form series representations when their payoff functions depend on the stock price at finitely many dates. Truncating these series at some finite order entails accurate option price approximations. This method builds on the series expansion of the ratio between the log price density and an auxiliary density, with respect to an orthonormal basis of polynomials in a weighted Lebesgue space. When the payoff functions can be similarly expanded, the method is particularly efficient computationally. In the second chapter, more flexible choices of weighted spaces are studied in order to obtain new series representations for option prices with faster convergence rates. The option price approximation method can then be applied to various stochastic volatility models. The second half of this thesis is concerned with modelling the default times of firms, and with the pricing of credit risk securities. A new class of credit risk models in which the firm default probability is linear in the factors is presented in the third chapter. The prices of defaultable bonds and credit default swaps have explicit linear-rational expressions in the factors. A polynomial model with compact support and bounded default intensities is developed. This property is exploited to approximate credit derivatives prices by interpolating their payoff functions with polynomials. In the fourth chapter, the joint term structure of default probabilities is flexibly modelled using factor copulas. A generic static framework is developed in which the prices of high dimensional and complex credit securities can be efficiently and exactly computed. Dynamic credit risk models with significant default dependence can in turn be constructed by combining polynomial factor copulas and linear credit risk models

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

Codes of Finance. Engineering Derivatives in a Global Bank

Codes of Finance is an ethnography of a global bank inventing new derivative products. It describes the multiple languages invented to describe and control these new products. It analyzes the recent discussions about financial derivatives and offers a new framework to understand financial innovation