Frame and Fourier methods for exotic option pricing and hedging
Kirkby, Justin L.
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Numerical option pricing has been revolutionized with the advent of fast transform methods. While lattice and Monte Carlo based approaches remain the most generally applicable, transform based approaches, when available, provide astonishing gains in accuracy and efficiency. Problems that may take a Monte Carlo routine hours or even days can be reduced to mere seconds (or fractions thereof) by a well designed transform method. This work is concerned primarily with a new option pricing framework, called the PROJ method (option pricing by frame projection), which was introduced in [Kirkby (2015)] and [Kirkby and Deng (2015)] as an efficient means of pricing large European option portfolios, as arises in the context of calibration, as well a tool for static portfolio hedging. The method has been successfully applied to pricing and hedging exotic contracts under exponential Levy processes (including Black-Scholes and jump diffusions), Regime-Switching Levy processes, and general stochastic volatility jump diffusions. The method is derived from theory of frames and Riesz bases, and applies more generally as a method of function approximation in problems for which a function's Fourier transform is known, or can can be estimated. This is the case for many random variables (and stochastic processes) which admit a closed-form characteristic function. Of particular interest are the B-splines, which offer a flexible range of Riesz bases. After an accessible introduction to the transform-based literature in Chapter 2, the third chapter of this work develops the PROJ method, which applies frame theory to price large baskets of European options and geometric Asian options, most commonly used as control variates in Monte Carlo schemes. This framework is refined and extended in Chapter 4, where the convergence order of various B-splines are studied, along with alternative methods for truncation interval selection, and a method for forward-starting option pricing. Numerical studies confirm the faster theoretical convergence of higher order bases, although low orders are often convenient for pricing path-dependent options, and contracts with short maturity or frequent monitoring. Chapter 5 considers the problem of static hedging, and presents a general framework. New approximation results are obtained for the calculation of dual coefficients and have the potential for various applications. Generalized arithmetic Asian (averaging) options are considered in Chapter 6. These contracts are highly path dependent, and the resulting method provides a substantial cost reduction over existing alternatives, often providing a several hundred fold time reduction. Chapter 7 considers the pricing of generalized barrier options using a fast Toeplitz-based convolution scheme. This includes standard single and double barrier options (which are weakly path dependent) as well as their more complex counterparts, namely Parisian options, Par-asian options, and step options. While several methods exist for the Black-Scholes model and recent work has extended to Kou's jump diffusion, the PROJ method is the first to consider Parisian options for general exponential Levy processes of an arbitrary form. Finally, Chapter 8 concludes the work with a discussion of future research objectives. Currently, the method is being extended to more exotic contract types and more general stochastic models, such as stochastic local volatility.
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