Options Pricing and Hedging in a Regime-Switching Volatility Model

Both deterministic and stochastic volatility models have been used to price and hedge options. Observation of real market data suggests that volatility, while stochastic, is well modelled as alternating between two states. Under this two-state regime-switching framework, we derive coupled pricing partial differential equations (PDEs) with the inclusion of a state-dependent market price of volatility risk (MPVR) term. Since there is no closed-form solution for this pricing problem, we apply and compare two approaches to solving the coupled PDEs, assuming constant Poisson intensities. First we solve the problem using numerical solution techniques, through the application of the Crank-Nicolson numerical scheme. We also obtain approximate solutions in terms of known Black-Scholes formulae by reformulating our problem and applying the Cauchy-Kowalevski PDE theorem. Both our pricing equations and our approximate solutions give way to the analysis of the impact of our state-dependent MPVR on theoretical option prices. Using financially intuitive constraints on our option prices and Deltas, we prove the necessity of a negative MPVR. An exploration of the regime-switching option prices and their implied volatilities is given, as well as numerical results and intuition supporting our mathematical proofs. Given our regime-switching framework, there are several different hedging strategies to investigate. We consider using an option to hedge against a potential regime shift. Some practical problems arise with this approach, which lead us to set up portfolios containing a basket of two hedging options. To be more precise, we consider the effects of an option going too far in- and out-of-the-money on our hedging strategy, and introduce limits on the magnitude of such hedging option positions. A complementary approach, where constant volatility is assumed and investor\u27s risk preferences are taken into account, is also analysed. Analysis of empirical data supports the hypothesis that volatility levels are affected by upcoming financial events. Finally, we present an extension of our regime-switching framework with deterministic Poisson intensities. In particular, we investigate the impact of time and stock varying Poisson intensities on option prices and their corresponding implied volatilities, using numerical solution techniques. A discussion of some event-driven hedging strategies is given

An operator-customized wavelet-finite element approach for the adaptive solution of second-order partial differential equations on unstructured meshes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Civil and Environmental Engineering, 2005.Includes bibliographical references (p. 139-142).Unlike first-generation wavelets, second-generation wavelets can be constructed on any multi-dimensional unstructured mesh. Instead of limiting ourselves to the choice of primitive wavelets, effectively HB detail functions, we can tailor the wavelets to gain additional qualities. In particular, we propose to customize our wavelets to the problem's operator. For any given linear elliptic second-order PDE, and within a Lagrangian FE space of any given order, we can construct a basis of compactly supported wavelets that are orthogonal to the coarser basis functions with respect to the weak form of the PDE. We expose the connection between the wavelet's vanishing moment properties and the requirements for operator-orthogonality in multiple dimensions. We give examples in which we successfully eliminate all scale-coupling in the problem's multi-resolution stiffness matrix. Consequently, details can be added locally to a coarser solution without having to re-compute the coarser solution.The Finite Element Method (FEM) is a widely popular method for the numerical solution of Partial Differential Equations (PDE), on multi-dimensional unstructured meshes. Lagrangian finite elements, which preserve C⁰ continuity with interpolating piecewise-polynomial shape functions, are a common choice for second-order PDEs. Conventional single-scale methods often have difficulty in efficiently capturing fine-scale behavior (e.g. singularities or transients), without resorting to a prohibitively large number of variables. This can be done more effectively with a multi-scale method, such as the Hierarchical Basis (HB) method. However, the HB FEM generally yields a multi-resolution stiffness matrix that is coupled across scales. We propose a powerful generalization of the Hierarchical Basis: a second-generation wavelet basis, spanning a Lagrangian finite element space of any given polynomial order.by Stefan F. D'Heedene.Ph.D

Automatic generation of high-throughput systolic tree-based solvers for modern FPGAs

Tree-based models are a class of numerical methods widely used in financial option pricing, which have a computational complexity that is quadratic with respect to the solution accuracy. Previous research has employed reconfigurable computing with small degrees of parallelism to provide faster hardware solutions compared with general-purpose processing software designs. However, due to the nature of their vector hardware architectures, they cannot scale their compute resources efficiently, leaving them with pricing latency figures which are quadratic with respect to the problem size, and hence to the solution accuracy. Also, their solutions are not productive as they require hardware engineering effort, and can only solve one type of tree problems, known as the standard American option. This thesis presents a novel methodology in the form of a high-level design framework which can capture any common tree-based problem, and automatically generates high-throughput field-programmable gate array (FPGA) solvers based on proposed scalable hardware architectures. The thesis has made three main contributions. First, systolic architectures were proposed for solving binomial and trinomial trees, which due to their custom systolic data-movement mechanisms, can scale their compute resources efficiently to provide linear latency scaling for medium-size trees and improved quadratic latency scaling for large trees. Using the proposed systolic architectures, throughput speed-ups of up to 5.6X and 12X were achieved for modern FPGAs, compared to previous vector designs, for medium and large trees, respectively. Second, a productive high-level design framework was proposed, that can capture any common binomial and trinomial tree problem, and a methodology was suggested to generate high-throughput systolic solvers with custom data precision, where the methodology requires no hardware design effort from the end user. Third, a fully-automated tool-chain methodology was proposed that, compared to previous tree-based solvers, improves user productivity by removing the manual engineering effort of applying the design framework to option pricing problems. Using the productive design framework, high-throughput systolic FPGA solvers have been automatically generated from simple end-user C descriptions for several tree problems, such as American, Bermudan, and barrier options.Open Acces

Interest rate models with Markov chains

Imperial Users onl

Numerical Methods for Real Options in Telecommunications

This thesis applies modern financial option valuation methods to the problem of telecommunication network capacity investment decision timing. In particular, given a cluster of base stations (wireless network with a certain traffic capacity per base station), the objective of this thesis is to determine when it is optimal to increase capacity to each of the base stations of the cluster. Based on several time series taken from the wireless and bandwidth industry, it is argued that capacity usage is the major uncertain component in telecommunications. It is found that price has low volatility when compared to capacity usage. A real options approach is then applied to derive a two dimensional partial integro-differential equation (PIDE) to value investments in telecommunication infrastructure when capacity usage is uncertain and has temporary sudden large variations. This real options PIDE presents several numerical challenges. First, the integral term must be solved accurately and quickly enough such that the general PIDE solution is reasonably accurate. To deal with the integral term, an implicit method is suggested. Proofs of timestepping stability and convergence of a fixed point iteration scheme are presented. The correlation integral is computed using a fast Fourier transform (FFT) method. Techniques are developed to avoid wrap-around effects. This method is tested on option pricing problems where the underlying asset follows a jump diffusion process. Second, the absence of diffusion in one direction of the two dimensional PIDE creates numerical challenges regarding accuracy and timestep selection. A semi-Lagrangian method is presented to alleviate these issues. At each timestep, a set of one dimensional PIDEs is solved and the solution of each PIDE is updated using semi-Lagrangian timestepping. Crank-Nicolson and second order backward differencing timestepping schemes are studied. Monotonicity and stability results are derived. This method is tested on continuously observed Asian options. Finally, a five factor algorithm that captures many of the constraints of the wireless network capacity investment decision timing problem is developed. The upgrade decision for different upgrade decision intervals (e. g. monthly, quarterly, etc. ) is studied, and the effect of a safety level (i. e. the maximum allowed capacity used in practice on a daily basis—which differs from the theoretical maximum) is investigated

A consistent framework for valuation under collateralization, credit risk and funding costs

We develop a consistent, arbitrage-free framework for valuing derivative trades with collateral, counterparty credit risk, and funding costs. This is achieved by modifying the payout cash-flows for the trade position. The framework is flexible enough to accommodate actual trading complexities such as asymmetric collateral and funding rates, replacement close-out, and rehypothecation of posted collateral. We show also how the traditional self-financing condition is adjusted to reflect the new market realities. The generalized valuation equation takes the form of a forward-backward SDE or semi-linear PDE. Nevertheless, it may be recast as a set of iterative equations which can be efficiently solved by our proposed least-squares Monte Carlo algorithm. We numerically implement the case of an equity option and show how its valuation changes when including the above effects. We also discuss the financial impact of the proposed valuation framework and of nonlinearity more generally. This is fourfold: Firstly, the valuation equation is only based on observable market rates, leaving the value of a derivatives transaction invariant to any theoretical risk-free rate. Secondly, the presence of funding costs and default close-out makes the valuation problem a recursive and nonlinear one. Thus, credit and funding risks are non-separable in general, and despite common practice in banks, the related CVA, DVA, and FVA cannot be treated as purely additive adjustments without running the risk of double counting. To quantify the valuation error that can be attributed to double counting, we introduce a nonlinearity valuation adjustment (NVA) and show that its magnitude can be significant under asymmetric funding rates and replacement close-out at default. Thirdly, as trading parties cannot observe each others liquidity policies nor their respective funding costs, the bilateral nature of a derivative price breaks down. Finally, valuation becomes aggregation-dependent and portfolio values cannot simply be added up. This has operational consequences for banks, calling for a holistic, consistent approach across trading desks and asset classes.Open Acces

The convolution method for pricing american options under Lévy processes

Tese de mestrado em Matemática Financeira, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2014Um método flexível, rápido e exacto para avaliação de opções, desde as mais simples às mais complexas com provisões de exercício antecipado, é apresentado. Este método baseia-se na Fast Fourier Transform (FFT) e depende, naturalmente, das transformadas de Fourier. A ideia principal baseia-se em reconhecer que a fórmula usual de avaliação neutra ao risco pode ser calculada como uma convolução. Esta característica, é extremamente útil, dado que convoluções no domínio do tempo podem ser transformadas facilmente em multiplicações no domínio de Fourier, o que permite aplicar a FFT e beneficiar da sua capacidade computacional. Este recente método de avaliação, proposto por Lord et al. (2008), foi apelidado de método da convolução, e é aplicável a uma grande variedade de payoffs necessitando apenas do conhecimento da função característica do modelo. Desta forma, o método é aplicável a vários modelos afins, entre os quais está a classe de modelos exponenciais de Levy. O método apresentado é capaz de estender os métodos anteriores, baseados na FFT para o cálculo de opções Europeias, ao conseguir avaliar opções com provisões de exercício antecipado. Considerando-se uma opção Bermuda M vezes exercível, a complexidade global do método é O(MN log(N)), em que N é número de pontos da grelha utilizados na discretização do preço do activo subjacente. No contexto das opções Americanas, que são os contratos de opções em bolsa mais transaccionados, uma técnica eficiente, baseada na aplicação da extrapolação de Richardson aos preços de opções Bermudas, é apresentada.A flexible, fast and accurate method for pricing options, from plain vanilla to the more complex ones with early-exercise features, is presented. The method is based on the Fast Fourier Transform (FFT) which relies, naturally, on Fourier transformations. The key idea is to recognize that the usual risk-neutral valuation formula can be calculated as a convolution. This feature, is highly useful, since convolutions in the time domain can be translated easily to the Fourier domain, enabling one to apply the FFT and benefit from its computational power. This recent pricing method, proposed by Lord et al. (2008), was dubbed the convolution method, and is applicable to a wide variety of payoffs requiring only the knowledge of the characteristic function of the model. As such, the method is applicable within many regular affine models, among which is the class of exponential Levy models. The presented method is able to extend previous methods, based on the calculation of the FFT for pricing European options, by pricing options with early-exercise features. Considering an M-times exercisable Bermudan option, the overall complexity of the method is O(MN log(N)), with N grid points used to discretize the price of the underlying asset. In the context of American options, which are the most exchange traded option contracts, a highly efficient technique, based on the application of the Richardson extrapolation to the prices of Bermudan options, is presented

Numerical schemes and Monte Carlo techniques for Greeks in stochastic volatility models

The main objective of this thesis is to propose approximations to option sensitivities in stochastic volatility models. The first part explores sequential Monte Carlo techniques for approximating the latent state in a Hidden Markov Model. These techniques are applied to the computation of Greeks by adapting the likelihood ratio method. Convergence of the Greek estimates is proved and tracking of option prices is performed in a stochastic volatility model. The second part defines a class of approximate Greek weights and provides high-order approximations and justification for extrapolation techniques. Under certain regularity assumptions on the value function of the problem, Greek approximations are proved for a fully implementable Monte Carlo framework, using weak Taylor discretisation schemes. The variance and bias are studied for the Delta and Gamma, when using such discrete-time approximations. The final part of the thesis introduces a modified explicit Euler scheme for stochastic differential equations with non-Lipschitz continuous drift or diffusion; a strong rate of convergence is proved. The literature on discretisation techniques for stochastic differential equations has been motivational for the development of techniques preserving the explicitness of the algorithm. Stochastic differential equations in the mathematical finance literature, including the Cox-Ingersoll-Ross, the 3/2 and the Ait-Sahalia models can be discretised, with a strong rate of convergence proved, which is a requirement for multilevel Monte Carlo techniques.Open Acces