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Option Pricing Under Regime Switching Analytical Pde And Fft Methods
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Book Synopsis Option Pricing Under Regime Switching (analytical, PDE, and FFT Methods) by : Mohammad Yousef Akhavein Sohrabi
Download or read book Option Pricing Under Regime Switching (analytical, PDE, and FFT Methods) written by Mohammad Yousef Akhavein Sohrabi and published by . This book was released on 2011 with total page 83 pages. Available in PDF, EPUB and Kindle. Book excerpt: Although globally used in option pricing, the Black-Scholes model has not been able to reflect the evolution of stocks in the real world. A regime-switching model which allows jumps in the underlying asset prices and the parameters of the corresponding stochastic process is more accurate. We evaluate the analytical solution for pricing of European options under a two-state regime switching model. Both the convergence of the analytical solution and the feature of implied volatility are investigated through numerical examples.
Book Synopsis Modular Pricing of Options by : Jianwei Zhu
Download or read book Modular Pricing of Options written by Jianwei Zhu and published by Springer. This book was released on 2014-03-12 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: From a technical point of view, the celebrated Black and Scholes option pricing formula was originally developed using a separation of variables technique. However, already Merton mentioned in his seminal 1973 pa per, that it could have been developed by using Fourier transforms as well. Indeed, as is well known nowadays, Fourier transforms are a rather convenient solution technique for many models involving the fundamental partial differential equation of financial economics. It took the community nearly another twenty years to recognize that Fourier transform is even more useful, if one applies it to problems in financial economics without seeking an explicit analytical inverse trans form. Heston (1993) probably was the first to demonstrate how to solve a stochastic volatility option pricing model quasi analytically using the characteristic function of the problem, which is nothing else than the Fourier transform of the underlying Arrow /Debreu-prices, and doing the inverse transformation numerically. This opened the door for a whole bunch of new closed form solutions in the transformed Fourier space and still is one of the most active research areas in financial economics.
Book Synopsis Pricing American Options Under Regime Switching Using Method of Lines by : Carl Chiarella
Download or read book Pricing American Options Under Regime Switching Using Method of Lines written by Carl Chiarella and published by . This book was released on 2016 with total page 19 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper considers the American option pricing problem under regime-switching by using the method-of-lines (MOL) scheme. American option prices in each regime involve prices in all other regimes. We treat the prices from other regimes implicitly, thus guaranteeing consistency. Iterative procedures are required but very few iterative steps are needed in practice. Numerical tests demonstrate the robustness, accuracy and efficiency of the proposed numerical scheme. We compare our results with Buffington and Elliott (2002)'s analytical approximation under two regimes. Our MOL scheme provides improved results especially for out-of-the money options, possibly because they use a separation of variable approach to the PDEs which cannot hold around the early exercise region. We also compare our results with those of Khaliq and Liu (2009) and suggest that their implicit scheme can be improved.
Book Synopsis Option Valuation Under a Regime-Switching Model Using the Fast Fourier Transform by : Elham Sohrabi
Download or read book Option Valuation Under a Regime-Switching Model Using the Fast Fourier Transform written by Elham Sohrabi and published by . This book was released on 2018 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Pricing American Options in Regime-Switching Models by : Svetlana Boyarchenko
Download or read book Pricing American Options in Regime-Switching Models written by Svetlana Boyarchenko and published by . This book was released on 2008 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The pricing problem for American options in Markov-modulated Lévy models is solved. The early exercise boundaries and prices are calculated using a generalization of Carr's randomization procedure for regime-switching models. The pricing procedure is efficient even if the number of states is large provided the transition rates are not large w.r.t. the riskless rates. The payoffs and riskless rates may depend on a state. Special cases are stochastic volatility models and models with stochastic interest rate; both must be modeled as finite-state Markov chains. In contrast with the earlier version of the method, an explicit algorithm is formulated for wide classes of Lévy processes, and FFT and iFFT are used.
Book Synopsis Options Pricing and Hedging in a Regime-Switching Volatility Model by : Melissa Anne Mielkie
Download or read book Options Pricing and Hedging in a Regime-Switching Volatility Model written by Melissa Anne Mielkie and published by . This book was released on 2014 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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's risk preferences are taken into account, is also analysed. Analysis of empirical data supports the hypothesis that volatility levels are a effected 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.
Book Synopsis Option Pricing Using Fourier Space Time-stepping Framework by : Vladimir Surkov
Download or read book Option Pricing Using Fourier Space Time-stepping Framework written by Vladimir Surkov and published by . This book was released on 2009 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis develops a generic framework based on the Fourier transform for pricing and hedging of various options in equity, commodity, currency, and insurance markets. The pricing problem can be reduced to solving a partial integro-differential equation (PIDE). The Fourier Space Time-stepping (FST) framework developed in this thesis circumvents the problems associated with the existing finite difference methods by utilizing the Fourier transform to solve the PIDE. The FST framework-based methods are generic, highly efficient and rapidly convergent. The Fourier transform can be applied to the pricing PIDE to obtain a linear system of ordinary differential equations that can be solved explicitly. Solving the PIDE in Fourier space allows for the integral term to be handled efficiently and avoids the asymmetrical treatment of diffusion and integral terms, common in the finite difference schemes found in the literature. For path-independent options, prices can be obtained for a range of stock prices in one iteration of the algorithm. For exotic, path-dependent options, a time-stepping methodology is developed to handle barriers, free boundaries, and exercise policies. The thesis includes applications of the FST framework-based methods to a wide range of option pricing problems. Pricing of single- and multi-asset, European and path-dependent options under independent-increment exponential Levy stock price models, common in equity and insurance markets, can be done efficiently via the cornerstone FST method. Mean-reverting Levy spot price models, common in commodity markets, are handled by introducing a frequency transformation, which can be readily computed via scaling of the option value function. Generating stochastic volatility, to match the long-term equity options market data, and stochastic skew, observed in currency markets, is addressed by introducing a non-stationary extension of multi-dimensional Levy processes using regime-switching. Finally, codependent jumps in multi-asset models are introduced through copulas. The FST methods are computationally efficient, running in O(MNd log2 N) time with M time steps and N space points in each dimension on a d-dimensional grid. The methods achieve second-order convergence in space; for American options, a penalty method is used to attain second-order convergence in time. Furthermore, graphics processing units are utilized to further reduce the computational time of FST methods.
Book Synopsis Option Pricing and Hedging Analysis Under Regime-switching Models by : Chao Qiu
Download or read book Option Pricing and Hedging Analysis Under Regime-switching Models written by Chao Qiu and published by . This book was released on 2013 with total page 181 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis explores option pricing and hedging in a discrete time regime-switching environment. If the regime risk cannot be hedged away, then we cannot ignore this risk and use the Black-Scholes pricing and hedging framework to generate a unique pricing and hedging measure. We develop a risk neutral pricing measure by applying an Esscher Transform to the real world asset price process, with the focus on the issue of incompleteness of the market. The Esscher transform turns out to be a convenient and effective tool for option pricing under the discrete time regime switching models. We apply the pricing measure to both single variate European options and multivariate options. To better understand the effect of the pricing method, we also compared the results with those generated from two other risk neutral methods: the Black-Scholes model, and the natural equivalent martingale method. We further investigate the difference in hedging associated with different pricing measures. This is of interest when the choice of pricing method is uncertain under regime switching models. We compare four hedging strategies: delta hedging for the three risk neutral pricing methods under study, and mean variance hedging. We also develop a more general tool of tail ordering for hedging analysis in a general incomplete market with the uncertainty of the risk neutral measures. As a result of the analysis, we propose that pricing and hedging using the Esscher transform may be an effective strategy for a market where the regime switching process brings uncertainty.
Book Synopsis Option Pricing Under Regime-Switching Models by : Frédéric Godin
Download or read book Option Pricing Under Regime-Switching Models written by Frédéric Godin and published by . This book was released on 2018 with total page 46 pages. Available in PDF, EPUB and Kindle. Book excerpt: Although option pricing schemes in regime-switching frameworks were extensively explored in the literature, many models developed disregard the unobservability of regimes. In such a context, the traditional pricing approach pioneered by Hardy (2001) applied to vanilla options exhibits path-dependence even if the underlying asset price process can be embedded in a Markov process. This property is deemed counterintuitive and puzzling, warranting explanations and alternatives. The current work develops novel risk-neutral measures which remove the path-dependence issue. Pricing approaches based on dynamic programming and Monte-Carlo simulations which rely on the latter measures are illustrated.
Book Synopsis Efficient Option Pricing by Frame Duality with the Fast Fourier Transform by : Justin Kirkby
Download or read book Efficient Option Pricing by Frame Duality with the Fast Fourier Transform written by Justin Kirkby and published by . This book was released on 2016 with total page 31 pages. Available in PDF, EPUB and Kindle. Book excerpt: We develop a method for efficiently inverting analytic characteristic functions using frame projection, as in the case of Heston's model and exponential Levy models. Utilizing the duality theory of Riesz bases, we derive analytical formulas for coefficients of the orthogonally projected density, which are computed numerically with exponential convergence by the FFT. Convergence is demonstrated for geometric Asian options as well as the pricing of baskets of European options. The method is compared to state-of-the-art procedures to demonstrate its efficiency and robustness, without requiring any user-supplied "control parameters." Even greater improvement is observed for the method's extension to arithmetic Asian option pricing, as well as for Bermudan and barrier options, and credit default swaps, which will appear in follow up papers that expand on the foundations developed in this work.
Book Synopsis An Introduction to the Mathematics of Financial Derivatives by : Salih N. Neftci
Download or read book An Introduction to the Mathematics of Financial Derivatives written by Salih N. Neftci and published by Academic Press. This book was released on 2000-05-19 with total page 550 pages. Available in PDF, EPUB and Kindle. Book excerpt: A step-by-step explanation of the mathematical models used to price derivatives. For this second edition, Salih Neftci has expanded one chapter, added six new ones, and inserted chapter-concluding exercises. He does not assume that the reader has a thorough mathematical background. His explanations of financial calculus seek to be simple and perceptive.
Book Synopsis Fourier Space Time-Stepping for Option Pricing With Levy Models by : Kenneth R. Jackson
Download or read book Fourier Space Time-Stepping for Option Pricing With Levy Models written by Kenneth R. Jackson and published by . This book was released on 2009 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt: Jump-diffusion and Levy models have been widely used to partially alleviate some of the biases inherent in the classical Black-Scholes-Merton model. Unfortunately, the resulting pricing problem requires solving a more difficult partial-integro differential equation (PIDE) and although several approaches for solving the PIDE have been suggested in the literature, none are entirely satisfactory. All treat the integral and diffusive terms asymmetrically and are difficult to extend to higher dimensions. We present a new, efficient algorithm, based on transform methods, which symmetrically treats the diffusive and integrals terms, is applicable to a wide class of path-dependent options (such as Bermudan, barrier, and shout options) and options on multiple assets, and naturally extends to regime-switching Levy models.
Book Synopsis Non-Gaussian Merton-Black-Scholes Theory by : Svetlana I. Boyarchenko
Download or read book Non-Gaussian Merton-Black-Scholes Theory written by Svetlana I. Boyarchenko and published by World Scientific. This book was released on 2002 with total page 421 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces an analytically tractable and computationally effective class of non-Gaussian models for shocks (regular Lvy processes of the exponential type) and related analytical methods similar to the initial Merton-Black-Scholes approach, which the authors call the Merton-Black-Scholes theory.The authors have chosen applications interesting for financial engineers and specialists in financial economics, real options, and partial differential equations (especially pseudodifferential operators); specialists in stochastic processes will benefit from the use of the pseudodifferential operators technique in non-Gaussian situations. The authors also consider discrete time analogues of perpetual American options and the problem of the optimal choice of capital, and outline several possible directions in which the methods of the book can be developed further.Taking account of a diverse audience, the book has been written in such a way that it is simple at the beginning and more technical in further chapters, so that it is accessible to graduate students in relevant areas and mathematicians without prior knowledge of finance or economics.
Book Synopsis Multivariate Bayesian Statistics by : Daniel B. Rowe
Download or read book Multivariate Bayesian Statistics written by Daniel B. Rowe and published by CRC Press. This book was released on 2002-11-25 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: Of the two primary approaches to the classic source separation problem, only one does not impose potentially unreasonable model and likelihood constraints: the Bayesian statistical approach. Bayesian methods incorporate the available information regarding the model parameters and not only allow estimation of the sources and mixing coefficients, but
Book Synopsis The Heston Model and its Extensions in Matlab and C# by : Fabrice D. Rouah
Download or read book The Heston Model and its Extensions in Matlab and C# written by Fabrice D. Rouah and published by John Wiley & Sons. This book was released on 2013-08-01 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tap into the power of the most popular stochastic volatility model for pricing equity derivatives Since its introduction in 1993, the Heston model has become a popular model for pricing equity derivatives, and the most popular stochastic volatility model in financial engineering. This vital resource provides a thorough derivation of the original model, and includes the most important extensions and refinements that have allowed the model to produce option prices that are more accurate and volatility surfaces that better reflect market conditions. The book's material is drawn from research papers and many of the models covered and the computer codes are unavailable from other sources. The book is light on theory and instead highlights the implementation of the models. All of the models found here have been coded in Matlab and C#. This reliable resource offers an understanding of how the original model was derived from Ricatti equations, and shows how to implement implied and local volatility, Fourier methods applied to the model, numerical integration schemes, parameter estimation, simulation schemes, American options, the Heston model with time-dependent parameters, finite difference methods for the Heston PDE, the Greeks, and the double Heston model. A groundbreaking book dedicated to the exploration of the Heston model—a popular model for pricing equity derivatives Includes a companion website, which explores the Heston model and its extensions all coded in Matlab and C# Written by Fabrice Douglas Rouah a quantitative analyst who specializes in financial modeling for derivatives for pricing and risk management Engaging and informative, this is the first book to deal exclusively with the Heston Model and includes code in Matlab and C# for pricing under the model, as well as code for parameter estimation, simulation, finite difference methods, American options, and more.
Book Synopsis Chebyshev and Fourier Spectral Methods by : John P. Boyd
Download or read book Chebyshev and Fourier Spectral Methods written by John P. Boyd and published by Courier Corporation. This book was released on 2001-12-03 with total page 690 pages. Available in PDF, EPUB and Kindle. Book excerpt: Completely revised text focuses on use of spectral methods to solve boundary value, eigenvalue, and time-dependent problems, but also covers Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions, as well as cardinal functions, linear eigenvalue problems, matrix-solving methods, coordinate transformations, methods for unbounded intervals, spherical and cylindrical geometry, and much more. 7 Appendices. Glossary. Bibliography. Index. Over 160 text figures.
Book Synopsis Quantitative Finance by : Maria C. Mariani
Download or read book Quantitative Finance written by Maria C. Mariani and published by John Wiley & Sons. This book was released on 2019-11-06 with total page 496 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents a multitude of topics relevant to the quantitative finance community by combining the best of the theory with the usefulness of applications Written by accomplished teachers and researchers in the field, this book presents quantitative finance theory through applications to specific practical problems and comes with accompanying coding techniques in R and MATLAB, and some generic pseudo-algorithms to modern finance. It also offers over 300 examples and exercises that are appropriate for the beginning student as well as the practitioner in the field. The Quantitative Finance book is divided into four parts. Part One begins by providing readers with the theoretical backdrop needed from probability and stochastic processes. We also present some useful finance concepts used throughout the book. In part two of the book we present the classical Black-Scholes-Merton model in a uniquely accessible and understandable way. Implied volatility as well as local volatility surfaces are also discussed. Next, solutions to Partial Differential Equations (PDE), wavelets and Fourier transforms are presented. Several methodologies for pricing options namely, tree methods, finite difference method and Monte Carlo simulation methods are also discussed. We conclude this part with a discussion on stochastic differential equations (SDE’s). In the third part of this book, several new and advanced models from current literature such as general Lvy processes, nonlinear PDE's for stochastic volatility models in a transaction fee market, PDE's in a jump-diffusion with stochastic volatility models and factor and copulas models are discussed. In part four of the book, we conclude with a solid presentation of the typical topics in fixed income securities and derivatives. We discuss models for pricing bonds market, marketable securities, credit default swaps (CDS) and securitizations. Classroom-tested over a three-year period with the input of students and experienced practitioners Emphasizes the volatility of financial analyses and interpretations Weaves theory with application throughout the book Utilizes R and MATLAB software programs Presents pseudo-algorithms for readers who do not have access to any particular programming system Supplemented with extensive author-maintained web site that includes helpful teaching hints, data sets, software programs, and additional content Quantitative Finance is an ideal textbook for upper-undergraduate and beginning graduate students in statistics, financial engineering, quantitative finance, and mathematical finance programs. It will also appeal to practitioners in the same fields.
