Download or read book Analysis of Markov Chain Approximation for Option Pricing and Hedging written by Lingfei Li and published by . This book was released on 2017 with total page 38 pages. Available in PDF, EPUB and Kindle. Book excerpt: Continuous time Markov chain (CTMC) approximation is an intuitive and powerful method for pricing options in general Markovian models. This paper analyzes how grid design affects the convergence behavior of barrier and European options in general diffusion models. Using the spectral method, we obtain sharp estimates for the convergence rate of option price for non-uniform grids. We propose to calculate an option's delta and gamma by taking central difference of option prices on the grid. For this simple method, we prove that, surprisingly, delta and gamma converge at the same rate as option price does. Our analysis allows us to develop principles that are sufficient and necessary for designing nonuniform grids that can achieve second order convergence for option price, delta and gamma. Based on these principles, we propose a novel class of non-uniform grids, which ensures that convergence is not only second order, but also smooth. This further allows extrapolation to be applied to achieve even higher convergence rate. Our grids enable the CTMC approximation method to price and hedge a large number of options with different strikes fast and accurately. Applicability of our results to jump models is discussed through numerical examples.
Download or read book Error Analysis of Finite Difference and Markov Chain Approximations for Option Pricing written by Lingfei Li and published by . This book was released on 2017 with total page 39 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mijatovic and Pistorius (Math. Finance, 2013) proposed an efficient Markov chain approximation method for pricing European and barrier options in general one-dimensional Markovian models. However, sharp convergence rates of this method for realistic financial payoffs, which are non-smooth, are rarely available. In this paper, we solve this problem for general one-dimensional diffusion models, which play a fundamental role in financial applications. For such models, the Markov chain approximation method is equivalent to the method of lines using the central difference. Our analysis is based on the spectral representation of the exact solution and the approximate solution. By establishing the convergence rate for the eigenvalues and the eigenfunctions, we obtain sharp convergence rates for the transition density and the price of options with non-smooth payoffs. In particular, we show that for call-/put-type payoffs, convergence is second order, while for digital-type payoffs, convergence is generally only first order. Furthermore, we provide theoretical justification for two well-known smoothing techniques that can restore second-order convergence for digital-type payoffs and explain oscillations observed in the convergence for options with non-smooth payoffs. As an extension, we also establish sharp convergence rates for European options for a rich class of Markovian jump models constructed from diffusions via subordination. The theoretical estimates are confirmed using numerical examples.
Download or read book Option Pricing with Transaction Costs Using a Markov Chain Approximation written by Michael Monoyios and published by . This book was released on 2001 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book A General Continuous Time Markov Chain Approximation for Multi Asset Option Pricing With Systems of Correlated Diffusions written by Justin Kirkby and published by . This book was released on 2020 with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt: Continuous time Markov Chain (CTMC) approximation techniques have received increasing attention in the option pricing literature, due to their ability to solve complex pricing problems, although existing approaches are mostly limited to one or two dimensions. This paper develops a general methodology for modeling and pricing financial derivatives which depend on systems of stochastic diffusion processes. This is accomplished with a general de-correlation procedure, which reduces the system of correlated diffusions to an uncorrelated system. This enables simple and efficient approximation of the driving processes by uni-variate CTMC approximations. Weak convergence of the approximation is demonstrated, with second order convergence in space. Numerical experiments demonstrate the accuracy and efficiency of the method for various European and early-exercise options in two and three dimensions.
Download or read book Derivatives Pricing and Model Calibration Using Continuous Time Markov Chain Approximation Model written by Chia Lo and published by . This book was released on 2014 with total page 43 pages. Available in PDF, EPUB and Kindle. Book excerpt: We propose a non-equidistant Q rate matrix setting formula such that a well-defined continuous time Markov chain can lead to excellent approximations to jump-diffusions with affine or non-affine functional specifications. This approach also accommodates state-dependent jump intensity and jump distribution, a fexibility that is very hard to achieve with traditional numerical methods. Our approach not only satisfies Kushner (1990) local consistency conditions but also resolves the approximation errors induced by Piccioni (1987) scheme. European stock option pricing examples based on jump-diffusions illustrate the ease of implementation of our model. The proposed algorithm for pricing American options highlights the speed and accuracy. Finally the empirical analysis using daily VIX data shows that the maximum likelihood estimates of the underlying jump-diffusions can be efficiently computed by the model proposed in this article.
Download or read book American Option Pricing Under GARCH by a Markov Chain Approximation written by Duan, Jin-Chuan and published by Montréal : École des hautes études commerciales, Groupe de recherche en finance. This book was released on 1997 with total page 52 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Continuous Time Markov Chain and Regime Switching Approximations with Applications to Options Pricing written by Zhenyu Cui and published by . This book was released on 2019 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this chapter, we present recent developments in using the tools of continuous-time Markov chains for the valuation of European and path-dependent financial derivatives. We also survey results on a newly proposed regime switching approximation to stochastic volatility, and stochastic local volatility models. The presented framework is part of an exciting recent stream of literature on numerical option pricing, and offers a new perspective that combines the theory of diffusion processes, Markov chains, and Fourier techniques. It is also elegantly connected to partial differential equation (PDE) approaches.
Download or read book Modeling Stochastic Control Optimization and Applications written by George Yin and published by Springer. This book was released on 2019-07-16 with total page 599 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume collects papers, based on invited talks given at the IMA workshop in Modeling, Stochastic Control, Optimization, and Related Applications, held at the Institute for Mathematics and Its Applications, University of Minnesota, during May and June, 2018. There were four week-long workshops during the conference. They are (1) stochastic control, computation methods, and applications, (2) queueing theory and networked systems, (3) ecological and biological applications, and (4) finance and economics applications. For broader impacts, researchers from different fields covering both theoretically oriented and application intensive areas were invited to participate in the conference. It brought together researchers from multi-disciplinary communities in applied mathematics, applied probability, engineering, biology, ecology, and networked science, to review, and substantially update most recent progress. As an archive, this volume presents some of the highlights of the workshops, and collect papers covering a broad range of topics.
Download or read book Option Pricing and Estimation of Financial Models with R written by Stefano M. Iacus and published by John Wiley & Sons. This book was released on 2011-02-23 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents inference and simulation of stochastic process in the field of model calibration for financial times series modelled by continuous time processes and numerical option pricing. Introduces the bases of probability theory and goes on to explain how to model financial times series with continuous models, how to calibrate them from discrete data and further covers option pricing with one or more underlying assets based on these models. Analysis and implementation of models goes beyond the standard Black and Scholes framework and includes Markov switching models, Lévy models and other models with jumps (e.g. the telegraph process); Topics other than option pricing include: volatility and covariation estimation, change point analysis, asymptotic expansion and classification of financial time series from a statistical viewpoint. The book features problems with solutions and examples. All the examples and R code are available as an additional R package, therefore all the examples can be reproduced.
Download or read book Weak Convergence of Financial Markets written by Jean-Luc Prigent and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive overview of weak convergence of stochastic processes and its application to the study of financial markets. Split into three parts, the first recalls the mathematics of stochastic processes and stochastic calculus with special emphasis on contiguity properties and weak convergence of stochastic integrals. The second part is devoted to the analysis of financial theory from the convergence point of view. The main problems, which include portfolio optimization, option pricing and hedging are examined, especially when considering discrete-time approximations of continuous-time dynamics. The third part deals with lattice- and tree-based computational procedures for option pricing both on stocks and stochastic bonds. More general discrete approximations are also introduced and detailed. Includes detailed examples.
Download or read book American Type Options written by Dmitrii S. Silvestrov and published by Walter de Gruyter GmbH & Co KG. This book was released on 2015-03-03 with total page 672 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book gives a systematical presentation of stochastic approximation methods for discrete time Markov price processes. Advanced methods combining backward recurrence algorithms for computing of option rewards and general results on convergence of stochastic space skeleton and tree approximations for option rewards are applied to a variety of models of multivariate modulated Markov price processes. The principal novelty of presented results is based on consideration of multivariate modulated Markov price processes and general pay-off functions, which can depend not only on price but also an additional stochastic modulating index component, and use of minimal conditions of smoothness for transition probabilities and pay-off functions, compactness conditions for log-price processes and rate of growth conditions for pay-off functions. The volume presents results on structural studies of optimal stopping domains, Monte Carlo based approximation reward algorithms, and convergence of American-type options for autoregressive and continuous time models, as well as results of the corresponding experimental studies.
Download or read book Convergence Rate Analysis for the Continuous Time Markov Chain Approximation of Occupation Time Derivatives and Asian Option Greeks written by Jingtang Ma and published by . This book was released on 2019 with total page 43 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper establishes the second-order convergence rates of the continuous-time Markov chain (CTMC) approximation method for pricing continuously monitored occupation time derivatives (step options, conditional Asian options) and arithmetic Asian options and their Greeks. We fill the gap in the current literature on the analysis of CTMC approximation errors for pricing Asian options by not only rigorously proving the exact second order convergence rate but also developing corresponding error and convergence analysis for the Greeks through the novel use of pathwise method and Malliavin calculus techniques. We further extend the scope of the analysis of the CTMC approximation method to the case of general occupation time derivatives (e.g. step options) and the recently introduced conditional Asian options, and then propose a novel CTMC scheme for their valuation. We carry out a detailed error and convergence analysis of the algorithms and numerical experiments substantiate the theoretical findings.
Download or read book Numerical Methods in Finance written by René Carmona and published by Springer Science & Business Media. This book was released on 2012-03-23 with total page 478 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical methods in finance have emerged as a vital field at the crossroads of probability theory, finance and numerical analysis. Based on presentations given at the workshop Numerical Methods in Finance held at the INRIA Bordeaux (France) on June 1-2, 2010, this book provides an overview of the major new advances in the numerical treatment of instruments with American exercises. Naturally it covers the most recent research on the mathematical theory and the practical applications of optimal stopping problems as they relate to financial applications. By extension, it also provides an original treatment of Monte Carlo methods for the recursive computation of conditional expectations and solutions of BSDEs and generalized multiple optimal stopping problems and their applications to the valuation of energy derivatives and assets. The articles were carefully written in a pedagogical style and a reasonably self-contained manner. The book is geared toward quantitative analysts, probabilists, and applied mathematicians interested in financial applications.
Download or read book Principles Of Infinitesinal Stochastic Financial Analysis written by Imme Van Den Berg and published by World Scientific. This book was released on 2000-07-27 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: There has been a tremendous growth in the volume of financial transactions based on mathematics, reflecting the confidence in the Nobel-Prize-winning Black-Scholes option theory. Risks emanating from obligatory future payments are covered by a strategy of trading with amounts not determined by guessing, but by solving equations, and with prices not resulting from offer and demand, but from computation. However, the mathematical theory behind that suffers from inaccessibility. This is due to the complexity of the mathematical foundation of the Black-Scholes model, which is the theory of continuous-time stochastic processes: a thorough study of mathematical finance is considered to be possible only at postgraduate level.The setting of this book is the discrete-time version of the Black-Scholes model, namely the Cox-Ross-Rubinstein model. The book gives a complete description of its background, which is now only the theory of finite stochastic processes. The novelty lies in the fact that orders of magnitude — in the sense of nonstandard analysis — are imposed on the parameters of the model. This not only makes the model more economically sound (such as rapid fluctuations of the market being represented by infinitesimal trading periods), but also leads to a significant simplification: the fundamental results of Black-Scholes theory are derived in full generality and with mathematical rigour, now at graduate level. The material has been repeatedly taught in a third-year course to econometricians.
Download or read book Option Pricing and Hedging for Discrete Time Autoregressive Hidden Markov Model written by Massimo Caccia and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Markov functional and Stochastic Volatility Modelling written by Duy Pham and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we study two practical problems in applied mathematical fi nance. The first topic discusses the issue of pricing and hedging Bermudan swaptions within a one factor Markov-functional model. We focus on the implications for hedging of the choice of instantaneous volatility for the one-dimensional driving Markov process of the model. We find that there is a strong evidence in favour of what we term \parametrization by time" as opposed to \parametrization by expiry". We further propose a new parametrization by time for the driving process which takes as inputs into the model the market correlations of relevant swap rates. We show that the new driving process enables a very effective vega-delta hedge with a much more stable gamma profile for the hedging portfolio compared with the existing ones. The second part of the thesis mainly addresses the topic of pricing European options within the popular stochastic volatility SABR model and its extension with mean reversion. We investigate some effcient approximations for these models to be used in real time. We first derive a probabilistic approximation for three different versions of the SABR model: Normal, Log-Normal and a displaced diffusion version for the general constant elasticity of variance case. Specifically, we focus on capturing the terminal distribution of the underlying process (conditional on the terminal volatility) to arrive at the implied volatilities of the corresponding European options for all strikes and maturities. Our resulting method allows us to work with a variety of parameters which cover long dated options and highly stress market condition. This is a different feature from other current approaches which rely on the assumption of very small total volatility and usually fail for longer than 10 years maturity or large volatility of volatility. A similar study is done for the extension of the SABR model with mean reversion (SABR-MR). We first compare the SABR model with this extended model in terms of forward volatility to point out the fundamental difference in the dynamics of the two models. This is done through a numerical example of pricing forward start options. We then derive an effcient probabilistic approximation for the SABRMR model to price European options in a similar fashion to the one for the SABR model. The numerical results are shown to be still satisfactory for a wide range of market conditions.
Download or read book Option Pricing with Continuous time Markov Chain Regime Switching written by Craig Steven Edwards and published by . This book was released on 2004 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt:
