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 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 Analysis of Markov Chain Approximation for Diffusion Models with Non Smooth Coefficients written by Gongqiu Zhang and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Diffusion models with non-smooth coefficients often appear in financial applications, with examples including but not limited to threshold models for financial variables, the pricing of occupation time derivatives and shadow rate models for interest rate dynamics. To calculate the expected value of a discounted payoff under general state-dependent discounting and monitoring of barrier crossing, continuous time Markov chain (CTMC) approximation can be applied. In a recent work, Zhang and Li (2018, Operations Research, forthcoming) established sharp convergence rates of CTMC approximation for diffusion models with smooth coefficients but non-smooth payoff functions, and proposed grid design principles to ensure nice convergence behaviors. However, their theoretical analysis fails to obtain sharp convergence rates when model coefficients lack smoothness. Moreover, it is unclear how to design the grid of CTMC to remedy the inferior convergence behaviors resulting from non-smooth model coefficients. In this paper, we introduce new ways for the theoretical analysis of CTMC approximation for general diffusion models with non-smooth coefficients. We prove that convergence of option price is only first order in general. However, strikingly, if all the discontinuous points of the model coefficients and the payoff function are in the midway between two grid points, second order convergence in the maximum norm is restored and in this case, delta and gamma have second order convergence at almost all grid points except those next to the discontinuous points. Numerical experiments are conducted that confirm the validity of our theoretical results. We also compare the CTMC approximation approach with properly designed grids to a classical numerical PDE scheme for diffusion models with non-smooth coefficients, where the finite difference method is applied separately in each region with smooth coefficients and continuous pasting of the value function is enforced at the discontinuities. We show that our approach is superior to the latter in terms of both the convergence rate and the simplicity of implementation.

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 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 Discrete Time Continuous State Interest Rate Models written by and published by DIANE Publishing. This book was released on with total page 22 pages. Available in PDF, EPUB and Kindle. Book excerpt:

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 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 Continuous Time Speed for Discrete Time Models written by Ivo Bakota and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We propose a Markov-chain approximation method for discrete-time control problems, showing how to reap the speed gains from continuous-time algorithms in this class of models. Our approach specifies a discrete Markov chain on a grid, taking a first-order approximation of conditional distributions in their first and second moments around a reference point. Standard dynamic-programming results guarantee convergence. We show how to apply our method to standard consumption-savings problems with and without a portfolio choice, realizing speed gains of up to two orders of magnitude (a factor 100) with respect to state-of-the-art methods, when using the same number of grid points. This is without significant loss of precision. We show how to avoid the curse of dimensionality and keep computation times manageable in high-dimensional problems with independent shocks. Finally, we show how our approach can substantially simplify the computation of dynamic games with a large state space, solving a discrete-time version of the altruistic savings game studied by Barczyk & Kredler (2014).German abstract:Wir schlagen eine Markov-Ketten-Approximationsmethode für zeitdiskrete Steuerungsprobleme vor und zeigen, wie man die Geschwindigkeitsvorteile von zeitstetigen Algorithmen in dieser Modellklasse nutzen kann. Unser Ansatz spezifiziert eine diskrete Markov-Kette auf einem Gitter, wobei eine Approximation erster Ordnung der bedingten Verteilungen in ihren ersten und zweiten Momenten um einen Referenzpunkt herum verwendet wird. Standardergebnisse der dynamischen Optimierung garantieren Konvergenz. Wir zeigen, wie unsere Methode auf kanonische Sparprobleme mit und ohne Portfoliowahl angewandt werden kann, wobei Geschwindigkeitsgewinne von bis zu zwei Größenordnungen (ein Faktor 100) im Vergleich zu modernsten Methoden erzielt werden, wenn dieselbe Anzahl von Gitterpunkten verwendet wird. Dies geschieht ohne signifikanten Verlust an Präzision. Wir zeigen, wie man den Fluch der Dimensionalität vermeidet und die Berechnungszeiten bei hochdimensionalen Problemen mit unabhängigen Schocks überschaubar hält. Schließlich zeigen wir, wie unser Ansatz die Berechnung von dynamischen Spielen mit einem großen Zustandsraum erheblich vereinfachen kann, indem wir eine zeitdiskrete Version des altruistischen Sparspiels lösen, das in Barczyk & Kredler (2014) untersucht wurde.

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 Application of Continuous Time Markov Chain Models written by Chia Chun Lo and published by . This book was released on 2009 with total page 454 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Correlated Continuous Time Markov Chains and Derivatives Pricing written by and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Essentials of Stochastic Processes written by Richard Durrett and published by Springer. This book was released on 2016-11-07 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: Building upon the previous editions, this textbook is a first course in stochastic processes taken by undergraduate and graduate students (MS and PhD students from math, statistics, economics, computer science, engineering, and finance departments) who have had a course in probability theory. It covers Markov chains in discrete and continuous time, Poisson processes, renewal processes, martingales, and option pricing. One can only learn a subject by seeing it in action, so there are a large number of examples and more than 300 carefully chosen exercises to deepen the reader’s understanding. Drawing from teaching experience and student feedback, there are many new examples and problems with solutions that use TI-83 to eliminate the tedious details of solving linear equations by hand, and the collection of exercises is much improved, with many more biological examples. Originally included in previous editions, material too advanced for this first course in stochastic processes has been eliminated while treatment of other topics useful for applications has been expanded. In addition, the ordering of topics has been improved; for example, the difficult subject of martingales is delayed until its usefulness can be applied in the treatment of mathematical finance.

Download or read book An Introduction to Stochastic Modeling written by Howard M. Taylor and published by Academic Press. This book was released on 2014-05-10 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: An Introduction to Stochastic Modeling provides information pertinent to the standard concepts and methods of stochastic modeling. This book presents the rich diversity of applications of stochastic processes in the sciences. Organized into nine chapters, this book begins with an overview of diverse types of stochastic models, which predicts a set of possible outcomes weighed by their likelihoods or probabilities. This text then provides exercises in the applications of simple stochastic analysis to appropriate problems. Other chapters consider the study of general functions of independent, identically distributed, nonnegative random variables representing the successive intervals between renewals. This book discusses as well the numerous examples of Markov branching processes that arise naturally in various scientific disciplines. The final chapter deals with queueing models, which aid the design process by predicting system performance. This book is a valuable resource for students of engineering and management science. Engineers will also find this book useful.

Download or read book Introduction to Applied Linear Algebra written by Stephen Boyd and published by Cambridge University Press. This book was released on 2018-06-07 with total page 477 pages. Available in PDF, EPUB and Kindle. Book excerpt: A groundbreaking introduction to vectors, matrices, and least squares for engineering applications, offering a wealth of practical examples.

Download or read book Statistical Rethinking written by Richard McElreath and published by CRC Press. This book was released on 2018-01-03 with total page 488 pages. Available in PDF, EPUB and Kindle. Book excerpt: Statistical Rethinking: A Bayesian Course with Examples in R and Stan builds readers’ knowledge of and confidence in statistical modeling. Reflecting the need for even minor programming in today’s model-based statistics, the book pushes readers to perform step-by-step calculations that are usually automated. This unique computational approach ensures that readers understand enough of the details to make reasonable choices and interpretations in their own modeling work. The text presents generalized linear multilevel models from a Bayesian perspective, relying on a simple logical interpretation of Bayesian probability and maximum entropy. It covers from the basics of regression to multilevel models. The author also discusses measurement error, missing data, and Gaussian process models for spatial and network autocorrelation. By using complete R code examples throughout, this book provides a practical foundation for performing statistical inference. Designed for both PhD students and seasoned professionals in the natural and social sciences, it prepares them for more advanced or specialized statistical modeling. Web Resource The book is accompanied by an R package (rethinking) that is available on the author’s website and GitHub. The two core functions (map and map2stan) of this package allow a variety of statistical models to be constructed from standard model formulas.

Download or read book The Best of Wilmott 2 written by Paul Wilmott and published by Wiley. This book was released on 2005-11-18 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Team at Wilmott is very proud to present this compilation of Wilmott magazine articles and presentations from our second year. We have selected some of the very best in cutting-edge research, and the most illuminating of our regular columns. The technical papers include state-of-the-art pricing tools and models. You'll notice there's a bias towards volatility modelling in the book. Of course, it's one of my favourite topics, but volatility is also the big unknown as far as pricing and hedging is concerned. We present research in this area from some of the best newcomers in this field. You'll see ideas that make a mockery of 'received wisdom,' ideas that are truly paradigm shattering - for we aren't content with a mere 'shift.' We know you'll enjoy it! The Best of Wilmott will return again next year...