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Complete Analytical Solution Of The Sabr Model For Fixed Income Option Pricing And Value At Risk Problems A Probability Density Function Approach
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Book Synopsis Complete Analytical Solution of the SABR Model for Fixed-Income Option Pricing and Value-at-Risk Problems - A Probability Density Function Approach by : Alexander Izmailov
Download or read book Complete Analytical Solution of the SABR Model for Fixed-Income Option Pricing and Value-at-Risk Problems - A Probability Density Function Approach written by Alexander Izmailov and published by . This book was released on 2016 with total page 11 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first ever explicit f ...
Book Synopsis Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities by : Alexander Izmailov
Download or read book Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities written by Alexander Izmailov and published by . This book was released on 2015 with total page 17 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first ever explicit formulation of the concept of an option's probability density functions has been introduced in our publications “Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options - True Value-at-Risk and Option Based Hedging Strategies”, “Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems. A Probability Density Function Approach” and “Complete Analytical Solution of the Heston Model for Option Pricing and Value-At-Risk Problems. A Probability Density Function Approach.” Please see links 'http://ssrn.com/abstract=2489601 ' http://ssrn.com/abstract= 2489601, 'http://ssrn.com/abstract=2546430 ' http://ssrn.com/abstract= 2546430, 'http://ssrn.com/abstract=2549033 ' http://ssrn.com/abstract= 2549033). In this paper we report unique analytical results for pricing American Style Options in the presence of both constant and stochastic volatility (Heston model), enabling complete analytical resolution of all problems associated with American Style Options considered within the Heston Model. Our discovery of the probability density function for American and European Style Options with constant and stochastic volatilities enables exact closed-form analytical results for their expected values (prices) for the first time without depending on approximate numerical methods. Option prices, i.e. their expected values, are just the first moments. All higher moments are as easily represented in closed form based on our probability density function, but are not calculable by extensions of other numerical methods now used to represent the first moment. Our formulation of the density functions for options with American and European Style execution rights with constant and stochastic volatility (Heston model) is expressive enough to enable derivation for the first time ever of corollary closed-form analytical results for such Value-At-Risk characteristics as the probabilities that options with different execution rights, with constant or stochastic volatility, will be below or above any set of thresholds at termination. Such assessments are absolutely out of reach of current published methods for treating options.All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.
Book Synopsis Complete Analytical Solution of the Heston Model for Option Pricing and Value-at-Risk Problems by : Alexander Izmailov
Download or read book Complete Analytical Solution of the Heston Model for Option Pricing and Value-at-Risk Problems written by Alexander Izmailov and published by . This book was released on 2015 with total page 12 pages. Available in PDF, EPUB and Kindle. Book excerpt: • The first ever explicit formulation of the concept of an option's probability density functions has been introduced in our publications "Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options -- True Value-at-Risk and Option Based Hedging Strategies" and "Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems. A Probability Density Function Approach." (See links 'http://ssrn.com/abstract=2489601' http://ssrn.com/abstract=2489601 and 'http://ssrn.com/abstract=2546430' http://ssrn.com/abstract=2546430). • In this paper we report similar unique results for pricing options in the presence of stochastic volatility (Heston model), enabling complete analytical resolution of all problems associated with options considered within the Heston Model. • Our discovery of the probability density function for options with stochastic volatility enables exact closed-form analytical results for their expected values (prices) for the first time without depending on approximate numerical methods and a Fourier transform that only abbreviates complex numerical integration procedure. • Expected value is the first moment. All higher moments are as easily represented in closed form based on our probability density function, but are not calculable by extensions of other numerical methods, such as a Fourier transform, now used to represent the first moment. • Our formulation of the density function for options with stochastic volatility within the Heston model is expressive enough to enable derivation for the first time ever of corollary closed-form analytical results for such Value-At-Risk characteristics as the probabilities that options with stochastic volatility will be below or above any set of thresholds at termination. Such assessments are absolutely out of reach of current published methods for treating options within the Heston model.• All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.
Book Synopsis Modern SABR Analytics by : Alexandre Antonov
Download or read book Modern SABR Analytics written by Alexandre Antonov and published by Springer. This book was released on 2019-04-23 with total page 127 pages. Available in PDF, EPUB and Kindle. Book excerpt: Focusing on recent advances in option pricing under the SABR model, this book shows how to price options under this model in an arbitrage-free, theoretically consistent manner. It extends SABR to a negative rates environment, and shows how to generalize it to a similar model with additional degrees of freedom, allowing simultaneous model calibration to swaptions and CMSs. Since the SABR model is used on practically every trading floor to construct interest rate options volatility cubes in an arbitrage-free manner, a careful treatment of it is extremely important. The book will be of interest to experienced industry practitioners, as well as to students and professors in academia. Aimed mainly at financial industry practitioners (for example quants and former physicists) this book will also be interesting to mathematicians who seek intuition in the mathematical finance.
Book Synopsis The SABR/LIBOR Market Model by : Riccardo Rebonato
Download or read book The SABR/LIBOR Market Model written by Riccardo Rebonato and published by John Wiley & Sons. This book was released on 2011-03-01 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a major innovation in the interest rate space. It explains a financially motivated extension of the LIBOR Market model which accurately reproduces the prices for plain vanilla hedging instruments (swaptions and caplets) of all strikes and maturities produced by the SABR model. The authors show how to accurately recover the whole of the SABR smile surface using their extension of the LIBOR market model. This is not just a new model, this is a new way of option pricing that takes into account the need to calibrate as accurately as possible to the plain vanilla reference hedging instruments and the need to obtain prices and hedges in reasonable time whilst reproducing a realistic future evolution of the smile surface. It removes the hard choice between accuracy and time because the framework that the authors provide reproduces today's market prices of plain vanilla options almost exactly and simultaneously gives a reasonable future evolution for the smile surface. The authors take the SABR model as the starting point for their extension of the LMM because it is a good model for European options. The problem, however with SABR is that it treats each European option in isolation and the processes for the various underlyings (forward and swap rates) do not talk to each other so it isn't obvious how to relate these processes into the dynamics of the whole yield curve. With this new model, the authors bring the dynamics of the various forward rates and stochastic volatilities under a single umbrella. To ensure the absence of arbitrage they derive drift adjustments to be applied to both the forward rates and their volatilities. When this is completed, complex derivatives that depend on the joint realisation of all relevant forward rates can now be priced. Contents THE THEORETICAL SET-UP The Libor Market model The SABR Model The LMM-SABR Model IMPLEMENTATION AND CALIBRATION Calibrating the LMM-SABR model to Market Caplet prices Calibrating the LMM/SABR model to Market Swaption Prices Calibrating the Correlation Structure EMPIRICAL EVIDENCE The Empirical problem Estimating the volatility of the forward rates Estimating the correlation structure Estimating the volatility of the volatility HEDGING Hedging the Volatility Structure Hedging the Correlation Structure Hedging in conditions of market stress
Book Synopsis A Complete Analytical Solution of the Asian Option Pricing Within the Heston Model for Stochastic Volatility by : Alexander Izmailov
Download or read book A Complete Analytical Solution of the Asian Option Pricing Within the Heston Model for Stochastic Volatility written by Alexander Izmailov and published by . This book was released on 2015 with total page 11 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first ever explicit formulation of the concept of the option's probability density functions has been introduced in our publications “Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options - True Value-at-Risk and Option Based Hedging Strategies” and “Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems. A Probability Density Function Approach.” See links: 'http://ssrn.com/abstract=2489601' http://ssrn.com/abstract=2489601 and 'http://ssrn.com/abstract=2546430' http://ssrn.com/abstract=2546430.The first ever explicit formulation of the concept of the options' probability density functions within the framework of stochastic volatility (Heston model) has been introduced in our publications “Complete Analytical Solution of the Heston Model for Option Pricing and Value-at-Risk Problems: A Probability Density Function Approach”, “Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach” and “A Complete Analytical Resolution of the Double Barrier Option's Pricing Within the Heston Model. A Probability Density Approach.” See links:'http://ssrn.com /abstract=2549033' http://ssrn.com/abstract=2549033 and 'http://ssrn.com/abstract=2554038' http://ssrn.com/abstract=2554038 and 'http://ssrn.com/abstract=2605948' http://ssrn.com/abstract=2605948.In this paper we report complete analytical closed-form results for the European style Asian Options considered within the Heston model for Stochastic Volatility (SV). Our discovery of the probability density function of the European style Asian Options with SV enables exact closed-form representation of its expected value (price) for the first time ever. Our formulation of the probability density function for the European style Asian Options with SV is expressive enough to enable derivation for the first time ever of corollary analytical closed-form results for such Value-At-Risk characteristics as the probabilities that an Asian Option with SV will be below or above any threshold at any future time before or at termination. Such assessments are absolutely out of reach of the current published methods for treating Asian Options even in the framework of constant volatility.All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.
Book Synopsis Interest Rate Risk Modeling by : Sanjay K. Nawalkha
Download or read book Interest Rate Risk Modeling written by Sanjay K. Nawalkha and published by John Wiley & Sons. This book was released on 2005-05-09 with total page 436 pages. Available in PDF, EPUB and Kindle. Book excerpt: The definitive guide to fixed income valuation and risk analysis The Trilogy in Fixed Income Valuation and Risk Analysis comprehensively covers the most definitive work on interest rate risk, term structure analysis, and credit risk. The first book on interest rate risk modeling examines virtually every well-known IRR model used for pricing and risk analysis of various fixed income securities and their derivatives. The companion CD-ROM contain numerous formulas and programming tools that allow readers to better model risk and value fixed income securities. This comprehensive resource provides readers with the hands-on information and software needed to succeed in this financial arena.
Book Synopsis Pricing Continuously Monitored Barrier Options Under the Sabr Model by : Nian Yang
Download or read book Pricing Continuously Monitored Barrier Options Under the Sabr Model written by Nian Yang and published by . This book was released on 2019 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt: The stochastic alpha beta rho (SABR) model introduced by Hagan et al. (2002) is widely used in both fixed income and the foreign exchange (FX) markets. Continuously monitored barrier option contracts are among the most popular derivative contracts in the FX markets. In this paper, we develop closed-form formulas to approximate various types of barrier option prices (down-and-out/in, up-and-out/in) under the SABR model. We first derive an approximate formula for the survival density. The barrier option price is the one-dimensional integral of its payoff function and the survival density, which can be easily implemented and quickly evaluated. The approximation error of the survival density is also analyzed. To the best of our knowledge, it is the first time that analytical (approximate) formulas for the survival density and the barrier option prices for the SABR model are derived. Numerical experiments demonstrate the validity and efficiency of these formulas.
Book Synopsis Stochastic Interest Rate Modeling With Fixed Income Derivative Pricing (Third Edition) by : Nicolas Privault
Download or read book Stochastic Interest Rate Modeling With Fixed Income Derivative Pricing (Third Edition) written by Nicolas Privault and published by World Scientific. This book was released on 2021-09-02 with total page 373 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the mathematics of stochastic interest rate modeling and the pricing of related derivatives, based on a step-by-step presentation of concepts with a focus on explicit calculations. The types of interest rates considered range from short rates to forward rates such as LIBOR and swap rates, which are presented in the HJM and BGM frameworks. The pricing and hedging of interest rate and fixed income derivatives such as bond options, caps, and swaptions, are treated using forward measure techniques. An introduction to default bond pricing and an outlook on model calibration are also included as additional topics.This third edition represents a significant update on the second edition published by World Scientific in 2012. Most chapters have been reorganized and largely rewritten with additional details and supplementary solved exercises. New graphs and simulations based on market data have been included, together with the corresponding R codes.This new edition also contains 75 exercises and 4 problems with detailed solutions, making it suitable for advanced undergraduate and graduate level students.
Book Synopsis Approximate and PDE Solution to the Boundary Free SABR Model - Applications to Pricing and Calibration by : Joerg Kienitz
Download or read book Approximate and PDE Solution to the Boundary Free SABR Model - Applications to Pricing and Calibration written by Joerg Kienitz and published by . This book was released on 2015 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt: Considering the current interest rate environment it has become necessary to extend option pricing models for 0 and negative strikes. We consider the recently proposed free boundary SABR model, Antonov A., Konikov, M., and Spector, M. (2015). In their paper the authors provide a pricing formula for European Call options based on numerical integration and Markovian projection. Since it is necessary for practitioners to calibrate the model to market data fast approximation methods together with benchmark methods for their performance are essential. In this note we consider the PDE solution for pricing European Call options as well as two approximation formulas for the Bachelier, aka Normal volatility, produced by this model. The latter numbers can then be plugged into the Bachelier pricing formula to get the corresponding option prices.We have to stress two facts. First, the PDE method can be seen as a benchmark for the approximate solutions and, second, the approximation formulas can serve for calibration purposes, where fast calculation methods are essential, especially, if one wishes to calibrate to implied Bachelier volatilities. In the approach proposed by Antonov A., Konikov, M., and Spector, M. (2015) the implied volatilities have to be inferred from option prices.Finally, we stress the fact that the PDE or approximate solutions can be used to effciently apply a mixing approach to control the shape of the surface, especially the wings.
Book Synopsis Pricing Interest Rate Risk Derivatives Using Binomial Trees with MATLAB by : Alexander Esse
Download or read book Pricing Interest Rate Risk Derivatives Using Binomial Trees with MATLAB written by Alexander Esse and published by GRIN Verlag. This book was released on 2018-06-14 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt: Seminar paper from the year 2017 in the subject Business economics - Investment and Finance, grade: 1,00, University of Tubingen, language: English, abstract: In this assignment we approximate Oldrich Vasicek's (1977) term structure model with a binomial approach and show that it is convenient to use a recombining binomial tree to value interest rate derivatives in the Vasicek model. First, we illustrate that our applied binomial approximations converge to the dynamic continuous-time Vasicek model with an increasing number of time steps (subperiods). Furthermore, we apply the binomial approach to value a Discount Bond, Coupon Bond and a Futures Contract on both a Discount and Coupon Bond. The resulting approximations will be compared to the respective analytical solution, which we use as a benchmark. Thirdly, we determine the fair value of both an European and American Call and Put on a Discount Bond and Coupon Bond, respectively. We demonstrate that our estimated binomial prices converge with an increasing number of time steps. Moreover, we analyze both the behaviour of a Sraddle on a Discount Bond and the Early Exercise Premium of the considered American Options as a function of spot interest rates. We obtain all results shown in this report from the software "Matlab". Hence, the submitted "m.files" should be taken as a reference for a better understanding of the calculation procedures described in this report (Relevant Code is depicted in the Appendices). Furthermore, to reduce computational effort and required time to run our code we apply a joint calculation of specific approximations rather than run a code individually for each Task. This is mainly because some specific securities and interest rate derivatives require the same underlying and identical matrices of the interest rates and transition probabilities from the binomial trees for the approximation procedure. This approach is suitable because we apply the identical number of subperiods for specific Tasks and, thus, for the respective securities and or derivatives.
Book Synopsis Fixed Income Relative Value Analysis, + website by : Doug Huggins
Download or read book Fixed Income Relative Value Analysis, + website written by Doug Huggins and published by John Wiley & Sons. This book was released on 2024-05-13 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: An invaluable guide for fixed income practitioners, fully updated to incorporate the shift from LIBOR to SOFR Since its first edition in 2013, Fixed Income Relative Value Analysis: A Practitioner’s Guide to the Theory, Tools, and Trades has become the gold standard for guides linking financial theories with practical analysis tools. The newly revised second edition reflects both the progress in statistical tools over the last decade and the impact of the transition to SOFR on swap spreads. You’ll find a set of statistical and financial tools, a multitude of actual trades resulting from the application of these tools, as well as access to a companion website featuring spreadsheets illustrating some of the models contained in the book. This book covers: Statistical models for quantitative market analysis, in particular mean reversion models and principal component analysis, now including the multivariate Ornstein-Uhlenbeck model. An in-depth approach to understanding swap spreads in theory and practice. A comprehensive discussion of the various basis swaps and their combinations. The incorporation of credit default swaps in yield curve analysis. A classification of option trades into three types and the appropriate analysis tools. Fitted curve techniques for identifying relative value among different bonds. A multi-factor delivery option model for bond future contracts. Fixed Income Relative Value Analysis has proven to be an indispensable desk reference for buy- and sell-side fixed income professionals, including traders, quantitative analysts, portfolio managers, financial engineers, fixed income salespeople with sophisticated clientele and risk managers.
Book Synopsis Functional Analytic (Ir-)Regularity Properties of SABR-type Processes by : Leif Döring
Download or read book Functional Analytic (Ir-)Regularity Properties of SABR-type Processes written by Leif Döring and published by . This book was released on 2017 with total page 33 pages. Available in PDF, EPUB and Kindle. Book excerpt: The SABR model is a benchmark stochastic volatility model in interest rate markets, which has received much attention in the past decade. Its popularity arose from a tractable asymptotic expansion for implied volatility, derived by heat kernel methods. As markets moved to historically low rates, this expansion appeared to yield inconsistent prices. Since the model is deeply embedded in market practice, alternative pricing methods for SABR have been addressed in numerous approaches in recent years. All standard option pricing methods make certain regularity assumptions on the underlying model, but for SABR these are rarely satisfied. We examine here regularity properties of the model from this perspective with view to a number of (asymptotic and numerical) option pricing methods. In particular, we highlight delicate degeneracies of the SABR model (and related processes) at the origin, which deem the currently used popular heat kernel methods and all related methods from (sub-) Riemannian geometry ill-suited for SABR-type processes, when interest rates are near zero. We describe a more general semigroup framework, which permits to derive a suitable geometry for SABR-type processes (in certain parameter regimes) via symmetric Dirichlet forms. Furthermore, we derive regularity properties (Feller-properties and strong continuity properties) necessary for the applicability of popular numerical schemes to SABR-semigroups, and identify suitable Banach - and Hilbert spaces for these. Finally, we comment on the short time and large time asymptotic behaviour of SABR-type processes beyond the heat-kernel framework.
Book Synopsis New Methods in Fixed Income Modeling by : Mehdi Mili
Download or read book New Methods in Fixed Income Modeling written by Mehdi Mili and published by Springer. This book was released on 2018-08-18 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents new approaches to fixed income modeling and portfolio management techniques. Taking into account the latest mathematical and econometric developments in finance, it analyzes the hedging securities and structured instruments that are offered by banks, since recent research in the field of fixed incomes and financial markets has raised awareness for changes in market risk management strategies. The book offers a valuable resource for all researchers and practitioners interested in the theory behind fixed income instruments, and in their applications in financial portfolio management.
Book Synopsis Approximate Arbitrage-Free Option Pricing Under the SABR Model by : Nian Yang
Download or read book Approximate Arbitrage-Free Option Pricing Under the SABR Model written by Nian Yang and published by . This book was released on 2017 with total page 22 pages. Available in PDF, EPUB and Kindle. Book excerpt: The stochastic-alpha-beta-rho (SABR) model introduced by Hagan et al. (2002) provides a popular vehicle to model the implied volatilities in the interest rate and foreign exchange markets. To exclude arbitrage opportunities, we need to specify an absorbing boundary at zero for this model, which the existing analytical approaches to pricing derivatives under the SABR model typically ignore. This paper develops closed-form approximations to the prices of vanilla options to incorporate the effect of such a boundary condition. Different from the traditional normal distribution-based approximations, our method stems from an expansion around a one-dimensional Bessel process. Extensive numerical experiments demonstrate its accuracy and efficiency. Furthermore, the explicit expression yielded from our method is appealing from the practical perspective because it can lead to fast calibration, pricing, and hedging.
Book Synopsis Fixed Income Mathematics, Fifth Edition: Analytical and Statistical Techniques by : Frank J. Fabozzi
Download or read book Fixed Income Mathematics, Fifth Edition: Analytical and Statistical Techniques written by Frank J. Fabozzi and published by McGraw Hill Professional. This book was released on 2022-09-06 with total page 636 pages. Available in PDF, EPUB and Kindle. Book excerpt: The standard reference for fixed income portfolio managers—fully updated with new analytical frameworks Fixed Income Mathematics is known around the world as the leading guide to understanding the concepts, valuation models for bonds with embedded option, mortgage-backed securities, asset-backed securities, and other fixed income instruments, and portfolio analytics. Fixed Income Mathematics begins with basic concepts of the mathematics of finance, then systematically builds on them to reveal state-of-the-art methodologies for evaluating them and managing fixed-income portfolios. Concepts are illustrated with numerical examples and graphs, and you need only a basic knowledge of elementary algebra to understand them. This new edition includes several entirely new chapters―Risk-Adjusted Returns, Empirical Duration, Analysis of Floating-Rate Securities, Holdings-Based Return Attribution Analysis, Returns-Based Style Attribution Analysis, Measuring Bond Liquidity, and Machine Learning―and provides substantially revised chapters on: Interest rate modeling Probability theory Optimization models and applications to bond portfolio management Historical return measures Measuring historical return volatility The concepts and methodologies for managing fixed income portfolios has improved dramatically over the past 15 years. This edition explains these changes and provides the knowledge you need to value fixed-income securities and measure the various types of risks associated with individual securities and portfolios.
Book Synopsis Modeling Fixed Income Securities and Interest Rate Options by : Robert Jarrow
Download or read book Modeling Fixed Income Securities and Interest Rate Options written by Robert Jarrow and published by CRC Press. This book was released on 2019-09-17 with total page 385 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modeling Fixed Income Securities and Interest Rate Options, Third Edition presents the basics of fixed-income securities in a way that, unlike competitive texts, requires a minimum of prerequisites. While other books focus heavily on institutional details of the bond market, all of which could easily be learned "on the job," the third edition of this classic textbook is more focused with presenting a coherent theoretical framework for understanding all basic models. The author’s unified approach—the Heath Jarrow Morton model—under which all other models are presented as special cases, enhances understanding of the material. The author’s pricing model is widely used in today’s securities industry. This new edition offers many updates to align with advances in the research and requires a minimum of prerequisites while presenting the basics of fixed-income securities. Highlights of the Third Edition Chapters 1-16 completely updated to align with advances in research Thoroughly eliminates out-of-date material while advancing the presentation Includes an ample amount of exercises and examples throughout the text which illustrate key concepts .
