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Approximate Arbitrage Free Option Pricing Under The Sabr Model
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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 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 Soft Computing in Data Analytics by : Janmenjoy Nayak
Download or read book Soft Computing in Data Analytics written by Janmenjoy Nayak and published by Springer. This book was released on 2018-08-21 with total page 848 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume contains original research findings, exchange of ideas and dissemination of innovative, practical development experiences in different fields of soft and advance computing. It provides insights into the International Conference on Soft Computing in Data Analytics (SCDA). It also concentrates on both theory and practices from around the world in all the areas of related disciplines of soft computing. The book provides rapid dissemination of important results in soft computing technologies, a fusion of research in fuzzy logic, evolutionary computations, neural science and neural network systems and chaos theory and chaotic systems, swarm based algorithms, etc. The book aims to cater the postgraduate students and researchers working in the discipline of computer science and engineering along with other engineering branches.
Book Synopsis Barrier Option Pricing Under SABR Model Using Monte Carlo Methods by : Junling Hu
Download or read book Barrier Option Pricing Under SABR Model Using Monte Carlo Methods written by Junling Hu and published by . This book was released on 2013 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: The project investigates the prices of barrier options from the constant underlying volatility in the Black-Scholes model to stochastic volatility model in SABR framework. The constant volatility assumption in derivative pricing is not able to capture the dynamics of volatility. In order to resolve the shortcomings of the Black-Scholes model, it becomes necessary to find a model that reproduces the smile effect of the volatility. To model the volatility more accurately, we look into the recently developed SABR model which is widely used by practitioners in the financial industry. Pricing a barrier option whose payoff to be path dependent intrigued us to find a proper numerical method to approximate its price. We discuss the basic sampling methods of Monte Carlo and several popular variance reduction techniques. Then, we apply Monte Carlo methods to simulate the price of the down-and-out put barrier options under the Black-Scholes model and the SABR model as well as compare the features of these two models.
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 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 Mixing SABR Models for Negative Rates by : Alexandre Antonov
Download or read book Mixing SABR Models for Negative Rates written by Alexandre Antonov and published by . This book was released on 2015 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the current low-interest-rate environment, extending option models to negative rates has become an important issue. In our previous paper, we introduced the Free SABR model, which is a natural and an attractive extension to the classical SABR model. In spite of its advantages over the Shifted SABR, the Free SABR option pricing formula is based on an approximation. Although this approximation is very good, it cannot guarantee the absence of arbitrage.In this article, we build on an exact option pricing formula for the normal SABR with a free boundary and an arbitrary correlation. First, we derive this formula in terms of a 1D integral, which is suitable for fast calibration. Next, we apply the formula as a control variate to the Free SABR to improve the accuracy of its approximation, especially for high correlations. Finally, we come up with a Mixture SABR model, which is a weighted sum of the normal and free zero-correlation models. This model is guaranteed to be arbitrage free and has a closed-form solution for option prices. Added degrees of freedom also allow the Mixture SABR model to be calibrated to a broader set of trades, in particular, to a joint set of swaptions and CMS payment. We demonstrate this capability with a wide set of numerical examples.
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 Arbitrage-Free Evaluation of American-Style Options on Assets with Stochastics Variance Characteristics by : Andrew M.K. Wu
Download or read book Arbitrage-Free Evaluation of American-Style Options on Assets with Stochastics Variance Characteristics written by Andrew M.K. Wu and published by . This book was released on 2001 with total page 45 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper presents an arbitrage-free framework for contingent claim valuation under stochastic volatility that does not hinge on the market price of volatility risk. It is contrary to the traditional literature, in which some restrictive equilibrium assumptions about investors' preferences must be imposed if one allows arbitrary correlation between the asset price and volatility increments. Our approach to stochastic volatility modelling relies on specifying the forward variance as a function of the market option prices under the no-arbitrage condition. The model is implemented by the Ho-Stapleton-Subrahmanyam (1995) multivariate binomial approximation procedure, but with an extension to permit the multiplicative factor to be a function of stochastic volatility within a recombining context. In conjunction with the lattice-based algorithm, the generalised Geske-Johnson (1984) technique is employed to accelerate the computational efficiency when valuing American options. The key to the model is that it exactly matches the volatility structure inferred from a portfolio of actively traded options, yet is simple enough to be used for pricing a wide class of derivative securities within a reasonable time frame. We investigate how stochastic volatility influences the early exercise premium of an American option. The magnitude of this effect depends upon the moneyness of the option, the time to maturity, the volatilities of the state variables, as well as the correlation between them.
Book Synopsis Modern Computational Finance by : Antoine Savine
Download or read book Modern Computational Finance written by Antoine Savine and published by John Wiley & Sons. This book was released on 2018-11-13 with total page 592 pages. Available in PDF, EPUB and Kindle. Book excerpt: Arguably the strongest addition to numerical finance of the past decade, Algorithmic Adjoint Differentiation (AAD) is the technology implemented in modern financial software to produce thousands of accurate risk sensitivities, within seconds, on light hardware. AAD recently became a centerpiece of modern financial systems and a key skill for all quantitative analysts, developers, risk professionals or anyone involved with derivatives. It is increasingly taught in Masters and PhD programs in finance. Danske Bank's wide scale implementation of AAD in its production and regulatory systems won the In-House System of the Year 2015 Risk award. The Modern Computational Finance books, written by three of the very people who designed Danske Bank's systems, offer a unique insight into the modern implementation of financial models. The volumes combine financial modelling, mathematics and programming to resolve real life financial problems and produce effective derivatives software. This volume is a complete, self-contained learning reference for AAD, and its application in finance. AAD is explained in deep detail throughout chapters that gently lead readers from the theoretical foundations to the most delicate areas of an efficient implementation, such as memory management, parallel implementation and acceleration with expression templates. The book comes with professional source code in C++, including an efficient, up to date implementation of AAD and a generic parallel simulation library. Modern C++, high performance parallel programming and interfacing C++ with Excel are also covered. The book builds the code step-by-step, while the code illustrates the concepts and notions developed in the book.
Book Synopsis Mathematical Modeling and Methods of Option Pricing by : Lishang Jiang
Download or read book Mathematical Modeling and Methods of Option Pricing written by Lishang Jiang and published by World Scientific. This book was released on 2005 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the perspective of partial differential equations (PDE), this book introduces the Black-Scholes-Merton's option pricing theory. A unified approach is used to model various types of option pricing as PDE problems, to derive pricing formulas as their solutions, and to design efficient algorithms from the numerical calculation of PDEs.
Book Synopsis Preference-Free Option Pricing with Path-Dependent Volatility by : Steven L. Heston
Download or read book Preference-Free Option Pricing with Path-Dependent Volatility written by Steven L. Heston and published by . This book was released on 2015 with total page 12 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper shows how one can obtain a continuous-time preference-free option pricing model with a path-dependent volatility as the limit of a discrete-time GARCH model. In particular, the continuous-time model is the limit of a discrete-time GARCH model of Heston and Nandi (1997) that allows asymmetry between returns and volatility. For the continuous-time model, one can directly compute closed-form solutions for option prices using the formula of Heston (1993). Toward that purpose, we present the necessary mappings, based on Foster and Nelson (1994), such that one can approximate (arbitrarily closely) the parameters of the continuous-time model on the basis of the parameters of the discrete-time GARCH model. The discrete-time GARCH parameters can be estimated easily just by observing the history of asset prices.Unlike most option pricing models that are based on the absence of arbitrage alone, a parameter related to the expected return/risk premium of the asset does appear in the continuous-time option formula. However, given other parameters, option prices are not at all sensitive to the risk premium parameter, which is often imprecisely estimated.
Book Synopsis Novel Methods in Computational Finance by : Matthias Ehrhardt
Download or read book Novel Methods in Computational Finance written by Matthias Ehrhardt and published by Springer. This book was released on 2017-09-19 with total page 599 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book discusses the state-of-the-art and open problems in computational finance. It presents a collection of research outcomes and reviews of the work from the STRIKE project, an FP7 Marie Curie Initial Training Network (ITN) project in which academic partners trained early-stage researchers in close cooperation with a broader range of associated partners, including from the private sector. The aim of the project was to arrive at a deeper understanding of complex (mostly nonlinear) financial models and to develop effective and robust numerical schemes for solving linear and nonlinear problems arising from the mathematical theory of pricing financial derivatives and related financial products. This was accomplished by means of financial modelling, mathematical analysis and numerical simulations, optimal control techniques and validation of models. In recent years the computational complexity of mathematical models employed in financial mathematics has witnessed tremendous growth. Advanced numerical techniques are now essential to the majority of present-day applications in the financial industry. Special attention is devoted to a uniform methodology for both testing the latest achievements and simultaneously educating young PhD students. Most of the mathematical codes are linked into a novel computational finance toolbox, which is provided in MATLAB and PYTHON with an open access license. The book offers a valuable guide for researchers in computational finance and related areas, e.g. energy markets, with an interest in industrial mathematics.
Book Synopsis Market-Consistent Prices by : Pablo Koch-Medina
Download or read book Market-Consistent Prices written by Pablo Koch-Medina and published by Springer Nature. This book was released on 2020-07-16 with total page 448 pages. Available in PDF, EPUB and Kindle. Book excerpt: Arbitrage Theory provides the foundation for the pricing of financial derivatives and has become indispensable in both financial theory and financial practice. This textbook offers a rigorous and comprehensive introduction to the mathematics of arbitrage pricing in a discrete-time, finite-state economy in which a finite number of securities are traded. In a first step, various versions of the Fundamental Theorem of Asset Pricing, i.e., characterizations of when a market does not admit arbitrage opportunities, are proved. The book then focuses on incomplete markets where the main concern is to obtain a precise description of the set of “market-consistent” prices for nontraded financial contracts, i.e. the set of prices at which such contracts could be transacted between rational agents. Both European-type and American-type contracts are considered. A distinguishing feature of this book is its emphasis on market-consistent prices and a systematic description of pricing rules, also at intermediate dates. The benefits of this approach are most evident in the treatment of American options, which is novel in terms of both the presentation and the scope, while also presenting new results. The focus on discrete-time, finite-state models makes it possible to cover all relevant topics while requiring only a moderate mathematical background on the part of the reader. The book will appeal to mathematical finance and financial economics students seeking an elementary but rigorous introduction to the subject; mathematics and physics students looking for an opportunity to get acquainted with a modern applied topic; and mathematicians, physicists and quantitatively inclined economists working or planning to work in the financial industry.
Book Synopsis Arbitrage-Free Prediction of the Implied Volatility Smile by : Petros Dellaportas
Download or read book Arbitrage-Free Prediction of the Implied Volatility Smile written by Petros Dellaportas and published by . This book was released on 2014 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper gives an arbitrage-free prediction for future prices of an arbitrary co-terminal set of options with a given maturity, based on the observed time series of these option prices. The statistical analysis of such a multi-dimensional time series of option prices corresponding to n strikes (with n large, e.g. n ≥ 40) and the same maturity, is a difficult task due to the fact that option prices at any moment in time satisfy non-linear and non-explicit no-arbitrage restrictions. Hence any n-dimensional time series model also has to satisfy these implicit restrictions at each time step, a condition that is impossible to meet since the model innovations can take arbitrary values. We solve this problem for any n ∈ N in the context of Foreign Exchange (FX) by first encoding the option prices at each time step in terms of the parameters of the corresponding risk-neutral measure and then performing the time series analysis in the parameter space. The option price predictions are obtained from the predicted risk neutral measure by effectively integrating it against the corresponding option payoffs. The non-linear transformation between option prices and the risk-neutral parameters applied here is not arbitrary: it is the standard mapping used by market makers in the FX option markets (the SABR parameterisation) and is given explicitly in closed form. Our method is not restricted to the FX asset class nor does it depend on the type of parameterisation used. Statistical analysis of FX market data illustrates that our arbitrage-free predictions outperform the naive random walk forecasts, suggesting a potential for building management strategies for portfolios of derivative products, akin to the ones widely used in the underlying equity and futures markets.
Book Synopsis Option Pricing and Estimation of Financial Models with R by : Stefano M. Iacus
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.
Book Synopsis Analysis, Geometry, and Modeling in Finance by : Pierre Henry-Labordere
Download or read book Analysis, Geometry, and Modeling in Finance written by Pierre Henry-Labordere and published by CRC Press. This book was released on 2008-09-22 with total page 403 pages. Available in PDF, EPUB and Kindle. Book excerpt: Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing is the first book that applies advanced analytical and geometrical methods used in physics and mathematics to the financial field. It even obtains new results when only approximate and partial solutions were previously available.Through the problem of option pricing, th
