Download or read book Pricing and Calibration in Local Volatility Models Via Fast Quantization written by Giorgia Callegaro and published by . This book was released on 2017 with total page 13 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we propose the first calibration exercise based on quantization methods. Pricing and calibration are typically difficult tasks to accomplish: pricing should be fast and accurate, otherwise calibration cannot be performed efficiently. We apply in a local volatility context the recursive marginal quantization methodology to the pricing of vanilla and barrier options. A successful calibration of the Quadratic Normal Volatility model is performed in order to show the potentiality of the method in a concrete example, while a numerical exercise on barrier options shows that quantization overcomes Monte-Carlo methods.
Download or read book Fitting Local Volatility Analytic And Numerical Approaches In Black scholes And Local Variance Gamma Models written by Andrey Itkin and published by World Scientific. This book was released on 2020-01-22 with total page 205 pages. Available in PDF, EPUB and Kindle. Book excerpt: The concept of local volatility as well as the local volatility model are one of the classical topics of mathematical finance. Although the existing literature is wide, there still exist various problems that have not drawn sufficient attention so far, for example: a) construction of analytical solutions of the Dupire equation for an arbitrary shape of the local volatility function; b) construction of parametric or non-parametric regression of the local volatility surface suitable for fast calibration; c) no-arbitrage interpolation and extrapolation of the local and implied volatility surfaces; d) extension of the local volatility concept beyond the Black-Scholes model, etc. Also, recent progresses in deep learning and artificial neural networks as applied to financial engineering have made it reasonable to look again at various classical problems of mathematical finance including that of building a no-arbitrage local/implied volatility surface and calibrating it to the option market data.This book was written with the purpose of presenting new results previously developed in a series of papers and explaining them consistently, starting from the general concept of Dupire, Derman and Kani and then concentrating on various extensions proposed by the author and his co-authors. This volume collects all the results in one place, and provides some typical examples of the problems that can be efficiently solved using the proposed methods. This also results in a faster calibration of the local and implied volatility surfaces as compared to standard approaches.The methods and solutions presented in this volume are new and recently published, and are accompanied by various additional comments and considerations. Since from the mathematical point of view, the level of details is closer to the applied rather than to the abstract or pure theoretical mathematics, the book could also be recommended to graduate students with majors in computational or quantitative finance, financial engineering or even applied mathematics. In particular, the author used to teach some topics of this book as a part of his special course on computational finance at the Tandon School of Engineering, New York University.
Download or read book Multiple Time Scales Stochastic Volatility Modeling Method in Stochastic Local Volatility Model Calibration written by Fan Wang and published by . This book was released on 2013 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis we study carefully the stochastic local volatility (SLV) model for pricing barrier options in foreign exchange or equity market. We first discuss the advantage and disadvantage of popular models such as stochastic volatility and local volatility that have been used for pricing the same products, then introduce the necessities to build a hybrid SLV model. We classified the calibration process of SLV model into two major parts according to parameters' different nature, and point out the slowness of the calibration procedure is mainly caused by solving the lever-age surface from Kolmogorov forward equation via the iteration method. Our major contribution is to apply the fast mean reversion volatility modeling technique and singular/regular perturbation analysis developed by Fouque, Papanicolaou, Sircar and Sølna in [24, 27, 26] to the forward equation, which gives a starting point which is proved to be close to the true solution, so that the iteration time is significantly reduced. Besides, we developed target functions specifically designed for processing exotic option quotes and give suitable numerical methods for each step of the calibration.
Download or read book Calibrating and Pricing with a Stochastic Local Volatility Model written by Yu Tian and published by . This book was released on 2015 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we present our study on using the hybrid stochastic-local volatility (SLV) model for option pricing. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In this paper, we provide detailed discussion on the implementation of the calibration and pricing procedures. The implemented SLV model is used for pricing exotic options in the FX market. Pricing results are presented for the SLV model in direct comparison with the pure local volatility and pure stochastic volatility models. The SLV model is shown to match market traded implied volatility surfaces very well and to improve the pricing accuracy for market traded barrier options.
Download or read book The Hybrid Stochastic Local Volatility Model with Applications in Pricing FX Options written by Yu Tian and published by . This book was released on 2016 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.
Download or read book Pricing Via Quantization in Stochastic Volatility Models written by Giorgia Callegaro and published by . This book was released on 2015 with total page 27 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we apply a new methodology based on quantization to price options in stochastic volatility models. This method can be applied to any model for which an Euler scheme is available for the underlying process and it allows for pricing vanillas, as well as exotics, thanks to the knowledge of the transition probabilities for the discretized stock process. We apply the methodology to some celebrated stochastic volatility models, including the Stein and Stein (1991) model and the SABR model introduced in Hagan and Woodward (2002). A numerical exercise shows that the pricing of vanillas turns out to be accurate; in addition, when applied to some exotics like equity-volatility options, the quantization-based method overperforms by far the Monte Carlo simulation.
Download or read book Marginal and Functional Quantization of Stochastic Processes written by Harald Luschgy and published by Springer Nature. This book was released on 2023-12-06 with total page 918 pages. Available in PDF, EPUB and Kindle. Book excerpt: Vector Quantization, a pioneering discretization method based on nearest neighbor search, emerged in the 1950s primarily in signal processing, electrical engineering, and information theory. Later in the 1960s, it evolved into an automatic classification technique for generating prototypes of extensive datasets. In modern terms, it can be recognized as a seminal contribution to unsupervised learning through the k-means clustering algorithm in data science. In contrast, Functional Quantization, a more recent area of study dating back to the early 2000s, focuses on the quantization of continuous-time stochastic processes viewed as random vectors in Banach function spaces. This book distinguishes itself by delving into the quantization of random vectors with values in a Banach space—a unique feature of its content. Its main objectives are twofold: first, to offer a comprehensive and cohesive overview of the latest developments as well as several new results in optimal quantization theory, spanning both finite and infinite dimensions, building upon the advancements detailed in Graf and Luschgy's Lecture Notes volume. Secondly, it serves to demonstrate how optimal quantization can be employed as a space discretization method within probability theory and numerical probability, particularly in fields like quantitative finance. The main applications to numerical probability are the controlled approximation of regular and conditional expectations by quantization-based cubature formulas, with applications to time-space discretization of Markov processes, typically Brownian diffusions, by quantization trees. While primarily catering to mathematicians specializing in probability theory and numerical probability, this monograph also holds relevance for data scientists, electrical engineers involved in data transmission, and professionals in economics and logistics who are intrigued by optimal allocation problems.
Download or read book Numerical Probability written by Gilles Pagès and published by Springer. This book was released on 2018-07-31 with total page 591 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook provides a self-contained introduction to numerical methods in probability with a focus on applications to finance. Topics covered include the Monte Carlo simulation (including simulation of random variables, variance reduction, quasi-Monte Carlo simulation, and more recent developments such as the multilevel paradigm), stochastic optimization and approximation, discretization schemes of stochastic differential equations, as well as optimal quantization methods. The author further presents detailed applications to numerical aspects of pricing and hedging of financial derivatives, risk measures (such as value-at-risk and conditional value-at-risk), implicitation of parameters, and calibration. Aimed at graduate students and advanced undergraduate students, this book contains useful examples and over 150 exercises, making it suitable for self-study.
Download or read book Local Volatility written by Adil Reghai and published by . This book was released on 2015 with total page 14 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper explores a powerful calibration technique of local volatility models based on the fixed point algorithm. It proves to be more robust and generic than the standard Dupire Approach. We also show how to dramatically increase the performance of Monte Carlo simulations by means of techniques borrowed from quantum physics. In particular, we use operator theory combined with fast discrete random generation to construct fast, efficient and robust algorithms for production purposes. This contribution is an engineering piece of work.
Download or read book A Portable and Fast Stochastic Volatility Model Calibration Using Multi and Many Core Processors written by Matthew Francis Dixon and published by . This book was released on 2014 with total page 6 pages. Available in PDF, EPUB and Kindle. Book excerpt: Financial markets change precipitously and on-demand pricing and risk models must be constantly recalibrated to reduce risk. However, certain classes of models are computationally intensive to robustly calibrate to intraday prices- stochastic volatility models being an archetypal example due to the non-convexity of the objective function. In order to accelerate this procedure through parallel implementation, financial application developers are faced with an ever growing plethora of low-level high-performance computing frameworks such as OpenMP, OpenCL, CUDA, or SIMD intrinsics, and forced to make a trade-off between performance versus the portability, flexibility and modularity of the code required to facilitate rapid in-house model development and productionization.This paper describes the acceleration of stochastic volatility model calibration on multi-core CPUs and GPUs using the Xcelerit platform. By adopting a simple dataflow programming model, the Xcelerit platform enables the application developer to write sequential, high-level C code, without concern for low-level high-performance computing frameworks. This platform provides the portability, flexibility and modularity required by application developers. Speedups of up to $30$x and $293$x are respectively achieved on an Intel Xeon CPU and NVIDIA Tesla K40 GPU, compared to a sequential CPU implementation. The Xcelerit platform implementation is further shown to be equivalent in performance to a low-level CUDA version. Overall, we are able to reduce the entire calibration process time of the sequential implementation from 6,189 seconds to 183.8 and 17.8 seconds on the CPU and GPU respectively without requiring the developer to reimplement in low-level high performance computing frameworks.
Download or read book A Local Volatility Model to Price and Calibrate Year on Year and Zero Coupon Inflation Options written by Diana Ribeiro and published by . This book was released on 2013 with total page 21 pages. Available in PDF, EPUB and Kindle. Book excerpt: We present a one-factor local volatility model in discrete time to price and calibrate year-on-year and zero-coupon inflation options. This model provides an exact fit to year-on-year implied volatilities and to year-on-year forward convexity adjustments, for all strikes and maturities. It is simple to implement, accurate and provides fast calibration and pricing. The model primitive is the year-on-year inflation ratio, which follows an exponential mean reverting process. The pricing of year-on-year options is described by two one dimensional (1D) partial differential equations (PDEs). The first PDE provides the discrete time probability density function for the year-on-year ratio at pre-defined annual maturities. The other PDE ensures that the model hits the market inflation index forward curve. The model is then extended to price and calibrate zero-coupon options, whereby we derive a 2D PDE that provides the discrete time joint probability density function for the year-on-year ratio and the zero-coupon inflation index. This density is used to calibrate and simultaneously price zero-coupon and year-on-year options. By construction, zero-coupon and year-on-year options are priced consistently. We demonstrate the practical application and properties of this model by calibrating it to UK RPI and EUR HICPx data.
Download or read book Multi Currency Local Volatility Model written by Daniel Alexandre Bloch and published by . This book was released on 2008 with total page 37 pages. Available in PDF, EPUB and Kindle. Book excerpt: We establish the need for local volatility coupled with domestic and foreign stochastic interest rates to properly manage some exotic hybrid options. We then compute such a local volatility and identify a bias with respect to the local volatility with deterministic rates. Performing variance-covariance analysis on the logarithm of the underlying price together with the domestic and foreign spot rates we estimate that bias by calculating the variances of the logarithm of the underlying price with and without stochastic rates at fixed points in time and in space. Equating the resulting variances we express the local volatility with stochastic rates in terms of the one with deterministic rates plus a bias obtaining an exact, fast and robust way of calibrating any local volatility with stochastic rates to market prices. We calculate it by using a bootstrapping method requiring solving a quadratic equation at each maturity and strike and present results on the Japanese market.
Download or read book Local Volatility Under Stochastic Interest Rates Using Mixture Models written by Mark S. Joshi and published by . This book was released on 2016 with total page 22 pages. Available in PDF, EPUB and Kindle. Book excerpt: A key requirement of any equity hybrid derivatives pricing model is the ability to rapidly and accurately calibrate to vanilla option prices. To this end, we present two methods for calibrating a local volatility model under correlated stochastic interest rates. This is achieved by first fitting a mixture model to market prices, and then determining the local volatility function that is consistent with this mixture model.
Download or read book Calibration of local volatility models and proper orthogonal decomposition reduced order modeling for stochastic volatility models written by Jian Geng and published by . This book was released on 2013 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book On Calibration and Simulation of Local Volatility Model with Stochastic Interest Rate written by Mingyang Xu and published by . This book was released on 2019 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Local volatility model is a relatively simple way to capture volatility skew/smile. In spite of its drawbacks, it remains popular among practitioners for derivative pricing and hedging. For long-dated options or interest rate/equity hybrid products, in order to take into account the effect of stochastic interest rate on equity price volatility stochastic interest rate is often modelled together with stochastic equity price. Similar to local volatility model with deterministic interest rate, a forward Dupire PDE can be derived using Arrow-Debreu price method, which can then be shown to be equivalent to adding an additional correction term on top of Dupire forward PDE with deterministic interest rate. Calibrating a local volatility model by the forward Dupire PDE approach with adaptively mixed grids ensures both calibration accuracy and efficiency. Based on Malliavin calculus an accurate analytic approximation is also derived for the correction term incorporating impacts from both interest rate volatility and correlation, which integrates along a more likely straight line path for better accuracy. Eventually, the hybrid local volatility model can be calibrated in a two-step process, namely, calibrate local volatility model with deterministic interest rate and add adjustment for stochastic interest rate. Due to the lack of analytic solution and path-dependency nature of some products, Monte Carlo is a simple but flexible pricing method. In order to improve its convergence, we develop a scheme to combine merits of different simulation schemes and show its effectiveness.
Download or read book Local Stochastic Volatility Models written by Cristian Homescu and published by . This book was released on 2014 with total page 57 pages. Available in PDF, EPUB and Kindle. Book excerpt: We analyze in detail calibration and pricing performed within the framework of local stochastic volatility LSV models, which have become the industry market standard for FX and equity markets. We present the main arguments for the need of having such models, and address the question whether jumps have to be included. We include a comprehensive literature overview, and focus our exposition on important details related to calibration procedures and option pricing using PDEs or PIDEs derived from LSV models. We describe calibration procedures, with special attention given to usage and solution of corresponding forward Kolmogorov PDE/PIDE, and outline powerful algorithms for estimation of model parameters. Emphasis is placed on presenting practical details regarding the setup and the numerical solution of both forward and backward PDEs/PIDEs obtained from the LSV models. Consequently we discuss specifics (based on our experience and best practices from literature) regarding choice of boundary conditions, construction of nonuniform spatial grids and adaptive temporal grids, selection of efficient and appropriate finite difference schemes (with possible enhancements), etc. We also show how to practically integrate specific features of various types of financial instruments within calibration and pricing settings. We consider all questions and topics identified as most relevant during the selection, calibration and pricing procedures associated with local stochastic volatility models, providing answers (to the best of our knowledge), and present references for deeper understanding and for additional perspectives. In a nutshell, it is our intention to present here an effective roadmap for a successful LSV journey.
Download or read book Smooth Calibration in Local Volatility with Jumps written by Gilles Boya and published by . This book was released on 2016 with total page 11 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this article is to provide tools to calibrate a smooth local volatility surface in the presence of jumps. First we provide techniques to approximate the value of European options in a local volatility model with jumps, then we propose a quick and robust fixed point algorithm combined with this method to build smooth local volatility surfaces.
