Local Variance Gamma Revisited

Local Variance Gamma Revisited
Author: Markus Falck
Publisher:
Total Pages: 35
Release: 2017
Genre:
ISBN:

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In this paper we develop a new method for implied volatility surface construction for FX options. The methodology is based on the local variance gamma model developed by Carr (2008). Our approach is to solve a simplified "one-step" version of the Dupire equation analytically under the assumption of a continuous five parameter diffusion function. The unique solution to this equation can be interpreted as a continuous representation of option prices, defined for strikes in an arbitrarily large range. The derived price functions are C^2 -positive, arbitrage-free by construction, and they do not depend on the strike discretization. By using a least-square approach, we calibrate price functions to Reuters quoted FX volatility smiles. Our results suggest that the model allows for very rapid calibration; using a Levenberg-Marquardt algorithm we measure the average calibration time to less than 1 ms for one expiry on a standard personal computer.We also extend our model to allow for interpolation between maturities and present sufficient conditions for absence of calendar spread arbitrage. In order to generate the whole implied volatility surface, we suggest a simple, fast and yet market-consistent technique allowing for arbitrage-free interpolation of calibrated price functions in the maturity dimension.The methodology is tested against EURUSD and EURSEK options, where we show that the model has the capability to produce volatility surfaces which fit market quotes with an error of few volatility basis points. We then apply the methodology to pricing variance swaps.

Fitting Local Volatility: Analytic And Numerical Approaches In Black-scholes And Local Variance Gamma Models

Fitting Local Volatility: Analytic And Numerical Approaches In Black-scholes And Local Variance Gamma Models
Author: Andrey Itkin
Publisher: World Scientific
Total Pages: 205
Release: 2020-01-22
Genre: Business & Economics
ISBN: 9811212783

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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.

Model-free Hedging

Model-free Hedging
Author: Pierre Henry-Labordere
Publisher: CRC Press
Total Pages: 115
Release: 2017-05-25
Genre: Mathematics
ISBN: 1351666223

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Model-free Hedging: A Martingale Optimal Transport Viewpoint focuses on the computation of model-independent bounds for exotic options consistent with market prices of liquid instruments such as Vanilla options. The author gives an overview of Martingale Optimal Transport, highlighting the differences between the optimal transport and its martingale counterpart. This topic is then discussed in the context of mathematical finance.

Volatility and Correlation

Volatility and Correlation
Author: Riccardo Rebonato
Publisher: John Wiley & Sons
Total Pages: 864
Release: 2005-07-08
Genre: Business & Economics
ISBN: 0470091401

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In Volatility and Correlation 2nd edition: The Perfect Hedger and the Fox, Rebonato looks at derivatives pricing from the angle of volatility and correlation. With both practical and theoretical applications, this is a thorough update of the highly successful Volatility & Correlation – with over 80% new or fully reworked material and is a must have both for practitioners and for students. The new and updated material includes a critical examination of the ‘perfect-replication’ approach to derivatives pricing, with special attention given to exotic options; a thorough analysis of the role of quadratic variation in derivatives pricing and hedging; a discussion of the informational efficiency of markets in commonly-used calibration and hedging practices. Treatment of new models including Variance Gamma, displaced diffusion, stochastic volatility for interest-rate smiles and equity/FX options. The book is split into four parts. Part I deals with a Black world without smiles, sets out the author’s ‘philosophical’ approach and covers deterministic volatility. Part II looks at smiles in equity and FX worlds. It begins with a review of relevant empirical information about smiles, and provides coverage of local-stochastic-volatility, general-stochastic-volatility, jump-diffusion and Variance-Gamma processes. Part II concludes with an important chapter that discusses if and to what extent one can dispense with an explicit specification of a model, and can directly prescribe the dynamics of the smile surface. Part III focusses on interest rates when the volatility is deterministic. Part IV extends this setting in order to account for smiles in a financially motivated and computationally tractable manner. In this final part the author deals with CEV processes, with diffusive stochastic volatility and with Markov-chain processes. Praise for the First Edition: “In this book, Dr Rebonato brings his penetrating eye to bear on option pricing and hedging.... The book is a must-read for those who already know the basics of options and are looking for an edge in applying the more sophisticated approaches that have recently been developed.” —Professor Ian Cooper, London Business School “Volatility and correlation are at the very core of all option pricing and hedging. In this book, Riccardo Rebonato presents the subject in his characteristically elegant and simple fashion...A rare combination of intellectual insight and practical common sense.” —Anthony Neuberger, London Business School

The Laplace Distribution and Generalizations

The Laplace Distribution and Generalizations
Author: Samuel Kotz
Publisher: Springer Science & Business Media
Total Pages: 358
Release: 2012-12-06
Genre: Mathematics
ISBN: 146120173X

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This book describes the inferential and modeling advantages that this distribution, together with its generalizations and modifications, offers. The exposition systematically unfolds with many examples, tables, illustrations, and exercises. A comprehensive index and extensive bibliography also make this book an ideal text for a senior undergraduate and graduate seminar on statistical distributions, or for a short half-term academic course in statistics, applied probability, and finance.

Mathematical Reviews

Mathematical Reviews
Author:
Publisher:
Total Pages: 940
Release: 2006
Genre: Mathematics
ISBN:

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