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.

Fitting Local Volatility

Fitting Local Volatility
Author: Andrey Itkin
Publisher:
Total Pages: 205
Release: 2020
Genre: Finance
ISBN: 9789811212772

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Generalized Integral Transforms In Mathematical Finance

Generalized Integral Transforms In Mathematical Finance
Author: Andrey Itkin
Publisher: World Scientific
Total Pages: 508
Release: 2021-10-12
Genre: Business & Economics
ISBN: 9811231753

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This book describes several techniques, first invented in physics for solving problems of heat and mass transfer, and applies them to various problems of mathematical finance defined in domains with moving boundaries. These problems include: (a) semi-closed form pricing of options in the one-factor models with time-dependent barriers (Bachelier, Hull-White, CIR, CEV); (b) analyzing an interconnected banking system in the structural credit risk model with default contagion; (c) finding first hitting time density for a reducible diffusion process; (d) describing the exercise boundary of American options; (e) calculating default boundary for the structured default problem; (f) deriving a semi-closed form solution for optimal mean-reverting trading strategies; to mention but some.The main methods used in this book are generalized integral transforms and heat potentials. To find a semi-closed form solution, we need to solve a linear or nonlinear Volterra equation of the second kind and then represent the option price as a one-dimensional integral. Our analysis shows that these methods are computationally more efficient than the corresponding finite-difference methods for the backward or forward Kolmogorov PDEs (partial differential equations) while providing better accuracy and stability.We extend a large number of known results by either providing solutions on complementary or extended domains where the solution is not known yet or modifying these techniques and applying them to new types of equations, such as the Bessel process. The book contains several novel results broadly applicable in physics, mathematics, and engineering.

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.

Option Pricing Models and Volatility Using Excel-VBA

Option Pricing Models and Volatility Using Excel-VBA
Author: Fabrice D. Rouah
Publisher: John Wiley & Sons
Total Pages: 456
Release: 2012-06-15
Genre: Business & Economics
ISBN: 1118429206

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This comprehensive guide offers traders, quants, and students the tools and techniques for using advanced models for pricing options. The accompanying website includes data files, such as options prices, stock prices, or index prices, as well as all of the codes needed to use the option and volatility models described in the book. Praise for Option Pricing Models & Volatility Using Excel-VBA "Excel is already a great pedagogical tool for teaching option valuation and risk management. But the VBA routines in this book elevate Excel to an industrial-strength financial engineering toolbox. I have no doubt that it will become hugely successful as a reference for option traders and risk managers." —Peter Christoffersen, Associate Professor of Finance, Desautels Faculty of Management, McGill University "This book is filled with methodology and techniques on how to implement option pricing and volatility models in VBA. The book takes an in-depth look into how to implement the Heston and Heston and Nandi models and includes an entire chapter on parameter estimation, but this is just the tip of the iceberg. Everyone interested in derivatives should have this book in their personal library." —Espen Gaarder Haug, option trader, philosopher, and author of Derivatives Models on Models "I am impressed. This is an important book because it is the first book to cover the modern generation of option models, including stochastic volatility and GARCH." —Steven L. Heston, Assistant Professor of Finance, R.H. Smith School of Business, University of Maryland

Applied Conic Finance

Applied Conic Finance
Author: Dilip Madan
Publisher: Cambridge University Press
Total Pages: 205
Release: 2016-10-13
Genre: Mathematics
ISBN: 1316776778

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This is a comprehensive introduction to the brand new theory of conic finance, also referred to as the two-price theory, which determines bid and ask prices in a consistent and fundamentally motivated manner. Whilst theories of one price classically eliminate all risk, the concept of acceptable risks is critical to the foundations of the two-price theory which sees risk elimination as typically unattainable in a modern financial economy. Practical examples and case studies provide the reader with a comprehensive introduction to the fundamentals of the theory, a variety of advanced quantitative models, and numerous real-world applications, including portfolio theory, option positioning, hedging, and trading contexts. This book offers a quantitative and practical approach for readers familiar with the basics of mathematical finance to allow them to boldly go where no quant has gone before.

Applied Quantitative Finance

Applied Quantitative Finance
Author: Wolfgang Karl Härdle
Publisher: Springer
Total Pages: 369
Release: 2017-08-02
Genre: Business & Economics
ISBN: 3662544865

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This volume provides practical solutions and introduces recent theoretical developments in risk management, pricing of credit derivatives, quantification of volatility and copula modeling. This third edition is devoted to modern risk analysis based on quantitative methods and textual analytics to meet the current challenges in banking and finance. It includes 14 new contributions and presents a comprehensive, state-of-the-art treatment of cutting-edge methods and topics, such as collateralized debt obligations, the high-frequency analysis of market liquidity, and realized volatility. The book is divided into three parts: Part 1 revisits important market risk issues, while Part 2 introduces novel concepts in credit risk and its management along with updated quantitative methods. The third part discusses the dynamics of risk management and includes risk analysis of energy markets and for cryptocurrencies. Digital assets, such as blockchain-based currencies, have become popular b ut are theoretically challenging when based on conventional methods. Among others, it introduces a modern text-mining method called dynamic topic modeling in detail and applies it to the message board of Bitcoins. The unique synthesis of theory and practice supported by computational tools is reflected not only in the selection of topics, but also in the fine balance of scientific contributions on practical implementation and theoretical concepts. This link between theory and practice offers theoreticians insights into considerations of applicability and, vice versa, provides practitioners convenient access to new techniques in quantitative finance. Hence the book will appeal both to researchers, including master and PhD students, and practitioners, such as financial engineers. The results presented in the book are fully reproducible and all quantlets needed for calculations are provided on an accompanying website. The Quantlet platform quantlet.de, quantlet.com, quantlet.org is an integrated QuantNet environment consisting of different types of statistics-related documents and program codes. Its goal is to promote reproducibility and offer a platform for sharing validated knowledge native to the social web. QuantNet and the corresponding Data-Driven Documents-based visualization allows readers to reproduce the tables, pictures and calculations inside this Springer book.

Derivatives in Financial Markets with Stochastic Volatility

Derivatives in Financial Markets with Stochastic Volatility
Author: Jean-Pierre Fouque
Publisher: Cambridge University Press
Total Pages: 222
Release: 2000-07-03
Genre: Business & Economics
ISBN: 9780521791632

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This book, first published in 2000, addresses pricing and hedging derivative securities in uncertain and changing market volatility.