Download or read book Asymptotic Methods for Computing Implied Volatilities Under Stochastic Volatility written by Alexey Medvedev and published by . This book was released on 2008 with total page 38 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we propose an analytical formula for computing implied volatilities of European options based on their short term asymptotics. The analysis is performed in a general framework with local and stochastic volatility. Assuming CEV volatility of volatility we first obtain a quasi-analytical solution for the limit of implied volatilities as time-to-maturity goes to zero (instanteneous implied volatility). Then we develop our analytical formula in the form of a local transformation of the instanteneous implied volatility. Numerical experiments suggests that this approximation is extremely accurate at short maturities (one or two month). We further introduce a class of models under which this method is accurate even for long maturity options. In the particular case of SABR model we improve the formula derived in Hagan et al. (2002).

Download or read book Asymptotic Chaos Expansions in Finance written by David Nicolay and published by Springer. This book was released on 2014-11-25 with total page 503 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic instantaneous volatility models such as Heston, SABR or SV-LMM have mostly been developed to control the shape and joint dynamics of the implied volatility surface. In principle, they are well suited for pricing and hedging vanilla and exotic options, for relative value strategies or for risk management. In practice however, most SV models lack a closed form valuation for European options. This book presents the recently developed Asymptotic Chaos Expansions methodology (ACE) which addresses that issue. Indeed its generic algorithm provides, for any regular SV model, the pure asymptotes at any order for both the static and dynamic maps of the implied volatility surface. Furthermore, ACE is programmable and can complement other approximation methods. Hence it allows a systematic approach to designing, parameterising, calibrating and exploiting SV models, typically for Vega hedging or American Monte-Carlo. Asymptotic Chaos Expansions in Finance illustrates the ACE approach for single underlyings (such as a stock price or FX rate), baskets (indexes, spreads) and term structure models (especially SV-HJM and SV-LMM). It also establishes fundamental links between the Wiener chaos of the instantaneous volatility and the small-time asymptotic structure of the stochastic implied volatility framework. It is addressed primarily to financial mathematics researchers and graduate students, interested in stochastic volatility, asymptotics or market models. Moreover, as it contains many self-contained approximation results, it will be useful to practitioners modelling the shape of the smile and its evolution.

Download or read book Short Term At the Money Asymptotics Under Stochastic Volatility Models written by Omar El Euch and published by . This book was released on 2019 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt: A small-time Edgeworth expansion of the density of an asset price is given under a general stochastic volatility model, from which asymptotic expansions of put option prices and at-the-money implied volatilities follow. A limit theorem for at-the-money implied volatility skew and curvature is also given as a corollary. The rough Bergomi model is treated as an example.

Download or read book Atomic Implied Volatilities written by Marc Decamps and published by . This book was released on 2009 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this note, we present a novel approach to derive asymptotics for Black implied volatilities under the same generic model as proposed in Antonov and Misirpashaev (2009). We perform a time substitution as used by Duru and Kleinert (1979) to calculate the path integral formulation of the H-atom. We demonstrate that the method provides asymptotic implied volatility formula comparable to the result of Hagan and Woodward (1999) for local volatility models and Hagan et al. (2001) for stochastic volatility models. We also discuss possible application to the pricing of basket options. The method is presented as an alternative to Markov projection as introduced by Piterbarg (2006) and is claimed to be applicable to a wide range of numerical problems arising in finance.

Download or read book Approximation and Calibration of Short Term Implied Volatilities Under Jump Diffusion Stochastic Volatility written by Alexey Medvedev and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We derive an asymptotic expansion formula for option implied volatility under a two-factor jump-diffusion stochastic volatility model when time-to-maturity is small. We further propose a simple calibration procedure of an arbitrary parametric model to short-term near-the-money implied volatilities. An important advantage of our approximation is that it is free of the unobserved spot volatility. Therefore, the model can be calibrated on option data pooled across different calendar dates to extract information from the dynamics of the implied volatility smile. An example of calibration to a sample of Samp;P 500 option prices is provided. (JEL G12).

Download or read book A General Asymptotic Implied Volatility for Stochastic Volatility Models written by Pierre Henry-Labordere and published by . This book was released on 2005 with total page 35 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we derive a general asymptotic implied volatility at the first-order for any stochastic volatility model using the heat kernel expansion on a Riemann manifold endowed with an Abelian connection. This formula is particularly useful for the calibration procedure. As an application, we obtain an asymptotic smile for a SABR model with a mean-reversion term, called lambda-SABR, corresponding in our geometric framework to the Poincare hyperbolic plane. When the lambda-SABR model degenerates into the SABR-model, we show that our asymptotic implied volatility is a better approximation than the classical Hagan-al expression. Furthermore, in order to show the strength of this geometric framework, we give an exact solution of the SABR model with beta=0 or 1. In a next paper, we will show how our method can be applied in other contexts such as the derivation of an asymptotic implied volatility for a Libor market model with a stochastic volatility.

Download or read book Multiscale Stochastic Volatility for Equity Interest Rate and Credit Derivatives written by Jean-Pierre Fouque and published by Cambridge University Press. This book was released on 2011-09-29 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: Building upon the ideas introduced in their previous book, Derivatives in Financial Markets with Stochastic Volatility, the authors study the pricing and hedging of financial derivatives under stochastic volatility in equity, interest-rate, and credit markets. They present and analyze multiscale stochastic volatility models and asymptotic approximations. These can be used in equity markets, for instance, to link the prices of path-dependent exotic instruments to market implied volatilities. The methods are also used for interest rate and credit derivatives. Other applications considered include variance-reduction techniques, portfolio optimization, forward-looking estimation of CAPM 'beta', and the Heston model and generalizations of it. 'Off-the-shelf' formulas and calibration tools are provided to ease the transition for practitioners who adopt this new method. The attention to detail and explicit presentation make this also an excellent text for a graduate course in financial and applied mathematics.

Download or read book Implied and Local Volatilities Under Stochastic Volatility written by Roger W. Lee and published by . This book was released on 2000 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Large Deviations and Asymptotic Methods in Finance written by Peter K. Friz and published by Springer. This book was released on 2015-06-16 with total page 590 pages. Available in PDF, EPUB and Kindle. Book excerpt: Topics covered in this volume (large deviations, differential geometry, asymptotic expansions, central limit theorems) give a full picture of the current advances in the application of asymptotic methods in mathematical finance, and thereby provide rigorous solutions to important mathematical and financial issues, such as implied volatility asymptotics, local volatility extrapolation, systemic risk and volatility estimation. This volume gathers together ground-breaking results in this field by some of its leading experts. Over the past decade, asymptotic methods have played an increasingly important role in the study of the behaviour of (financial) models. These methods provide a useful alternative to numerical methods in settings where the latter may lose accuracy (in extremes such as small and large strikes, and small maturities), and lead to a clearer understanding of the behaviour of models, and of the influence of parameters on this behaviour. Graduate students, researchers and practitioners will find this book very useful, and the diversity of topics will appeal to people from mathematical finance, probability theory and differential geometry.

Download or read book Asymptotic Skew Under Stochastic Volatility written by Antoine (Jack) Jacquier and published by . This book was released on 2007 with total page 9 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this paper is to improve and discuss the asymptotic formula of the implied volatility (when maturity goes to infinity) derived by A.Lewis. Indeed, we are here able to provide more accurate at-the-money asymptotics. Such analytic formulas are useful for calibration.

Download or read book Asymptotic Implied Volatility at the Second Order with Application to the SABR Model written by Louis Paulot and published by . This book was released on 2016 with total page 27 pages. Available in PDF, EPUB and Kindle. Book excerpt: We provide a general method to compute a Taylor expansion in time of implied volatility for stochastic volatility models, using a heat kernel expansion. Beyond the order 0 implied volatility which is already known, we compute the first order correction exactly at all strikes from the scalar coefficient of the heat kernel expansion. Furthermore, the first correction in the heat kernel expansion gives the second order correction for implied volatility, which we also give exactly at all strikes. As an application, we compute this asymptotic expansion at order 2 for the SABR model.

Download or read book Asymptotic Methods in the Theory of Stochastic Differential Equations written by A. V. Skorokhod and published by American Mathematical Soc.. This book was released on 2009-01-07 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: Ergodic theorems: General ergodic theorems Densities for transition probabilities and resolvents for Markov solutions of stochastic differential equations Ergodic theorems for one-dimensional stochastic equations Ergodic theorems for solutions of stochastic equations in $R^d$ Asymptotic behavior of systems of stochastic equations containing a small parameter: Equations with a small right-hand side Processes with rapid switching Averaging over variables for systems of stochastic differential equations Stability. Linear systems: Stability of sample paths of homogeneous Markov processes Linear equations in $R^d$ and the stochastic semigroups connected with them. Stability Stability of solutions of stochastic differential equations Linear stochastic equations in Hilbert space. Stochastic semigroups. Stability: Linear equations with bounded coefficients Strong stochastic semigroups with second moments Stability Bibliography

Download or read book Malliavin Calculus in Finance written by Elisa Alos and published by CRC Press. This book was released on 2021-07-14 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: Malliavin Calculus in Finance: Theory and Practice aims to introduce the study of stochastic volatility (SV) models via Malliavin Calculus. Malliavin calculus has had a profound impact on stochastic analysis. Originally motivated by the study of the existence of smooth densities of certain random variables, it has proved to be a useful tool in many other problems. In particular, it has found applications in quantitative finance, as in the computation of hedging strategies or the efficient estimation of the Greeks. The objective of this book is to offer a bridge between theory and practice. It shows that Malliavin calculus is an easy-to-apply tool that allows us to recover, unify, and generalize several previous results in the literature on stochastic volatility modeling related to the vanilla, the forward, and the VIX implied volatility surfaces. It can be applied to local, stochastic, and also to rough volatilities (driven by a fractional Brownian motion) leading to simple and explicit results. Features Intermediate-advanced level text on quantitative finance, oriented to practitioners with a basic background in stochastic analysis, which could also be useful for researchers and students in quantitative finance Includes examples on concrete models such as the Heston, the SABR and rough volatilities, as well as several numerical experiments and the corresponding Python scripts Covers applications on vanillas, forward start options, and options on the VIX. The book also has a Github repository with the Python library corresponding to the numerical examples in the text. The library has been implemented so that the users can re-use the numerical code for building their examples. The repository can be accessed here: https://bit.ly/2KNex2Y.

Download or read book Financial Signal Processing and Machine Learning written by Ali N. Akansu and published by John Wiley & Sons. This book was released on 2016-05-31 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: The modern financial industry has been required to deal with large and diverse portfolios in a variety of asset classes often with limited market data available. Financial Signal Processing and Machine Learning unifies a number of recent advances made in signal processing and machine learning for the design and management of investment portfolios and financial engineering. This book bridges the gap between these disciplines, offering the latest information on key topics including characterizing statistical dependence and correlation in high dimensions, constructing effective and robust risk measures, and their use in portfolio optimization and rebalancing. The book focuses on signal processing approaches to model return, momentum, and mean reversion, addressing theoretical and implementation aspects. It highlights the connections between portfolio theory, sparse learning and compressed sensing, sparse eigen-portfolios, robust optimization, non-Gaussian data-driven risk measures, graphical models, causal analysis through temporal-causal modeling, and large-scale copula-based approaches. Key features: Highlights signal processing and machine learning as key approaches to quantitative finance. Offers advanced mathematical tools for high-dimensional portfolio construction, monitoring, and post-trade analysis problems. Presents portfolio theory, sparse learning and compressed sensing, sparsity methods for investment portfolios. including eigen-portfolios, model return, momentum, mean reversion and non-Gaussian data-driven risk measures with real-world applications of these techniques. Includes contributions from leading researchers and practitioners in both the signal and information processing communities, and the quantitative finance community.

Download or read book Applied Conic Finance written by Dilip Madan and published by Cambridge University Press. This book was released on 2016-10-13 with total page 205 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Asymptotic Behavior of Stochastic Volatility Models written by Max Souza and published by . This book was released on 2005 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Partial Differential Equations of Parabolic Type written by Avner Friedman and published by Courier Corporation. This book was released on 2013-08-16 with total page 369 pages. Available in PDF, EPUB and Kindle. Book excerpt: With this book, even readers unfamiliar with the field can acquire sufficient background to understand research literature related to the theory of parabolic and elliptic equations. 1964 edition.