Implied and Local Volatilities Under Stochastic Volatility PDF Download

Author: Roger W. Lee
Publisher:
ISBN:
Category :
Languages : en
Pages : 186

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Author: Matthew Lorig
Publisher:
ISBN:
Category :
Languages : en
Pages : 36

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We consider an asset whose risk-neutral dynamics are described by a general class of local-stochastic volatility models and derive a family of asymptotic expansions for European-style option prices and implied volatilities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under five different model dynamics: CEV local volatility, quadratic local volatility, Heston stochastic volatility, 3/2 stochastic volatility, and SABR local-stochastic volatility.

Author: Lorenzo Bergomi
Publisher: CRC Press
ISBN: 1482244071
Category : Business & Economics
Languages : en
Pages : 520

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Packed with insights, Lorenzo Bergomi's Stochastic Volatility Modeling explains how stochastic volatility is used to address issues arising in the modeling of derivatives, including:Which trading issues do we tackle with stochastic volatility? How do we design models and assess their relevance? How do we tell which models are usable and when does c

Author: Lorenzo Bergomi
Publisher:
ISBN:
Category :
Languages : en
Pages : 86

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This is Chapter 2 of Stochastic Volatility Modeling, published by CRC/Chapman & Hall.In this chapter the local volatility model is surveyed as a market model for the underlying together with its associated vanilla options.First, relationships of implied to local volatilities are derived, as well as approximations for skew and curvature. Exact and approximate techniques for taking dividends into account are presented.We then turn to the dynamics of the local volatility model. We introduce the Skew Tickiness Ratio (SSR) and derive approximate formulas for the SSR and volatilities of volatilities in the local volatility model.We also examine future skews.We then consider the delta and carry P&L of a hedged option position. We derive the expression of the market-model delta of the local volatility model and discuss the relationship between sticky-strike and market-model deltas. We characterize the gamma/theta break-even levels of a hedged position and show that the local volatility model is indeed a market model.We then derive the expression of the vega-hedge portfolio.Markov-functional models are considered next.Finally, we survey the Uncertain Volatility Model and its usage.A digest summarizes key points.

Author: Ricardo Baeza-Yates
Publisher: Springer Science & Business Media
ISBN: 0387233946
Category : Mathematics
Languages : en
Pages : 497

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Applied probability is a broad research area that is of interest to scientists in diverse disciplines in science and technology, including: anthropology, biology, communication theory, economics, epidemiology, finance, geography, linguistics, medicine, meteorology, operations research, psychology, quality control, sociology, and statistics. Recent Advances in Applied Probability is a collection of survey articles that bring together the work of leading researchers in applied probability to present current research advances in this important area. This volume will be of interest to graduate students and researchers whose research is closely connected to probability modelling and their applications. It is suitable for one semester graduate level research seminar in applied probability.

Author: Alan L. Lewis
Publisher:
ISBN:
Category : Business & Economics
Languages : en
Pages : 372

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Author: Carol Alexander
Publisher:
ISBN:
Category :
Languages : en
Pages : 25

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There are two unique volatility surfaces associated with any arbitrage-free set of standard European option prices, the implied volatility surface and the local volatility surface. Several papers have discussed the stochastic differential equations for implied volatilities that are consistent with these option prices but the static and dynamic no-arbitrage conditions are complex, mainly due to the large (or even infinite) dimensions of the state probability space. These no-arbitrage conditions are also instrument-specific and have been specified for some simple classes of options. However, the problem is easier to resolve when we specify stochastic differential equations for local volatilities instead. And the option prices and hedge ratios that are obtained by making local volatility stochastic are identical to those obtained by making instantaneous volatility or implied volatility stochastic. After proving that there is a one-to-one correspondence between the stochastic implied volatility and stochastic local volatility approaches, we derive a simple dynamic no-arbitrage condition for the stochastic local volatility model that is model-specific. The condition is very easy to check in local volatility models having only a few stochastic parameters.

Author: Alexey Medvedev
Publisher:
ISBN:
Category :
Languages : en
Pages : 38

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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).

Author: Andrey Itkin
Publisher: World Scientific
ISBN: 9811212783
Category : Business & Economics
Languages : en
Pages : 205

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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.

Author: Elisa Alos
Publisher: CRC Press
ISBN: 1000403513
Category : Mathematics
Languages : en
Pages : 350

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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.