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Asymptotic Skew Under Stochastic Volatility
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Book Synopsis Asymptotic Skew Under Stochastic Volatility by : Antoine (Jack) Jacquier
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.
Book Synopsis Short-Term At-the-Money Asymptotics Under Stochastic Volatility Models by : Omar El Euch
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.
Book Synopsis How Leverage Shifts and Scales a Volatility Skew by : Roger Lee
Download or read book How Leverage Shifts and Scales a Volatility Skew written by Roger Lee and published by . This book was released on 2015 with total page 15 pages. Available in PDF, EPUB and Kindle. Book excerpt: To model leveraged investments such as leveraged ETFs, define the beta-leveraged product on a positive semimartingale S to be the stochastic exponential of beta times the stochastic logarithm of S.In various asymptotic regimes, we relate rigorously the implied volatility surfaces of the beta-leveraged product and the underlying S, via explicit shifting/scaling transformations. In particular, a family of regimes with jump risk admit a shift coefficient of -3/2, unlike the previously conjectured 1/2 shift. The 1/2, we prove, holds in a family of continuous (including fBm-driven) stochastic volatility regimes at short expiry and at small volatility-of-volatility. In another regime, which does not admit a simple spatial shifting/scaling rule, we find an expiry scaling together with a spatial transformation.
Book Synopsis Asymptotic Chaos Expansions in Finance by : David Nicolay
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.
Book Synopsis Asymptotic Behavior of Stochastic Volatility Models by : Max Souza
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:
Book Synopsis Analytically Tractable Stochastic Stock Price Models by : Archil Gulisashvili
Download or read book Analytically Tractable Stochastic Stock Price Models written by Archil Gulisashvili and published by Springer. This book was released on 2012-09-05 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Asymptotic analysis of stochastic stock price models is the central topic of the present volume. Special examples of such models are stochastic volatility models, that have been developed as an answer to certain imperfections in a celebrated Black-Scholes model of option pricing. In a stock price model with stochastic volatility, the random behavior of the volatility is described by a stochastic process. For instance, in the Hull-White model the volatility process is a geometric Brownian motion, the Stein-Stein model uses an Ornstein-Uhlenbeck process as the stochastic volatility, and in the Heston model a Cox-Ingersoll-Ross process governs the behavior of the volatility. One of the author's main goals is to provide sharp asymptotic formulas with error estimates for distribution densities of stock prices, option pricing functions, and implied volatilities in various stochastic volatility models. The author also establishes sharp asymptotic formulas for the implied volatility at extreme strikes in general stochastic stock price models. The present volume is addressed to researchers and graduate students working in the area of financial mathematics, analysis, or probability theory. The reader is expected to be familiar with elements of classical analysis, stochastic analysis and probability theory.
Book Synopsis Asymptotic Methods for Computing Implied Volatilities Under Stochastic Volatility by : Alexey Medvedev
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).
Book Synopsis Recent Advances in Applied Probability by : Ricardo Baeza-Yates
Download or read book Recent Advances in Applied Probability written by Ricardo Baeza-Yates and published by Springer Science & Business Media. This book was released on 2006-02-28 with total page 497 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Derivatives in Financial Markets with Stochastic Volatility by : Jean-Pierre Fouque
Download or read book Derivatives in Financial Markets with Stochastic Volatility written by Jean-Pierre Fouque and published by Cambridge University Press. This book was released on 2000-07-03 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book, first published in 2000, addresses pricing and hedging derivative securities in uncertain and changing market volatility.
Book Synopsis Asymptotic Analysis of New Stochastic Volatility Models by : Fangwei Shi
Download or read book Asymptotic Analysis of New Stochastic Volatility Models written by Fangwei Shi and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A General Asymptotic Implied Volatility for Stochastic Volatility Models by : Pierre Henry-Labordere
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.
Book Synopsis Pricing Volatility Options Under Stochastic Skew with Application to the VIX Index by : Jacinto Marabel Romo
Download or read book Pricing Volatility Options Under Stochastic Skew with Application to the VIX Index written by Jacinto Marabel Romo and published by . This book was released on 2015 with total page 31 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent years there has been a remarkable growth of volatility options. In particular, VIX options are among the most actively trading contracts at CBOE. These options exhibit upward sloping volatility skew and the shape of the skew is largely independent of the volatility level. To take into account these stylized facts, this article introduces a novel two-factor stochastic volatility model with mean reversion that accounts for stochastic skew consistent with empirical evidence. Importantly, the model is analytically tractable. In this sense, I solve the pricing problem corresponding to standard-start, as well as to forward-start European options through the Fast Fourier Transform.To illustrate the practical performance of the model, I calibrate the model parameters to the quoted prices of European options on the VIX index. The calibration results are fairly good indicating the ability of the model to capture the shape of the implied volatility skew associated with VIX options.
Book Synopsis Modelling and Simulation of Stochastic Volatility in Finance by : Christian Kahl
Download or read book Modelling and Simulation of Stochastic Volatility in Finance written by Christian Kahl and published by Universal-Publishers. This book was released on 2008 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt: The famous Black-Scholes model was the starting point of a new financial industry and has been a very important pillar of all options trading since. One of its core assumptions is that the volatility of the underlying asset is constant. It was realised early that one has to specify a dynamic on the volatility itself to get closer to market behaviour. There are mainly two aspects making this fact apparent. Considering historical evolution of volatility by analysing time series data one observes erratic behaviour over time. Secondly, backing out implied volatility from daily traded plain vanilla options, the volatility changes with strike. The most common realisations of this phenomenon are the implied volatility smile or skew. The natural question arises how to extend the Black-Scholes model appropriately. Within this book the concept of stochastic volatility is analysed and discussed with special regard to the numerical problems occurring either in calibrating the model to the market implied volatility surface or in the numerical simulation of the two-dimensional system of stochastic differential equations required to price non-vanilla financial derivatives. We introduce a new stochastic volatility model, the so-called Hyp-Hyp model, and use Watanabe's calculus to find an analytical approximation to the model implied volatility. Further, the class of affine diffusion models, such as Heston, is analysed in view of using the characteristic function and Fourier inversion techniques to value European derivatives.
Book Synopsis Implied Volatility Asymptotics Under Affine Stochastic Volatility Models by : Antoine Jacquier
Download or read book Implied Volatility Asymptotics Under Affine Stochastic Volatility Models written by Antoine Jacquier and published by . This book was released on 2010 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Asymptotic Behaviour of Stochastic Volatility Models by : Max Souza
Download or read book Asymptotic Behaviour of Stochastic Volatility Models written by Max Souza and published by . This book was released on 2005 with total page 5 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis On Error Estimates for Asymptotic Expansions with Malliavin Weights -- Application to Stochastic Volatility Model by : Akihiko Takahashi
Download or read book On Error Estimates for Asymptotic Expansions with Malliavin Weights -- Application to Stochastic Volatility Model written by Akihiko Takahashi and published by . This book was released on 2014 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper proposes a unified method for precise estimates of the error bounds in asymptotic expansions of an option price and its Greeks (sensitivities) under a stochastic volatility model. More generally, we also derive an error estimate for an asymptotic expansion around a partially elliptic diffusion under a multi-dimensional setting, which includes various important models in finance as the special cases.In particular, we take the Malliavin calculus approach, and estimate the error bounds for the Malliavin weights of both the coefficient and the residual terms in the expansions by effectively applying the properties of Kusuoka-Stroock functions.
Book Synopsis A Stochastic Volatility Model with GH Skew Student's T-distribution by :
Download or read book A Stochastic Volatility Model with GH Skew Student's T-distribution written by and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
