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Author: Alexey Medvedev Publisher: ISBN: Category : Languages : en Pages : 38
Book Description
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: Alexey Medvedev Publisher: ISBN: Category : Languages : en Pages : 38
Book Description
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: David Nicolay Publisher: Springer ISBN: 1447165063 Category : Mathematics Languages : en Pages : 503
Book Description
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.
Author: Omar El Euch Publisher: ISBN: Category : Languages : en Pages : 20
Book Description
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.
Author: Marc Decamps Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
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.
Author: Alexey Medvedev Publisher: ISBN: Category : Languages : en Pages :
Book Description
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).
Author: Pierre Henry-Labordere Publisher: ISBN: Category : Languages : en Pages : 35
Book Description
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.
Author: Peter K. Friz Publisher: Springer ISBN: 3319116053 Category : Mathematics Languages : en Pages : 590
Book Description
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.
Author: Antoine (Jack) Jacquier Publisher: ISBN: Category : Languages : en Pages : 9
Book Description
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.
Author: Louis Paulot Publisher: ISBN: Category : Languages : en Pages : 27
Book Description
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.
Author: Alexey Medvedev
Author: Alexey Medvedev
Author: David Nicolay
Author: Omar El Euch
Author: Marc Decamps
Author: Alexey Medvedev
Author: Pierre Henry-Labordere
Author: Roger W. Lee
Author: Peter K. Friz
Author: Antoine (Jack) Jacquier
Author: Louis Paulot