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Can the implied volatility surface move by parallel shifts? Finance and Stochastics
"... Abstract. This note explores the analogy between the dynamics of the interest rate term structure and the implied volatility surface of a stock. In particular, we prove an impossibility theorem conjectured by Steve Ross. Implied volatility and smile asymptotics and long rates JEL Classification: G1 ..."
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Abstract. This note explores the analogy between the dynamics of the interest rate term structure and the implied volatility surface of a stock. In particular, we prove an impossibility theorem conjectured by Steve Ross. Implied volatility and smile asymptotics and long rates JEL Classification: G13 Mathematics Subject Classification
From Implied to Spot Volatilities
, 2004
"... This paper is concerned with the link between spot and implied volatilities. The main result is the derivation of the stochastic differential equation driving the spot volatility based on the shape of the implied volatility surface. This equation is a consequence of noarbitrage constraints on the ..."
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This paper is concerned with the link between spot and implied volatilities. The main result is the derivation of the stochastic differential equation driving the spot volatility based on the shape of the implied volatility surface. This equation is a consequence of noarbitrage constraints on the implied volatility surface right before expiry. We investigate the regularity of this surface at maturity in the case of the Constant Elasticity of Variance and Heston models. We also show that a simple way to link spot and implied volatilities is to relate the coefficients of the implied volatility surface Taylor expansion to the coefficients of a certain chaos expansion of the spot volatility process. As a byproduct, we give expansions for the implied volatility surface for a general stochastic volatility model. 1
LOCAL VOLATILITY DYNAMIC MODELS
, 2007
"... This paper is concerned with the characterization of arbitrage free dynamic stochastic models for the equity markets when Itô stochastic differential equations are used to model the dynamics of a set of basic instruments including, but not limited to, the underliers. We study these market models i ..."
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Cited by 9 (3 self)
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This paper is concerned with the characterization of arbitrage free dynamic stochastic models for the equity markets when Itô stochastic differential equations are used to model the dynamics of a set of basic instruments including, but not limited to, the underliers. We study these market models in the framework of the HJM philosophy originally articulated for Treasury bond markets. The approach to dynamic equity models which we follow was originally advocated by Derman and Kani in a rather informal way. The present paper can be viewed as a rigorous development of this program, with explicit formulae, rigorous proofs and numerical examples.
A Hull and White Formula for a General Stochastic Volatility JumpDiffusion Model with Applications to the Study of the ShortTime Behavior of the Implied Volatility
, 2008
"... We obtain a Hull and White type formula for a general jumpdiffusion stochastic volatility model, where the involved stochastic volatility process is correlated not only with the Brownian motion driving the asset price but also with the asset price jumps. Towards this end, we establish an anticipati ..."
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We obtain a Hull and White type formula for a general jumpdiffusion stochastic volatility model, where the involved stochastic volatility process is correlated not only with the Brownian motion driving the asset price but also with the asset price jumps. Towards this end, we establish an anticipative Itô’s formula, using Malliavin calculus techniques for Lévy processes on the canonical space. As an application, we show that the dependence of the volatility process on the asset price jumps has no effect on the shorttime behavior of the atthemoney implied volatility skew.
Asymptotic and non asymptotic approximations for option valuation
"... We give a broad overview of approximation methods to derive analytical formulas for accurate and quick evaluation of option prices. We compare different approaches, from the theoretical point of view regarding the tools they require, and also from the numerical point of view regarding their perform ..."
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We give a broad overview of approximation methods to derive analytical formulas for accurate and quick evaluation of option prices. We compare different approaches, from the theoretical point of view regarding the tools they require, and also from the numerical point of view regarding their performances. In the case of local volatility models with general timedependency, we derive new formulas using the local volatility function at the midpoint between strike and spot: in general, our approximations outperform previous ones by Hagan and HenryLabordère. We also provide approximations of the option delta. 1.
The implied volatility surface does not move by parallel shifts. Working Paper
, 2006
"... Abstract. This note explores the analogy between the dynamics of the interest rate term structure and the implied volatility surface of a stock. In particular, we prove an impossibility theorem conjectured by Steve Ross. 1. ..."
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Abstract. This note explores the analogy between the dynamics of the interest rate term structure and the implied volatility surface of a stock. In particular, we prove an impossibility theorem conjectured by Steve Ross. 1.
CAN THERE BE AN EXPLICIT FORMULA FOR IMPLIED VOLATILITY?
, 2012
"... It is “well known” that there is no explicit expression for the BlackScholes implied volatility. We prove that, as a function of underlying, strike, and call price, implied volatility does not belong to the class of Dfinite functions. This does not rule out all explicit expressions, but shows that ..."
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It is “well known” that there is no explicit expression for the BlackScholes implied volatility. We prove that, as a function of underlying, strike, and call price, implied volatility does not belong to the class of Dfinite functions. This does not rule out all explicit expressions, but shows that implied volatility does not belong to a certain large class, which contains many elementary functions and classical special functions.
Small time central limit theorems for semimartingales with applications
, 2012
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ArbitrageFree Models for VIX and Equity Derivatives
, 2015
"... This thesis must be used in accordance with the provisions of the Copyright Act 1968. Reproduction of material protected by copyright may be an infringement of copyright and copyright owners may be entitled to take legal action against persons who infringe their copyright. Section 51 (2) of the Copy ..."
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This thesis must be used in accordance with the provisions of the Copyright Act 1968. Reproduction of material protected by copyright may be an infringement of copyright and copyright owners may be entitled to take legal action against persons who infringe their copyright. Section 51 (2) of the Copyright Act permits an authorized officer of a university library or archives to provide a copy (by communication or otherwise) of an unpublished thesis kept in the library or archives, to a person who satisfies the authorized officer that he or she requires the reproduction for the purposes of research or study. The Copyright Act grants the creator of a work a number of moral rights, specifically the right of attribution, the right against false attribution and the right of integrity. You may infringe the author’s moral rights if you: fail to acknowledge the author of this thesis if you quote sections from the work attribute this thesis to another author subject this thesis to derogatory treatment which may prejudice the author’s reputation
Implied Volatility Surface Simulation with Tangent Lévy Models
, 2014
"... With the recent developments of a liquid derivative market, as well as the demands for an improved risk management framework post the financial crisis, it is becoming increasingly important to consistently model the implied volatility dynamics of an asset. Many attempts have been made on this front, ..."
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With the recent developments of a liquid derivative market, as well as the demands for an improved risk management framework post the financial crisis, it is becoming increasingly important to consistently model the implied volatility dynamics of an asset. Many attempts have been made on this front, but few manage to exclude arbitrage opportunities with reasonable tractability. In this thesis, we present two approaches based on tangent Lévy models to achieve the task. One of the biggest advantages of tangent Lévy models is that, by using the tangent process ’ jump density as the codebook to describe the option price dynamics, it enables an explicit expression of the noarbitrage conditions, hence allows for tractable implementation. Our first approach is based on the tangent Lévy model with tangent processes being derived from the double exponential process. This approach is easy to implement given the small number of parameters and the availability of an analytical pricing formula. In the second approach, the tangent process takes only finitely many jump sizes. With