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Asymptotics for exponential Lévy processes and their volatility smile: survey and new results
 Int. J. Theor. Appl. Finance
, 2013
"... Abstract. Exponential Lévy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, etc. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes, and the corresponding implied volatility surfac ..."
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Abstract. Exponential Lévy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, etc. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes, and the corresponding implied volatility surfaces have been analyzed in some detail. In the nonasymptotic regimes, option prices are described by the LewisLipton formula which allows one to represent them as Fourier integrals; the prices can be trivially expressed in terms of their implied volatility. Recently, attempts at calculating the asymptotic limits of the implied volatility have yielded several expressions for the shorttime, longtime, and wing asymptotics. In order to study the volatility surface in required detail, in this paper we use the FX conventions and describe the implied volatility as a function of the BlackScholes delta. Surprisingly, this convention is closely related to the resolution of singularities frequently used in algebraic geometry. In this framework, we survey the literature, reformulate some known facts regarding the asymptotic behavior of the implied volatility, and present several
Arbitragefree SVI volatility surfaces
 Quantitative Finance
"... In this article, we show how to calibrate the widelyused SVI parameterization of the implied volatility smile in such a way as to guarantee the absence of static arbitrage. In particular, we exhibit a large class of arbitragefree SVI volatility surfaces with a simple closedform representation. We ..."
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Cited by 8 (4 self)
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In this article, we show how to calibrate the widelyused SVI parameterization of the implied volatility smile in such a way as to guarantee the absence of static arbitrage. In particular, we exhibit a large class of arbitragefree SVI volatility surfaces with a simple closedform representation. We demonstrate the high quality of typical SVI fits with a numerical example using recent SPX options data. 1
Shapes of implied volatility with positive mass at zero, available at arXiv:1310.1020
, 2013
"... Abstract We study the shapes of the implied volatility when the underlying distribution has an atom at zero and analyse the impact of a mass at zero on atthemoney implied volatility and the overall level of the smile. We further show that the behaviour at small strikes is uniquely determined by t ..."
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Cited by 4 (1 self)
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Abstract We study the shapes of the implied volatility when the underlying distribution has an atom at zero and analyse the impact of a mass at zero on atthemoney implied volatility and the overall level of the smile. We further show that the behaviour at small strikes is uniquely determined by the mass of the atom up to high asymptotic order, under mild assumptions on the remaining distribution on the positive real line. We investigate the structural difference with the nomassatzero case, showing how one cantheoreticallydistinguish between mass at the origin and a heavylefttailed distribution. We numerically test our modelfree results in stochastic models with absorption at the boundary, such as the CEV process, and in jumptodefault models. Note that while Lee's moment formula
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.
Modelfree implied volatility: From surface to index, forthcoming
 in International Journal of Theoretical and Applied Finance
, 2010
"... Abstract. We propose a new method for approximating the expected quadratic variation of an asset based on its option prices. The quadratic variation of an asset price is often regarded as a measure of its volatility, and its expected value under pricing measure can be understood as the market’s expe ..."
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Cited by 3 (3 self)
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Abstract. We propose a new method for approximating the expected quadratic variation of an asset based on its option prices. The quadratic variation of an asset price is often regarded as a measure of its volatility, and its expected value under pricing measure can be understood as the market’s expectation of future volatility. We utilize the relation between the asset variance and the BlackScholes implied volatility surface, and discuss the merits of this new modelfree approach compared to the CBOE procedure underlying the VIX index. The interpolation scheme for the volatility surface we introduce is designed to be consistent with arbitrage bounds. We show numerically under the Heston stochastic volatility model that this approach significantly reduces the approximation errors, andwe further provide empirical evidence from the Nikkei 225 options that the new implied volatility index is more accurate in predicting future volatility. Key words: modelfree implied volatility index; volatility forecasting; volatility surface; variance swaps JEL classification: C02 1.
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
English. <NNT: 2010PEST1017>. <tel00588686>
, 2011
"... probability distributions of diffusions and financial models with nonglobally smooth coefficients ..."
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probability distributions of diffusions and financial models with nonglobally smooth coefficients
of Diffusions and Financial Models
, 2011
"... Docteur de l’Université ParisEst ..."
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DEVELOPPEMENT STOCHASTIQUE ET FORMULES FERMEES DE PRIX POUR LES OPTIONS EUROPEENNES
"... pour obtenir le grade de ..."