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Smalltime asymptotics of option prices and first absolute moments
 Journal of Applied Probability
, 2011
"... We study the leading term in the smalltime asymptotics of atthemoney call option prices when the stock price process 푆 follows a general martingale. This is equivalent to studying the first centered absolute moment of 푆. We show that if 푆 has a continuous part, the leading term is of order 푇 in t ..."
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We study the leading term in the smalltime asymptotics of atthemoney call option prices when the stock price process 푆 follows a general martingale. This is equivalent to studying the first centered absolute moment of 푆. We show that if 푆 has a continuous part, the leading term is of order 푇 in time 푇 and depends only on the initial value of the volatility. Furthermore, the term is linear in 푇 if and only if 푆 is of finite variation. The leading terms for purejump processes with infinite variation are between these two cases; we obtain their exact form for stablelike small jumps. To derive these results, we use a natural approximation of 푆 so that calculations are necessary only for the class of Lévy processes.
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
HJM: A Unified Approach to Dynamic Models for Fixed Income, Credit and Equity Markets
"... Summary. The purpose of this paper is to highlight some of the key elements of the HJM approach as originally introduced in the framework of fixed income market models, to explain how the very same philosophy was implemented in the case of credit portfolio derivatives and to show how it can be exten ..."
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Summary. The purpose of this paper is to highlight some of the key elements of the HJM approach as originally introduced in the framework of fixed income market models, to explain how the very same philosophy was implemented in the case of credit portfolio derivatives and to show how it can be extended to and used in the case of equity market models. In each case we show how the HJM approach naturally yields a consistency condition and a noarbitrage conditions in the spirit of the original work of Heath, Jarrow and Morton. Even though the actual computations and the derivation of the drift condition in the case of equity models seems to be new, the paper is intended as a survey of existing results, and as such, it is mostly pedagogical in nature. 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.
An infinite dimensional stochastic analysis approach to local volatility dynamic models
 COMMUNICATIONS ON STOCHASTIC ANALYSIS
, 2008
"... The difficult problem of the characterization of arbitrage free dynamic stochastic models for the equity markets was recently given a new life by the introduction of market models based on the dynamics of the local volatility. Typically, market models are based on Itô stochastic differential equatio ..."
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The difficult problem of the characterization of arbitrage free dynamic stochastic models for the equity markets was recently given a new life by the introduction of market models based on the dynamics of the local volatility. Typically, market models are based on Itô stochastic differential equations modeling the dynamics of a set of basic instruments including, but not limited to, the option underliers. These market models are usually recast in the framework of the HJM philosophy originally articulated for Treasury bond markets. In this paper we streamline some of the recent results on the local volatility dynamics by employing an infinite dimensional stochastic analysis approach as advocated by the pioneering work of L. Gross and his students.
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.
The smallmaturity implied volatility slope for Lévy models. Available at arXiv:1310.3061
, 2014
"... Abstract. We consider the atthemoney strike derivative of implied volatility as the maturity tends to zero. Our main results quantify the growth of the slope for infinite activity exponential Lévy models. As auxiliary results, we obtain the limiting values of short maturity digital call options, ..."
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Abstract. We consider the atthemoney strike derivative of implied volatility as the maturity tends to zero. Our main results quantify the growth of the slope for infinite activity exponential Lévy models. As auxiliary results, we obtain the limiting values of short maturity digital call options, using Mellin transform asymptotics. Finally, we discuss when the atthemoney slope is consistent with the steepness of the smile wings, as given by Lee’s moment formula. 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
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"... model calibration in finance Calibration non paramétrique de modèles en finance Thèse présentée par ..."
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model calibration in finance Calibration non paramétrique de modèles en finance Thèse présentée par