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Dynamics of Implied Volatility Surfaces.
, 2001
"... The prices of index options at a given date are usually represented via the corresponding implied volatility surface, presenting skew/smile features and term structure which several models have attempted to reproduce. ..."
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The prices of index options at a given date are usually represented via the corresponding implied volatility surface, presenting skew/smile features and term structure which several models have attempted to reproduce.
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.
Tangent Lévy Market Models
"... In this paper, we introduce a new class of models for the time evolution of the prices of call options of all strikes and maturities. We capture the information contained in the option prices in the density of some timeinhomogeneous Lévy measure (an alternative to the implied volatility surface), ..."
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In this paper, we introduce a new class of models for the time evolution of the prices of call options of all strikes and maturities. We capture the information contained in the option prices in the density of some timeinhomogeneous Lévy measure (an alternative to the implied volatility surface), and we set this static codebook in motion by means of stochastic dynamics of Itôs type in a function space, creating what we call a tangent Lévy model. We then provide the consistency conditions, namely, we show that the call prices produced by a given dynamic codebook (dynamic Lévy density) coincide with the conditional expectations of the respective payoffs if and only if certain restrictions on the dynamics of the codebook are satisfied (including a drift condition à la HJM). We then provide an existence result, which allows us to construct a large class of tangent Lévy models, and describe a specific example for the sake of illustration.
The dynamics of the volatility skew: A Kalman filter approach
 Journal of Banking and Finance
, 2009
"... Preliminary draft: please do not quote ..."
TANGENT MODELS AS A MATHEMATICAL FRAMEWORK FOR DYNAMIC CALIBRATION
"... ABSTRACT. Motivated by the desire to integrate repeated calibration procedures into a single dynamic market model, we introduce the notion of tangent market model in an abstract set up, and we show that this new mathematical paradigm accommodates all the recent attempts to study consistency and abse ..."
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ABSTRACT. Motivated by the desire to integrate repeated calibration procedures into a single dynamic market model, we introduce the notion of tangent market model in an abstract set up, and we show that this new mathematical paradigm accommodates all the recent attempts to study consistency and absence of arbitrage in market models. For the sake of illustration, we concentrate on equity models and we assume that market quotes provide the prices of European call options for a specific set of strikes and maturities. While reviewing our recent results on dynamic local volatility and tangent L’evy models, we provide new results on the short timetomaturity asymptotics which shed new light on the dichotomy between these two disjoint classes of models, with and without jumps, helping choose in practice, which class of models is most appropriate to the market characteristics at hand. 1.
Deltahedging vega risk
 Journal Of Quantitative Finance
"... In this article we compare the Profit and Loss arising from the deltaneutral dynamic hedging of options, using two possible values for the delta of the option. The first one is the Black– Scholes implied delta, while the second one is the local delta, namely the delta of the option in a generalized ..."
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In this article we compare the Profit and Loss arising from the deltaneutral dynamic hedging of options, using two possible values for the delta of the option. The first one is the Black– Scholes implied delta, while the second one is the local delta, namely the delta of the option in a generalized Black–Scholes model with a local volatility, recalibrated to the market smile every day. We explain why in negatively skewed markets the local delta should provide a better hedge than the implied delta during slow rallies or fast selloffs, and a worse hedge, though to a lesser extent, during fast rallies or slow selloffs. Since slow rallies and fast selloffs are more likely to occur than fast rallies or slow selloffs in negatively skewed markets (provided we have physical as well as implied negative skewness), we conclude that on average the local delta provides a better hedge than the implied delta in negatively skewed markets. We obtain the same conclusion in the case of positively skewed markets. We illustrate these results by using both simulated and real timeseries of equityindex data, that have had a large negative implied skew since the stock market crash of October 1987. Moreover we check numerically that the conclusions we draw are true when transaction costs are taken into account. In the last section we discuss the case of barrier options.
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