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14
Smalltime expansions of the distributions, densities, and option prices of stochastic volatility models with Lévy Jumps
, 2010
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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
2012): “Asymptotic and exact pricing of options on variance,”Finance and Stochastics, forthcoming
"... Abstract. We consider the pricing of derivatives written on the discretely sampled realized variance of an underlying security. In the literature, the realized variance is usually approximated by its continuoustime limit, the quadratic variation of the underlying logprice. Here, we characterize ..."
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Abstract. We consider the pricing of derivatives written on the discretely sampled realized variance of an underlying security. In the literature, the realized variance is usually approximated by its continuoustime limit, the quadratic variation of the underlying logprice. Here, we characterize the smalltime limits of options on both objects. We find that the difference between them strongly depends on whether or not the stock price process has jumps. Subsequently, we propose two new methods to evaluate the price of options on the discretely sampled realized variance. One of the methods is approximative; it is based on correcting prices of options on quadratic variation by our asymptotic results. The other method is exact; it uses a novel randomization approach and applies FourierLaplace techniques. We compare the methods and illustrate our results by some numerical examples. 1.
Smalltime expansions for local jumpdiffusion models with infinite jump activity
, 2011
"... Abstract. We consider a Markov process X, which is the solution of a stochastic differential equation driven by a Lévy process Z and an independent Wiener process W. Under some regularity conditions, including nondegeneracy of the diffusive and jump components of the process as well as smoothness o ..."
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Abstract. We consider a Markov process X, which is the solution of a stochastic differential equation driven by a Lévy process Z and an independent Wiener process W. Under some regularity conditions, including nondegeneracy of the diffusive and jump components of the process as well as smoothness of the Lévy density of Z outside any neighborhood of the origin, we obtain a smalltime secondorder polynomial expansion for the tail distribution and the transition density of the process X. Our method of proof combines a recent regularizing technique for deriving the analog smalltime expansions for a Lévy process with some new tail and density estimates for jumpdiffusion processes with small jumps based on the theory of Malliavin calculus, flow of diffeomorphisms for SDEs, and timereversibility. As an application, the leading term for outofthemoney option prices in short maturity under a local jumpdiffusion model is also derived. 1.
A new look at shortterm implied volatility in asset price models with jumps
 IN MATHEMATICAL FINANCE
, 2013
"... We analyse the behaviour of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalisation of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, w ..."
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We analyse the behaviour of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalisation of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (BlumenthalGetoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant modelindependent slope. This result gives a theoretical justification for the preference of the infinite variation Lévy models over the finite variation ones in the calibration based on shortmaturity option prices.
The smallmaturity Heston forward smile
 SIAM J. on Financial Mathematics
, 2013
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Asymptotic power utilitybased pricing and hedging
 Mathematics and Financial Economics
, 2013
"... Abstract Kramkov and Sîrbu ..."
Shorttime asymptotics for marginal distributions of semimartingales
, 2012
"... We study the shorttime aymptotics of conditional expectations of smooth and nonsmooth functions of a (discontinuous) Ito semimartingale; we compute the leading term in the asymptotics in terms of the local characteristics of the semimartingale. We derive in particular the asymptotic behavior of ca ..."
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We study the shorttime aymptotics of conditional expectations of smooth and nonsmooth functions of a (discontinuous) Ito semimartingale; we compute the leading term in the asymptotics in terms of the local characteristics of the semimartingale. We derive in particular the asymptotic behavior of call options with short maturity in a semimartingale model: whereas the behavior of outofthemoney options is found to be linear in time, the short time asymptotics of atthemoney options is
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.
A note on highorder shorttime expansions for ATM option prices under the CGMY model
, 2013
"... The shorttime asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In the present work, a novel thirdorder approximation for ATM option prices under the CGMY Lévy model is derived, and extended to a model with an additional independen ..."
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The shorttime asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In the present work, a novel thirdorder approximation for ATM option prices under the CGMY Lévy model is derived, and extended to a model with an additional independent Brownian component. Our results shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration.