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14
The smallmaturity smile for exponential Lévy models
, 2012
"... We derive a smalltime expansion for outofthemoney call options under an exponential Lévy model, using the smalltime expansion for the distribution function given in FigueroaLópez&Houdré[FLH09], combined with a change of numéraire via the Esscher transform. In particular, we find that the e ..."
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Cited by 18 (6 self)
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We derive a smalltime expansion for outofthemoney call options under an exponential Lévy model, using the smalltime expansion for the distribution function given in FigueroaLópez&Houdré[FLH09], combined with a change of numéraire via the Esscher transform. In particular, we find that the effect of a nonzero volatility σ of the Gaussian component of the driving Lévy process is to increase the call price by 1 2σ2t 2 e k ν(k)(1+o(1)) as t → 0, where ν is the Lévy density. Using the smalltime expansion for call options, we then derive a smalltime expansion for the implied volatility ˆσ 2 t(k) at logmoneyness k, which sharpens the first order estimate ˆσ 2 t(k) ∼ 1 2 k2 tlog(1/t) given in [Tnkv10]. Our numerical results show that the second order approximation can significantly outperform the first order approximation. Our results are also extended to a class of timechanged Lévy models. We also consider a smalltime, small logmoneyness regime for the CGMY model, and apply this approach to the smalltime pricing of atthemoney call options; we show that for Y ∈ (1,2), limt→0t −1/Y E(St−S0)+ = S0E ∗ (Z+) and the corresponding atthemoney implied volatility ˆσt(0) satisfies limt→0 ˆσt(0)/t 1/Y −1/2 = √ 2πE ∗ (Z+), where Z is a symmetric Ystable random variable under P ∗ and Y is the usual parameter for the CGMY model appearing in the Lévy density ν(x) = Cx −1−Y e −Mx 1{x>0} +Cx  −1−Y e −Gx  1{x<0} of the process.
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
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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Cited by 6 (1 self)
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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.
The smallmaturity Heston forward smile
 SIAM J. on Financial Mathematics
, 2013
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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.
Asymptotic Behavior of the Stochastic Rayleighvan der Pol Equations with Jumps
"... We study the stability, attractors, and bifurcation of stochastic Rayleighvan der Pol equations with jumps. We first established the stochastic stability and the large deviations results for the stochastic Rayleighvan der Pol equations. We then examine the existence limit circle and obtain some n ..."
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We study the stability, attractors, and bifurcation of stochastic Rayleighvan der Pol equations with jumps. We first established the stochastic stability and the large deviations results for the stochastic Rayleighvan der Pol equations. We then examine the existence limit circle and obtain some new random attractors. We further establish stochastic bifurcation of random attractors. Interestingly, this shows the effect of the Poisson noise which can stabilize or unstabilize the system which is significantly different from the classical Brownian motion process.
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.
ThirdOrder ShortTime Expansions for ClosetotheMoney Option Prices Under the CGMY Model
, 2015
"... 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, novel thirdorder approximations for closetothemoney European option prices under an infinitevariation CGMY Lévy model are derived, and are th ..."
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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, novel thirdorder approximations for closetothemoney European option prices under an infinitevariation CGMY Lévy model are derived, and are then extended to a model with an additional independent Brownian component. The asymptotic regime considered, in which the strike is made to converge to the spot stock price as the maturity approaches zero, is relevant in applications since the most liquid options have strikes that are close to the spot price. Our results shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of option prices near expiration when the strike is close to the spot price. In particular, a new type of transition phenomenon is uncovered in which the third order term exhibits two distinct asymptotic regimes depending on whether