Results 1  10
of
16
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 ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
(Show Context)
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.
Smalltime expansions of the distributions, densities, and option prices of stochastic volatility models with Lévy Jumps
, 2010
"... ..."
(Show Context)
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 ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
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.
Smalltime asymptotics for fast meanreverting stochastic volatility models
, 2010
"... In this paper, we study stochastic volatility models in regimes where the maturity is small but large compared to the meanreversion time of the stochastic volatility factor. The problem falls in the class of averaging/homogenization problems for nonlinear HJB type equations where the “fast variable ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
In this paper, we study stochastic volatility models in regimes where the maturity is small but large compared to the meanreversion time of the stochastic volatility factor. The problem falls in the class of averaging/homogenization problems for nonlinear HJB type equations where the “fast variable” lives in a noncompact space. We develop a general argument based on viscosity solutions which we apply to the two regimes studied in the paper. We derive a large deviation principle and we deduce asymptotic prices for OutofTheMoney call and put options, and their corresponding implied volatilities. The results of this paper generalize the ones obtained in [11] (J. Feng, M. Forde and J.P. Fouque, Short maturity asymptotic for a fast mean reverting Heston stochastic volatility model, SIAM Journal on Financial Mathematics, Vol. 1, 2010) by a moment generating function computation in the particular case of the Heston model.
From Smile Asymptotics to Market Risk Measures
, 2012
"... The left tail of the implied volatility skew, coming from quotes on outofthemoney put options, can be thought to reflect the market’s assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary m ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
The left tail of the implied volatility skew, coming from quotes on outofthemoney put options, can be thought to reflect the market’s assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations (BSDEs), to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear PDE and provide a small timetomaturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parametrized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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
Efficient Importance Sampling Estimation for Joint Default Probability: the First Passage Time Problem,” Stochastic Analysis with Financial Applications. Editors
 Progress in Probability,
, 2011
"... Abstract This paper aims to estimate joint default probabilities under the structuralform model with a random environment; namely stochastic correlation. By means of a singular perturbation method, we obtain an asymptotic expansion of a twoname joint default probability under a fast meanrevertin ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract This paper aims to estimate joint default probabilities under the structuralform model with a random environment; namely stochastic correlation. By means of a singular perturbation method, we obtain an asymptotic expansion of a twoname joint default probability under a fast meanreverting stochastic correlation model. The leading order term in the expansion is a joint default probability with an effective constant correlation. Then we incorporate an efficient importance sampling method used to solve a first passage time problem. This procedure constitutes a homogenized importance sampling to solve the full problem of estimating the joint default probability with stochastic correlation models.
Small time central limit theorems for semimartingales with applications
, 2012
"... ..."
(Show Context)
Maximum Likelihood Estimation for Small Noise Multiscale Diffusions
, 2013
"... ar ..."
(Show Context)