1,435 research outputs found
Option Pricing in the Moderate Deviations Regime
We consider call option prices in diffusion models close to expiry, in an
asymptotic regime ("moderately out of the money") that interpolates between the
well-studied cases of at-the-money options and out-of-the-money fixed-strike
options. First and higher order small-time moderate deviation estimates of call
prices and implied volatility are obtained. The expansions involve only simple
expressions of the model parameters, and we show in detail how to calculate
them for generic local and stochastic volatility models. Some numerical
examples for the Heston model illustrate the accuracy of our results
Short-time asymptotics for marginal distributions of semimartingales
We study the short-time asymptotics of conditional expectations of smooth and
non-smooth 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
\textit{out-of-the-money} options is found to be linear in time, the short time
asymptotics of \textit{at-the-money} options is shown to depend on the fine
structure of the semimartingale
Small-time asymptotics for fast mean-reverting stochastic volatility models
In this paper, we study stochastic volatility models in regimes where the
maturity is small, but large compared to the mean-reversion 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 out-of-the-money call and put options, and their corresponding implied
volatilities. The results of this paper generalize the ones obtained in Feng,
Forde and Fouque [SIAM J. Financial Math. 1 (2010) 126-141] by a moment
generating function computation in the particular case of the Heston model.Comment: Published in at http://dx.doi.org/10.1214/11-AAP801 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
High-order short-time expansions for ATM option prices of exponential L\'evy models
In the present work, a novel second-order approximation for ATM option prices
is derived for a large class of exponential L\'{e}vy models with or without
Brownian component. The results hereafter 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. In the presence of a
Brownian component, the second-order term, in time-, is of the form
, with only depending on , the degree of jump
activity, on , the volatility of the continuous component, and on an
additional parameter controlling the intensity of the "small" jumps (regardless
of their signs). This extends the well known result that the leading
first-order term is . In contrast, under a
pure-jump model, the dependence on and on the separate intensities of
negative and positive small jumps are already reflected in the leading term,
which is of the form . The second-order term is shown to be of
the form and, therefore, its order of decay turns out to be
independent of . The asymptotic behavior of the corresponding Black-Scholes
implied volatilities is also addressed. Our approach is sufficiently general to
cover a wide class of L\'{e}vy processes which satisfy the latter property and
whose L\'{e}vy densitiy can be closely approximated by a stable density near
the origin. Our numerical results show that the first-order term typically
exhibits rather poor performance and that the second-order term can
significantly improve the approximation's accuracy, particularly in the absence
of a Brownian component.Comment: 35 pages, 8 figures. This is an extension of our earlier submission
arXiv:1112.3111. To appear in Mathematical Financ
From Smile Asymptotics to Market Risk Measures
The left tail of the implied volatility skew, coming from quotes on
out-of-the-money 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
time-to-maturity 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.Comment: 24 pages, 4 figure
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