id: 06130636
dt: j
an: 
au: Figueroa-López, José E.; Forde, Martin
ti: The small-maturity smile for exponential Lévy models.
so: SIAM J. Financ. Math. 3, No. 1, 33-65, electronic only (2012).
py: 2012
pu: Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA
la: EN
cc: 60G51 60F99 91G20 91G60
ut: exponential Lévy models; time-changed Lévy models; option pricing;
    short-time asymptotics; implied volatility
ci: Zbl 1179.60026; Zbl 1205.91161
li: doi:10.1137/110820658
ab: Summary: We derive a small-time expansion for out-of-the-money call options
    under an exponential Lévy model, using the small-time expansion for
    the distribution function given in [{\it J. Figueroa-López} and {\it
    C. Houdré}, Stochastic Process. Appl. 119, No. 11, 3862‒3889 (2009;
    Zbl 1179.60026)], 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 $\frac{1}{2}σ^2 t^2
    e^{k}ν(k)(1+o(1))$ as $t \to 0$, where $ν$ is the Lévy density.
    Using the small-time expansion for call options, we then derive a
    small-time expansion for the implied volatility $\hatσ_{t}^{2}(k)$ at
    log-moneyness k, which sharpens the first order estimate
    $\hatσ_{t}^{2}(k)\sim \frac{\frac{1}{2}k^2}{t\log (1/t)}$ given in
    [{\it P. Tankov}, “Pricing and hedging in exponential Lévy models:
    review of recent results”, Lecture Notes in Mathematics 2003,
    319‒359 (2011; Zbl 1205.91161)]. 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 time-changed
    Lévy models. We also consider a small-time, small log-moneyness regime
    for the CGMY model and apply this approach to the small-time pricing of
    at-the-money call options; we show that for $Y\in(1,2)$,
    $\lim_{t\to0}t^{-1/Y}\mathbb{E}(S_t-S_0)_{+}=S_{0}\mathbb{E}^{*}(Z_{+})$
    and the corresponding at-the-money implied volatility $\hatσ_t(0)$
    satisfies $\lim_{t \to
    0}\hatσ_t(0)/t^{1/Y-1/2}=\sqrt{2π}\,\mathbb{E}^{*}(Z_{+})$, where Z
    is a symmetric Y-stable random variable under $\mathbb{P}^*$ and Y is
    the usual parameter for the CGMY model appearing in the Lévy density
    $ν(x)=C x^{-1-Y}e^{-M x}\bold{1}_{\{x>0\}}+C
    |x|^{-1-Y}e^{-G|x|}\bold{1}_{\{x<0\}}$ of the process.
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