Option pricing in the moderate deviations regime

Abstract

We consider call option prices close to expiry in diffusion models, in an asymptotic regime (“moderately out of the money”) that interpolates between the well‐studied cases of at‐the‐money and out‐of‐the‐money regimes. First and higher order small‐time moderate deviation estimates of call prices and implied volatilities are obtained. The expansions involve only simple expressions of the model parameters, and we show how to calculate them for generic local and stochastic volatility models. Some numerical computations for the Heston model illustrate the accuracy of our results.

Pathwise moderate deviations for option pricing

“…as t tends to zero, from Proposition 2.4 and Corollary 2.5, with k t := k √ th(t) > 0. This behaviour in fact follows directly from our main result above; in the model (6), the pathwise moderate deviations principle for the first component (Corollary 2.5) generalises (being pathwise and with weaker assumptions) the results from [17], and directly yields (26). More precisely, from the proposed scaling (X ε t , Y ε t ) := (X εt , Y εt ), Corollary 2.5 and Remark 2.6 imply, as t tends to zero, for k > x 0 = 0,…”

“…We develop a unifying framework for pathwise moderate deviations of (multiscale) Itô diffusions, with applications to small-time, large-time and tail asymptotics for the diffusions and related integrated functionals. The original motivation is a pathwise extension of the moderate deviations proved in [17] in the context of small-time option pricing only. More specifically, we consider a generic two-dimensional stochastic volatility model X = (X, Y ) of the form (1)…”

The Randomized Heston Model

“…Theorem 4.8, however, does not assume a thin-tail behaviour as in Theorem 4.6 in Section 4.2. One of the striking feature of moderate deviations is that, contrary to classical large deviations, the rate function is usually available analytically, and is often of quadratic form [31,39,40].…”

Short-Term at-the-Money Asymptotics Under Stochastic Volatility Models

“…The skew behavior is especially important in order to argue the consistency of a model to the empirically observed power law. (5) with q θ of the form (7). Then, for any z ∈ R,…”