Pathwise moderate deviations for option pricing

Abstract

We provide a unifying treatment of pathwise moderate deviations for models commonly used in financial applications, and for related integrated functionals. Suitable scaling enables us to transfer these results into small‐time, large‐time, and tail asymptotics for diffusions, as well as for option prices and realized variances. In passing, we highlight some intuitive relationships between moderate deviations rate functions and their large deviations counterparts; these turn out to be useful for numerical purposes, as large deviations rate functions are often difficult to compute.

Large and moderate deviations for importance sampling in the Heston model

“…We do not consider the stochastic drift change here, since a rigorous treatment is out of scope of this paper. Similar problem has been studied in [21,Theorem 2.1] and [17,Theorem 3.3], where authors propose fewer conditions, although in a simpler setting (which happens to include the Heston model as well). We now introduce a theorem, which is a direct application of [17,Theorem 3.3] and [21,Theorem 2.1], that provides the desired MDP.…”

Moderate deviations for systems of slow-fast stochastic reaction-diffusion equations

“…Remark 10. The estimates derived in this section do not require any regularity for the initial conditions of the controlled system (25). Such considerations have to be taken into account in the next section.…”

“…Here, X ǫ,u corresponds to a controlled slow-fast system (X ǫ,u , Y ǫ,u ) (see (25) below) which results from (1) by perturbing the paths of the noise by an appropriately re-scaled control. It is due to the latter that this representation is called variational.…”

“…In light of ( 6), we see that in order to obtain a limit as ǫ → 0 of the Laplace functional (i.e. to prove an MDP), one needs to analyze the limiting behavior of η ǫ,u and, before doing so, obtain a priori estimates for the underlying controlled slow-fast system given in (25). The latter is the first technical part of the current work (Section 4).…”

Small-time moderate deviations for the randomised Heston model

“…Introduced in [27], they have become an increasingly useful tool in probability and in statistical physics, as discussed in [3, 8, 9, 24]. They have also recently appeared in mathematical finance in order to provide a different, yet somehow more useful, view on asymptotics, and important results in this direction can be studied in [5, 13, 22].…”