Abstract:
In this work we show Malliavin differentiability of strong solutions of SDE's in dimension one for very general drift coefficients, namely, discontinuous unbounded coefficients with some certain growth rate.
Moreover, we discuss other properties of the solution, such as, Sobolev differentiability and give an explicit formula for both the Sobolev and Malliavin derivative of the solution, independent of the derivative of the drift coefficient.
Then we are able to derive the so-called Bismut-Elworthy-Li formula for such general coefficients which will be an essential tool for the sensitivity analysis of options. The latter result can be considered a generalization of results obtained by Fournier et al. (see below) to the case of SDE's with discontinuous coefficients.
Finally, we use all the previous results for the sensitivity analysis of options whose underlying price process is modelled by an SDE with irregular drift. Such class of drift coefficients allow us to consider more general models, as for instance, Ornstein-Uhlenbeck interest rate models with regime switching, that is discontinuous unbounded drift. We derive some formulas for the greeks of European and Asian options.