Oliver Menokeu Pamen: Computing Greeks without derivatives
Oliver Menokeu Pamen, CMA, holder et seminar med tittelen: Computing Greeks without derivatives
In life insurance, the primary objective is to provide ﬁnancial security to policy-holders. One way to do this is to manage investment guarantees whose prices can be obtained by Actuarial and Mathematical Finance methods. In Mathematical Finance the so-called ”Greeks” are quantities that measure the sensitivity of an option price or investment guarantee with respect to some model parameters involved. One of the most prominent applications of Malliavin calculus in Mathematical Finance is the representation of these Greeks as expectation functionals that does not involve the derivative of the pay-oﬀ function of the option. Since most ﬁnancial pay-oﬀ functions are not smooth this representation yields a numerically eﬃcient way to compute the Greeks. However, in the above mentioned representation of the Greeks the Ito diﬀusion modeling the price process of the underlying is assumed to have diﬀerentiable coeﬃcients. For example an extended Ornstein-Uhlenbeck process with switching mean reversion rate, an important model in electricity price modeling, is not included in this class of diﬀusions. In this presentation we demonstrate how to generalize this application of Malliavin calculus to price processes driven by Ito diﬀusions with irregular drift coeﬃcients. To this end, we study the general theoretical question of existence and Malliavin diﬀerentiability of strong solutions of stochastic diﬀerential equations (SDE’s) with irregular drift coeﬃcients. This approach yields the additional important result that the constructed strong solutions are even Malliavin diﬀerentiable. This insight ﬁnally enables us to represent greeks based on Ito diﬀusions with irregular drift coeﬃcients as expectation functionals that neither involve the derivative of the pay-oﬀ function nor the derivatives of the diﬀusion coeﬃcients.