Oliver Menokeu Pamen: 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.