Dynamic greeks

The sensitivity of a price (or premium or reserve) to changes in its arguments is given by its derivatives, in finance known as “greeks”. Differential equations for sensitivities are obtained by simply differentiating the differential equation and the side condition that uniquely determine the price function. The device opens up prospects of efficient computation of greeks for a wide range of price functions in parametric models. It is applied here to examples in the Black–Merton–Scholes model and in a Markov chain model. Mathematical issues arising are, firstly, to construct the differential equation for the primary function and, secondly, to prove that the sensitivities actually exist. General resolutions to these problems seem not to be in reach, so only some special situations are discussed here.