Options and Partial Differential Equations

The aim of this chapter is to show how partial differential equations appear in financial models and to present succinctly numerical methods used for effective computations of prices and hedging of options. In the first part, we take up the reasoning which allowed Louis Bachelier to bring out a relationship between the heat equation and a modelling of the evolution of share prices. In the second part, the equations satisfied by options prices are introduced. The third part is dedicated to numerical methods and, in particular, to the methods based on the simulation of hazard (the so-called Monte Carlo methods). The understanding of this text does not necessitate mathematical knowledge beyond that of an undergraduate level. Thus, we hope that it may be read by non-mathematical scientists.