Martingales and Wavelets: Applications to Hedging Financial Derivatives

We will describe a new discretization of financial instruments in terms of a martingale expansion constructed using Haar wavelets systems. In particular, expansions on these bases give the pointwise convergence needed for hedging derivatives. Examples of these systems are constructed which illustrate the discrete, spacewise, nature of the approximations. We describe natural conditions under which our Haar hedging strategy can be realized by means of a self financing portfolio consisting of binary options. We will explain how basic ideas from wavelet theory allows us to construct efficient approximations to a given financial portfolio X . This allows to reduce transaction costs in the financial implementation of Haar hedging.

Little background will be assumed, in particular, the basic ideas behind hedging financial derivatives and the role of martingales will be explained.

Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).