OMI

: Recently strict local martingales have been used to model

exchange rates. In such models, put-call parity does not hold if one

assumes minimal superreplicating costs as contingent claim prices. I

will illustrate how put-call parity can be restored by changing the

definition of a contingent claim price.

More precisely, I will discuss a change of numeraire technique when the

underlying is only a local martingale. Then, the new measure is not

necessarily equivalent to the old measure. If one now defines the price

of a contingent claim as the minimal superreplicating costs under both

measures, then put-call parity holds. I will discuss properties of this

new pricing operator.

To illustrate this techniques, I will discuss the class of "Quadratic

Normal Volatility" models, which have drawn much attention in the

financial industry due to their analytic tractability and flexibility.

This talk is based on joint work with Peter Carr and Travis Fisher.

Most financial models introduced for the purpose of pricing and hedging derivatives concentrate

on the dynamics of the underlying stocks, or underlying instruments on which the derivatives

are written. However, as certain types of derivatives became liquid, it appeared reasonable to model

their prices directly and use these market models to price or hedge exotic derivatives. This framework

was originally advocated by Heath, Jarrow and Morton for the Treasury bond markets.

We discuss the characterization of arbitrage free dynamic stochastic models for the markets with

infinite number of European Call options as the liquid derivatives. Subject to our assumptions on the

presence of jumps in the underlying, the option prices are represented either through local volatility or

through local L´evy measure. Each of the latter ones is then given dynamics through an Itˆo stochastic

process in infinite dimensional space. The main thrust of our work is to characterize absence of arbitrage

in this framework and address the issue of construction of the arbitrage-free models.