Pricing Intervals, Super-replication and Fully Nonlinear PDE's
In mathematical finance a contingent claim or an option is priced by considering an auxiliary problem for a rational investor. In this problem, one imagines that the investor instead of buying an option invests her wealth in the financial market available to her. Then, in an ideal (or complete) market, the price of the option is exactly the initial wealth that allows her to have a terminal wealth which is exactly (with probability one)equal to the future random value of the option at the terminal time. Hence the investor replicates the option exactly. This is called the arbitrage pricing and with this price of the option there is no arbitrage possibility neither for the buyer nor the seller. In certain financial markets, however, exact replication is not possible. By considering, investment strategies that are super-replicating rather than replicating one finds a minimal super-replicating price for the seller. Also in a dual approach one finds the maximal super-replicated price for the buyer. The interval from the price of the buyer to the price of the seller is the range of prices for which there is a no arbitrage possibility.
In this talk, I will first briefly explain the above pricing techniques in markets with portfolio or gamma constraints. Then, I will show how these techniques are related to ceratin partial differentail equations. Finally, I will outline a theory of Cheredito, myself, Touzi and Victoir extending the theory of Backward stohastic differential equations (BSDE) to obtain a representation result for all second order parabolic partial differential equations. This extension, called 2BSDE, has an additional stochastic differential equation for the portfolio (or the gradient) process. The uniqueness of the 2BSDE is proved using the method of super-replication.
H. Mete Soner, Koc University, Istanbul, Turkey