10 May 2004

14:15

Touzi Nizar

Abstract

We formulate a problem of super-hedging under gamma constraint by
taking the portfolio process as a controlled state variable. This
leads to a non-standard stochastic control problem. An intuitive
guess of the associated Bellman equation leads to a non-parabolic
PDE! A careful analysis of this problem leads to the study of the
small time behaviour of double stochastic integrals. The main result
is a characterization of the value function of the super-replication
problem as the unique viscosity solution of the associated Bellman
equation, which turns out to be the parabolic envelope of the above
intuitive guess, i.e. its smallest parabolic majorant. When the
underlying stock price has constant volatility, we obtain an
explicit solution by face-lifting the pay-off of the option.