10 May 2004
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.
- Stochastic Analysis Seminar