Small time behaviour of double stochastic integrals and hedging under gamma constraints

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.