  In a recent formulation of a quantum field theory of forward rates, the
volatility of the forward rates was taken to be deterministic. The field theory
of the forward rates is generalized to the case of stochastic volatility. Two
cases are analyzed, firstly when volatility is taken to be a function of the
forward rates, and secondly when volatility is taken to be an independent
quantum field. Since volatiltiy is a positive valued quantum field, the full
theory turns out to be an interacting nonlinear quantum field theory in two
dimensions. The state space and Hamiltonian for the interacting theory are
obtained, and shown to have a nontrivial structure due to the manifold moving
with a constant velocity. The no arbitrage condition is reformulated in terms
of the Hamiltonian of the system, and then exactly solved for the nonlinear
interacting case.
