  We obtain new closed-form pricing formulas for contingent claims when the
asset follows a Dupire-type local volatility model. To obtain the formulas we
use the Dyson-Taylor commutator method that we have recently developed in [5,
6, 8] for short-time asymptotic expansions of heat kernels, and obtain a family
of general closed-form approximate solutions for both the pricing kernel and
derivative price. A bootstrap scheme allows us to extend our method to large
time. We also perform analytic as well as a numerical error analysis, and
compare our results to other known methods.
