Before revision:  To obtain the formulas we use the Dyson-Taylor commutator method re- cently developed in [ 7, 8, 10 ] for short time asymptotic expansions of heat kernels, and obtain a family of general explicit closed form approx- imate solutions for both the pricing kernel and derivative price  .
After revision:  To obtain the formulas we use the Dyson-Taylor commutator method re- cently developed in [ 7, 8, 10 ] 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 .