An inverse problem arisen in the zero-coupon bond pricing

The zero-coupon bond ( ZCB) is a special kind of bond without coupons, which is purchased today at a certain price, while at maturity the bond is redeemed for a fixed price. In the ZCB world there is the important quantity λ ( t ) which is often called the market price of risk and cannot be observed directly but has a major impact on the ZCB value. In this paper, the inverse problem of determining the market price of risk from the current market prices of ZCB is discussed. Being different from ordinary parameter identification problems in parabolic equations, the mathematical model in the paper belongs to the second order parabolic equations with non-negative characteristic form, i.e., there exists degeneracy on the lateral boundaries in the model. Based on the optimal control framework, the existence, uniqueness and stability of the minimizer for the cost functional are established. A necessary condition which is a coupled system of a parabolic equation and a parabolic variational inequality is deduced. The results obtained in the paper are interesting and useful, and may be applied to a variety of derivatives pricing problems.