General Restrictions on Prices of Financial Derivatives Written on Underlying Diffusions

We apply probabilistic solutions of parabolic PDEs with terminal and boundary conditions to obtain restrictions on contingent claims written on one-dimensional diffusions. For term structure derivatives, we obtain monotonicity and convexity results with respect to the short-term interest rate. We apply them to bonds, calls on bonds, and puts on interest rates and we find a condition for the price of these derivatives to be convex in that rate. We find that yield curves corresponding to higher short-term rates lie uniformly above curves with lower rates. Regarding options on assets with local volatility, we obtain probabilistic representations, bounds, and asymptotic results for delta, rho, and theta. Similar results are obtained for Asian options.