Theory of Rational Option Pricing: II (Revised: 1-96)

This paper investigates the properties of contingent claim prices in a one dimensional diffusion world and establishes that (i) the delta of any claim is bounded above (below) by the sup (inf) of its delta at maturity, and (ii), if its payoff is convex (concave) then its current value is convex (concave) in the current value of the underlying. These properties are used as the foundation for a detailed study of the properties of option prices. Interestingly, although an upward shift in the term structure of interest rates will always increase a call’s value, a decline in the present value of the exercise price can be associated with a decline in the call price. We provide a new bound on the values of calls on dividend-paying assets. We establish that when the underlying’s instantaneous volatility is bounded above (below), the call price is bounded above (below) by its Black-Scholes value evaluated at the bounding volatility level. This leads to a new bound on a call’s delta. We also show that if changes in the value of the underlying follow a multidimensional diffusion (i.e., a stochastic volatility world), or are discontinuous or non-Markovian, then call option prices can exhibit properties very different from those of a Black-Scholes world: they can be decreasing, concave functions of the value of the underlying.