General Properties of Option Prices

Abstract

This paper examines general properties of prices of contingent claims. When the underlying follows a one- dimensional diffusion and interest rates are deterministic, a claim's delta is bounded by the infimum and the supremum of its delta at maturity. Similar bounds hold for the bond position in a claim's replicating portfolio. If the claim's payoff at maturity is convex (concave), then prior to expiration the claim's price is convex (concave). We also establish conditions under which the bounds on delta and the inherited convexity result extend to a multi-dimensional diffusion (stochastic volatility) setting. These results allow us to undertake comparative static analyses of the effects of changes in interest rates, in dividend rates, and in volatility on the prices of call options in a one- dimensional diffusion setting. The bounds on a claim's delta are shown to be a reflection of the fact that, for a given realization of the Brownian motion driving the risk- neutralized stock price process, the realized value of the process at the claim's maturity date is increasing in its starting value. We dub this the 'no-crossing' property. We demonstrate that if we relax either the continuity or Markovian properties inherent in a diffusion, or consider an unrestricted stochastic volatility setting, then the 'no-crossing' property can be violated. The price of a call option can then be decreasing or concave over some underlying price range, increasing in the passage of time, and decreasing in the level of interest rates.