Theory of Rational Option Pricing: Ii
Posted: 2 Oct 1999
The bulk of the option pricing properties established in Merton's Classic Theory when the option price is homogeneous of degree one in the underlying's value and the exercise price, are shown to extend to any Markovian diffusion world. The most important result is that calls are increasing convex functions of the value of the underlying. Still, some caveats are in order: Although an upward shift in the term structure of interest rates will 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; and a call's elasticity need not be everywhere increasing with the passage of time, or everywhere decreasing in the level of the stock price. As a direct implication of convexity, we are able to undertake a comparative static analysis of the effects of shifts in the term structure, in dividend policy, and in the underlying asset's instantaneous volatility function. We provide a new bound on the relative values of calls on otherwise equivalent dividend- and non-dividend- paying assets. With respect to volatility, we present two fascinating results. First, an equivalence between (i) a comparison of two different functional forms for the relation between instantaneous volatility and the contemporaneous stock price and time and (ii) increasing risk in the Rothschild-Stiglitz sense. Second, when the instantaneous volatility is bounded above (below), the call price is bounded above (below) by its Black-Scholes value evaluated at the bounding volatility level, and we can place upper and lower bounds on the stock positions necessary to hedge a given option position. We also show that if we relax either the continuity assumption or the Markovian assumption, then call options can be 'bloating' (not 'wasting') assets, whose value over some range is a decreasing, concave function of the value of the underlying. We argue that when considering the valuation of long-dated options on the stock of a firm, it is both intuitive and correct to view the dynamics of the underlying stock price as Non-Markovian.
JEL Classification: G12, G13
Suggested Citation: Suggested Citation