On Bounding Option Prices in Paretian Stable Markets

This article establishes bounds on option prices when the distribution of the underlying asset is Paretian stable. Modeling prices by Paretian stable processes has the advantage that return distributions will have fatter tails than the normal and display jumps, results that are consistent wi th empirical evidence in most markets. Unfortunately, i n a Paretian stable security market, options cannot be replicated by self-financing trading strategies. Rather than making very spec@ assumptions on beliefs, preferences, and jo in t distributions between stock prices and aggregate wealth, this article bounds prices under weak assumptions regarding preferences. For the special case where the price process is geometric Wiener, the bounds collapse rapidly to the BlackScholes price. For the more general processes, the bounds tighten as the number o f trading opportunities increases, and as the degree of kurtosis relative to a normal distribution diminishes. For shorter-term options, the bounds are identified to be within a bid-ask spread. Therefore, for all practical purposes, the bounds are as good as prices. The resulting pricing relationships allow us to obtain insight that may explain the volatility smile efect.