Conic Option Pricing

Abstract

Derivatives pricing models based on arbitrage produce a single no-arbitrage option value. In theory, trading by arbitrageurs will force the market price to equal this value. However, the real world is full of frictions and uncertainty, and there is always a range in which the market price may wander without becoming sufficiently mispriced to touch off arbitrage trades. Under weaker assumptions, as explored in this article, such a range should encompass a convex set of prices such that the net payout has positive expected value under a broad set of supporting probability distributions and is closed under scaling, which makes it a mathematical cone. Madan and Schoutens use this principle to develop a theory of conic option pricing, in which an upper and lower bound on an option’s price are obtained by applying two deformations to the probability density for returns. Where the market price is expected to fall within these bounds is based on the principle that order flow for buying and selling should balance out on average. The resulting valuation does not require a specific density, such as lognormal (this article uses the variance-gamma process). Rather, it specifies how the density chosen by the user should be modified to produce the bounds. An empirical demonstration of the approach using seven years of options data on 208 underlying assets shows that market prices obey the pricing bounds quite well. An interesting and valuable extension is the valuation of delta and delta-gamma hedged positions, which naturally produce much tighter bounds.