Delta-Hedging Works: On Market Completeness in Diffusion Models

Abstract

This article provides new criteria for the completeness of market driven by diffusion processes. The main result is that if the value functions for smooth payoffs with compact support are weakly differentiable in the direction of where the driving process is volatile, then the market of payoffs dependent on the process is complete.

The intuitive meaning is that as soon as there is some weak concept of "delta" for very smooth payoff functions, then we can approximate all measurable functions and the market becomes complete.

In particular, we show that if the coefficients of the SDE are C1 almost surely, the the market of payoffs measurable with respect to the market process is complete.

Our approach is in marked contrast to the classic requirement that the volatility matrix of the SDE is invertible in order to retrieve the background driving motion which is much stronger and often violated in practice due to differing trading times for underlyings in different time zones. It is also not a very natural approach since a period of zero volatility 'in one direction' should not impede replicability in another risk factor.

Keywords: Complete Market, Diffusion, Predictable Representation Property, Extremal Martingale, Second Fundamental Theorem of Asset Pricing, Replication, Delta-Hedging

JEL Classification: G12