The pricing of contingent claims and optimal positions in asymptotically complete markets

TL;DR

This paper investigates how utility-based prices and optimal claim positions behave in incomplete markets as hedging errors or risk aversion vanish, revealing their asymptotic growth patterns and connections to large deviations theory.

Contribution

It establishes the asymptotic growth rate of optimal positions in incomplete markets under vanishing errors or risk aversion, linking to large deviations principles.

Findings

Optimal positions grow large at a specific rate as errors or risk aversion vanish.

The limiting behavior of prices connects to large deviations theory, especially the Gärtner-Ellis theorem.

The results apply to various models, including Black-Scholes with vanishing default or transaction costs.

Abstract

We study utility indifference prices and optimal purchasing quantities for a contingent claim, in an incomplete semi-martingale market, in the presence of vanishing hedging errors and/or risk aversion. Assuming that the average indifference price converges to a well defined limit, we prove that optimally taken positions become large in absolute value at a specific rate. We draw motivation from and make connections to Large Deviations theory, and in particular, the celebrated G\"{a}rtner-Ellis theorem. We analyze a series of well studied examples where this limiting behavior occurs, such as fixed markets with vanishing risk aversion, the basis risk model with high correlation, models of large markets with vanishing trading restrictions and the Black-Scholes-Merton model with either vanishing default probabilities or vanishing transaction costs. Lastly, we show that the large claim regime could naturally arise in partial equilibrium models.