## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 3 (2017), 1778-1830.

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

Michail Anthropelos, Scott Robertson, and Konstantinos Spiliopoulos

#### Abstract

We study utility indifference prices and optimal purchasing quantities for a contingent claim, in an incomplete semimartingale 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 Gärtner–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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 3 (2017), 1778-1830.

**Dates**

Received: September 2015

Revised: September 2016

First available in Project Euclid: 19 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1500451244

**Digital Object Identifier**

doi:10.1214/16-AAP1246

**Mathematical Reviews number (MathSciNet)**

MR3678485

**Zentralblatt MATH identifier**

06775362

**Subjects**

Primary: 91G99: None of the above, but in this section 60F10: Large deviations 60H10: Stochastic ordinary differential equations [See also 34F05]

**Keywords**

Indifference pricing incomplete markets utility functions large position size

#### Citation

Anthropelos, Michail; Robertson, Scott; Spiliopoulos, Konstantinos. The pricing of contingent claims and optimal positions in asymptotically complete markets. Ann. Appl. Probab. 27 (2017), no. 3, 1778--1830. doi:10.1214/16-AAP1246. https://projecteuclid.org/euclid.aoap/1500451244