Risk-hedging a European option with a convex risk measure and without no-arbitrage condition

Abstract

In this article, we revisit the discrete-time problem of pricing a contingent claim with respect to a dynamic risk measure defined by its acceptance sets. Without any no-arbitrage condition, we show that it is possible to characterize the prices of a European claim. Our analysis reveals a natural weak no-arbitrage condition that we study. This is a condition formulated in terms of the (risk) hedging prices instead of the attainable claims. Our approach is not based on a robust representation of the risk measure and we do not suppose the existence of a risk-neutral probability measure.

Keywords:

• Risk-hedging prices
• dynamic risk measures
• absence of instantaneous profit aip
• random sets
• conditional essential infimum and supremum

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 49J53
• 60D05
• 91G20
• 91G80

JEL Classifications:

• C02
• C61
• G13

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 This means that the σ-algebra ${\mathcal{F}}_t$, for each $0\le t\le T$, contains the negligible sets so that an equality between two random variables is understood up to a negligible set.

2 $(C_t^n{)}_{n\ge 1}$ is called a Castaing representation of ${\mathcal{A}}_t^X$.

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant number 11871275] and Grant-in-Aid for Scientific Research from Nanjing University of Science and Technology [grant number KN11008].