Integrability Requirements for Contingent Claim Payoffs

Summary

The document asks why mathematical finance texts impose different integrability conditions on contingent claim payoffs. One definition requires the discounted payoff to be integrable under the risk-neutral measure, which permits the conditional expectation used in valuation. Another requires the payoff itself to be square-integrable. The question situates this distinction between Brownian market models and more general semimartingale frameworks.

Its central issue is whether square-integrability is needed for pricing or instead serves technical purposes in results about replication, stochastic integration, or the spaces used to formulate a model. The text does not include an answer or resolve the distinction. It therefore provides a useful statement of the modeling question, but no conclusion, example, or evidence for preferring one definition. In applying either condition, the payoff’s units relative to the bank account and the assumptions made about the market model need to be kept clear.

Key ideas

• A risk-neutral pricing expectation requires suitable integrability of the discounted payoff.
• Some formulations require the payoff to be square-integrable rather than merely integrable after discounting.
• The question connects differing conventions to Brownian and more general semimartingale models.
• The document raises, but does not answer, whether square-integrability is required for pricing or for technical results.

Tags

# Complete Financial Market: Integrability condition for Contingent Claims


# Complete Financial Market: Integrability condition for Contingent Claims












Consider an arbitrage-free and complete financial market with underlying filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\,\in\,[0,T]},\mathbb{Q})$, where $T\in(0,\infty)$ is some terminal time horizon and $\mathbb{Q}$ is the (unique) risk-neutral measure. Moreover, let $B=\{B_{t}\}_{t\,\in\,[0,T]}$ denote the (risk-less) bank account.

I have seen two definitions for a contingent claim:

- A contingent claim with payoff at time $t\in[0,T]$ is an $\mathcal{F}_{t}$-measureable random variable $H$ with $\frac{H}{B_{t}}\in\mathcal{L}^{1}$ (see e.g. here)

- A contingent claim with payoff at time $t\in[0,T]$ is an $\mathcal{F}_{t}$-measureable random variable $H$ with $H\in\mathcal{L}^{2}$ (see e.g. here, page 4).

I understand that integrability is required for the existence of the (conditional) expectation under the risk-neutral measure, but why do some authors require square-integrability of $H$ (and not even $\frac{H}{B_{t}}$)?

To add some context: the first definition is usually given in texts that consider a Brownian financial market model. The second definition is given in more general semi-martingale models. I guess the reason for the two different definitions is a technical one, but I don't see where this is needed.

Thanks for your time and effort!

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.