Risk-Neutral Expectations in Bounds for Contingent Claims

Summary

The document asks which probability measure should be used to calculate the expected discounted payoff of a replicable claim that lies below a possibly nonreplicable contingent claim. It presents an equality between a supremum over such replicable claims and an infimum over risk-free probability measures, then gives a brief resolution based on a theorem about expectations of replicable payoffs.

The key point is that the expected value of the replicable payoff is constant across the relevant equivalent martingale measures, so the choice among those measures does not change the bound. The note does not explain the theorem’s assumptions or prove the equality, and it does not address how the bound would be calculated in a particular market. Its value is therefore as a concise conceptual clarification, rather than a full treatment of pricing or superhedging contingent claims.

Key ideas

• The document considers a lower price bound for a contingent claim that may not be replicable.
• It relates the bound to expectations of replicable payoffs and risk-free probability measures.
• A replicable payoff has the same expectation under all measures in the specified set.
• Consequently, choosing among those measures does not change the stated bound.

Tags

# Contingent Claim Bounds


# Contingent Claim Bounds












In my course on discrete-time finance we derived the following equality for a lower bound for the value of a not necessarily replicable contingent claim $D$. Here we are looking at a single period market with risk-free interest rate $r$ and $E$ is the expectation operator. $$\sup\{E[Y(1)/(1+r)]:Y \text{ replicable, }\text Y\le D\} \\ = \inf\{E_{Q'}[D(1)/(1+r))]:Q' \text{ risk free P-measure}\text \}$$ My question: Which probability measure do we use to compute the first expectation on the left? Is it a risk-neutral one, or is it the real one?

## Answer by Winger 14 (score 0, accepted)

https://quant.stackexchange.com/a/57440

Coming back to this question way later, I managed to resolve my confusion. Using a well-known theorem we have that $E_{Q}[Y(1)]$ is constant on $\mathbb{M}$. Thus, it is irrelevant which measure we are using, since the resulting bound will be the same.

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.