Consistent Price Functionals and Admissible Dividend Streams

Summary

The document raises a question from mathematical finance about consistent price functionals. In the stated framework, a positive linear functional assigns prices to adapted processes, and consistency requires it to assign zero value to dividend streams generated by trading strategies. The notes also allow the process that pays one unit initially and zero thereafter to have value one.

The apparent tension is whether that initial-payment process can itself be generated as a trading strategy’s dividend stream. The document provides no resolution, definitions beyond this brief description, or worked example. Answering the question would require the precise model assumptions, including the traded assets, admissible strategies, and how dividend streams are defined. It is therefore a conceptual prompt about the boundary between attainable cash flows and the normalization of a pricing functional, rather than a developed pricing method.

Key ideas

• A consistent price functional is described as a positive linear map from adapted processes to real-valued prices.
• The stated consistency condition assigns zero value to dividend streams generated by trading strategies.
• The process paying one unit initially and nothing later is separately assigned value one.
• Whether that process is attainable depends on the model’s definitions of strategies and dividend streams.
• The document poses the issue but does not give a solution or supporting derivation.

Tags

# Is $(1,0,0,0,...,0)$ a legitimate dividend stream?


# Is $(1,0,0,0,...,0)$ a legitimate dividend stream?












A book I am reading defines a positive linear functional as a "price functional" from a set of adapted processes to the real numbers. Specifically, it defines a "consistent price functional" as one that gives the value zero to every dividend stream corresponding to any trading strategy. However, it also says that the value of the process $(1,0,0,0,....)$ can be set to $1$.

Then that means this process is NOT a dividend stream corresponding to some trading strategy. Why not?

EDIT: It's from page 59 of this notes.

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