No-Arbitrage Prices of Cash Flows and Forward Contracts as Choquet Representations

TL;DR

This paper characterizes no-arbitrage prices of deterministic cash flows and forward contracts using Choquet representations, showing how market conditions influence valuation formulas and extending to multi-currency settings.

Contribution

It provides a novel characterization of arbitrage-free pricing as Choquet integrals over zero-coupon bonds, including counterexamples and extensions to forward markets.

Findings

Unique arbitrage-free prices are given by Choquet integrals under mild conditions.

Counterexamples show the limits of the Choquet representation when assumptions are relaxed.

Valuation of foreign cash flows aligns with forward FX rates under certain assumptions.

Abstract

In a market of deterministic cash flows, given as an additive, symmetric relation of exchangeability on the finite signed Borel measures on the non-negative real time axis, it is shown that the only arbitrage-free price functional that fulfills some additional mild requirements is the integral of the unit zero-coupon bond prices with respect to the payment measures. For probability measures, this is a Choquet representation, where the Dirac measures, as unit zero-coupon bonds, are the extreme points. Dropping one of the requirements, the Lebesgue decomposition is used to construct counterexamples, where the Choquet price formula does not hold despite of an arbitrage-free market model. The concept is then extended to deterministic streams of assets and currencies in general, yielding a valuation principle for forward markets. Under mild assumptions, it is shown that a foreign cash flow's worth in local currency is identical to the value of the cash flow in local currency for which the Radon-Nikodym derivative with respect to the foreign cash flow is the forward FX rate.