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Forward Pricing, No-Arbitrage Conditions, and Market Existence

Article Quant Q&A · Author: Evan Aad
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Summary

The document examines a logical gap in a standard forward-pricing result: a theorem gives the forward price under assumptions of a perfect, arbitrage-free market, but does not itself establish that such a market containing a forward contract exists. It asks whether such a market can be constructed and whether a forward contract can be introduced into an existing market without changing those properties.

The response points toward the Fundamental Theorem of Asset Pricing and the role of a risk-neutral measure in connecting no-arbitrage pricing, market completeness, and replication. In this framing, a forward’s value can be determined from an underlying asset when the market structure supports replication. The document does not provide a proof, construction, or detailed account of the theorem’s assumptions, so it serves chiefly as a pointer to the relevant theoretical framework. Readers should consult the cited result and the underlying asset-pricing theory to assess existence and model-specific conditions.

Key ideas

  • A forward-pricing formula is conditional on assumptions about the market and contract.
  • The stated pricing result does not establish the existence of a market satisfying those assumptions.
  • Risk-neutral pricing connects no-arbitrage conditions with the valuation of financial instruments.
  • Replication by an underlying asset can determine the price of a forward in a suitable market.
  • The response points to foundational theory but does not provide a proof or specify its assumptions.

Tags

Derivatives pricingArbitrageFuturesStatistics
Full text
# On the existence of a perfect market with no arbitrage that contains a forward contract


# On the existence of a perfect market with no arbitrage that contains a forward contract












Consider the following theorem from p. 31 of Steven Roman's "Introduction to the Mathematics of Finance Arbitrage and Option Pricing" (Undergraduate Texts in Mathematics, 2012), giving the forward price of a forward contract.

The theorem is a conditional: IF the market is perfect and has no arbitrage AND IF $F$ is a forward contract in this market, THEN the forward price of $F$ is $F_{0, T} = S_0e^{rT}$. However, the theorem says nothing about the existence of such a market. Which brings me to the following questions.

- Does a perfect market with no arbitrage exists (mathematically), in which a forward contract is defined?

- Is it possible to define a forward contract in any given perfect market with no arbitrage without changing the market's properties of being perfect and free of arbitrage, regardless of what other financial instruments are already defined in it?

## Answer by GWD (score 1, accepted)

https://quant.stackexchange.com/a/16981

You should dive into: Fundamental Theorem of Asset Pricing: If there is a risk-neutral measure, then there is no arbitrage. This then drives the existence of arbitrage, market perfection and replication by which one instrument (here S) is the basis for the price of another (here F).

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.