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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 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.

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

  2. 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?

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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).

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