Forward Contract Value at Inception and the Fair Delivery Price

Summary

The document examines whether setting a newly entered forward contract’s value to zero is circular when deriving its fair delivery price. It distinguishes the contract’s value from its contractual delivery price: the value depends on the difference between the underlying asset’s spot price and the present value of the agreed payment. Under the stated no-arbitrage argument, the long forward position can be compared with holding the asset while borrowing the discounted delivery payment. This gives a value of spot price minus discounted delivery price, which is zero at inception when the delivery price equals spot grown at the risk-free rate over the term.

The post also points out that a contract with a nonfair delivery price can have nonzero value, using a zero-payment purchase obligation as an intuitive example. Its reasoning assumes a simple asset with no income or carrying costs and deterministic financing at the stated rate. The discussion does not extend the pricing relation to dividends, storage costs, or other market conventions.

Key ideas

• A forward’s contractual delivery price is distinct from the market value of the contract.
• The long forward value is derived by comparing it with an equivalent financed spot position.
• At inception, the fair delivery price makes the contract value zero under the stated assumptions.
• A contract with a delivery price different from fair value can have a nonzero value.

Tags

# value of forward contract at inception


# value of forward contract at inception












I am reading a derivation of the forward price $F$ of a forward contract. I think the author uses a circular argument to assume that "the value of the forward at inception is equal to 0" because the value of the forward at inception is equal to 0 only when the forward price is $S(0)e^{rT}$. When the forward price is not $S(0)e^{rT}$, the forward value should not be zero to prevent from an arbitrage opportunity. Can someone confirm it?

Edit: Here is my derivation. Consider two portfolios; one is long one forward contract for the price K at time T and short 1 unit of the underlying asset. The other one is made of borrowing $Ke^{-rT}$ cash. Then two portfolios have the same value at the maturity. Assuming no arbitrage opportunity, they should, by LOOP, have the same value at time t $\leq$ T, thus we can write the equation. $$F(t) = S(t) - Ke^{-r(T-t)}$$ where F(t) is the value of the forward contract, S(t) is the price of the underlying asset. By taking t = 0, $F(0) = S(0) - Ke^{-r(T)}$. That shows $F(0)$ is equal to zero only when $K = s(0)e^{r(T)}$. When the contractual price $K$ is not equal to $s(0)e^{r(T)}$, the forward price $F(0)$ should not be zero to prevent an arbitrage opportunity. A simple example is the value of a contract that requires to buy one unit of the asset for price 0 at time T is definitely not zero.

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