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# Pricing and valuation of forward contracts

In case of traditional assets such as equities and bonds, and options, value and price are sometimes used interchangeably because they both represent the same thing. But in the case of forwards, futures and swaps, value and price have fundamentally different meanings. Price in case of a forward commitment is the amount at which the underlying is transacted to be exchanged while their value is the net benefit that they would generate for the buyer.

Forward commitments are created such that they have zero value at inception, and as the underlying’s price changes, its value becomes positive or negative. We typically calculate value from the perspective of the buyer (i.e. the long position).

## General expression for value of a forward contract

A forward contract obligates two parties to exchange the underlying at time T at a price agreed at inception, called the forward price F0(T). The value of a forward contract (VT(T)) at expiry (to the long position) equals the spot price of the underlying at expiration (ST) minus the agreed price (i.e. the forward price).

$V_T(T)=S_T-F_0(T)$

The value to the short position is exactly opposite to value to the long position i.e. − VT(T) = F0(T) − ST.

## Forward price and value at initiation

Since no exchange takes place at time 0, a forward contract has zero value at inception. As the forward contracts are valued using arbitrage, if someone holds an asset whose current spot price is S0 and he sells it forward, he should not earn more than the risk-free rate (because there is no risk). Hence, the forward price must equal the current spot price compounded at the risk-free rate r.

$F_0(T)=S_0\times{(1+r)}^T$

## Benefits and costs of the underlying

If the asset has benefits (such as dividends) represented by γ and/or costs such as storage costs, represented by θ, such that net benefits are γ − θ, forward price F0(T) is determined by after adjusting for them:

$F_0(T)=S_0\times{(1+r)}^T-(\gamma\ – θ)×(1+r)^T$ $F_0(T)={(S}_0-\gamma\ +\ \theta)\times{(1+r)}^T$

If the asset has net benefits, the forward price is lower and vice versa.

## Value of forward contract after initiation but before expiry

At any time after inception but before expiration, the value of a forward contract equals the then spot price minus the present value (at the risk-free rate) of the forward price:

$V_t(T)=S_t-\frac{F_0(T)}{{(1+r)}^{T-t}}$

Where T is the original time to expiry and t is the time which has lapsed so far.

If the underlying has cost to carry, we need to subtract the compounded value of net benefits from the then spot price St which equals (γ – θ) × (1+ r)t.

$V_t(T)=S_t-(\gamma\ – θ)×(1+r)^{t}-F_0(T)(1+r)^{T-t}$