###### Put-Call Parity

Put-call parity is a no-arbitrage concept. It involves a combination of cash and... **Read More**

This learning outcome covers how to differentiate forward price and forward value and how these are affected differently during the initiation, life cycle, and expiration of the contract. It is crucial to understand the difference between forward price and forward value first before moving on to calculating a forward contract value throughout the different stages of its life cycle.

The price of a forward contract is fixed, meaning that it does not change throughout the contract’s life cycle because the underlying will be purchased at a later date. We can consider the price of the forward contract “embedded” into the contract. The forward value is the opposite and fluctuates as the market conditions change. At initiation, the forward contract value is zero and then either becomes positive or negative throughout the life-cycle of the contract. Value and price are completely different from each other, and that is crucial to understand.

At the initiation of the forward contract, no money is exchanged, and the contract at initiation is valueless (V_{0}(T)). The forward price that the parties have agreed at the initiation is a special price that results in the contract having zero value and thus no arbitrage opportunities. The forward price at initiation is the spot price of the underlying compounded at the risk-free rate over the contract’s life.

$$ V_0 (T)=0 $$

$$ F_0 (T)=S_0 (1+r)^T $$

The value of the forward contract is the spot price of the underlying asset minus the present value of the forward price:

$$ V_T (T)=S_T-F_0 (T)(1+r)^{-(T-r)}$$

Remember that this is a zero-sum game: The value of the contract to the short position is the negative value of the long position.

At expiration, when discounting, we would normally compute that the forward price does not take place since the time remaining on the contract is zero. Therefore, the value of the forward contract (long position) will be:

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

QuestionConsider a forward contract that has a term of 2 years. The price of the asset underlying the contract is currently $200 and the risk-free rate is 9%. Given the forward price of $220, the value of the forward contract is

closest to:A. $14.83

B. -$1.83

C. $31.66

SolutionThe correct answer is A.

In this scenario, the value of the forward contract at initiation is the difference between the price of the underlying asset today and the forward price discounted at the risk-free rate:

200 – [220 / (1 + 0.09)

^{2}] = $14.83Note that the forward price at contract initiation is the unique price that would induce traders to participate in arbitrage until the price of the forward contract equals the non-arbitrage forward price.