**FORMULA AND DERIVATION **

Notations used:-

\(K_0\) is the forward price agreed at the time \(t=0\) .

\(K_r\) is the forward price agreed at the time \(r\).

\(r\) is the time at which the value of the forward contract is to be found.

\(T\) is the time at which the forward contract will mature.

\(S_r\) is the price of the underlying asset at the time \(r\).

\(\delta \) is the risk-free force of interest.

\(V_l\) is the value of the forward contract to the party in the long position.

\(V_s\) is the value of the forward contract to the party in the short position.

Suppose an investor holds a long forward contract at time \(r\). The long forward contract was entered into at the time \(t=0\) to buy an asset \(S\) at the time \(T\) at the forward price \(K_0\) .
The investor needs to know the value of the forward at time \(r\left(0<r<T\right)\) , i.e. at some point of time before \(T\).

Consider the following portfolios at time \(r\):-

Portfolio 1- Consists of the existing forward contract bought at the time \(t=0\). Its current value is \(V_l\). A simultaneous investment of amount \(K_0e^{-\delta \left(T-r\right)}\) is made at time \(r\), in the risk free investment for \(T-r\) years.

Portfolio 2- A new forward contract is bought at time \(r\), with maturity at time \(T\) and forward price \(K_r=S_re^{\delta \left(T-r\right)}\). Also, an investment of amount \(K_re^{-\delta \left(T-r\right)}\) is made in the risk free investment for \(T-R\) years.

Now, let us consider the payout of the two portfolios at the time \(T\).

Portfolio 1- The investments of \(K_0e^{-\delta \left(T-r\right)}\) accumulates to \(K_0e^{-\delta \left(T-r\right)}\times e^{\delta \left(T-r\right)}=K_0\) . This\(K_0\) is paid on the forward contract to receive one unit of asset \(S\).

Portfolio 2- The investment of amount \(K_re^{-\delta \left(T-r\right)}\) accumulates to \(K_re^{-\delta \left(T-r\right)}\times e^{\delta \left(T-r\right)}=K_r\) . This \(K_r\) is paid on the forward contract to receive one unit of \(S\).

Since both the portfolios gives the same payout at the time \(T\), their prices can be equated [according to the no arbitrage assumption and law of one price].

The price of portfolio 1 is \({V_l+K}_0e^{-\delta \left(T-r\right)}\)

The price of portfolio 2 is \(K_re^{-\delta \left(T-r\right)}\)

Equating the two prices, we get:

\[{V_l+K}_0e^{-\delta \left(T-r\right)}=K_re^{-\delta \left(T-r\right)}\]
\[\Rightarrow V_l=K_re^{-\delta \left(T-r\right)}{-K}_0e^{-\delta \left(T-r\right)}\]
\[V_l=\left(K_r-K_0\right)e^{-\delta \left(T-r\right)}\]

The value of the contract to the party in the short position would be \(-V_l\) . This is because if the long party earns a profit of some amount, the party in the short position incurs an equivalent amount of loss.

Therefore,

\[V_l=\left(K_r-K_0\right)e^{-\delta \left(T-r\right)}\]
\[V_s=\left(K_0-K_r\right)e^{-\delta \left(T-r\right)}=-V_l\]

Using the above formula would pose several restrictions, as it would be difficult to ascertain \(K_r\), which is the price of a forward contract entered at time \(r\). We can therefore derive another formula in terms of the asset price when the forward contract was entered into at time \(t=0\) and the asset price at the time when the value needs to be found, i.e. at time \(r\).

This can be done by substituting the price of forward contract at time \(r\), in the above formula.

When the underlying asset is a security with no income, the forward price is given as follows:

\[K_0=S_0e^{\delta T}\]
\[K_r=S_re^{\delta \left(T-r\right)}\]

Substituting these in the above formula for \(V_l\), we get:

\[V_l=\left(K_r-K_0\right)e^{-\delta \left(T-r\right)}\]
\[\ \ \ =(S_re^{\delta \left(T-r\right)}-S_0e^{\delta T})e^{-\delta (T-r)}\]
\[\ \ \ =S_re^{\delta \left(T-r\right)}e^{-\delta (T-r)}-S_0e^{\delta T}e^{-\delta (T-r)}\]
\[\ \ \ =S_r-S_0e^{\delta T-\delta T+\delta r}\]
\[\ \ \ =S_r-S_0e^{(\delta r)}\]

and

\[V_s=-V_l\]
\[\ \ \ \ =S_0e^{\delta r}-S_r\]

When the underlying asset is a security with a fixed continuously payable dividend yield, the forward price is:

\[K_0=S_0e^{\left(\delta -D\right)T}\]
\[K_r=S_re^{\left(\delta -D\right)(T-r)}\]

Substituting these in the formula for \(V_l\) , we get

\[V_l=S_re^{-D(T-r)}-S_0e^{(\delta r-DT)}\]

and

\[V_s=-V_l\]
\[\ \ \ \ =S_0e^{\left(\delta r-DT\right)}-S_re^{-D\left(T-r\right)}\]

**EXAMPLES**

Example 1

Two months ago, an investor entered into a six month forward contract to sell a zero coupon bond worth \(\$480\) at the time of entering the contract, at a forward price of \(\$489.7\). The asset is currently worth \(\$486\). The investor wishes to find the value of this short forward contract. The six-month risk free force interest on government bonds was \(4\%\) p.a.

In this case,

\[S_0=480\]
\[S_r=S=486\]
\[\delta =4\%=0.04\]

\[{V_s=S}_0e^{\delta r-S_r}\]
\[\ \ \ \ =480e^{0.04\times {2}/{12}}-486\]
\[\ \ \ \ =483.21069-486\]
\[\ \ \ \ =-2.79\]

This means that if the forward contract was settled now (two months after entering the contract), the investor would incur a loss of \(\$2.79\). This is because even though the market price of the zero coupon bond is \(\$486\), the investor would have to sell it at a lesser price of \(\$483.21069\) (which is the value of the first term in the equation). Therefore, the forward contract has a negative value to the investor (party in the short position).

On the other hand, the value of the forward contract to the party in the long position(buyer of the bond) will be \(\$2.79\). This is because he would earn a profit of \(\$2.79\) by paying only \(\$483.21069\) for the bond which is actually worth \(\$486\) at that point of time