A forward contract for 4 months is entered into when a stock index is at 1000. If the risk free interest rate is 3% per year (with continuous compounding) and the dividend yield on the index is 2% per year, what is the futures price? After 1 month the index price is 980, what is the value of the futures agreement assuming the risk free rate with continuous compounding is now 5% (but the original dividend rate remains 2%) and the contract price is equal to $50 times the index value?
For the first part of the question I used the following equation to find the futures price F0 = 1,000*e(0.03-0.02)*(4/12) = $1,003.3389. However, I'm not sure what to do with the second part.
The value $V_{t,T}$ of a long forward contract at time $t$ with expiry $T$ is given by the discounted change in the forward index times the contract price per index point, $P$. We can express the value as
$$\begin{align} V_{t,T}&=P\left(F_{t,T}-F_{0,T}\right)e^{-r_{t}(T-t)}\\\\ &=P(S_te^{(r_t-\delta_{t})(T-t)}-S_0e^{(r_{0}-\delta_{0})T})e^{-r_{t}(T-t)} \end{align}$$
where $S_0$ and $S_t$ are the spot indices on the purchase date and valuation date, respectively, $r_{0}$ and $r_t$ are the interest rates on the purchase date and valuation date, respectively, $\delta_0$ and $\delta_t$ are the dividend yield of the underlying stock on the purchase date and valuation date, respectively, $t$ is the number of years between the purchase and valuation dates, and $T$ is the number of years between the purchase and expiry dates.
We are given $P=\$50.00$, $S_0=1000$, $S_t=980$, $r_{0}=3%$, $r_{t}=5%$, $\delta_{0}=\delta_{t}=2%$, $T=1/3$, and $t=1/12$.
The numerical evaluation is left as an exercise for the interested reader.