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Finance Questions: Stock Prices, Gold Value

A stock is expected to pay a dividend of $2 per share in one month and in four months. The stock price is $40 and the risk-free rate of interest is 6% per annum with continuous compounding for all maturities. An investor has just taken a short position in a five-month forward contract on the stock.
What are the forward price and the initial value of the forward contract?
Three months later, the price of the stock is $48 and the risk-free rate of interest is still 6% per annum. What are the forward price and the value of the short position in the forward contract?
Make sure that you show or explain all calculations. Make sure you answer all questions above.

Suppose the current spot price for gold is $800 per ounce. The risk-free interest rate available to all investors for borrowing or lending is 0.50% per month (monthly compounding). Forward contracts are available to buy or sell gold for delivery in 1 year; the forward price for gold is $890 per ounce. You have a large inventory of gold.
Assume that storage costs for gold are zero. Is there an arbitrage opportunity? If you answer "YES," then show step by step how you would make a profit and calculate the profit per ounce of gold. If you answer "NO," then show why there is no arbitrage opportunity.

Now assume that the present value of the storage cost for gold is $100 per ounce for one year of storage. Is there an arbitrage opportunity? If you answer "YES," then show step by step how you would make a profit and calculate the profit per ounce of gold. If you answer "NO," then show why there is no arbitrage opportunity.
Make sure that you show or explain all calculations. Make sure you answer all questions above.

Solution Preview

A stock is expected to pay a dividend of $2 per share in one month and in four months. The stock price is $40 and the risk-free rate of interest is 6% per annum with continuous compounding for all maturities. An investor has just taken a short position in a five-month forward contract on the stock.
What are the forward price and the initial value of the forward contract?
The present value of the dividends is

I = $2e^(−0.06×1/12) + $2e^(−0.06×4/12) = $1.9900+$1.9604 = $3.9504
The forward price is therefore
F0 = (40 − 3.9504)e(0.06×5/12) = $47.22
The value at origination of a forward contract is zero.
Three months later, the ...

$2.19