Question

The value of an index on shares is currently at 350. The risk-free interest rate is 8% per year (continuous compound) and the dividend rate of the index is 4% per year (continuous compound).

Calculate the price of the future for a four-month contract. Reply

The value of a futures contract at the moment signed is zero, but at a later time its value can be positive or negative. The value depends on the exercise price and the underlying price according to the expression: f = S − Ke − rT (for assets that do not pay income)

What will be the expression for assets that pay a known dividend rate? f = Answer

If in the previous year the exercise price is K = 450, what is the initial value of the contract? Reply

Answer 1

Part 1: Price of futures for a four-month contract

To arrive at the futures price, we use the formula: F= S₀(e)^(r-d)T where:

F: futures price, S₀= value of the index today (spot price), r= risk-free rate, d= dividend yield, T= time

Now, plug in the values into the formula:

Step 1: F= 350(e)^{(0.08-0.04)*4/12}

Step 2: F= 350(2.71828)^0.0133

Step 3: F= 354.6861 (approx)

Final answer: The price of the futures is pegged at 354.6861 approximately. (Note: It would be useful to use a business calculator to make the adjustments quickly).

Part 2: Valuation of futures with dividend yield

In case of continuously compounded rates, we can use the above formula to derive the futures price i.e. F= S₀(e)^(r-d)T

However, we can also derive the futures price using F= S₀(1+r-d), where d denotes the dividend yield. (The logic is to subtract the dividend yield while calculating the futures price)

Note: Value of the futures contract is zero at the time of inception.