Final answer:
The value of the derivative can be calculated using the Black-Scholes formula, which takes into account the current level of the index, the risk-free rate, the dividend yield, and the volatility of the index. The value of the derivative is determined by calculating the expected future value of the index and comparing it to the strike price of 1,000. If the expected future value is greater than 1,000, the payoff is $100. Otherwise, the payoff is zero.
Step-by-step explanation:
The value of the derivative can be calculated using the Black-Scholes formula, which takes into account the current level of the index, the risk-free rate, the dividend yield, and the volatility of the index. In this case, the current level of the index is 960, the risk-free rate is 8% per annum, the dividend yield on the index is 3% per annum, and the volatility of the index is 20%. Using these inputs, we can calculate the value of the derivative as follows:
First, we calculate the expected future value of the index after six months. Assuming the index follows a log-normal distribution, we can use the following formula:
Expected Future Value = Current Value * exp((Risk-Free Rate - Dividend Yield - (Volatility^2/2)) * Time + Volatility * sqrt(Time) * Random)
Where Time is the time period in years (6 months = 0.5 years), and Random is a random number from a standard normal distribution. Plugging in the values, we get:
Expected Future Value = 960 * exp((0.08 - 0.03 - (0.2^2/2)) * 0.5 + 0.2 * sqrt(0.5) * Random)
Next, we calculate the payoff of the derivative. If the expected future value of the index is greater than 1,000, the payoff is $100. Otherwise, the payoff is zero.
Finally, we discount the payoff to the present value using the risk-free rate. The present value is calculated as:
Present Value = Payoff * exp(-Risk-Free Rate * Time)
Plugging in the values, we get:
Present Value = $100 * exp(-0.08 * 0.5)
Therefore, the value of the derivative is the present value of the payoff, which equals $100 * exp(-0.08 * 0.5).