Question

Suppose that the dividend yield on a stock is 2% every quarter with continuous com- pounding. The stock price is standing at 40, and the futures price for a futures contract deliverable in three months is 42. What risk-free interest rate guarantees no-arbitrage opportunities?

Answer 1

To find the risk-free interest rate that guarantees no-arbitrage opportunities, use the concept of present discounted value. The risk-free interest rate is approximately 19.87%.

Step-by-step explanation:

To find the risk-free interest rate that guarantees no-arbitrage opportunities, we can use the concept of present discounted value. The present discounted value of the future dividends is equal to the futures price. We can use the formula:

FV = PV * e^(r * t)

where:

FV is the futures price

PV is the stock price

r is the risk-free interest rate

t is the time to maturity in years

In this case, FV = 42, PV = 40, and t = 3 months, which is equal to 0.25 years. Plugging these values into the formula, we can solve for r:

42 = 40 * e^(r * 0.25)

Simplifying the equation:

e^(r * 0.25) = 1.05

Using natural logarithms:

r * 0.25 = ln(1.05)

Solving for r:

r = ln(1.05) / 0.25

Using a calculator, we find that r is approximately 0.1987. Therefore, the risk-free interest rate that guarantees no-arbitrage opportunities is approximately 19.87%.