Question

A financial derivative is guaranteed to be worth $110.00 in 20 months. Assume the risk-free rate is 5.9%.

(a) What is the derivative worth today?

(b) Describe an arbitrage opportunity if the derivative is trading at $70.00 today. This should only involve one of the derivatives. What is your guaranteed profit in 20 months from the arbitrage?

Answer 1

a) Guaranteed Worth of derivative in 20 months = F = $110, Time period = t = 20 months = (20/12) years, Risk free rate = r = 5.9%

Worth of derivative today = F / (1+r)^t = 110 / (1+5.6%)^(20/12) = 110 / (1.056)^(20/12) = 110 / 1.095064 = 100.4507

Hence Worth of derivative today = $100.4507

b) Current price of derivative = $70

Since the current price of derivative is less than the current worth(no arbitrage price) of derivative. So derivative contract is underpriced today. It is know that we always buy or take long position underpriced security.or contract. Due to this mispricing there exists a arbitrage opportunity.

So Steps in arbitrage

i) Borrow $70 at risk free rate of 5.6% for 20 months to buy (long position) in derivative contract.

ii) Amount owed after 20 months = Amount borrowed x (1+r)^20/12 = 70 x (1+5.6%)^20/12 = 70 x 1.095064 = $76.6544

iii) After 20 months derivative has a guaranteed worth of $110, Hence it can be sold or we can settle(close) long for $110, Proceeds from sale of derivative contract or closing of long position in derivative = $110

iv) After 20 months, Guaranteed Profit = Proceeds from closing of long position - Amount owed after 20 months = 110 - 76.6544 = $33.3456

Hence Guaranteed profit = $33.3456