Question

Consider a single-stock futures contract on Apple stock. Consider the following scenario:

• Annualized, continuously compounded risk-free interest rate for 2-month period: r = 2.95%.
• Annualized, continuously compounded risk-free interest rate for 4-month period: r = 5.65%.
• Current spot price of Apple stock: $562 per share.
• Dividend per share of $0.62 in 2 months.
• Contract expiration: T = 4 months.
• Futures price on Apple single-stock futures 4 months from now: $600 per share.

An arbitrage opportunity exists. What is the net profit per share when the futures contract expires? Use a strategy that has zero net cash flows today and zero net cash flows in two months.

Answer 1

Theoretical Futures Price = [Spot Price + PV of Cost of Carry - PV of Dividends]*Future Value Factor = [S + (C*e^-rt) + (D*e^-rt)]*e^rt

Where S = Spot Rate, e = constant (2.71828), r = Risk Free Rate, t = years to expiry

Applying the above formula,

S = 562, r = 0.0565 & 0.0295, t = 4/12 = 1/3 & 2/12 = 1/6, C = 0, D = 0.62

Therefore, Theoretical Futures Price = [562 + 0 + (0.62*e^-0.0295/6)]*e^0.0565/3 = [562 + 0 + (0.62*e^-0.0049167)]*e^0.018833 = [562+(0.62*0.9951)]*1.019 = $573.3

Actual Futures Price is $600 i.e Greater than Theoretical Futures Price

Therefore, Future is Overvalued

Therefore, There IS an Arbitrage Opportunity

To make an Arbitrage Gain, Buy Spot & Sell under Futures Contract

Steps to Arbitrage:

Now,

1. Borrow $562 for 4 months @5.65%
2. Buy Stock @ current price i.e. $562
3. Sell Futures contact, expiring in 4 months to sell at $600

Net Cash Flow = 562-562 = 0

After 2 months,

4. Receive Dividend of $0.62
5. Invest 0.62 for 2 months @2.95%

Net Cash Flow = 0.62-0.62 = 0         

After 4 months,

(6) Realize Invested Dividend i.e. 0.62*e^0.0295/6 = 0.62*e^0.0049167 = 0.62*1.0049167 = 0.623

7. Sell under Futures Contract and receive $600
8. Repay loan with interest 562*e^0.0565/3 = 562*e^0.018833 = 562*1.019 = 572.678

Net Cash Flow = 0.623+600-572.678 = $27.945

Therefore, Arbitrage Gain = $27.945