Question

On November 8, the S&P 500 index is at 1,305, the continuously compounded dividend yield is 3 percent, and the continuously compounded risk-free rate is 5.2 percent. The December futures contract, which expires in 40 days, is priced at 1,316.30. What is the fair value of the futures contract here? What arbitrage trade is possible given this prevailing pricing and what is the net profit if transaction costs are 0.5% of dollars invested and we invest $20mm?

Answer 1

Theoretical Futures Price = Spot Price*e^[(risk free rate-dividend yield)*t]

Where e = constant (2.71828), t = years to expiry

Applying the above formula,

Spot Price = 1305, Risk Free Rate = 0.052, Dividend Yield = 0.03, t = 40/360 = 1/9

Therefore, Theoretical Futures Price = 1305*e^[(0.052-0.03)*1/9] = 1305*e^0.00244   = 1305*1.00244(from table) = $1308.1842

Actual Futures Price is $1316.3 i.e Greater than Theoretical Futures Price

Therefore, Future is Overvalued.

Therefore, to make an Arbitrage Gain, Buy Spot & Sell under Futures Contract.

Steps to Arbitrage:

Now,

1. Borrow for 40 days @ Risk Free Rate
2. Buy Stock @ current price i.e. $1305
3. Sell Futures contact, expiring in 40 days

(4) Receive Dividends

After 40 days,

5. Sell under Futures Contract
6. Repay loan with interest

Arbitrage Gain:

Amount Realized – Amount Repaid for loan – Transaction Costs

= [Investment + Capital Gain + Dividends] - Amount Repaid for loan – [Amount Invested*0.005]

= [20000000 + {(1316.3-1305)*20000000/1305}+{20000000*0.03)}] – [20000000*e^(0.052*1/9)] – [20000000*0.005]

= [20000000 + 176245.2107 + 6000000] - [20000000*e^0.00578] - 100000

= 20776245.2107 - [20000000*1.00578] - 100000

= 20776245.2107 - 20115600 - 100000

= $560645.2107

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