Question

Suncor Energy Inc. (SU) shares are listed on the New York Stock Exchange. At 9:30 a.m. on January 14, 2016, these shares sold for $21.85 per share. The volatility on the returns of Suncor shares is approximately 24%. The following call and put option contracts were available for the months of January, February, and March:

	CALLS
Strike/Expiry	January 22, 2016	February 19, 2016	March 18, 2016
23	0.34	0.72	0.96
24	0.13	0.41	0.69
25	0.25	0.26	0.40

	PUTS
Strike/Expiry	January 22, 2016	February 19, 2016	March 18, 2016
23	1.28	2.01	2.14
24	2.63	2.80	2.92
25	3.60	3.70	3.95

Each option contract involves 100 shares. The risk-free rates for these three expiration dates are 0.6%, 1%, and 1.2%. All three rates are continuously compounded.  

a. Construct a box-spread using the March option contracts with exercise prices of 24 and 25.

b. Construct a profitable riskless arbitrage opportunity using this box-spread, with the requirement of $0 investment today. Calculate the NPV of the riskless profit.

Answer 1

Box Spread consists of following-
1. Bull CALL Spread with exercise price K1 and K2
2. Bear PUT Spread with same exercise price K1 and K2

For a Bull Call Spread -
1. Long CALL Option with strike price K1
2. Short CALL Option with strike price K2
3. K1 < K2, so price of 1st CALL is higher than price of 2nd CALL, hence setup of this spread requires initial investment.

For a Bear PUT Spread -
1. Short PUT Option with strike price K1
2. Long PUT Option with strike price K2
3. K1 < K2, so price of 1st PUT is lower than the price of 2nd PUT, hence setup of this spread also requires initial investment.

Final Pay-off from the Box Spread = (K2-K1) in all circumstances.
Hence present value of box spread = (K2-K1)*e^-rT

Part A:

So, our box spread as required in the question will have following options -
1. Long March CALL Option with strike price = 24, option price = 0.69
2. Short March CALL Option with strike price = 25, option price = 0.40
3. Short March PUT Option with strike price = 24, option price = 2.92
4. Long March PUT Option with strike price = 25, option price = 3.95

Hence,
Box-spread setup cost =
Premium paid for LONG CALL and Long PUT - Premium Received for Short CALL and Short PUT
= 0.69 + 3.95 - 0.40 - 2.92
= 1.32

Part B:
It will be a riskless arbitrage if PV of Box Spread is greater than the set-up cost of box spread.
Risk Free Rate for March Option = 1.2%

Today's Date = 14 January 2016
Expiry Date = 18 March 2016
No. of Days Passed = 64
No. of Days in 2016 = 366
T = 64/366 = 0.17486

Hence PV of Box Spread = (K2 - K1)*e^-rT
= (25 - 24)*e^{-0.012*0.17486}
= 0.9979

Hence, setup cost of 1.32 is higher than PV of Box Spread of 0.9979
Loss from this strategy = 1.32 - 0.9979 = 0.3221

So lets reverse the strategy to make a riskless arbitrage, lets call it Reverse Box Spread.

Strategy would be to short the box spread initially-
1. Short March CALL Option with strike price = 24, option price = 0.69
2. Long March CALL Option with strike price = 25, option price = 0.40
3. Long March PUT Option with strike price = 24, option price = 2.92
4. Short March PUT Option with strike price = 25, option price = 3.95
5. Invest the (K2-K1)*e^-rT at Risk Free Rate for 64 days in risk free bond.

At the time of expiry -
1. Bond will mature to give (K2-K1)
2. Pay-off the options utilized in setting-up of Reverse Box Spreads from Bond's maturity, with total pay-off = 0

Reverse Box-spread setup cost =
Premium paid for LONG CALL and Long PUT - Premium Received for Short CALL and Short PUT
= +0.40 + 2.92 - 0.69 - 3.95-
= -1.32

Amount invested in risk free bond = (K2 - K1)/e^-rT = (K2 - K1)e^-rT   = (25 - 24)*e^{-0.012*0.17486} = 0.9979

Hence Net Income from the staregy = 1.32 - 0.9979 = 0.3221 at t = 0
At expiry Net Pay-off = 0.

Hence, we have made a riskless arbitrage of 0.3221 using this strategy. This is also NPV of pay-off of this strategy.

Alternatively, we could have invested total 1.32 received by setting up reverse box spread in the riskless bond, which will give 1.32*e^rT = 1.32277 at expiry, use amount 1 from it to pay-off the box-spread Options, saving 0.32277 at the time of expiry.
PV of 0.32277 = 0.32277*e^-rT = 0.32277* = 0.3221, which is same as previous case.