Question

Given the following data as of the close of trading on 10/26/18:

Quest Diagnostics (DGX): 91.47

T-bill: Asked 2.28, Days to Maturity 84.

a.Imagine now that the call option struck at $80 trades at a premium of $11.00. Show that you can earn an arbitrage profit from buying the option and exercising it immediately.

b.Continue to assume that the call option struck at $80 trades at a premium of $11.00. This time, take the following positions: (1) buy the call; (2) short the stock; (3) invest in the T-bill at the risk-free rate. Show the value of these positions at expiration. Fill out the following table and show you make arbitrage profit.

Position	Cash Flow Today	Expiration, DGX < 80	Expiration, DGX > 80
Buy call	-11
Short DGX	+91.47
Invest in T-bill	-80.47
TOTAL	0

c. Calculate present value of the arbitrage profit made in part d and compare it to the arbitrage profit you made in part b. Which strategy do you choose: immediate exercise or holding positions until expiration?

Answer 1

a) On Purchasing the call option for $11 and Immediately Exercising it to buy the stock for $80, total amount spent =$91 and the market value of the stock =$91.47

So arbitrage profit from buying the option and exercising it immediately = $0.47

b) Here steps are

1. Buy the call for $11

2. Short the stock for $91.47

3. Invest the remaining $81.47 in T-bills for 84 days to get $80.47*(1+0.0228*84/365) = $80.89 at expiration

4. At Expiration

if DGX<80, Call option is worthless, cashflow = 0

Short Share's cashflow = -DGX (on closing the short position)

T-Bill Cashflow = $80.89

Total Cashflow = 80.89 -DGX  

As DGX<80, the total cashflow is positive and the minimum value is $0.89 when DGX = $80

if DGX>80, Call option can be exercised, cashflow = DGX-80

Short Share's cashflow = -DGX (on closing the short position)

T-Bill Cashflow = $80.89

Total Cashflow = $0.89

So, the completed table looks like

Position	Cash Flow Today	Expiration, DGX < 80	Expiration, DGX > 80
Buy call	-11	0	DGX-80
Short DGX	91.47	-DGX	-DGX
Invest in T-bill	-80.47	80.89	80.89
TOTAL	0	80.89-DGX	0.89

c) The value of Arbitrage profit is at least $0.89 at expiration in all situation

So, present value of the Arbitrage profit is at least= 0.89/(1+0.0228*84/365)  =0.8853

which is much greater than the Arbitrage profit by Immediate profit.

Hence,Holding position until expiration must be chosen over Immediate Exercise