Question

Apple stock sells at $186.18 and is not expected to pay any dividend over the next six months. The 6-month $180-strike call is selling at $27.90. The risk-free rate is 3% per annum continuously compounded. Options considered here are assumed to be of European style.
a) What is the theoretical price of the 6-month put?
b) The 6-month put is selling at $17.80 in the market. How would you undertake this arbitrage opportunity? Show the cash flow table.
c) How would your answer to b) change if the 6-month put is selling at $20?

Answer 1

(a)

According to the put-call parity, the following must stand true:

Call Premium + PV of Strike = Spot price of underlying + Put Premium

With the information above, we can calculate the Put Premium as:

Put Premium (Theoretical price of Put) = Call Premium + PV of Strike - Spot price of Apple stock

Put Premium (Theoretical price of Put) = $ 27.90 + (180*e^(-0.03*6/12)) - $186.18

Put Premium (Theoretical price of Put) = $ 205.22 - $186.18

Put Premium (Theoretical price of Put) = $ 19.04

(b)

As calculated in (a) above, the Theoretical price of Put should be $ 19.04. As the put is currently selling for $ 17.80 in the market, it is currently undervalued and we must follow the following steps to create an arbitrage:

• Long the Put (Pay Premium $ 17.80)
• Short the Call (Receive Premium $ 27.90
• Long the Apple Stock (Pay $ 186.18)
• Borrow Amount required at the risk-free rate of 3% p.a. for 6 months.

Calculating amount to be borrowed:

Long Put (Premium Paid)	$ 17.80
Long Apple Stock (Amount paid)	$ 186.18
Short Call (Premium Received)	- $ 27.90
Amount required to be borrowed at risk free rate	$ 176.08

After 6 months, we will have to pay back: $ 176.08* e^(0.03*6/12)

After 6 months, we will have to pay back: $ 178.84

On the date of expiry, either of two things can take place

• Situation 1: Either the Spot price > Strike Price (Call is in the money, Put out of money)
• Situation 2: Else the Spot price < Strike Price (Put is in the money, Call out of money)

Let's assume in the first situation that Spot price at expiry = $ 185, thus ( > Strike of $180), and

Assume that in the second situation that Spot price at expiry = $ 175, thus ( < Strike of $180)

Note: In situation 1, as the call is in the money, put is out of money and in situation 2, the put is in the money and call is out of money. Thus, the Long call counter-party would exercise the option in situation 1 and not in situation 2. Similarly, we (the long put party) would exercise the option in situation 2 and not in situation 1.

Situation 1 (Cash Settlement)

Payment to Long Call counter-party	- $ 5 ($185-$180)
Proceeds from Selling stock in open market	$185
Payment to Bank (Borrowed money)	- $ 178.74
Arbitrage Profit	$ 1.26

Situation 2 (Cash Settlement)

Proceeds from Short Put counter-party	$ 5 ($180-$175)
Proceeds from Selling stock in open market	$175
Payment to Bank (Borrowed money)	- $ 178.74
Arbitrage Profit	$ 1.26

Thus, in both the possible situations we have generated an arbitrage profit of $ 1.26.

(c)

As calculated in (a) above, the Theoretical price of Put should be $ 19.04. As the put is currently selling for $ 20 in the market, it is currently overvalued and we must follow the following steps to create an arbitrage:

• Short the Put (Receive Premium $ 20)
• Long the Call (Pay Premium $ 27.90)
• Short-sell the Apple Stock (Receive $ 186.18)
• Lend/ Invest excess amount at the risk-free rate of 3% p.a. for 6 months.

Calculating excess amount to be lend/ invested:

Short Put (Premium Received)	$ 20
Short-sell Apple Stock (Amount received)	$ 186.18
Long Call (Premium Paid)	- $ 27.90
Amount to be lend/ invested at risk free rate	$ 178.28

After 6 months, we shall receive: $ 178.28* e^(0.03*6/12)

After 6 months, we shall receive: $ 180.94

On the date of expiry, either of two things can take place

• Situation 1: Either the Spot price > Strike Price (Call is in the money, Put out of money)
• Situation 2: Else the Spot price < Strike Price (Put is in the money, Call out of money)

Let's assume in the first situation that Spot price at expiry = $ 185, thus ( > Strike of $180), and

Assume that in the second situation that Spot price at expiry = $ 175, thus ( < Strike of $180)

Note: In situation 1, as the call is in the money, put is out of money and in situation 2, the put is in the money and call is out of money. Thus, we, the Long call party would exercise the option in situation 1 and not in situation 2. Similarly, the  long put counter-party would exercise the option in situation 2 and not in situation 1.

Situation 1 (Cash Settlement)

Proceeds from Investing/ Lending	$ 180.94
Proceeds from Gain from Call counter-party	$ 5
Payment to Purchase and return stock	- $ 185
Arbitrage Profit	$ 0.94

Situation 2 (Cash Settlement)

Proceeds from Investing/ Lending	$ 180.94
Payment to Put counter-party	-$ 5
Payment to Purchase and return stock	- $ 175
Arbitrage Profit	$ 0.94

Thus, in both the possible situations we have generated an arbitrage profit of $ 0.94.