Question

1.The price of a three-month European put option on a stock with a strike price of $60 is $5. There is a $1.0067 dividend expected in one month. The current stock price is $58 and the continuously compounded risk-free rate (all maturities) is 8%. What is the price of a three-month European call option on the same stock with a strike price of $60?

Select one:

a. $5.19

b. $1.81

c. $2.79

d. $3.19

2.For the above question, if the price of a three-month European call option on the same stock with a strike price of $60 is $4.50, are there any arbitrage opportunities? If so, what are the transactions to achieve risk-free profits?

Select one:

a. There are arbitrage opportunities: write the put, short sell the share, buy the call, and invest the rest.

b. There are arbitrage opportunities: write the put, write the call, short sell the share, and invest the rest.

c. There are NO arbitrage opportunities

d. There are arbitrage opportunities: write the call, borrow some money, buy the put, and buy the share.

3.For the previous question where the price of a three-month European call option on the same stock with a strike price of $60 is $4.50, are there any arbitrage opportunities? If so, how much risk-free profits can an arbitrageur make in three months?

Select one:

a. There are arbitrage opportunities. The risk free profit in three months would be $1.34

b. There are arbitrage opportunities. The risk free profit in three months would be $2.11

c. There are NO arbitrage opportunities

d. There are arbitrage opportunities. The risk free profit in three months would be $0.70

Answer 1

S0 = 58
Put = $5
X = $60
Time to expiration = 3 months
R = 8% p.a.
Div = 1.0067 in 1 month
PV of Div = Div * (e(-r * t))
= 1.0067 * [e(-0.08*1/12)]
PV of Div = $1.000010988

Put-Call Parity
Put + S0 - PV(div) = Call + X * e(-r*t)
5 + 58 - 1.000010988 = Call + 60 * e(-0.08*3/12)
5 + 56.99998901 = Call + 58.8119204
Call = 61.99998901 - 58.8119204
Call = $ 3.188068613 ~ $3.19 i.e. Option D

2) Consider 2 separate Portfolios
Portfolio 1 = Put Option + Shares
Portfolio 2 = Call Option + X * e(-r*t)

Put - Call Parity
Put + S0 - PV(div) = Call + X * e(-r*t)
5 + 56.99998901 = 4.5 + 58.8119204
61.99998901 is not equal to 63.3119204

Since the Call + investment in risk free bond (Portfolio 2) is overvalued when compared to Put + Share purchased (Portfolio 1). In order to satisfy put call parity equation, both the LHS (Put + Stock) and RHS (Call + Risk Free Bond) should be equal.
Since RHS = 63.3119204 and Call price ($4.5) is more than no-arbitrage price of $3.19, it means that the call option is overvalued.

Hence we need to sell the overvalued portfolio i.e Portfolio 2 (Call + Risk-Free Bond) and buy the undervalued portfolio which is Portfolio 1 (Put + Shares).
Option D is correct
We need to write the call option, borrow the money, buy the put option and buy the shares in order to make arbitrage profits.

3) The Risk-free Proft in 3 months earned is $1.34

ST (Spot Price at the end of expiration) = $65 is assumed for calculation purpose
Amount to be Borrowed = Call Premium Received + Put Premium Paid + Strik Price Amount
= -4.5 + 5 + 60
Amount to be Borrowed = $57.5
Amount to be Repaid after 3 months = Amount Borrowed * e(r*t)
=  58.66
Interest Amount = Amount Repaid - Amount Borrowed
=  $ 1.16

Since we have written a call option and ST > X, therefore, we will a negative payoff of $5 on the day of expiration. However, the total loss would be Premium received - Payoff (4.5 - 5) = $(0.5)
Since ST > X, therefore put option will expire worthless and hence the max loss on put option will be premium paid i.e. $(5)
Share Purchase Amount = S0 - PV of Div
=  57.00
Profit On Shares Purchased = ST - Share Purchased
= 65 - 57
Profit On Shares Purchased = $8