Question

David Murphy observes that a stock, Twin Inc., which is not expected to pay dividends in the next year,
is currently trading at $38.56/share. A put option with a strike price of 40 and 6-month expiration date
costs $5.15 and a call option with the same strike and expiration date is priced at $4.32 for each share
Twin's stock. The current risk-free rate is 0.2% per month. After checking the prices with the put-call
parity, David decides to take an arbitrage opportunity by

A. Taking short positions in the call option and the underlying stock and holding long
positions in the risk-free zero-coupon bond with a par of $40 and in the put option.
B. Taking short positions in the put option and the underlying stock and holding long
positions in the risk-free zero-coupon bond with a par of $40 and in the call option.
C. Taking long positions in the call option and the underlying stock and holding short
positions in the risk-free zero-coupon bond with a par of $40 and in the put option.
D. Taking long positions in the put option and the underlying stock and holding short
positions in the risk-free zero-coupon bond with a par of $40 and in the call option.

Answer 1

D. Taking long positions in the put option and the underlying stock and holding short
positions in the risk-free zero-coupon bond with a par of $40 and in the call option.

• PUT CALL PARITY THEOREM

There is a relationship between the price of the call option and the price of a put option on the same underlying instrument with same strike price and the same expiration date. The relationship commonly referred to as the put call parity theorem. We have been considering put call parity relationship applicable only for European options.

Where,

E = Exercise Price

S = Current underlying asset price

P = Put Premium

C = Call Premium

r = risk free rate

T =Time to maturity

Provided,

Current price of Twin's Stock (S) = $38.56

Strike price (E) = $40

Put premium (P) = $5.15

Call premium (C) = $4.32

risk free rate(r) = 0.2% per month

Time to maturity (month)(T) =6

And,

thus, Put Call Parity does not hold here, Arbitrage opportunity exist.

Arbitrage Strategy :

According Put call parity the payoff of the Stock and Put option is same as risk free bond and Call option, we can write this in following manner -

Bond Par value is $40, thus Zero coupon bond current price would be -

putting in above equation -

We can see Stock and Put option is cheaper than Bond and Call and we know, we should always buy cheaper stocks thus, Arbitrage strategy should be -

Long Position in Put option and underlying stock and Short position in Zero coupon bond and Call option

Hope this will help, please do comment if you need any further explanation. Your feedback would be highly appreciated.