Question

A European call option and put option on a stock both have a strike price of $21 and an expiration date in 4 months. The call sells for $2 and the put sells for $1.5. The risk-free rate is 10% per annum for all maturities, and the current stock price is $20. The next dividend is expected in 6 months with the value of $1 per share.

(a) In your own words, describe the meaning of “put-call parity”.

(b) Check whether the put-call parity holds.

(c) If the put-call parity does not hold, describe step-by-step how an investor can take the advantage of this arbitrage opportunity to make a profit.

(d) If these two options are American options rather than European options, briefly explain whether the put-call parity still holds here.

Answer 1

A) Put call parity theory (PCPT) defines the relation between a European call option and a European put option. For the put call parity theory to be effective the securities should be of same class with same underlying securities, same strike price and same explanation date.

B) Equation of PCPT

Vc + PV of SP = Vp + CMP

Vc =Value of call option (call premium ie. $2)

PV of SP =Present value of strike price (refer below)

Vp =Value of put option (put premium ie. $1.5)

CMP = Current Market Price ( current stock price ie $20)

PV of SP = Strike price ÷ e^rt

r= rate t= term

rt = 10(rf given) × 4 (months) ÷ 12 (months)

rt = 10×4÷12 = 3.33%

e^rt = e^3.33%

e^x = (x×x)÷2 +1 +x

e^3.33% = (0.0333×0.0333)÷2 +1 +0.0333

= 1.03385

PV of SP = 21÷ 1.03385

= $20.3124

PCPT

Vc + PV of SP = Vp + CMP

2 + 20.3124 = 1.5 + 20

22.3124 = 21.5

Left hand side is not equal to right hand side. Hence put call parity theory does not hold.

C) arbitrage opportunity

Left hand side (that is the call option), is overpriced than the right-hand side (that is the put option). Hence we can take a short position in the call option ( can be sold) and can take a long position in the put option( can be purchased).

Let's understand this by shortening the call option and taking long position in the put option along with the funds required to be borrowed by the arbitrageur at the risk free rate.

ie, -2+1.5+20 = 19.5

Hence arbitrageur needs to borrow $19.5 at risk free rate of 10% for 4 months.

The repayment after 4 months will be

= 19.5 × e^rt

= 19.5 × 1.03385

= $20.16

After 4 months other call option or put option will be in the money and the stock will be sold for $21 and the arbitrageur will get $21.

Hence the profit will be

= 21 - 20.16

= $0.84

D) put call parity theory holds true only for European option and does not hold for American option this is because American option can be sold at any time before expiry and the security will not hold till expiry. Early exercise of option will result in a variation of the present value of the two portfolios.