Question

Consider a put and call option with the same strike price and non-dividend-paying stock. Explain how an arbitrage opportunity can be created if the put-call parity is not satisfied by giving a concrete example.

Answer 1

According to Put-Call parity,

P + S0 = C + K/e^{r*t}

where

S0 is current spot price

P is price of Put option

C is price of call option

K is strike price

r is risk free rate of interest

t is no. of periods remaining

e is the exponential function = 2.72

Suppose,

C = 3	S0= 31
t = 0.25 (3 months)	r = 10%
K =30	P = 2.25

Here, S0 + P = 31 + 2.25

= 33.25....(i)

C + K/e^{r*t} = 3 + 30/e^{0.1*0.25}

= 32.26....(ii)

Sicce, (i) is not equal to (ii)

Therefore, there exists an arbitrage opportunity.

Since, S0 + P is overvalued, short sell the stock and buy the call.

Let St be the spot price after 3 months

Action now: Buy call for $3 Short put to realize $ 2.25 Short the stock to realize $ 31 Net cash received = 2.25 + 31 - 3 = $30.25 Invest $ 30.25 for 3 months @ 10% r

Action in 3 months Invested amount = 30.25*e^{r*t} = 30.25*e^0.025 = 31.02 If St > 30 Receive $ 31.02 from investment Exercise call to buy stock for $ 30. Net profit = 1.02

If St < 30 Receive $ 31.02 from investment put exercised by other party, we have obligation to buy stock for $ 30. Net profit = 1.02