Question:

Assume Nortel is a non-dividend-paying stock trading at $9. The riskless rate is 4%. Assume both a 3-month European CALL and a 3-month European PUT. Both the call and the put have a strike price of $8.

(i) Characterize the two portfolios (one containing a call and another containing a put) that create equivalent payoffs, and thereby demonstrate put-call parity. Show both portfolios have the same payoff under two scenarios: the stock price increases to $12 and the stock price decreases to $6.

(ii) Assume the price of the call is $2. According to put-call parity, what is the price of the put?

(iii) What is the minimum value of the call option?

Answer:

(i)
The four components are the call (c), the put (p), the stock (S) and cash equal to the discounted exercise price. The discounted exercise price = (K)EXP[(-rate)(time)]; the idea is that the cash will grow such that, at then end of the period, cash will equal the exercise price.

The first portfolio (#1) is the CALL plus (+) cash equal to the discounted exercise price; i.e., (K)EXP[(-rate)(time)]
The second portfolio (#2) is the PUT plus one share

Under a scenario where the stock price increases:
Portfolio #1 = Gain on call + Cash (K)
Portfolio #2 = Share of stock [put is worthless]

Specifically, under a scenario where stock = $12
Portfolio #1 = ($12 - $8) + $8 = $12
Portfolio #2 = $12

Under a scenario where the stock price descreases:
Portfolio #1 = Cash (K) [call is worthless]
Portfolio #2 = Share of stock + Gain on put

Specifically, under a scenario where stock = $6
Portfolio #1 = $8 [cash]
Portfolio #2 = $6 + ($8 - $6) = $8

(ii)
Put call parity says: c + (K)EXP[(-r)(T)] = p + S. Therefore,
p = c + (K)EXP[(-r)(T)] - S. In this case,
p = $2 + $7.92 - $9 = $0.92

(iii)
Minimum value of call = S - (K)EXP[(-r)(T)]. In this case,
Minimum value of call = $9 - $7.92 = $1.08