Question

You are observing the following market prices. A put option that expires in six months with an exercise price of $45 sells for $5.80. The stock is currently priced at $40, and the risk-free rate is 3.6% per year, compounded continuously.

1. What is the price of a call option with the same exercise prices and maturity?
2. In the above example, suppose you form a portfolio consisting of selling a call option and buying a put option on the same stock. Is this portfolio risk free?
3. 1.    In the above example, suppose you form a portfolio consisting of buying a share of the stock, selling a call option and buying a put option on the same stock. Is this portfolio risk-free?

Answer 1

a) From put-call parity, we know that:

Call price + Strice price*e^(-r*T) = Stock price + Put price.

risk free rate, r= 3.6%, tenure = 6 months = 0.5 year

Stock price, S_o = $40. Strike Price, X = $45. Put price, P = $5.80

call price, c = $40 + $5.80 - $45*e^(-0.036*0.5) = $1.60

2) From put-call parity we have:

S + P = C + Xe^(-rT)

Buying a put option and selling a call option is equivalent to P-C

Since, P-C = Xe^(-rt) - S

The Xe^(-rt) portion is constant but the stock price is variable and so this portfolio is not risk - free.

3) Similarly if we construct a portfolio by buying the stock and the put option on the same stock and selling the call option on this stock, this portfolio is equivalent to S+P - C

From Put-Call parity, S+P-C = Xe^(-rT) = $45*e^(-0.036*0.5) = $44.20.

Hence, this portfolio has a constant value and is risk-free.