Question

The trading price of the ABC stock has been around $100 per share for the past months, and you believe it is going to stay in that range for the next 3 months. The price of a 3-month put option with a strike price of $100 is $10.

a) (4 points) If the risk-free interest rate is 5% p.a. with continuous compounding, what is the price of a 3-month call option on ABC stock at a strike price of $100 if it is at the money? (The stock pays no dividends.)

b) (6 points) Based on your belief about the future movement of the stock price, what options strategy using a put and a call would you adopt? What is the most money you can make on this position? Explain the reason.

c) (10 points) How can you create a position with a put, a call, and riskless lending (risk-free interest rate is 5% p.a. with continuous compounding) that would have the same payoff as the stock at the expiration date? What is the net cost of establishing that position now?

Answer 1

a). We will use the Put call parity equation to find the value of call

St + pt = ct + X * e^(-rt)

where,

St = Spot Price of the Underlying Asset

pt = Put Option Price

ct = Call Option Price

X* e^(-rt) =  Present Value of the Strike Price, discounted from the date of expiration

r = The Discount Rate, often the Risk-Free Rate

Here, Call option Price Ct = $10

Spot Price St = $100

Present Value of Strike Price = 100 x e^(-5%x 3/12) = 100 x 0.9876 = 98.76

Value of Put Pt = Ct + X*e^(-rt) - St

= 10+ 98.76 - 100

Value of Put = $8.76

b) If the price of stock remains at $100, a short strangle can deliver goof profit.

A short strangle involves Writing/ selling a Call & Put at same strike price. The investor will gain from both the put and the call, if the price remains same.

Maximum profit would be the premium received (from writing the Call & Put). If the stock remains at the current price, the Put & Call both would lose their time value at expiry. And since, the stock price has'nt moved the intrinsic value would also be zero. This would give the writer, the entire premium as profit.

Maximum Profit from short strangle strategy on stock ABC

= Premium on Call + Premium on Put

= $10 + $8.76

= $ 18.76

c)

The put-call parity relationship shows that a portfolio consisting of a long call option and a short put option at the same strike price should be equal to the value of stock at expiry.

St + pt = ct + X * e^(-rt)

The cost would be premium paid for call less the premium received from selling put

= 8.76 -10

= - 1.24 (to be paid)