Question

1. A 10-month European call option on a stock is currently selling for $5. The stock price is $64, the strike price is $60. The continuously-compounded risk-free interest rate is 5% per annum for all maturities.

1) Suppose that the stock pays no dividend in the next ten months, and that the price of a 10-month European put with a strike price of $60 on the same stock is trading at $1. Is there an arbitrage opportunity? If yes, how can you take advantage of it to make profit?

2) Now suppose instead that a dividend of $10 will be paid in six months. What price do you expect a 10-month European put with a strike price of $60 on the same stock to be trading at?

Answer 1

1. To find out arbitrage opportunity, we have to compare theoretical call option premium with actual call option premium $5. If both are different, there will be an arbitrage opportunity.

Put-Call Parity Model will be used

Rf = 5% per annum i.e. 4.17% for 10 months

Stock Price + Put Option Premium = Call Option Premium + PV of strike price

64 + 1 = Call Option Premium + 60/(1.0417)

Call option premium = 65 - 57.6

Theoretical Call option premium = 7.4

Actual Call option premium = 5

Call option is undervalued and there is an arbitrage opportunity which can result in arbitrage gain.

It can be done by selling Stock in cash market at $64 and buying call option with strike price of $60 for $5. Money received in net i.e. $59 will need to be invested in risk free investment. After 10 months invested amount will generate return of $2.46, making total amount $ 61.46. In all the cases at least $1.46 will be earned.

2. In case there is a dividend we have to deduct present value of dividend from the stock price.

Present value of dividend 10/{1+(0.05*6/12)} = 9.76

Stock price to be taken = 64 - 9.76 = 54.24

Put Call Parity Model

Stock Price + Put Option premium = Call Option Premium + PV of Strike price

54.24 + Put option premium = 5 + 60(1.0417)

Put option premium = 5 + 57.60 - 54.24

Put option premium = 8.36

So, if there is a dividend of $10 in 6 months, expected put option premium will be $8.36.