Question

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.

a) 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?

b) 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

A) Present Value of strike price = 60(e^-.05*10/12)

=60(0.959)

=57.54

So now stock price price - PV strike price=64 - 57.54

=6.46 and as 5< 6.46

So Yes,Arbitrage oppurtunity exists.

An arbitrageur should buy the option and short the stock. This generates 64 – 5 = $59. The arbitrageur invests 59 for 10 months at 5%. Regardless of what happens a profit will materialize.

B) Present Value of strike Price =. 60(e^-.05*10/12) + 10e^(-0.5*6/12)

=60(0.959) + 10(0.975)

=57.54 +9.75

=67.29

so as 64<67.29 so here no arbitrage exists and we can use put call parity equation to calculate value of put:

So, c+PV(x)=p + s

: 5 + 67.29 = 64 +p

p=72.29-64

p=8.29

so price of put will be 8.29.