Question

Derive the upper and lower bound for a six-month call option with strike price K=$75 on stock XYZ. The spot price is $80. The risk-free interest rate (annually compounded) is 10%. If the option price is below the lower bound, describe the arbitrage strategy.

Answer 1

The lower bound can be calculated as:

= Spot Price – (Strike price * (e^(-Rf*Time)))

= 80 – (75 * (e^(-0.1*0.5))) = $8.67

Now, the present value of strike price is

= (75 * (e^(-0.1*0.5)) = 71.33

Now, if the option price is below the lower bound, say for eg. It is $4.

Now, the arbitrageur will buy the option and short the stock.

This will generate 80 – 4 = $76

Which he will invest at 10% for 6 month.

So, now if stock price goes below 75, the arbitrageur will lose the premium of $4 but he will gain on the short position, more than that.

For eg: stock price becomes 74. Then gain = (80-74) – 4 + ((1.1)^0.5 -1) * 80, which is greater than 0.

If stock price goes above 75, the arbitrageur will exercise the option and buy the shares at 75 and close the short position.

If price is say 78, gain = -4 + 80 – 78 + (0.1)^0.5 * 80, which is greater than zero

So, there will always be gain