Question

6) A stock is selling for $18.50. The strike price on a call, maturing in 6 months, is $20. The possible stock prices at the end of 6 months are $22.50 and $15.00. Interest rates are 6.0%. How much money would you borrow to create an arbitrage on a call trading for $2.00?

A) $2.54

B) $4.85

C) $6.60

D) $8.85

Answer: B need details of solution

Answer 1

Let us start by buying 'x' quantity of the stock and writing 1 call. At the end of 6 months, the possible stock values are 22.50 and 15. Hence the value of this portfolio can be either:

Stock Price (22.50) : 22.50 x - 2.50 (which the loss on writing the call option)

Stock Price (15) : 15 x (call option expired worthless).

Now under no arbitrage condition, we should both these portfolio equate to each other. Hence

22.50 x -2.50 = 15 x which gives us the value of x = 1/3 (assuming fractional purchase)

Hence the portfolio value at the end of 6 months = 22.5 * 1/3 - 2.5 = 15 * 1/3 = 5

Now since we have a no arbitrage condition, the present value of this portfolio value is what the current price of x stock and 1 short call should be. We will discount the portfolio value of 5 at 6% for 6 months as below:

Present value = 5 * e^{?(-6%, 6/12)}? where is the 2.7183 and this value comes to = $4.85

Now the present value of the portfolio should be : stock price * x - sale price of 1 call = 4.85

18.5 * 1/3 - 1C = 4.85 which gives the call price to be $1.31 but since the call price is $ 2 , it means there is possible arbitrage in this case.

Hence we will borrow 4.85 (the present value of this portfolio from 6 month hence) and sell 1 unit of call and purchase 1/3 unit of stock. For no arbtrage the call price should be at 1.31 but since it is trading at 2, the excess is the arbitrgae profit.