Question

The current value of INDY index is 1,015 and a portfolio replicating this index has a value of $1,015. European options with a strike price K = $1,000 and maturing in T = 6 months trade on YBM. The continuously compounded, risk-free interest rate r is 5 percent per year. The stocks underlying INDY index have a dividend yield δ = 1.2 percent per year. A trader quotes a call price c = $80 and a put price p = $37.

Find the amount of arbitrage profits that you can make today by trading securities based on one share of INDY (Clearly show the arbitrage portfolio and all the steps involved)      

Answer 1

This sum has to be solved using the Put Call Parity Theorem which is as follows:

Put premium + Current Market price of Stock = Call premium + Present Value of Exercise price

i.e. 37 + 1015 = 80 +1000/(1+(Rate for 6m - Dividend yield)

i.e. 1052 = 80 + 1000/[e^(5%-1.2%)*6/12]....... since it is continuously compounded (e is a constant at 2.718281828)

i.e. 1052 = 80 + 1000/1.019182

i.e. 1052 = 80 + 981.18

i.e. 1052 = 1061.18 ......that is, they are not equal, there is an arbitrage opportunity, since 1052 is lesser, it is underpriced, buy a put option and buy 1 share at 1015. At the same time, sell the call option and borrow an amount since the portfolio is overpriced.

The amount to be borrowed is as follows:

Stock price+ put option purchased - call premium received on sale

=1015+37-80

=972

After 6 months, when the borrowing has to be repaid, the borrowing amount will be

=972*e^(5%-1.2%)*6/12

=990.645

The arbitrage profit would be as follows:

Steps involved in arbitrage position	Cost involved
Borrow $972 for six months and create a position by selling one call option for $80/- and buying one put option for $37/- along with a share for $1015/-	-990.645
After six months, if share price is more than the strike price, call option would be exercised and if it is below strike price then put option would be exercised, irrespectively 1000 would be received	1,000
Arbitrage Net Profit (+) / Net Loss (-)	9.355