Question

Suppose the call price is $14.10 and the put price is $9.30 for stock option where the exercise price is $100, the risk free rate is 5 percent (continuously compounded), and the time to expiration is one year. Explain how you would create a synthetic stock position and identify the cost. Suppose you observe a $100 stock price; identify any arbitrage opportunity.

Answer 1

To create a synthetic long stock position, we must buy the call and sell the put with the same strike price ($100 here). So if the price of the underlying stock goes above $ 100, the put option is worthless and will not be exercised by the buyer while we can exercise the call option which now behaves like a long stock in terms of the payoff. Likewise, if the price goes below $100, the buyer of the put option will exercise it while the long call option we hold is worthless and hence we stand to lose the difference between the price of the stock and the strike price and the payoff diagram thus behaves like a long stock bought at the strike price of $100.

The cost is $14.10 spent on buying the call minus $9.30 obtained on selling the put = $4.80

To spot an arbitrage, we use the put/call parity equation:

Price of Call + PV (strike price) = Price of put + Current price of stock

Current Price of stock = 14.10 - 9.30 +

= 4.8 +       (365 day convention)

=4.8 + 95.12

= 99.92

So we can sell the stock and buy the put, call and the risk free rate bond to give us a profit of $0.08 at expiry.

(Using a 360 day convention, T = 360/365 where we get an even lesser arbitrade profite of $0.01 at expiry)