Question

Google’s stock is trading at a price of $905. A call option with 3 months to maturity is available with an exercise price of $890 with a premium of $61.40. A put option with an exercise price of $890 trades at a premium of $48. The risk free rate of interest is 5% per year, create a long synthetic call position.

Answer 1

As per put-call parity, Call Premium + Present Value of Strike Price = Put Premium + Stock Price, for a call and put option with the same underlying, same expiry and same strike price.

Strike Price = $ 890, Current Stock Price = $ 905, Call Premium = $ 61,4 and Put Premium = $ 48, Risk-Free Rate = 5% and Expiry = 3 months or 0.25 years

Call Premium = Put Premium + Current Stock Price - PV of Strike Price

Hence, a synthetic call position can be created by going long on a put option & the underlying stock, accompanied with borrowing money at the risk-free rate for 3-months such that the borrowing matures to expend an amount equal to the strike price at maturity.

Call Premium = 48 + 905 - [890/e^(0.05 x 0.25)] = $ 74.06

Synthetic Call Payoff:

If Stock Price is S > 890, then Put Payoff = $ 0, Stock Payoff = S and Borrowing Payoff = - $ 890

Net Payoff = 0 + S - 890 = S - 890

If Stock Price is S < 890, then Put Payoff = (890 - S), Stock Payoff = S and Borrowing Payoff = - $ 890

Net Payoff = (890-S) + S - 890 = $ 0

As is observable, the payoff for the synthetic call position exactly matches the payoff for a normal call position, thereby proving that a synthetic call position can be created by buying a put and underlying asset while simultaneously borrowing money equal to the present value of the common strike price.