Question

European-style options on foreign currencies trade at the Philadelphia Exchange. A call option on sterling with an exercise price of 0.925 £/$ and a time to expiration of 3 months has a price of 0.0125 £/$. The current price of sterling is USD 0.94 £/$, the volatility is 3%, the risk-free rate of interest on U.S. dollars is 5.40% continuous and the risk-free rate of interest on sterling is 5.80% continuous. Is there an arbitrage opportunity? If there is such an opportunity, what is the arbitrage profit and what is the strategy to earn it?

Answer 1

We use the Black Scholes model to price the option and then compare it to the traded price of the call.

rf = 5.8% (risk-free rate of Sterling), vol = 3%, X(strike price) = 0.925, S(Spot Price) = 0.94, T-t =3/12=0.25

The Black-Scholes model is given as:

Hence, the option price as calculated from the above formula is:

Option Price = $0.03.

Hence, we see that the traded price of the call option is 0.0125 which is less than the price calculated from the Black-Scholes model. Therefore, there is an arbitrage opportunity.

To make use of this arbitrage opportunity, we should go long on the call option available in the market and short the synthetic call. Hence, we make use of the put call parity. Put call parity is given as:

Call + Borrowed Amount = Put + Underlying.

Hence to short a synthetic call, we will have to borrow the amount and short the put and the underlying.

The arbitrage profit from this would be =0.03-0.0125 = $0.0175 per call option.