Question

Currently, I owe euros (short position in euro). Currently, 1 euro = $1. There is a call option on euro with an exercise price of $1/euro and a premium of $0.02/euro There is also a put option on euro with an exercise price of $1/euro and a premium of $0.03/euro Is there an arbitrage opportunity here? (no risk and all profit) If so, explain. (Hint: think about combining your short position in euro with positions in both the call and the put options).

Answer 1

We are having a short position in Euro. This means we are afraid of Euro appreciating. To hedge the same we will buy a call option with exercise price of $1 to hedge risk of share price rising.
Hedging involves making risk less profits. We have a put option with exercise price of $1. Since we are afraid of Euro rising we will buy a call option which will have initial premium. To reduce the cost of strategy we will sell put option and receive initial premium.

Initial Premium paid on call option = $0.02/Euro
Initial Premium received on put option = $0.03/Euro

Net Initial Premium Received = $0.03/Euro - $0.02/Euro
                                          = $0.01/Euro

Since this is greater than 0 and combining this strategy with short position in Euro we will end up 0, arbitrage opportunity exist as there will be risk less profit involved in the strategy.
Profit from arbitrage strategy = $0.01/Euro

If share price went up, call option will be exercised and payoff from call option will set off loss from short stock.
If share price went down, put option will be exercised and the negative payoff from put option will set off profit from short stock.

Payoff Table :
Let S0 be the price of $/Euro at maturity

Particulars	$/Euro > 1	S/Euro = 1	$/Euro < 1
Payoff from Short Stock (a)	-(S0 – 1)	0	1 – S0
Payoff from Call Option (b)	(S0 – 1)	0	0
Payoff from Put Option ©	0	0	-(1 - S0)
Total Payoff (a) + (b) + (c)	0	0	0

So in any $/Euro share price at maturity outflow = 0 and profit will be net initial premium received = $0.01/Euro.