# Delta Hedging/ Exchange for Currency Options

I'm looking at 2 cases of hedging EURUSD, using call spread or range forward.

Lets say spot is 1.1300 and my buy call is at 1.1300 and sell call is at 1.1500. Hypothetically I'm assuming that this is delta neutral.

And in case of a range forward, buy call is at 1.1400 and sell put is at 1.1270. In this case, my delta is positive. Am I exposed to any risk here and upon closing of options do I need to hedge the delta, by buying EURUSD spot?

IS the call spread or range forward more effective, assuming premia for both are close to each other

I am assuming you are short EUR and long USD based on your description of your hedges. I am also assuming the size of your hedges and your fx position are the same.

In the first example of a hedge of this position, you cannot be delta neutral. A long at-the-money (ATM) EUR call will have a higher delta than the short out-of-the-money (OTM) call--you will be long delta on the hedge but not enough to offset the short EUR position. You are buying a call that is ATM, which should protect you from any appreciation of EUR from 1.13. To help defray the cost of this call, you are selling the 1.15 call. This call spread would leave you exposed to appreciation beyond 1.15 of EUR.

I modeled your trade up in Bloomberg to clarify my points. As you can expect the long ATM call is about 50% delta (49.8356%) and the short OTM call is -1.389% delta for a net delta on the hedge of 48.4466%. Your short EUR position will have a delta of -100%. As a result, your net delta will be -51.553%. You would pay 0.2884% to put this one week hedge on against your short EUR position.

The payoff diagram below of your first hedge should help you see where you are hedged. Any loss from your short EUR position resulting from an appreciation of the EUR will be offset by a gain on your hedge. This is demonstrated by the upward sloping between the spot levels of 1.13 (long call strike) and 1.15 (short call strike). You are flat above 1.15 and below 1.13 on your hedge. In other words, your hedge is not protecting your short EUR position in these ranges.

In your second example, this is commonly known as a risk reversal. Without knowing the maturity and vol of your hedges, it is impossible to know exactly what your delta is but you will be short delta overall if my assumption of you being short EUR is true. The hedge will be long delta (long calls and short puts are both long delta) but not enough to offset the short delta of your short EUR position. The hedge will protect you from EUR appreciation beyond 1.14, at the expense you not gaining on the depreciation of the EUR lower than 1.127. Most people do these risk reversals where the deltas on the long call leg are the same as that of the short put leg to try balance the upside and downside of the currency. 25 delta is a common choice. Others will put this on dollar neutral. In other words, they will pick a delta (and strike) on one leg of the trade and have the other option struck such that the premium for that option will exactly offset the premium of the first leg.

On your risk reversal, you would be long 12.2266% delta from your long call and long 36.0762% delta from your short put for a total long delta of 48.3028%. You will be short 100% delta on your short EUR position for a net delta of -51.6972%. You will receive 0.1367% to put this hedge on for 1 week against your short EUR position.

The payoff diagram below of your second hedge will help illustrate where you are protected from your hedge. You will be hedged on any appreciation above 1.14 from your long call as shown by the upward sloping line where spot is greater than 1.14. Your short put position will offset any gains from your short EUR position one for one below 1.127 as demonstrated by the upward sloping line below 1.127. In the range from 1.127 - 1.14, you will be fully exposed to your short position in EUR, losing from any appreciation and gaining from any depreciation of EUR as your hedge is flat in this range.

As for the effectiveness of the hedge, it depends on what you mean by effectiveness. What is your objective and view of the EUR. Both of these structures provide some protection but at different costs (or remaining exposures).

• What a difference, huh? In one case you are hedged only against small movements in the exchange rate, in the other you are hedged globally except for small changes in exchange rates. Of course "small" depends on the context. Mar 26, 2019 at 23:23