# Hedging FX exposure

If a trader sells the compo put on GSK, he will want to hedge himself.

However, the trader can only execute his delta on the local stock, which is not quoted in the compo currency.

Even if he delta hedges, he will still have an FX risk, as the payout on the compo option will be in the compo currency and his delta hedge is in the local currency of the underlying stock.

Example: the trader hedges the short compo put option by just delta hedging.

Suppose for the argument's sake that he hedges the compo option on a delta = 1, he makes 2£, which is worth 3\$ at maturity.

As the \$ value of GSK goes from 26\$ to 16.5\$, the trader loses 9.5\$ on the compo option and makes 3\$ on his delta hedge.

--> Just delta hedging is clearly not enough!

The trader would need to buy \$ on the notional of his delta hedge to be fully immune to all the risks of the compo option on GSK.

As an FX hedge, the trader would sell 13£ to receive 26\$ for every GSK share he shorted as part of his delta at inception of the trade.

Total profit on his FX hedge = 6.5\$    --> 6.5\$ + 3\$ = 9.5\$ = the loss on the compo option.

Summary

To hedge the FX exposure on a compo option, one just needs to realise that:

- if one sells stocks as a delta hedge --> one needs to sell the currency of the stock and buy the compo currency in the same notional as the delta hedge.

- if one buys stock as a delta hedge --> one needs to buy the currency of the stock and sell the compo currency in the same notional as the delta hedge.

Obviously, the FX hedge is not static and should be adjusted along with any delta adjustments.

In other words, to be perfectly hedged against both stock movements and FX movements, one need to have at any time the same notional of FX hedge as delta hedge.

All the previous did not take into account that one also needs to put an FX hedge in place for the premium paid for the compo option at inception of the trade.

For the call option, the FX hedge on the premium paid works in the opposite direction to the FX hedge on the notional of the delta hedge.

For the put option, the FX hedge on the premium paid works in the same direction as the FX hedge on the notional of the delta hedge.

The reason a trader wants to hedge the FX exposure on the premium is that in order to buy the compo option in the compo currency the trader would first need to sell the local currency to buy the compo currency. This effectively gives him an FX position which still needs to be hedged.

Counterintuitively, no FX hedge needs to be executed on the premium if the trader sells a compo option, as the trader receives the premium in the compo currency and the model used to determine the price and the delta of the option assumes financing in the compo currency.

TO SUM UP

1. Long Compo Call

FX hedge on delta notional --> Sell local currency to buy compo currency

FX hedge on premium paid --> Buy local currency to sell compo currency

2. Short Compo Call

FX hedge on delta notional --> Buy local currency to sell compo currency

FX hedge on premium paid --> None

3. Long Compo Put

FX hedge on delta notional --> Buy local currency to sell compo currency

FX hedge on premium paid --> Buy local currency to sell compo currency

4. Short Compo Put

FX hedge on delta notional --> Sell local currency to buy compo currency

FX hedge on premium paid --> None