# Chapter 11 Compo Options

This chapter has been written based on Frans de Weert's book - Exotic Option Trading (2008).

Compo options are also called composite options.

## 11.1 Description

As a quanto option, a compo option is also an option denominated in a currency other than the currency in which the underlying asset is traded. Unlike quanto option, the holder of a compo option has exposure to the FX rate. In a compo option, the payout and the strike are fixed in the compo's currency. Trading a compo option allows the investors to protect the value in their own currency on a foreign investment.

### 11.1.1 Example

A European investor holds some Glencore stocks. Assume that Glencore'stock price is 3£ and the FX rate (investor/foreign) is 0.86 EUR/GBP. This means the EUR value of one Glencore share is 2.58€.

To protect his holding, he decides to buy a 1Y ATM compo put option on GLEN.

Assume that after one year, GLEN decreased to 2.7£ and the EURGBP decreased to 0.8. This means that the EUR value of one Glencore share is 2.16€ (0.8*2.7).

However, this loss is offset by the payout of the compo put option because its strike is fixed in EUR (compo put payout = 2.58€ - 2.16€).

As you can see, compo options can be used to protect the underlying value in the investor currency to both FX rate and stock price movements.

## 11.2 Additional sensitivies : correlation risk and FX volatility

In comparison to a vanilla option, a compo option is sensitive to two new market variables:

- the correlation between the log of the underlying price and the log of the FX rate
- the FX volatility

As we did with quanto options, let us model the stock price difference in the compo currency for a small time interval:

\(\boxed{\frac{dF_t}{F_t} = r_{compo} \;d_t + \sigma_{compo} \; dW_t}\)

where

\(\sigma^{2}_{compo} = \sigma^{2}_{S} + 2\rho \sigma_S \sigma_{FX} + \sigma^{2}_{FX}\)

and \(r_{compo}\) is the RFR of the compo currency.

The dynamics of this formula is quite different from the quanto scenario. This time, the main divergence with vanilla option lies in the difference of implied volatility.

This formula makes it clear that the holder of a compo option is long correlation and long FX volatility.

## 11.3 Hedging FX Exposure

Whenever a trader sells a compo put option on GLEN, he will want to delta hedge himself. Since the drift part of the above formula is the same as the drift part of a normal option on a stock in the compo currency, the delta can easily be determined by setting strikes and stock price in the compo currency. However, the trader can only execute his delta on the underlying stock quoted in the foreign currency. Even if the trader delta hedges himself, he will still have an FX risk as the compo option payout and the delta hedge will not be in the same currency.

Let us illustrate this using the above example and assume that the trader only delta hedges his short position on a 1Y compo put on GLEN. Let us assume for the argument's sake he did so on a delta of 1. That means he made 0.30£ (3£ - 2.7£) on his delta hedge per compo put, which is worth 0.24€ at maturity as we assumed the EURGBP went down to 0.8 by that time. As the EUR value of GLEN went from 2.58€ to 2.16€, he lost 0.42€ on the compo option but only made 0.12€ on his delta hedge. He ends up with a 0.18€ loss.

*Only delta hedging is clearly not enough. What can the trader do to be fully hedged then?*

The trader would need to buy EUR on the notional of his delta hedge to be fully hedged on his compo put position. Let us make sure that it is indeed the case.

To delta hedge a short compo put position, the trader needs to sell shares. For every share that he sold, he received 3£. He would need to sell this 3£ to get 2.58€. At maturity, the trader can buy back these 3£ for 2.4€. The total profit on this FX hedge is 0.18€ (2.58€ - 2.4€). Adding this profit to the profit of his delta hedge offsets the loss on the compo put position.

Basically if the trader sells stock as a delta hedge, he needs to sell the stock's currency and buy the compo currency in the same notional as his delta hedge. If he buys stock as a delta hedge, he needs to buy the stock's currency and sell the compo currency in the same notional as his delta hedge.

This FX hedge is dynamic and will need to be adjusted all along with his delta adjustments. In other words, he will need to have, at any time, the same notional of FX hedge as delta hedge.

We forgot about something though! he will also need a FX hedge on the compo option premium paid at inception for a long position. The reason is clear: to buy a compo option, he will first need to sell the local currency to be able to buy the compo currency. That gives him an FX position that needs to be hedged. The hedge will then be to buy the local currency and sell the compo currency.

For a short position, such an FX hedge on the premium will not have to be put in place as the model used above assumes financing in the compo currency (EUR in our example).