# Chapter 10 Quanto Options

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

You can price and analyze the underlying risks of quanto options using our quanto options pricer. We used it to retrieve most of the graphs from this chapter.

Quanto options are also called quantity adjusting options.

## 10.1 Description

A quanto option is an option denominated in a currency other than the currency in which the underlying asset is traded. A quanto product converts underlying asset prices into units of the payoff currency by applying a fixed exchange rate. It is suited for investors who want to enter into an option strategy on a foreign underlying asset but are only interested in the % return of that strategy and want this return to be paid on their own currency with no FX rate exposure. The FX rate will be fixed to the rate prevailing at the inception of the deal and the payout of the quanto strategy will simply be this FX rate times the payout of the regular strategy. Cashflows are computed from the underlying in one currency but the payoff is made in another.

The payoff of a quanto call option is given by: \(\boxed{Quanto \; C_T = FX_0 * max(S_T - K, 0)}\)

Where the strike price, \(K\), is expressed in the underlying's currency.

### 10.1.1 Example

A European investor is long a 1Y ATM call on Glencore (GBP) and wants to get his return in EUR. Assume that Glencore'stock price is 3£, FX rate (investor/foreign) = 0.86 EUR/GBP and that in one year, Glencore'stock price is 3.3£.

At maturity, the payout of this quanto ATM call is 0.258€ (0.3 * 0.86). Since Glencore'stock has increased by 10%, the European investor also expects 10% return on his EUR investment.

As you can see, the payoff of quanto options is quite straightforward. Suprisingly, pricing them and understanding what market variables it depends on is a much harder task and will be the subject of the following section.

## 10.2 Additional sensitivies : correlation risk and FX volatility

In comparison to a vanilla option, a quanto 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

### 10.2.1 Correlation between underlying's price and FX rate

A European investor long an ATM EUR quanto call is short or long the FX correlation? The answer obviously depends how we express the FX rate. Let us express it as FX = investor/foreign.

Let us assume the correlation is positive. In this case, if the GBP gets more valuable against the EUR, Glencore's price tends to increase. The investor would then have been better of with a vanilla call. He is therefore short this correlation.

Like many correlations, it is very hard to obtain an implied quanto correlation from market data. If you have a liquid market for quanto options, you can back out this quanto correlation as all other parameters are known. Most of the time, you will have to estimate is using the realized correlation and taking some margin.

### 10.2.2 FX Volatility

Note that whathever how you expressed the exchange rate, its volatility will be the same. In other words, the volatility of EUR/GBP and the GBP/EUR are the same thing.

The sensitivity to this market variable is more tricky and less intuitive to grasp.

Is the European investor long or short the FX volatility?

To shed some light on the sensitivity to the FX volatility, it is time to introduce a small and intuitive model describing the stock price difference in the quanto currency for a small time interval.

\(\boxed{\frac{dF_t}{F_t} = (r_{local} - \rho \sigma_S \sigma_{FX})d_t + \sigma_S dW_t}\)

where

- \(F_t\) is the underlying stock quoted in the quanto currency
- \(\rho\) is the correlation between log(S) and log(FX)
- \(r_{local}\) is the risk-free rate of the underlying stock's own currency. In this case, it is the GBP risk-free rate.

\(F_t\) is defined so that a vanilla option on \(F_t\) is actually a quanto option on \(S_t\).

*Many things can be learned from this equation.*

**First of all**, it confirms our intuition about the correlation sensitivity as the holder of a quanto call (put) is always short (long) the correlation.

**Secondly**, it makes clear that the FX volatility sensitivity depends on the sign of the correlation. If the correlation is negative, then the holder of a quanto call (put) is long (short) FX volatility. If the correlation is positive, then the holder of a quanto call (put) is short (long) FX volatility.

**Thirdly**, it highlights the intuititive fact that the volatility used to price the quanto option should be the same as the IV of the underlying stock. This is obvious as the FX rate is fixed and therefore the option payout only depends on the actual stock variation.

**Then**, the drift part is slightly different as the delta hedge is affected by the FX rate movements. While this will be further discussed in the next section, we can already give a small intuition. If the underlying stock price doubles, it impacts positively your delta hedge financing as you will sell more stocks at a higher level and therefore receive more interest. However, if the correlation is such that a doubling in the underlying stock price results in a halving of the EUR value against the GBP, then your delta hedge financing is actually unaffected. With this in mind, it does not seem surprising to adjust the financing part of the model by \(- \rho \sigma_S \sigma_{FX}\). This term is called the *'quanto adjustment'* and can be added to the typical dividend yield term. Since it has a different sign, you can always think about the quanto adjustment as having an opposite effect as dividends.

**Finally**, notice that the volatility of the underlying stock also appears in the quanto adjustment. It can have an opposite effect as the usual volatility effect. Generally speaking, the quanto effect will be secondary.

## 10.3 Hedging FX Exposure

While hedging the FX exposure is not particularly intuitive, it is actually quite simple in practice.

As we have just seen in the above example, the FX hedge is captured by the delta hedge. The notional of a quanto option is agreed in the quanto currency. Therefore the notional in the local currency (underlying's currency) changes whenever the FX changes. For example, if the quanto currency doubles with respect to the local currency, the notional of the quanto option in the local currency also doubles and therefore the trader needs to double his delta hedge even though the stock price has not moved.

This example shows that there is no need for a trader to put an FX hedge in place for a quanto option. Well, if he sold a quanto option, he would need to swap the FX on the premium received. This is in line with the small and intuitive model of the previous section that prescribes financing in the local currency.

The majority of the exotic desks are long delta, therefore long dividends and short quanto correlations.