Composite & Quanto Options
Derivatives-based Portfolio Solutions
There are two types of option involving different currencies. The simplest is a composite option, where the strike (or payoff) currency is in a different currency to the underlying. A slightly more complicated option is a quanto option, which is similar to a composite option, but the exchange rate of the conversion is fixed.
COMPOSITE OPTIONS USE DIFFERENT VOLATILITY INPUT
A composite option is a cash or physical option on a security whose currency is different from the strike or payoff currency (i.e. Euro strike option on Apple). If an underlying is in a foreign currency, then its price in the payout (or strike) currency will usually be more volatile (and hence more expensive) due to the additional volatility associated with currency fluctuations. Only for significantly negative correlations will a composite option be less expensive than the vanilla option (if there is zero correlation the effect of FX still lifts valuations). The value of a composite option can be calculated using Black-Scholes as usual, by substituting the volatility of the asset with the volatility of the asset in payout currency terms. The payout (or strike) currency risk-free rate should be used rather than the (foreign) security currency risk-free rate. The dividend yield assumption is unchanged (as it has no currency) between a composite option and a vanilla option.
σPayout = volatility of asset in payout (strike) currency
σSecurity = volatility of asset in (foreign) security currency
σFX = volatility of FX rate (between payout currency and security currency)
ρ = correlation of FX rate (security currency in payoff currency terms) and security price
Composite options are long correlation (if FX is foreign currency in domestic terms)
The formula to calculate the volatility of the underlying is given above. As the payoff increases with a positive correlation between FX and the underlying, a composite option is long correlation (the positive payout will be higher due to FX, while FX moving against the investor is irrelevant when the payout is zero). Note that care has to be taken when considering the definition of the FX rate; it should be the (foreign) security currency given in (domestic) payoff currency terms.
For example, if we are pricing a euro option on a dollar-based security and assume an extreme case of ρ = 100%, the volatility of the USD underlying in EUR will be the sum of the volatility of the underlying and the volatility of USD.
QUANTO OPTIONS USE DIFFERENT DIVIDEND INPUT
Quanto options are similar to a composite option, except the payout is always cash settled and a fixed FX rate is used to determine the payout. Quanto options can be modeled using Black-Scholes. As the FX rate for the payout is fixed, quanto options are modeled using the normal volatility of the underlying (as FX volatility has no effect). The payout is simply the fixed FX rate multiplied by the price of a vanilla option with the same volatility, but a different carry. The carry (risk-free rate - dividend) to be used is shown below (the risk-free rate for quanto options is assumed to be the risk-free rate of the security currency, ie, it is not the same as for composite options).
cQuanto = carry for quanto pricing
dQuanto = dividend for quanto pricing
d = dividend yield
rfrSecurity = risk free rate of security currency
rfrPayout = risk free rate of payout currency
Quanto options are either long or short correlation depending on the sign of the delta
The correlation between the FX and the security has an effect on quanto pricing, the direction (and magnitude) of which depends on the delta of the option. This is because the dividend risk of an option is equal to its delta, and the dividend used in quanto pricing increases as correlation increases.
Quanto option calls are short correlation (if FX is foreign currency in domestic terms)
As a call option is short dividends (call is an option on the price of underlying, not the total return of the underlying), a quanto call option is short correlation. A quanto put option is therefore slightly long correlation. In both cases, we assume the FX rate is the foreign security currency measured in domestic payout terms.
Intuitively, we can see a quanto call option is short correlation by assuming the dividend yield and both currency risk-free rates are all zero and comparing its value to a vanilla call option priced in the (foreign) security currency. If correlation is high, the vanilla call option is worth more than the quanto call option (as FX moves in favor of the investor if the price of the security rises). The reverse is also true (negative correlation causes a vanilla call option to be worth less than a quanto call option). As the price of a vanilla (single currency) call does not change due to the correlation of the underlying with the FX rate, this shows a quanto call option is short correlation.