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Describe what correlation does to the prices of quanto options, and of exotic options written on several assets.

FRM® Part II · Market Risk Measurement and Management · Correlation Basics: Definitions, Applications, and Terminology · FRM2.MR.8.c

The idea

Correlation enters option prices whenever the payoff depends on more than one random variable. A quanto option pays in a currency other than that of its underlying asset, at a fixed exchange rate. Its price therefore depends on the correlation between the asset and the exchange rate, through an adjustment to the risk-neutral drift. Let S be the asset price, rF the foreign risk-free rate, σS and σFX the volatilities of the asset and the exchange rate, ρ their correlation and T the maturity in years. With the exchange rate quoted as domestic per foreign currency, the quanto forward price is F = S × exp((rF − ρ × σS × σFX) × T). Higher correlation lowers the quanto forward and the quanto call. For multi-asset exotics the direction depends on the payoff. Basket calls, calls on the minimum and joint-probability digitals gain from higher correlation. Best-of calls, spread options and exchange options lose. For a spread or an exchange option the relevant variance contains minus 2 × ρ × σ1 × σ2. Exam point: read the payoff before deciding the sign.

Formulas

Quanto-adjusted forward price

F = S × e^((rf − ρ × σS × σFX) × T)

S
the asset price today in its own currency
rf
the foreign risk-free rate, continuously compounded
ρ
the correlation between the asset and the exchange rate
σS
the asset's volatility
σFX
the exchange rate's volatility, quoted as domestic per foreign currency
T
the time to maturity in years

The correlation term is subtracted, so positive correlation lowers the forward. With the exchange rate quoted the other way round the sign of ρ flips; forgetting to multiply the adjustment by T is the other slip.

A practice question

From the question bank, with the reason each wrong option is wrong.

What distinguishes a quanto option from an ordinary option on a foreign asset?

  1. It is paid in the foreign currency and converted at the spot rate prevailing at expiry.Why not: That is an ordinary foreign-asset option with open FX exposure; the quanto fixes the conversion rate.
  2. Its payoff depends on two assets and pays the better performer.Why not: That is a rainbow (best-of) option; the second variable in a quanto is the exchange rate.
  3. Its payoff depends on a foreign asset but is paid in domestic currency at a fixed exchange rate. - correct
  4. It pays only if the exchange rate crosses a barrier.Why not: That is a barrier FX feature; a quanto involves no barrier, only a fixed conversion.

A quanto fixes the conversion rate in advance, so the price depends on the correlation between the underlying asset and the exchange rate.

A harder one

A foreign index trades at 100 in the foreign currency. A quanto forward pays the index level at maturity in domestic currency at a fixed 1:1 exchange rate. The foreign risk-free rate is 3%, index volatility is 25%, exchange rate volatility (domestic per foreign) is 10%, the correlation between the index and the exchange rate is +0.5 and maturity is 2 years, with continuous compounding. The quanto-adjusted forward price is closest to:

  1. 106.18.Why not: Omits the quanto adjustment and uses the plain foreign forward 100 x exp(0.03 x 2).
  2. 108.87.Why not: Adds the correlation term instead of subtracting it, using r + rho x sigma_S x sigma_FX.
  3. 101.77.Why not: Applies the correlation adjustment for one year instead of two, forgetting to multiply by T.
  4. 103.56. - correct

F = 100 x exp((0.03 - 0.5 x 0.25 x 0.10) x 2) = 100 x exp(0.035) = 103.56. Positive correlation lowers the quanto forward.

The hardest tier

Two assets, both priced at 100, have volatilities of 30% and 20%. A one-year option pays the amount by which asset 1 outperforms asset 2 (an exchange option). Use the approximation that, for equal prices, the option value is about 0.4 x 100 x sigma x sqrt(T), where sigma is the volatility of the ratio S1/S2, with sigma² = s1² + s2² - 2 x rho x s1 x s2. As correlation rises from 0.2 to 0.8, the value moves from:

  1. 13.0 to 7.4. - correct
  2. 15.7 to 19.0.Why not: Adds the cross term (plus 2 x rho x s1 x s2), so the value wrongly rises with correlation.
  3. 4.2 to 1.4.Why not: Uses the variance of the ratio instead of its standard deviation in the approximation.
  4. 12.0 to 12.0.Why not: Uses the higher single-asset volatility and treats correlation as irrelevant.

At rho = 0.2: sigma = sqrt(0.09 + 0.04 - 0.024) = 0.3256 and value = 13.0. At rho = 0.8: sigma = sqrt(0.034) = 0.1844 and value = 7.4. Value falls as correlation rises.

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