Characterise how currency returns are distributed, and trace what that implies for implied volatility and the pricing of FX options.
The idea
Exchange rate returns are not lognormal with constant volatility. Volatility changes randomly over time, and exchange rates sometimes jump, for example when a central bank intervenes or a political shock occurs. Both features put more probability than a lognormal in the tails and give a higher peak, which is positive excess kurtosis. There is little asymmetry for a major pair, because a fall in one currency is a rise in the other. The implied volatility smile is therefore a symmetric U, lowest near the at-the-money (forward) strike and rising for both out-of-the-money calls and puts. Black-Scholes-Merton with the at-the-money volatility underprices both wings, so a strangle is undervalued while a risk reversal is roughly right. The effect is more pronounced for short maturities and flattens as the maturity lengthens.
A practice question
What shape does the implied volatility smile for currency options usually have?
- A steady decline from low to high strikes, with the highest volatility at the lowest strike.Why not: That is the equity skew, not the currency smile.
- A hump, with the highest volatility at the money and lower volatility at extreme strikes.Why not: That is the inverted pattern associated with thin tails, such as an anticipated single jump.
- A fairly symmetric U shape, with the lowest volatility near the at-the-money (forward) strike. - correct
- A flat line, because exchange rates follow geometric Brownian motion in the model.Why not: A flat line is the model's benchmark; observed currency options show a clear U-shape.
Fat tails on both sides make out-of-the-money calls and puts dear, so implied volatility rises as the strike moves away from the forward in either direction.
A harder one
Which pair of features of exchange rates best explains why the volatility of currency options is not flat across strikes?
- Volatility changes randomly over time, and the exchange rate sometimes jumps. - correct
- The firm's leverage rises when the rate falls, and investors fear a crash.Why not: Those are the equity explanations (leverage and crashophobia); they do not apply to an exchange rate.
- Option holders may exercise early, and dividends are paid unpredictably.Why not: Early exercise concerns American options and there are no dividends on a currency; neither explains the shape.
- The two interest rates differ, and forward points move with the rate differential.Why not: The rate differential changes the forward but is already in the model through the forward price, so it does not create a smile.
A lognormal with constant volatility fails because FX volatility is stochastic and the rate jumps (for example on central bank action or political events). Both put more probability in the tails.
The hardest tier
EUR/USD options show an implied volatility of 9.5% for a put struck at 1.10 (USD per EUR); the at-the-money volatility is 8.0% and the spot rate is 1.20. A USD-based counterparty wants the equivalent option quoted on USD/EUR. The equivalent instrument is a call on USD struck at 1/1.10 = 0.9091 EUR per USD. What implied volatility applies to it?
- 8.00%Why not: Substitutes the at-the-money volatility; the equivalent call is the same option and shares the put's volatility.
- 10.45%Why not: Multiplies the volatility by the 1.10 strike; volatility is a percentage and does not scale with the strike.
- 8.64%Why not: Divides the volatility by the 1.10 strike; volatility is unchanged by quoting the rate inversely.
- 9.50% - correct
A EUR put struck at K pays the same as a USD call struck at 1/K once converted at the strike. If S is lognormal, so is 1/S with the same volatility, so the implied volatility is unchanged. A symmetric smile stays symmetric when the quote is inverted.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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- Link the form taken by a smile or skew to the price distribution it implies for the underlying, and to the resulting option prices.
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- Set out other ways in which the volatility smile can be described.
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- Show how an option's Greek letter risk measures are affected by the volatility smile.
- Show how one jump in an asset's price affects a volatility smile.