# Intuitive explanation for the smile in FX

What is the intuitive reason for the smile in FX?

For equities this usually down to crash risk.

The general explanations quoted by @AK88 are the economic reasons why one observes skew in FX option markets (along with most others).

The reason FX has symmetric smiles rather than a consistent pattern of one-sided "smirks" is because any FX price is really a ratio.

To expand on that, note first that (vanilla option) skew is basically about the terminal probability distribution of the returns being non-gaussian. That is to say, for some foreign exchange rate $$X$$, we have that the terminal distribution

$$\Phi\left(\log\left(\frac{X_T}{X_0}\right)\right) \notin G$$

(where $$G$$ represents the set of gaussian distributions), giving rise to some observed skew.

Now, observe that $$X$$ is itself a derived variable from some currency ratio, so that we really have

$$X_t = \frac{A_t}{B_t}$$

for example $$X$$ might be $$\frac{\mathrm{EUR}}{\mathrm{USD}}$$.

Now, since $$\log\left(\frac{X_t}{X_0}\right) = \log(X_t) - \log(X_0)$$ and $$\log(X_0)$$ is constant, all that really matters is the stochastic process for $$\log(X_t)$$. But we also have that

$$-\log(X_t) = -log(\frac{A_t}{B_t}) = log(\frac{B_t}{A_t}) = \log(1/X_t)$$

so the skew of, for example $$\frac{\mathrm{USD}}{\mathrm{EUR}}$$ on the left-hand side is the same as the skew of $$\frac{\mathrm{EUR}}{\mathrm{USD}}$$ on the right-hand side.

In other words, every skew that is high on one side (in $$X_t$$ terms) automatically has a companion skew that is high on the other side (in $$1/X_t$$ terms).

This doesn't mean that all FX skews will be perfectly symmetric, but it does mean you cannot have a general pattern of an asymmetric "smirk" to the high (or low) side in currency options in general. Any smirk to one side has a mirror if you just reverse the quotient of currencies.

• pointing out that currencies are always ratios was why i awarded this – Permian Dec 3 '19 at 19:12
• Skews do occur regularly in FX. Consider GBP during Brexit and JPY during the height of the carry trade (2006). It occurs when there is a lopsided event or lopsided positioning. – kdragger Dec 14 '19 at 19:58

I think Hull does a pretty good job explaining the smile in FX options:

In the mid-1980s, a few traders knew about the heavy tails of foreign exchange probability distributions. Everyone else thought that the lognormal assumption of Black–Scholes–Merton was reasonable. The few traders who were well informed followed the strategy we have described [to buy deep out of the money call and put options on a variety of different currencies] - and made lots of money. By the late 1980s everyone realized that foreign currency options should be priced with a volatility smile and the trading opportunity disappeared.

Also he notes that:

In practice, neither of these conditions (constant volatility and no jump assumption in the prices of assets) is satisfied for an exchange rate. The volatility of an exchange rate is far from constant, and exchange rates frequently exhibit jumps, sometimes in response to the actions of central banks. It turns out that both a nonconstant volatility and jumps will have the effect of making extreme outcomes more likely.

The impact of jumps and nonconstant volatility depends on the option maturity. As the maturity of the option is increased, the percentage impact of a nonconstant volatility on prices becomes more pronounced, but its percentage impact on implied volatility usually becomes less pronounced. The percentage impact of jumps on both prices and the implied volatility becomes less pronounced as the maturity of the option is increased. The result of all this is that the volatility smile becomes less pronounced as option maturity increases.

• I dont think this totally explain why it bends up on both sides – Permian Nov 29 '19 at 11:34
• I think the argument about jumps can easily be attributed to that. Recall Swiss national bank intervention, for example. – AK88 Nov 29 '19 at 15:59

The first thing to make sure that you have at the front of your mind are the assumptions of the Black-Scholes model, especially lognormal distribution behavior. Meaning that, if the underlying security does have a lognormal distribution, then options are correctly priced using the same volatility at every strike for a given expiry.

However, if the lognormal distribution does not describe the underlying's actual distribution, then either 1) use a different model or 2) adjust inputs. It turns out that FX (and really nearly all financial assets) do not have lognormal distributions. They have fat tails. Which is a fancy way of saying they have a stronger than predicted tendency to stay around the current price but their extreme moves are far more extreme than predicted.

All of the following are generalizations: Equities has a put skew because it is a "supply" market. That is, there is a bunch of supply that needs to be hedged. Commodities tend to be "demand" markets (especially smaller ones) because of the risk of not being able to get that commodity when needed. FX is different because there are two currencies and, presumably, a symmetry of potential action around either direction (currency A moving or currency B moving).

That symmetry is why there typically is not much skew and the fat tails cause the smile. The smile is a trader's way of nudging the model to match reality.

Also worth noting is the additional leverage one gets from far out of the money options (noted by Hull in the other answer but not really explained). One can construct a vega-neutral portfolio of options constructed as long out of the money options and short at-the-money options. Such a portfolio makes money if volatility moves and is extremely attractive in a flat volatility structure. As traders learned this and bought "wings" vs ATM options, the smile formed as they bid up the OTM pricing.