| Математика
Key words: FX, volatility smile,sticky strike ,sticky delta, broker fly, ATM, RR, BR fly
volatility smile

The famoust Black-Scholes model assumes a constant volatility. As we know, the implied volatility is not constant and is observed from a shape that is commonly known as “smile”. The reason for the smile is due to expectation of market participants that spot prices does not follow the Brownian motion and thus prices are not lognormal under more extreme fat-tailed circumstances.

If we expect sharp movements, the volatility will increase and in this case the demand for OTM options will be higher in order to hedge or speculate on the volatile market.

If the demand is lower for OTM call options, we may see a skew shape, especially for stock markets, instead of the smile shape. Why is that? Because it is believed that market will be more or less flat or will increase slowly. In this case, the volatility will not change sharply. In the same time, OTM puts can be quoted with higer volatility compared to ATM options because it is still possible that market will drop down sharply and volatility can increase drastically, i.e. jump risk is considered to be high. That is why we have the skew shape, OTM puts have higher volatility while OTM calls have lower volatility.
The third shape is a frown shape but it is very rare for FX and stock markets.

FX Smile

In FX market the main shape is a U-shaped smile.

In Black-Scholes world, the volatility is the only one parameter that is not directly observable in the market but can be derived from options prices and is called implied volatility. Of cause, using put-call parity, we may say that call and put options with the same maturity and strike have the same implied volatility.

As been stated, volatility depends on spot prices and therefore it changes with spot movements. We say about smile dynamics, i.e. smile movements are linked to spot movements.
In equity markets implied volatility is always plotted against strike, i.e. sticks to strike. In FX implied volatility is plotted against delta, i.e. sticks to delta. Therefore, there are sticky strike and sticky delta.

Sticky strike model assumes the smile to be the same when spot changes. Volatility of a particular strike is unaffected by a spot change. Sticky strike model is also known as absolute skew.

Sticky delta rule assumes the smile to move along the strike axis. This is due to moneyness, spot divided by strike. As soon as spot changes,moneyness changes as well. Sticky delta also known as a relative or floating skew.

Using two different approaches can be explained by the nature of markets:

1. FX options are mainly OTC market so there is no reason to use many different strikes, it is easily to price options with respect to the delta.

2. Equity markets are more standardized because they are exchange traded that makes possible to use only standard strikes and respective volatilities.

When we plot the implied volatility against strike and maturity, we get a volatility surface.

In order to build volatility smile we need implied volatilities for different delta points on the volatility surface. Brokers used to quote volatilities rather than the option price. By the official market convention the implied volatilities are quoted in terms of ATM Straddle, 25 delta and 10 delta Risk Reversal (RR), 25 delta and 10 delta Butterfly (BF or Fly). Using these implied volatility quotes it is possible to build volatility surface.

In practice, brokers use Market Strangle (Brokers Fly) that gives completely different results compared to BF convention quotes.

ATM volatilities indicate the level of the smile.
RR is a measure of skewness (the slope of the smile) and indicates which currency is favored by the market participants. If RR is positive, the market favors the foreign currency. By market convention, 25-delta RR is the difference between the call and put 25-delta volatilities.
RR volatility (25-delta) = Call volatility (25-delta) – Put volatility (25-delta)
BF volatility is simply defined as:
BF volatility (25 delta) = 0.5*(Call volatility (25-delta)+Put volatility (25-delta)) – ATM volatility.
Now, Using these two equations, it is possible to calculate volatility for calls and puts:
Call volatility (25-delta) = ATM volatility+0.5*RR volatility (25 delta)+BF volatility (25 delta)
Put volatility (25-delta) = ATM volatility – 0.5*RR volatility (25 delta)+BF volatility (25 delta)

Implied volatility for 10-delta puts and calls can be calculated in the same way. These quotes are reffered as Smile Fly.
Theoretically, it would be possible to use this simple math but market practice is to use different approach, namely Broker Fly.
Broker fly is the difference btw the unique volatility(one vol) and ATM volatility.