## FX Vol

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#### Four user interfaces:

**Data API**.**Excel Add-ins**.**Model Analytic API**.**GUI APP.**

**Data API**.**Excel Add-ins**.**Model Analytic API**.**GUI APP.**

FinPricing offers the following FX implied volatility surfaces via Excel or API:

- FX implied volatility surfaces
- Precious metal implied volatility surfaces
- Nordic electricity futures curve
- VIX futures curve
- S&P 500 futures curve

An implied volatility is the volatility implied by the market price of an option based on the Black-Scholes option pricing model. A volatility surface is derived from quoted volatilities that provides a way to interpolate an implied volatility at any strike and maturity.

Unlike in other markets that quote volatility versus strike directly, the FX smile is given implicitly as a set of restrictions implied by market instruments and as such a calibration procedure to construct a volatility-delta or volatility-strike smile is used.

The volatility surface in FX market is constructed based on the sticky delta rule. The sticky delta rule takes the implied volatility to be a function of delta only. This is equivalent to saying that the implied volatility is a function of the moneyness ratio S/K.

The underlying assumption is that options are valued depending on their delta, so that when the FX spot rate moves and the delta of an option changes accordingly, a different implied volatility has to be used into the option pricing formula.

Using the Black-Scholes formula, the implied volatility can be obtained from observed market prices for different strikes, thus defining a volatility smile.

There are four different types of delta are used in the FX volatility market quotes.

- Spot delta
- Forward delta
- Premium-adjusted spot delta
- Premium-adjusted forward delta

FX volatility smile is represented by three entities: at-the-money (ATM) volatility, risk reversal,
and butterfly.

The ATM volatilities quoted by brokers can have various interpretations depending on currency pairs. Here we introduced
the most popular one used by the FX brokers. The ATM volatility is the value from the smile curve where the strike is
such that the delta of the call equals, in absolute value, that of the put (this strike is called ATM “straddle”
or ATM “delta neutral”).

A risk reversal (RR) is a combination of a long call option and a short put option with the same maturity. This is a zero-cost product as one can finance a call option by selling a put option. Risk reversal volatility is the difference between the volatility of the call option and the put option at the same moneyness level, i.e.,

**25 RR = 25 Delta Call Vol – 25 Delta Put Vol**

A butterfly (BF) is a combination of a long call option, a long put option, a short ATM call option, and a short ATM put option. Butterfly volatility is the average of the difference between the volatility of the call option and put option minus the ATM volatility, i.e.,

**25 BF = (25 Delta Call Vol + 25 Delta Put Vol)/2 - ATM**

In liquid FX markets some of the most traded strategies include Strangle, Straddle, and Butterfly.

The FX specific delta and ATM conventions are important to understand the volatility construction procedure in FX markets. In FX option markets it is common to use the delta to measure the degree of moneyness. Consequently, volatilities are assigned to deltas (for any delta types), rather than strikes. However, the delta-volatility version of the smile is not directly quoted in the market but can be translated after using the simile construction procedure.

In FX markets, the precise meaning of the broker quotes depends on the details of the contract. For example, there are at least four different definitions for ATM strike (‘spot’, ‘forward’, ‘delta neutral’, ’50 delta call’) and for delta (‘spot delta’, ‘premium-adjusted spot delta’, ‘forward delta’, ‘premium-adjusted forward delta’). Using the wrong definition can lead to significant errors in the construction of the smile surface.

The key point in the construction of the FX volatility smile is to build σ(k) such that it matches the volatilities and prices implied by market quotes. Before starting the smile construction it is important to analyze the exact characteristics of the above quotes, such as which ATM convention is used and which delta type is used.

Smile fly is not directly quoted on the market. The market quotation is broker’s fly which need to convert into smile fly before it’s put into the simple formula.

there are three most useful methods to construct the volatility surface, such as interpolation/extrapolation method, Vanna –Volga (VV) method, stochastic volatility modeling method.

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