FX Option Delta and Volatility Quotes

Article Quant Q&A · Author: elypticla

Summary

The document explains why an FX option’s delta convention must match the convention used to build a volatility smile or surface. It lists spot and forward delta, each with premium-adjusted and unadjusted forms, and notes that converting between conventions can require solving for an implied strike or delta.

An example builds a smile using forward delta and shows that a put quoted at a 25-delta point retrieves the corresponding volatility when the option uses the same convention. Changing the option to spot premium-adjusted delta shifts the smile index and the returned volatility. The example illustrates convention sensitivity rather than prescribing one universal convention; users need to identify the quoting convention of their market data and option before looking up volatility.

Key ideas

FX volatility smiles may be indexed by spot or forward delta, with or without premium adjustment.
The option’s delta convention should match the convention used to construct the volatility smile.
Conversions between delta conventions can be nontrivial and may require numerical root solving.
The example shows that using a different convention changes the smile index and retrieved volatility.

Tags

Forex Options Volatility Derivatives pricing

Full text

# On what delta do I look up my vol for an fx Option?


# On what delta do I look up my vol for an fx Option?












I have a smile where I need to look up vols based on delta.

But in FX option space, you speak of deltas that are premium-adjusted or not.

So what delta do I need to use to look up on the vol surface?

## Answer by Attack68 (score 4)

https://quant.stackexchange.com/a/81258

As the above two comments from @AKdemy and @Andrea have made apparent you must be aware of what delta types you are using in order for this to make sense. There are generally 4 types:

- forward

- spot

- forward premium adjusted

- spot premium adjusted

In my library both the Smile/Surface are labelled as being constructed under one type and FXOptions can be labelled as being a different delta type. The library will make automatic conversions between the two, but this is non-trivial and in the slowest cases requires roots solvers (generally when going from non-premium adjusted to premium adjusted or vice-versa). Some of the mechanics of what is going on here uses the cited paper https://www.researchgate.net/publication/275905055_A_Guide_to_FX_Options_Quoting_Conventions

As an example, suppose one creates an FXDeltaVolSmile specified with a forward delta type:

```
from rateslib import *  # Python 3.12, rateslib 1.5.0

# Create an FX Forwards market from spot rates and interest rate curves
fxr = FXRates({"eurusd": 1.08}, settlement=dt(2024, 11, 26))
fxc = {
    "eureur": Curve({dt(2024, 11, 26): 1.0, dt(2025, 11, 26): 0.97}),
    "eurusd": Curve({dt(2024, 11, 26): 1.0, dt(2025, 11, 26): 0.97}),
    "usdusd": Curve({dt(2024, 11, 26): 1.0, dt(2025, 11, 26): 0.96}),
}
fxf = FXForwards(fx_rates=fxr, fx_curves=fxc)

# Create a Vol Smile
fxs = FXDeltaVolSmile(
    eval_date=dt(2024, 11, 26),
    expiry=dt(2025, 4, 26),
    nodes={0.25: 10.4, 0.5: 8.4, 0.75: 10.1},
    delta_type="forward"
)
```

Now if we construct a 25delta `Put` under the same convention we expect that the vol is exactly 10.4 (as that is an indexed point on the smile)

```
# Create an FXOption
fxo = FXPut(
    strike="-25d", 
    delta_type="forward", 
    expiry=dt(2025, 4, 26), 
    pair="eurusd"
)
fxo.analytic_greeks(
    curves=[None, fxf.curve("eur", "usd"), None, fxf.curve("usd", "usd")], 
    fx=fxf, 
    vol=fxs
)
####
{'delta': -0.250000,
 '_delta_index': 0.250000,
 '__delta_type': 'forward',
 '__vol': 0.104000,
 '__strike': 1.0391718351025454,
}
####
```

However, if one constructed a `Put` under a different delta type definition (such as spot premium adjusted) then the results would be different.

```
fxo = FXPut(
    strike="-25d", 
    delta_type="spot_pa", 
    expiry=dt(2025, 4, 26), 
    pair="eurusd"
)
fxo.analytic_greeks(
    curves=[None, fxf.curve("eur", "usd"), None, fxf.curve("usd", "usd")], 
    fx=fxf, 
    vol=fxs
)
####
{'delta': -0.250000,
 '_delta_index': 0.243168,
 '__delta_type': 'spot_pa',
 '__vol': 0.104959,
 '__strike': 1.0372647541668385,
}
####
```

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.

Matching FX Option Delta Conventions to Volatility Smile Quotes

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