Mapping Delta-Quoted Option Volatility Smiles to Call Delta

Summary

The document explains how to interpret a foreign-exchange implied volatility smile quoted using put and call delta labels. It asks how to find volatility for a call at a specified delta, and whether that requires extrapolating from listed call quotes or using the corresponding put-side delta. The answer recommends assigning a continuous delta coordinate across the listed quotes: put deltas map to the lower half of the range, the at-the-money quote to the midpoint, and call deltas to the upper half. This allows interpolation across the full smile rather than treating the two sides as unrelated sets.

The note then points toward interpolating the smile and transforming delta into forward moneyness. It gives no interpolation formula, convention details, numerical worked result, or discussion of alternative market delta conventions. Practitioners should therefore treat the coordinate mapping as a starting point and consult the relevant market quoting and pricing conventions before applying it.

Key ideas

• Put and call delta labels can be represented on one ordered delta scale.
• The at-the-money quote sits at the midpoint between the put and call wings.
• Interpolation can use quotes from both sides of the smile after mapping their delta coordinates.
• A further conversion from delta to forward moneyness may be needed for some smile representations.

Tags

# Volatility smile interpolation when using delta moneyness


# Volatility smile interpolation when using delta moneyness












I have an implied volatility smile in delta terms given by bloomberg:

```
5DP     71.49
10DP    69.92
15DP    68.82
25DP    66.94
35DP    65.44
50D     64.08
35DC    63.89
25DC    64.37
15DC    65.98
10DC    67.87
5DC     71.33
```

Say I want to get the volatility of a call with 70 delta.

Am I supposed to take the corresponding interpolated put delta at 100 - 70 = 30DP, or am I supposed to extrapolate the available call deltas going from 5 to 50 i.e. using only:

```
50D     64.08
35DC    63.89
25DC    64.37
15DC    65.98
10DC    67.87
5DC     71.33
```

More generally, how do I go from a smile in call / put delta terms like this:

`['5DP', '10DP', '15DP', '25DP', '35DP', '50D', '35DC', '25DC', '15DC', '10DC', '5DC']`

To a smile in deltas like this :

`[5, 10, 15, 25, 35, 50, 65, 75, 85, 90, 95]`

(e.g. this question for a smile in deltas)

## Answer by JejeBelfort (score 2)

https://quant.stackexchange.com/a/82177

Thanks to the comments I managed to achieve what I wanted:

From @KaiSqDist, I use directly:

`[5, 10, 15, 25, 35, 50, 65, 75, 85, 90, 95]`

as my deltas for

`['5DP', '10DP', '15DP', '25DP', '35DP', '50D', '35DC', '25DC', '15DC', '10DC', '5DC']`

to get:

Then I referred to this post to interpolate and transform delta into forward moneyness:

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.