Reconstructing FX Option Smiles from Delta Quotes with Garman-Kohlhagen
Summary
The article explains how to convert foreign-exchange option quotes from the market’s delta-based format into strikes and a volatility smile. It describes the at-the-money volatility, risk reversal, and butterfly quotes used to define smile pillars, with optional wing quotes for a more detailed shape. Garman-Kohlhagen pricing accounts for both domestic and foreign interest rates, and the article emphasizes that FX delta depends on market convention: spot or forward delta, with or without premium adjustment. These choices affect the strike recovered from a quoted delta.
The described tool includes pricing and Greeks, convention-specific delta and strike solvers, reconstruction from quote sheets, and a provider that summarizes listed option chains into delta-space inputs. Sample data and implementation details illustrate the workflow. The reconstruction is exact at supplied pillars but depends on interpolation between them. Rate compounding and currency-leg assignment must be handled consistently, and a chain summarized into a few quotes loses information. More advanced smile calibration and interpolation methods are presented as possible extensions.
Key ideas
- FX options are commonly quoted by delta, so strikes must be recovered from the market’s volatility quotes.
- Garman-Kohlhagen extends Black-Scholes by using separate domestic and foreign interest rates.
- At-the-money volatility, risk reversal, and butterfly quotes define the central smile pillars.
- Spot versus forward delta and premium adjustment are distinct conventions that can change solved strikes.
- Interpolation fills the smile between quoted pillars, while summarizing a listed chain into quote pillars discards detail.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.