Converting Delta-Based Option Volatility Surfaces to Strike Space

Summary

The document raises a practical construction question for equity option volatility surfaces. The input data are organized by delta and maturity, while the desired surface is organized by strike and maturity so volatility can be interpolated for a chosen strike and expiry. The author proposes inverting the Black–Scholes delta relationship at each observed delta and maturity, but finds that the resulting strike depends on maturity and asks whether this is an error.

The text contains no answer, method, calculation, or evidence resolving the issue. It therefore serves as a statement of the interpolation problem rather than guidance on how to solve it. In general, strike is tied to both delta and maturity through the underlying price, rates, dividends, and volatility assumptions, so a common strike axis does not mean each delta maps to the same strike across expiries. The appropriate conversion depends on consistent market conventions and inputs, which the document does not specify.

Key ideas

• The author wants to convert an equity option surface from delta and maturity coordinates to strike and maturity coordinates.
• Inverting a delta relationship separately by maturity can produce different strikes for the same delta.
• Delta-to-strike conversion depends on maturity and model inputs, so separate strike mappings can be expected.
• The document poses the question but provides no answer or validation of a conversion method.

Tags

# Equity options volatility surface: how to align strikes for different maturities?


# Equity options volatility surface: how to align strikes for different maturities?












I am currently trying to transform a volatility surface of equity options from delta space to strike space. This means I have a surface (delta, tenor) and want to convert it to (strike, tenor), so that I can then interpolate it and read a volatility for any given strike and maturity date. My naive approach was simply to convert the Black Schole formula and then numerically determine a strike value for each data point (delta, tenor). However, the result of this is different strikes for each tenor, which seems very counterintuitive to me. In books/on the internet, there are never two different strike axes for different maturities.

Therefore, I would be very grateful if someone could help me identify my error in reasoning. How can this be done correctly so that the same strike is obtained for all maturities?

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