Choosing Black or Black–Scholes Pricing with SABR Volatility

Summary

This question asks how to turn volatility from Hagan’s SABR approximation into a call price when interest rates are nonzero. It compares the Black formula, typically applied to a forward, with Black–Scholes, which prices an option on a spot asset while accounting for rates and carry. The choice depends on the underlying and quoting convention: a forward option is generally priced with the Black formula, while a spot option uses Black–Scholes with consistent discounting and carry inputs.

The document provides no answer or pricing example; it only poses the choice and asks whether the distinction depends on using a forward. It therefore identifies a useful model-selection issue but does not resolve details such as the asset’s carry, dividend treatment, or how the SABR volatility is defined. Those conventions must be aligned before comparing prices.

Key ideas

• SABR’s implied volatility approximation can be used as an input to an option pricing formula.
• The Black formula is commonly used for options on forwards, while Black–Scholes is formulated for spot assets with rates and carry.
• Nonzero rates and the underlying’s forward or spot convention affect how pricing inputs are applied.
• The source poses this choice but does not provide a worked answer.

Tags

# Black-Scholes vs Blacks model. Which one to use with SABR?


# Black-Scholes vs Blacks model. Which one to use with SABR?












Say I want to compute a call price for a given set of SABR parameters.

I use Hagans approximation and compute $\sigma_B$. The rate is not zero. Should I then compute the option price using

- Blacks formula (https://en.wikipedia.org/wiki/Black_model)

- or Black-Scholes formula? (https://en.wikipedia.org/wiki/Black–Scholes_model)

Does it matter whether the asset is a forward or not?

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