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Interpreting Black–Scholes Greeks for Option Risk and P/L Attribution

Article Quant Q&A · Author: mrdrralph
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Summary

The document examines a common tension in options practice: Black–Scholes is often treated as a way to express prices through implied volatility, even though its assumptions do not fully describe markets, while its Greeks are still used to measure portfolio risk and explain profit and loss. It asks whether delta, gamma, vega, vanna, and volga remain meaningful when derived from that model.

The text is a question rather than a resolved analysis. It does not provide evidence, an alternative pricing framework, or a procedure for testing attribution accuracy. Its central concern is whether local sensitivities computed under Black–Scholes reliably describe actual option dynamics, and how practitioners can distinguish using the model as a quoting convention from relying on it as a risk model. The limitation is that the document raises the issue without supplying an answer or comparing Greeks across models.

Key ideas

  • Black–Scholes implied volatility can serve as a standardized way to express option prices.
  • Portfolio risk and P/L attribution often use Greeks derived from Black–Scholes.
  • The document questions whether those sensitivities remain accurate when the model’s assumptions fail.
  • It raises, but does not answer, how model-based Greeks should be validated against observed option behavior.

Tags

OptionsDerivatives pricingVolatilityRisk management
Full text
# Does it make sense to use Black Scholes greeks to attribute P/L given the Black Scholes assumptions don't hold?


# Does it make sense to use Black Scholes greeks to attribute P/L given the Black Scholes assumptions don't hold?












I've seen some takes from experts in the industry (Benn Eifert for example) who say that we should treat Black Scholes as a translation mechanism for putting price into a more workable form (IV). They say that no one really uses Black Scholes as a "model" since clearly the assumptions are violated.

However, these same people use Black Scholes calculated greeks for understanding the risk of a portfolio of options and for P/L attribution. Vega, vanna, and volga are also sometimes added to account for the non-deterministic nature of implied volatility.

My question boils down to this: shouldn't our Black Scholes calculated greeks be incorrect measures of our risk since they're just partial derivatives of an equation that's based on false assumptions? For example, if our Black Scholes greeks say we earned 100k from delta, 50k from gamma, 50k from vega, etc. why should we believe this at all? Isn't it totally possible that our gain from gamma is completely different under a more "correct" formulation of the second derivative of price to spot? How can we say we aren't using black scholes as a model if we're using black scholes greeks?

To be clear I'm not asking about "how much of our price change does our taylor series with a few terms explain", but rather how do we know we're accurately capturing the dynamics of our option with our greeks if we know the equation they're derived from has an incomplete picture of the market.

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.