Interpreting Vega and Vomma Under Black–Scholes Volatility Assumptions
Summary
The document explores an apparent tension between Black–Scholes's constant-volatility assumption and the use of vega and vomma, which describe option-value sensitivity to volatility and changes in that sensitivity. The responses characterize Black–Scholes as a simplified pricing framework rather than a literal account of market volatility. They note that observed volatility smiles conflict with a single constant-volatility description, while the model remains useful for organizing intuition and for practical pricing after adapting volatility inputs.
The discussion distinguishes using a model's assumptions to derive a price from using its sensitivities to examine how that price responds when an input changes. It also points toward volatility surfaces and model-free implied-volatility approaches based on risk-neutral densities. These are brief forum comments rather than a derivation, and they do not fully define the Greeks or prescribe a model for realized volatility. The central lesson is that a Black–Scholes Greek remains a conditional sensitivity within a chosen pricing setup, not a claim that volatility is fixed in markets.
Key ideas
- Vega measures option price sensitivity to the volatility input, while vomma describes vega's sensitivity to volatility.
- A constant-volatility Black–Scholes setup simplifies market behavior and does not reproduce a volatility smile.
- Greeks can be interpreted as sensitivities within a pricing model even when its assumptions are imperfect.
- Market practice often uses volatility surfaces to accommodate varying implied volatilities.
- Model-free implied volatility approaches are mentioned as an alternative perspective based on risk-neutral densities.
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Full text
# Vega in a "constant volatility" Black-Scholes world? # Vega in a "constant volatility" Black-Scholes world? A little confused, I consulted the Wilmott forums for guidance on how I can interpret vega/vomma. Another user's post reminded me that the Black-Scholes model assumes that the underlying has constant volatility, so vega is an out-of-model concept. If Black-Scholes assumes that volatility does not change, then would this not render vega/vomma calculations useless since they are based from the Black-Scholes formula? The vega calculation seems odd because one gauges how the option value changes with volatility, but then Black-Scholes assumes volatility does not change. Going a step further, the idea of vomma in a Black-Scholes world seems odd because if volatility does not change, then surely vega would not change? I understand that Black-Scholes makes some questionable assumptions, but if we price options in a Black-Scholes world, then it seems peculiar that these metrics have come about as a result of the Black-Scholes formula. I would really appreciate any insight. ## Answer by Bob (score 0, accepted) https://quant.stackexchange.com/a/23070 On the topic of your second paragraph, the author below is the authority on precisely that topic. Start at page 19 https://www8.gsb.columbia.edu/leadership/sites/leadership/files/Is%20economics.pdf ## Answer by Moe (score 1) https://quant.stackexchange.com/a/23108 You are onto something, it is inconsistent to be calculating vega with Black-Scholes considering it assumes that volatility is constant. Black-Scholes is not a good for modeling option prices/implied volatility. It's a very good intuitive model (like the CAPM), and a good way of organizing thoughts, but it is not an accurate depiction of reality. If it were, you wouldn't be seeing volatility smiles (it would be a noisy horizontal line instead). If you're looking into modeling implied volatility, you should look into model free implied volatility which is based on risk neutral densities. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2220067 This is about implied volatility. Modeling realized volatility is another story. ## Answer by AKdemy (score 0) https://quant.stackexchange.com/a/63664 IMHO, the entire reason a vol surface exists and many OTC markets quote in vol is exactly this. Constant vol is a gross oversimplification and has a lot of shortcomings. However, once you adjust for vol, BS is robust, quick and widely used. What Greeks really are is generally an interesting question.
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