Why a Lognormal Pricing Model Produces Flat Implied Volatility

Summary

The document addresses why a lognormal model, such as the Black-Karasinski model mentioned in the question, produces a flat implied-volatility curve across strikes when options are priced consistently with that model. If the underlying follows the model’s assumed lognormal distribution with a single volatility parameter, options priced under the same assumptions recover that parameter as their implied Black volatility. Strike variation alone therefore does not create a skew within the model.

The answer contrasts this model result with observed markets, where implied volatilities may differ by strike. Such a pattern indicates that the market’s option prices are not fully described by the assumed lognormal distribution. The question also asks why a normal-rate model should behave differently, but the response does not explain that comparison. It offers no derivation or market evidence beyond the model’s definition, so its conclusion is limited to internally consistent lognormal pricing and does not establish that real-world implied volatility should be flat.

Key ideas

• A lognormal model with a single volatility parameter implies the same Black volatility across strikes when used consistently for pricing.
• Observed strike-dependent implied volatility signals a mismatch between market prices and the assumed lognormal distribution.
• The response does not explain the proposed contrast with normal-rate models.
• The flat-skew result is a model implication, not a claim that market volatility is actually flat.

Tags

# volatility skew for lognormal model is flat?


# volatility skew for lognormal model is flat?












Does anyone know why the volatility skew for lognormal model, such as BK, should be a flat line, meaning that implied black volatility for options will be same for those with different strike prices?

Why volatility skew for normal mode, such as Hull White, should not be flat? I assume that normal model assumes volatility is independent of rate level.

## Answer by dm63 (score 3)

https://quant.stackexchange.com/a/22042

The answer is that by definition, if the underlying stock obeys a lognormal distribution with std deviation parameter sigma, then the implied vol of options priced using this model will be sigma. Of course in the market we observe that options of different strikes have different vols - this just means that the underlying distribution is not perfectly lognormal.

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