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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.

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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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