\u201cAs we said earlier, the Black-Scholes model produces implied volatilities of traded options that can vary by exercise price for a given underlying asset. How should we respond to such a finding? First, we could suggest that the Black-Scholes model is not correct. Since there cannot be more than one volatility of the underlying asset, the model must be incorrect. Case closed.\u201d<\/p><\/blockquote>\n
I reply to my colleague\u2019s e-mail, saying this:<\/p>\n
\u201cOn the incorrectness of Black-Scholes and on the “smile” in particular (different “implied volatilities” for different strikes), I had shown on a few examples – and back in the early nineties (I should have the unpublished paper somewhere) that the smile vanishes if you split the “implied volatility” into two components a “true” implied volatility and an additional variable: the option writer’s “profit margin” – constant across strikes and easily calculated as the value that removes the smile, i.e. returns a single – correct – implied volatility.\u201d <\/p><\/blockquote>\n
Back home, I look for that paper: I remember it was short: 2 to 3 pages, with diagrams, I can\u2019t find it. The only thing I can find is an Excel spreadsheet with data and the diagrams. Here they are.<\/p>\n
I find also a reference to my piece of research in a lecture I gave at ABN Amro in Amsterdam in November 1996. Here is the relevant passage:<\/p>\n
\u201cActually, the fair prices <\/em>calculated this way (**) are lower than those observed on the markets. This is however the case as well for fair prices <\/em>calculated according to the theoretical models of Black-Scholes and Cox-Ross-Rubinstein. Prices closer to actual market prices obtain if one adds a particular constant to every one of them. This additional constant can be considered as the profit<\/em> of the seller<\/em> or writer<\/em> of options: to his\/her objective loss expectancy s\/he adds a margin of actual profit. This last parameter has been overlooked in all theoretical models. When introduced it removes a well-known anomaly of prices generated by the Black-Scholes model, the volatility skew <\/em>or \u00ab smile \u00bb.\u201d<\/p><\/blockquote>\n
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