It's easy to demonstrate that if European option prices are concave with strike, then an arbitrage exists. For example, the risk-neutral probability density is the second derivative of European put prices with respect to strike divided by the discount factor, and the existence of "negative probabilities" implies an arbitrage by the first Fundamental Theorem of Asset Pricing. It can also be demonstrated that a long butterfly spread with guaranteed non-negative payoff can be entered into at non-positive cost (i.e. sell 2 rich options in the body, buy 2 cheap options in the wings). It seems that the second derivative of undiscounted American option prices are not probability densities, but riskless profit can still be made through long butterfly spreads centered at local maxima.

I currently generate a volatility surface for American options by backing out the the Black Scholes implied volatilities from a binomial tree pricer, and I want to evaluate the quality of the fit. Since the Black Scholes formula is less computationally expensive and is easier to analytically manipulate (in particular, I can apply the Durrleman condition described by Aurell 2014), I was hoping to just evaluate a term of vanilla European options priced with the parameters of the American option. Will this curve reproduce the arbitrage at the same strikes? I doubt it since it seems this would impose some constraints on the second derivative of the the early exercise premium with respect to strike.


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