Using European Prices to Test American Option Butterfly Arbitrage

Summary

The document asks whether butterfly spread arbitrage in American options can be detected through prices of European options. It recalls that concavity of European option prices across strikes is tied to nonnegative risk-neutral probabilities, and that a violation can imply an arbitrage. A butterfly centered at a local price maximum is also raised as a possible trade with nonnegative payoff and nonpositive entry cost.

The motivating application is an American option volatility surface, built by backing out Black–Scholes implied volatilities from a binomial tree pricer. The author considers using European options priced with those parameters to assess the surface, including through a condition on the European price curve. However, the document provides no resolution to whether this reproduces American-option arbitrage at the same strikes. It specifically identifies the early exercise premium’s strike curvature as a possible source of differences, so the proposed European proxy remains an open question here.

Key ideas

• Concavity of European option prices across strike is associated with nonnegative risk-neutral probabilities.
• A butterfly spread can reveal a pricing inconsistency when it has a nonnegative payoff and costs no more than zero to enter.
• American option prices include an early exercise premium, so their curvature need not match the European price curve.
• The document proposes using European prices derived from American-option implied volatilities as a surface diagnostic, but does not establish that it detects the same arbitrage.

Tags

# Detecting butterfly spread arbitrage for American options through European option prices


# Detecting butterfly spread arbitrage for American options through European option prices












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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.