Detecting Butterfly Arbitrage from Implied Volatility

Summary

The document explains how to check an option volatility surface for smile arbitrage, understood here as butterfly arbitrage within a single expiry. The core condition is that call prices should be convex as a function of strike. A practical check is to convert the implied volatility curve into call prices and inspect their strike-wise convexity; a violation indicates arbitrage under the stated framework.

For a parameterized implied volatility curve, the answer also suggests deriving the risk-neutral density by taking the second strike derivative of the call price, applying the chain rule to account for volatility's dependence on strike. A negative density in some region signals the same problem. The document provides the conceptual test but no plotted data, worked calculation, or discussion of numerical stability. Results can depend on how the volatility curve is parameterized and differentiated, so the answer outlines a diagnostic rather than assessing any particular surface shown in the original question.

Key ideas

• Butterfly arbitrage corresponds to a failure of convexity in call prices across strike for a given maturity.
• Convert implied volatilities into call prices and inspect their strike-wise convexity to test for arbitrage.
• A negative second strike derivative of call prices implies a negative density in the affected region.
• When volatility varies with strike, apply the chain rule when differentiating call prices.

Tags

# Can we observe smile arbitrage from the implied and local volatility?


# Can we observe smile arbitrage from the implied and local volatility?












Here are graphs of implied volatility and local volatility. Our prof mentioned that we can observe that the short end low strike region has some smile arbitrage. I would like to know how?

Thanks

## Answer by BrownianBread (score 1)

https://quant.stackexchange.com/a/61450

Smile arbitrage is the presence of a butterfly spread arbitrage in a given maturity of your surface, i.e. if your call prices are non-convex leading to an arbitrage. An easy way to spot the arbitrage is to build the call prices and check for strictly convex prices in strike.

If you have a parametrisation of the implied volatility $\sigma(K)$ then you can derive the probability density function and show that it is negative in some regions to find the arbitrage. You can do this by using the formula $$p(K)=\frac{\partial^2C(\sigma(K))}{\partial K^2}$$ and apply the chain rule.

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