Why Cubic Spline Swaption Volatility Fits May Allow Arbitrage

Summary

The document asks whether cubic spline interpolation across swaption strikes guarantees an arbitrage-free implied volatility smile, and how that approach differs from a stochastic model. The answer says ordinary spline routines do not impose the necessary constraints. A smooth fit through observed volatility points can still violate option price relationships, including spread and local volatility conditions.

Stochastic volatility models may impose constraints that prevent call or put spread arbitrage for a given tenor. The answer distinguishes that model-based protection from arbitrage-free surface construction methods, which can guarantee the absence of static arbitrage. It points to further reading but gives no construction algorithm, equations, calibration details, or empirical comparison. The discussion is therefore a conceptual warning: interpolation smoothness alone is not an arbitrage test, and claims about guarantees depend on the model or surface-building procedure and the arbitrage conditions it enforces.

Key ideas

• Cubic spline interpolation does not inherently prevent arbitrage in a swaption volatility fit.
• A smooth implied volatility curve can still breach option spread or local volatility conditions.
• Some stochastic volatility models can enforce call and put spread constraints within each tenor.
• Dedicated arbitrage-free surface methods can guarantee the absence of static arbitrage when properly constructed.
• The document gives conceptual guidance but no implementation details or comparison evidence.

Tags

# Is Cubic spline Interpolation on swaption Volatility arbitrage free?


# Is Cubic spline Interpolation on swaption Volatility arbitrage free?












If I use interpolation technique such as cubic spline to estimate volatility of Swaption with different strike,(with a given forward rate, swap and option maturity) will this be arbitrage free? What is the main differences compared to using stochastic model?

Thanks.

## Answer by Atlas (score 6, accepted)

https://quant.stackexchange.com/a/10000

There is nothing in simple cubic spline fitting routines that would prevent arbitrage. Even with conscientious use of knot points and smoothing techniques you may end up with simple spread and local volatility arbitrage conditions. Stochastic volatility models on the other hand can explicitly constrain your solutions to prevent call/ put spread arbitrage at least on each tenor. Arbitrage free surface construction routines can guarantee lack of static arbitrage. This paper will help.

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