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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?

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up vote 6 down vote accepted

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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