Arbitrage-free smoothing of the implied volatility surface

Abstract

The pricing accuracy and pricing performance of local volatility models depends on the absence of arbitrage in the implied volatility surface. An input implied volatility surface that is not arbitrage-free can result in negative transition probabilities and consequently mispricings and false greeks. We propose an approach for smoothing the implied volatility smile in an arbitrage-free way. The method is simple to implement, computationally cheap and builds on the well-founded theory of natural smoothing splines under suitable shape constraints.

Keywords:

• Implied volatility surface
• Local volatility
• Cubic spline smoothing
• No-arbitrage constraints

Acknowledgements

The paper represents the author's personal opinion and does not reflect the views of Sal. Oppenheim. I thank Matthias Bode, Tom Christiansen, Enno Mammen, Christian Menn, Daniel Oeltz, Kay Pilz, Peter Schwendner, and the anonymous referees for their helpful suggestions. I am indebted to Eric Reiner for making his material available to me. Support by the Deutsche Forschungsgemeinschaft and by the SfB 649 is gratefully acknowledged.

Notes

†We stress that these properties do not depend on the existence of a density. In continuous-time models, they hold when the discounted stock price process is a martingale, but may fail for strict local martingales (Cox and Hobson 2005).

†It should be noted that the literature on the numerical treatment of splines also discusses other end conditions (Wahba 1990). A popular choice is to fix the first-order derivatives at the end points of the spline. We experimented with this solution. In this case, the smoothness penalty no longer has the convenient quadratic form (see Proposition 3.1) but could be approximated by the smoothness penalty given by the natural spline. Further, since in our application the two first-order derivatives are unknown, they must be estimated. As proxy we used the first-order BS derivative w.r.t. the strike evaluated at the strike implied volatility. In our simulations it turned out that the spline functions are very sensitive to a misspecification of the first-order derivatives and less robust than the natural spline solution.