Arbitrage-free Asset Class Independent Volatility Surface Interpolation on Probability Space using Normed Call Prices

Abstract

We consider arbitrage-free interpolation of arbitrage-free input data of European option prices. The method derived is independent of the underlying (equity, rates, FX, etc.). A particular contribution of the paper is that for the chosen coordinate system and a wide variety of interpolation methods, we prove that the method results in arbitrage-free interpolations. This result is achieved in two steps: first, we transform option prices to Normed Call Prices (NCP), which are independent of the application. Using NCPs as interpolants, the interpolation problem can be formulated independently of assumptions of deterministic rates. Besides, the constraints necessary for ensuring absence of arbitrage are much simpler in this interpolant system, as they do not involve any scaling at all. Having proposed our choice of the dependent variable, we turn to the independent variables, and transform the strike-maturity or moneyness-maturity coordinate system to a probability-maturity coordinate system, which results in an interpolation method called probability-space interpolation in maturity dimension on the proposed NCP system that we prove actually leads to arbitrage-free surfaces in both strike and maturity dimensions. The computational tractability is an added advantage of our interpolation scheme.

Keywords: NCP, Volatility Surface, Volatility Smile, Probability Space, Arbitrage Free Interpolation