Probability as the Criteria of Locality of Association in Interpolation in Time Dimension

Interpolation plays a non-trivial role in calibrating pricing models to market observed liquid instruments, because such instruments are limited in range. An immediate field of application is calibrating the Dupire's local volatilities to market observed European option prices. This is a classical problem, and although as a pricing model this is not state-of-the-art, this is often the model of choice for risk management and sensitivity calculation. There have been many calibration techniques which are standard in practice, but here we take a deep look at interpolation in time dimension, and consider qualitative and quantitative criteria that should determine the choice of interpolation method. However among the choices that are globally arbitrage-free, the importance of additional criteria such as locality of association is highlighted. In simple terms this may be categorized as the choice of a coordinate system based on which the interpolation is performed. We consider moneyness as a traditional criteria of locality, but suggest an alternative, more intuitive and abstract criteria, namely the probability (in the pricing measure) of the call option landing in the money at maturity as the locality of association, and show that the resulting interpolation method is globally arbitrage-free. A couple of numerical examples where this choice of locality results in smoother local volatility surfaces, which further leads to dramatic reduction in Monte Carlo errors, lends practical significance of our alternative choice of locality. Finally, an L2 norm as a metric of smoothness of the resulting interpolated surface is introduced, and it is shown how this metric theoretically justifies our choice of locality of association