An arbitrage-free method for smile extrapolation

A robust method for pricing options at strikes where there is not an observed price is a vital tool for the pricing, hedging, and risk management of derivatives. All institutions that trade derivatives will have an approach to this task. Typical examples might be a simple interpolation scheme across implied volatilities, or the use of a modelbased formula optimized to fit observed prices. It is our view that while these methods work well for interpolation across actively traded strikes, they often break down when used for extrapolation. We introduce in this paper a technique for smile extrapolation that is robust, simple, fast and offers control on the form of the tails in the distribution. Using this method allows distribution-sensitive products such as CMS rates or inverse-FX options to be priced consistently with the smile of traded vanilla options. The resulting arbitrage-free distributions are also key to the copula-based pricing of multi-asset products such as spread options, quantos and hybrids. Our approach fixes several problems currently seen in today’s stressed markets, e.g. with CMS rates. There are two distinct areas where smile extrapolations can be a problem: when an institution has legacy trades on its books that are struck far from today’s at-the-money strike (a common situation in the current market) and in the pricing of more exotic products that depend on the use of an entire risk-neutral probability density function (PDF) of the underlying. Examples of the second area include: