A Quadratic Gaussian Year-on-Year Inflation Model

Abstract

We introduce a new approach to model the market smile for inflation-linked derivatives by defining the Quadratic Gaussian Year-on-Year inflation model—the QGY model. We directly define the model in terms of a year-on-year ratio of the inflation index on a discrete tenor structure, which, along with the nominal discount bond, is driven by a log-quadratic function of a multi-factor Gaussian Markov process. We find closed-form expressions for the drift of the inflation index and for inflation-linked swaps. We get a Black-Scholes-type pricing formula for year-on-year inflation caplets in semi-analytical form. The formula contains an integral of a multivariate Gaussian density over a quadratic domain. In a two-dimensional case, we show how this integral reduces to a one-dimensional integration along the boundary of a conic section. In the case where the year-on-year inflation ratio is driven by two factors, we specify a spherical parameterisation. This gives an intuitive control over the curvature and the skew of the year-on-year inflation smile and shows the maximum curvature and skew obtainable with a particular three-factor version of the QGY model. Within this three-factor model, we identify a parameterisation to control the autocorrelation structure of the inflation index. We calibrate the model to year-on-year inflation options on the UK’s Retail Prices Index (RPI) and the eurozone’s Harmonised Index of Consumer Prices Excluding Tobacco (HICPxT) and get a good fit to the smile of implied volatilities. We use the calibrated model to price HICPxT zero-coupon inflation options and RPI limited price indices (LPIs). Furthermore, we provide methods to interpolate the process for the inflation index and the year-on-year inflation ratio between dates on the tenor structure. Keywords: Year-on-year inflation modelling; multi-factor log-quadratic Gaussian model; stochastic-volatility parameterisation; inflation autocorrelation; year-on-year inflation calibration; LPI pricing.