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
