Abstract
We introduce a new approach to model the market smile for inflation derivatives by defining a quadratic Gaussian year-on-year inflation model (the QGY model). We directly define the model in terms of ratios of the inflation index on a discrete tenor structure, which, along with the nominal discount bond, are driven by an exponential-quadratic function of a multivariate Gaussian process. The model provides an intuitive control of the dynamics of the inflation smile, the autocovariance structure of the inflation rates and the covariance structure between nominal and inflation rates.
We find closed-form expressions for the drift of the inflation index and for year-on-year inflation swaplets. We get a Black-Scholes-like pricing formula for year-on-year inflation caplets in semi-analytical form, which involves integrating a multivariate Gaussian density over a quadratic domain. In the bivariate case, we show how this reduces to a one-dimensional integration along the boundary of a conic section.
When the inflation index is driven by two factors, we specify a parameterization that mimics the parameters of the SABR model. For this parameterized QGY model, we calibrate the model to GBP RPI and EUR HICPxT year-on-year inflation options. We get a good qualitative fit to the smile/skew of implied volatilities, while being able to control the year-on-year convexity adjustment. Furthermore, we see how the calibrated model prices limited price indices (LPIs) and zero-coupon inflation options.
Main references:
- M. Trovato, D. Ribeiro and H. Gretarsson: A quadratic Gaussian year-on-year inflation model (working paper, Lloyds Banking Group, 2012).
- P. McCloud: Putting the smile back on the face of derivatives (Risk, Jan 2010).
- V. Piterbarg: Practical multi-factor quadratic Gaussian models of interest rates (Barclays Capital, 2008).
[PDF] Slides of the presentation.