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Inflation Derivatives: From Market Model to Foreign Currency Analogy

“Foreign currency analogy” has been the standard technology for modeling inflation-linked derivatives (Barone and Castagna, 1997; Bezooyen et al. 1997; Hughston, 1998; Jarrow and Yildirim, 2003). In this approach, real interest rate, defined as the difference between nominal interest rate and inflation rate, is treated as the interest rate of a foreign currency, while the consumer price index (CPI) is treated as the exchange rate between domestic and the foreign currency.

To price inflation derivatives, one needs to model nominal (domestic) interest rate, foreign (real) interest rate, and the exchange rate (CPI). A handy solution for modeling inflation derivatives is to adopt the Heath-Jarrow-Morton’s (1992) framework separately for both interest rates, and bridge them with a lognormal exchange-rate process. For a comprehensive yet succinct introduction of the pricing model under the so-called “HJM foreign currency analogy”, we refer readers to Manning and Jones (2003).

Although elegant in theory, a Heath-Jarrow-Morton type model is known to be inconvenient for derivatives pricing. The model takes unobservable instantaneous nominal and real forward rates as state variables, making it hard to be calibrated to most inflation derivatives, as their payoffs are written on CPI or simple compounding inflation rates.

Aimed at more convenient pricing and hedging of inflation derivatives, a number of alternative models have been developed over the years. These models typically adopt lognormal dynamics for certain observable inflationrelated variables, for examples, CPI index (Belgrade and Benhamou, 2004; Belgrade et al., 2004) or forward price of real zero-coupon bonds (Kazziha, 1999; Mercurio, 2005). Recently extensions of models along this line have incorporated more sophisticated driving dynamics like stochastic volatility (Mercurio and Moreni, 2006 and 2009) and a jump-diffusion (Hinnerich, 2008). Besides, there are also papers that address various issues in inflation rate modeling, like ensuring positive nominal interest rates by Cairns (2000), and estimating inflation risk premiums by Chen et al. (2006), among others.

Although most of these models achieve closed-form pricing for certain derivatives, they carry various drawbacks, from complexity of pricing to not being a proper term structure model that describes the co-evolution of nominal interest rates and inflation rates. In the meantime, market practitioners have generally adopted a model of their own, the so-called market model based on displaced diffusion dynamics for “forward inflation rates”. The market model of practitioners, however, has not appeared in literature available to public.

Inflation Derivatives: From Market Model to Foreign Currency Analogy