Rational Models for Inflation-Linked Derivatives

Abstract

We construct models for the pricing and risk management of inflation-linked derivatives. The model is rational in the sense that affine payoffs written on the consumer price index have prices that are rational functions of the state variables. The nominal pricing kernel is constructed in a multiplicative manner that allows for closed-form pricing of vanilla inflation products suchlike zero-coupon swaps, caps and floors, year-on-year swaps, caps and floors, and the exotic limited price index swap. The model retains the attractive features of a nominal multi-curve interest rate model such as closed-form pricing of nominal swaptions. We conclude with examples of how the model can be calibrated to EUR data. for inflation-linked bonds. These bonds are typically government-issued debt where the principal is indexed by the consumer price index (CPI) or similar. The bonds often have an embedded YoY floor protecting the principal from being adjusted down by deflation. Finally, Limited Price Index (LPI) products come with both a lower and upper bound on the adjustment on the principal creating a path-dependent collar on inflation. Despite its exotic nature LPIs have been in high demand by pension funds.All products should ideally be priced in a consistent manner using a tractable arbitragefree model. Cap/floor products naturally display volatility skews and non-flat term structures of volatility, both of which the model also should be able to capture. In addition, the model should yield closed-form solutions for the price of the most traded derivatives, here the YoY and the ZC cap/floor. Hughston (1998) develops a general arbitrage-free theory of interest rates and inflation in the case where the consumer price index and the real and nominal interest rate systems are jointly driven by a multi-dimensional Brownian motion. This approach is based on a foreign exchange analogy in which the CPI is treated like a foreign exchange rate, and the "real" interest rate system is treated as if it were the foreign interest rate system associated with the foreign currency. The often cited work by Jarrow and Yildirim (2003) makes use of such a setup and considers a three factor model (i.e., driven by three Brownian motions) in which the CPI is modelled as a geometric Brownian motion, with deterministic timedependent volatility, and the two interest rate systems are treated as extended Vasicek-type (or Hull-White) models. Similar to Jarrow and Yildirim (2003), Dodgson and Kainth (2006) use a short-rate approach where the nominal and the inflation rates are both modelled by Hull-White processes while they discard entirely the idea of a real economy. A GBM-based model for the CPI provides the baseline framework for how one might understand implied volatility in such a market, but any GBM model for the CPI does not, by construction, reproduce volatility smiles.Further development of inflation models has paralleled that of interest rates models. For example inflation counterparts to the nominal LIBOR Market Model, see for example Brigo and Me...

Stochastic modelling with randomized Markov bridges

“…If we were to look for similarities with existing financial instruments, then the GHG-contracts presented next would be related to inflation-linked or possibly insurance-linked securities, see e.g. Dam et al (2018) and Barrieu & Albertini (2009). Such contracts provide financial protection against losses due to, for example, currency depreciation or limit financial liabilities beyond a certain amount.…”

“…While we also use a no-arbitrage approach, though we prefer to work under the real-world measure, we think that the design of the here proposed market instruments written on GHG emissions is different. If we were to look for similarities with existing financial instruments, then the GHG-contracts presented next would be related to inflation-linked or possibly insurance-linked securities, see e.g Dam et al (2018). andBarrieu & Albertini (2009).…”

Consistent Valuation Across Curves Using Pricing Kernels

“…The derivations of such pricing formulae follow those for the forward rate agreement and the swap contracts presented in Section 3. Price models for inflationlinked securities, which are based on explicit pricing kernel models-hence, on explicit curve conversion factor processes-have been developed by Dam et al [13]. Such models feature a high degree of flexibility and good calibration properties.…”