On the Pricing of Inflation-Indexed Caps

Abstract

We consider the problem of pricing inflation-linked caplets in a Black-Scholes-type framework as well as in the presence of stochastic volatility. By using results on the pricing of forward starting options in Heston's Model on stochastic volatility, we derive closed-form solutions for year-on-year inflations caps which aim to receive smile-consistent option prices. Additionally we price options on the inflation development over a longer time horizon. In this paper we develop a new and more suitable formula for pricing inflation-linked options under the assumption of stochastic volatility. The formula in the presence of stochastic volatility allows to cover the smile effects observed in our Black-Scholes type environment, in which the exposure of year-on-year inflation caps to inflation volatility changes is ignored. The chosen diffusion processes reflect the macro-economic concept of Fisher making a connection between interest rates on the market and the expected inflation rate.