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 inflation 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.