Recent nancial crises have highlighted the sensitivity and vulnerability of nancial markets to in ation, which reduces the value of money and a ects the net returns of nancial instruments. In response to this, investors who are concerned with maintaining their investment's purchasing power rather than its market value are resorting to in ation linked (IL) products to hedge their in ation risk. Consequently, the in ation market has been rapidly growing for the last decade and has further great potential growth worldwide. It is highly probable that in ation linked derivatives will eventually be as common as conventional products. Another cause of the in ation market boost is the growing extension of the time frame of nancial transactions, which has generated an increase in in ation expectation; since 1980 the average time to maturity of long-dated transactions went from one decade to three decades. This is, in part, due to the ageing population in the developed world. This research investigates some alternative models in order to improve the match between model prices and observed prices in the American and South African in ation markets. It takes into account the relative illiquidity of IL products. The main tools used are L evy distributions, macroeconomic factors, no-arbitrage and pricing kernel models. L evy processes can replicate the behaviour of the return innovations of a wide range of nancial securities. Adding a stochastic time change to the L evy process randomises the market clock, thus generating stochastic volatilities, higher stochastic return moments and eventually stochastic skewness. These are observed stylised facts most conventional models do not achieve. Moreover, in contrast to the hidden factor approach, each L evy process component and its stochastic time change can readily be assigned an economic meaning. This explicit economic mapping facilitates the interpretation of current models and provides a more intuitive approach to building new models that capture other observed behaviours. Finally, L evy processes also provide tractable formulas for derivative pricing and market estimations. In general, in ation is a consequence of macroeconomic factors. Exogenous dynamics of the most signi cant of these factors are used to deduce the endogenous in ation dynamics in some of the considered models. In these cases, the calibration of the pricing kernel models requires little historical data on IL derivatives. In fact, the required macroeconomic historical data is easily available because of the current national and international legislation.