Risk‐neutral pricing techniques and examples

Tractability and flexibility are among the two most attractive features of models in mathematical finance. For the pricing of derivative products, to avoid arbitrage opportunities, the fundamental theorem of asset pricing requires the existence of an equivalent martingale measure under which the discounted price process is a (local) martingale. In the semimartingale setting, the traditional approach of risk‐neutral valuation uses a change of measure invoking Girsanov's theorem. This approach often destroys the tractability of the process which is undesirable for quantitative finance applications. To overcome these limitations, we employ a transformation based on the concept of intertwining relationships that allows us to convert Markovian semigroups into a pricing semigroup while keeping its tractability. In order to illustrate the usefulness of this approach, we apply it to exponential Lévy, positive self‐similar, and generalized CIR processes that all fall within the class of polynomial processes introduced by Cuchiero et al. (2012). Furthermore, for the jump CIR class, relying on recent work of Patie and Savov (2019), we provide an eigenvalues expansion of the pricing semigroup and carry out some numerical experiments.