Polynomial processes and their applications to mathematical finance

Abstract

We introduce a class of Markov processes, called m-polynomial, for which the calculation of (mixed) moments up to order m only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as Lévy-driven SDEs with affine vector fields. Thus, many popular models such as exponential Lévy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo methods.

Stationarity and Ergodicity for an Affine Two-Factor Model

“…The idea behind this formal representation is that the infinitesimal generator of the affine process (Y, X) formally maps the finite-dimensional linear space of all polynomials in (y, x) ∈ R + × R of degree less than or equal to k into itself, where k ∈ N. For a more general class of time-homogeneous Markov processes having this property, for the so-called polynomial processes, see Cuchiero et al [10].…”

Polynomial Jump-Diffusion Models

“…Examples include interest rates (Zhou, 2003;Delbaen and Shirakawa, 2002;Filipović et al, 2017b), stochastic volatility (Gourieroux and Jasiak, 2006;Ackerer et al, 2016), exchange rates (Larsen and Sørensen, 2007), life insurance liabilities (Biagini and Zhang, 2016), variance swaps (Filipović et al, 2016a), credit risk (Ackerer and Filipović, 2016), dividend futures (Filipović and Willem, 2017), commodities and electricity (Filipović et al, 2017a), and stochastic portfolio theory (Cuchiero, 2017). Properties of polynomial jump-diffusions can also be brought to bear on computational and statistical methods, such as generalized method of moments and martingale estimating functions (Forman and Sørensen, 2008), variance reduction (Cuchiero et al, 2012), cubature (Filipović et al, 2016b), and quantization (Callegaro et al, 2017). This recent body of research primarily relies on polynomial jump-diffusions that are not necessarily affine.…”

“…The theoretical literature in the jump-diffusion case is less abundant. The first systematic accounts are Cuchiero (2011) and Cuchiero et al (2012) in a Markovian framework. Gallardo and Yor (2006) study Dunkl processes, which are polynomial jump-diffusions; see also Dunkl (1992).…”

Infinitesimal generators for a class of polynomial processes

“…Such Markov processes enjoy many interesting properties and appeared in numerous references [2,3,10,12,15,16,17,22,23,24,25,26,27,28,29,30,31]. As observed by Cuchiero in [14] and Szab lowski [32], transition probabilities P s,t (x, dy) of such a process on an infinite state space define the family of linear transformations (P s,t ) 0≤s≤t that map the linear space P = P(R) of all polynomials in variable x into itself.…”

“…This will be the basis for our algebraic approach. Cuchiero in [14], see also [15] introduced the term "polynomial process" to denote such a process in the time-homogeneous case; we will use this term more broadly to denote the family of operators rather than a Markov process. That is, we adopt the point of view that the linear mappings P s,t of P can be analyzed "in abstract" without explicit reference to the underlying Markov process and the transition operators.…”