Correlators of Polynomial Processes

Abstract

A process is polynomial if its extended generator maps any polynomial to a polynomial of equal or lower degree. Then its conditional moments can be calculated in closed form, up to the computation of the exponential of the so-called generator matrix. In this article, we provide an explicit formula to the problem of computing correlators, that is, computing the expected value of moments of the process at different time points along its path. The strength of our formula is that it only involves linear combinations of the exponential of the generator matrix, as in the one-dimensional case. The framework developed allows then for easy-to-implement solutions when it comes to financial pricing, such as for path-dependent options or in a stochastic volatility models context.

Pricing Asian Options with Correlators

“…By considering the underlying spot price process from the family of jump-diffusion polynomial processes, we then obtain a fully explicit expression for the price functional thanks to the well-known moment formula for polynomial processes and to the correlator formula derived in [2]. This has the advantage to allow for sensitivity analysis, since Greeks are within reach, as we show in Section 4.1.…”

“…However the main details can be found in Appendix B. For more interested readers, we refer to [2]. Theorem 4.3.…”

“…However, we still need to compute moments of the average price process. We do this by the multinomial theorem and the correlator formula for polynomial processes derived in [2].…”

“…The price of the Asian option is then given by the discounted expected value of this infinite sum. By the multinomial theorem, we rewrite the terms of the sum as a linear combination of correlator-type terms in the sense of [2], that is, terms of the form…”

“…which we compute by the closed formula for correlators [2,Theorem 4.5]. This procedure gives an exact formula for pricing discrete Asian options.…”