Quantization Goes Polynomial

Abstract

Quantization algorithms have been recently successfully adopted in option pricing problems to speed up Monte Carlo simulations thanks to the high convergence rate of the numerical approximation. In particular, recursive marginal quantization has been proven a flexible and versatile tool when applied to stochastic volatility processes. In this paper we apply for the first time these techniques to the family of polynomial processes, by exploiting, whenever possible, their peculiar properties. We derive theoretical results to assess the approximation errors, and we describe in numerical examples practical tools for fast exotic option pricing. JEL classification codes: C63, G13.

Option Pricing with Orthogonal Polynomial Expansions

“…Notable progress has recently been made on the pricing of early-exercise and path-dependent options with stochastic volatility using recursive marginal quantization as studied in Pagès and Sagna (2015); McWalter et al (2017); Callegaro et al (2017a). In the context of affine and polynomial models, this approach has been shown to perform well when combined with Fourier transform techniques as in Callegaro et al (2017b), or polynomial expansion techniques as in Callegaro et al (2017), whose results could be further improved with the new expansions presented in our paper. The calculation of Greeks for stochastic volatility models is a difficulty task often adressed by Monte Carlo simulations, see for examples Broadie and Kaya (2004) for a discussion of different simulation based estimators in the Heston model, and Chan et al (2015) for more recent advances using algorithmic differentiation.…”

Polynomial Jump-Diffusion Models

“…Examples include interest rates (Delbaen and Shirakawa 2002, Zhou 2003, Filipović et al 2017, stochastic volatility (Gourieroux andJasiak 2006, Ackerer et al 2018), exchange rates (Larsen and Sørensen 2007), life insurance liabilities (Biagini and Zhang 2016), variance swaps , credit risk (Ackerer and Filipović 2020a), dividend futures (Filipović and Willems 2019), commodities and electricity (Filipović et al 2018), stochastic portfolio theory (Cuchiero 2019), and economic equilibrium (Guasoni and Wong 2018). 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 , and quantization (Callegaro et al 2017). This recent body of research primarily relies on polynomial jump diffusions that are not necessarily affine.…”

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.…”