Optimal Quantization for Finance: From Random Vectors to Stochastic Processes

Abstract

Abstract In this chapter, we present an overview of the recent developments of vector quantization and functional quantization and their applications as a numerical method in finance, with an emphasis on the quadratic case. Quantization is a way to approximate a random vector or a stochastic process, viewed as a Hilbert-valued random variable, using a nearest neighbor projection on a finite codebook. We make a review of cubature formulas to approximate expectation, an conditional expectation, including the introduction of a quantization-based Richardson—Romberg extrapolation method. The optimal quadratic quantization of the Brownian motion is presented in full detail. A special emphasis is made on the computational aspects and the numerical applications, in particular, the pricing of different kinds of options in various fields (swing options on gas and options in a Heston stochastic volatility model).