Robust Product Markovian Quantization

TL;DR

This paper improves product Markovian quantization by reformulating it as standard vector quantization, enabling the use of accelerated Lloyd's algorithm for more stable and efficient approximation of stochastic differential equations, especially in financial models.

Contribution

It reformulates PMQ as standard vector quantization, allowing robust and accelerated algorithms, and extends the method to stochastic volatility models with higher-order updates.

Findings

Enhanced stability and efficiency in quantization algorithms.

Successful application to European options with Heston model.

Extended to exotic products using SABR model.

Abstract

Recursive marginal quantization (RMQ) allows the construction of optimal discrete grids for approximating solutions to stochastic differential equations in d-dimensions. Product Markovian quantization (PMQ) reduces this problem to d one-dimensional quantization problems by recursively constructing product quantizers, as opposed to a truly optimal quantizer. However, the standard Newton-Raphson method used in the PMQ algorithm suffers from numerical instabilities, inhibiting widespread adoption, especially for use in calibration. By directly specifying the random variable to be quantized at each time step, we show that PMQ, and RMQ in one dimension, can be expressed as standard vector quantization. This reformulation allows the application of the accelerated Lloyd's algorithm in an adaptive and robust procedure. Furthermore, in the case of stochastic volatility models, we extend the PMQ algorithm by using higher-order updates for the volatility or variance process. We illustrate the technique for European options, using the Heston model, and more exotic products, using the SABR model.