Recursive marginal quantization of the Euler scheme of a diffusion process

TL;DR

This paper introduces a recursive quantization method for the Euler scheme of diffusion processes, improving computational efficiency and accuracy in pricing and simulation tasks, especially in local volatility models.

Contribution

The paper presents a novel recursive quantization approach for Euler diffusion marginals, with explicit formulas and numerical algorithms that outperform traditional methods in certain settings.

Findings

Error bounds are established for the quantization of Euler marginals.

The method reduces computational complexity in local volatility models.

Numerical tests show improved efficiency over Monte Carlo simulations.

Abstract

We propose a new approach to quantize the marginals of the discrete Euler diffusion process. The method is built recursively and involves the conditional distribution of the marginals of the discrete Euler process. Analytically, the method raises several questions like the analysis of the induced quadratic quantization error between the marginals of the Euler process and the proposed quantizations. We show in particular that at every discretization step $t\_k$ of the Euler scheme, this error is bounded by the cumulative quantization errors induced by the Euler operator, from times $t\_0=0$ to time $t\_k$. For numerics, we restrict our analysis to the one dimensional setting and show how to compute the optimal grids using a Newton-Raphson algorithm. We then propose a closed formula for the companion weights and the transition probabilities associated to the proposed quantizations. This allows us to quantize in particular diffusion processes in local volatility models by reducing dramatically the computational complexity of the search of optimal quantizers while increasing their computational precision with respect to the algorithms commonly proposed in this framework. Numerical tests are carried out for the Brownian motion and for the pricing of European options in a local volatility model. A comparison with the Monte Carlo simulations shows that the proposed method may sometimes be more efficient (w.r.t. both computational precision and time complexity) than the Monte Carlo method.