# Recursive Marginal Quantization of Higher-Order Schemes

## TL;DR

This paper extends recursive marginal quantization to higher-order schemes like Milstein, improving accuracy in stochastic process simulations and option pricing, with efficient implementation and boundary handling.

## Contribution

It introduces recursive marginal quantization for Milstein and weak order 2.0 schemes, enhancing numerical accuracy and efficiency in stochastic differential equation applications.

## Key findings

- Higher order schemes show improved weak order convergence.
- Numerical results demonstrate better approximation of analytical distributions.
- Enhanced option pricing accuracy for European, Bermudan, and Barrier options.

## Abstract

Quantization techniques have been applied in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and efficient calibration of large derivative books. Recursive Marginal Quantization of the Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This method involves recursively quantizing the conditional marginals of the discrete-time Euler approximation of the underlying process. By generalizing this approach, we show that it is possible to perform recursive marginal quantization for two higher-order schemes: the Milstein scheme and a simplified weak order 2.0 scheme. As part of this generalization a simple matrix formulation is presented, allowing efficient implementation. We further extend the applicability of recursive marginal quantization by showing how absorption and reflection at the zero boundary may be incorporated, when this is necessary. To illustrate the improved accuracy of the higher order schemes, various computations are performed using geometric Brownian motion and its generalization, the constant elasticity of variance model. For both processes, we show numerical evidence of improved weak order convergence and we compare the marginal distributions implied by the three schemes to the known analytical distributions. By pricing European, Bermudan and Barrier options, further evidence of improved accuracy of the higher order schemes is demonstrated.