Recursive Marginal Quantization of the Euler Scheme of a Diffusion Process

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.

Recursive Marginal Quantization of Higher-Order Schemes

“…In this paper, the recursive marginal quantization methodology of Pagès and Sagna [2015] has been extended from the standard Euler-Maruyama scheme to higher-order numerical schemes, specifically the Milstein scheme and the simplified weak order 2.0 scheme of Kloeden and Platen [1999]. This entailed introducing noncentral chi-squared updates and generalising the formulation of Recursive Marginal Quantization (RMQ).…”

“…The main result of Pagès and Sagna [2015] shows that if one uses the previously quantized distribution of X k , instead of the continuous distribution of X k , the resultant procedure converges. Furthermore, the error associated with this procedure is bounded by a constant, which is dependent on the parameters used.…”

“…The primary contribution is, however, the more general formulation of RMQ that enables the systematic extension of the work of Pagès and Sagna [2015]. The paper concludes with a discussion of ongoing work.…”

“…The innovation of Pagès and Sagna [2015] was to show that a recursive procedure based on quantizing these updates is possible.…”

“…Our formulation of the problem is presented in more generality than the original formulation by Pagès and Sagna [2015]. A matrix formulation, which demonstrates the connection with Markov chains, is provided, allowing easy and efficient implementation.…”