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Pricing of barrier options by marginal functional quantization

Abass Sagna

TL;DR

The paper tackles barrier option pricing under diffusion models by introducing a quadratic marginal functional quantization method that replaces Brownian motion with a finite, optimally quantized surrogate and builds a marginally quantized price process. A forward-recursive algorithm is developed to compute barrier option premia, with rigorous error bounds under Lipschitz assumptions. The approach is implemented and benchmarked against diffusion-bridge methods in Black-Scholes and a local-volatility (pseudo CEV) model, showing competitive speed especially for moderate discretization levels while highlighting trade-offs between grid size, accuracy, and computation time. These results indicate a fast, structure-preserving alternative to traditional path-simulation techniques for path-dependent options.

Abstract

This paper is devoted to the pricing of Barrier options by optimal quadratic quantization method. From a known useful representation of the premium of barrier options one deduces an algorithm similar to one used to estimate nonlinear filter using quadratic optimal functional quantization. Some numerical tests are fulfilled in the Black-Scholes model and in a local volatility model and a comparison to the so called Brownian Bridge method is also done.

Pricing of barrier options by marginal functional quantization

TL;DR

The paper tackles barrier option pricing under diffusion models by introducing a quadratic marginal functional quantization method that replaces Brownian motion with a finite, optimally quantized surrogate and builds a marginally quantized price process. A forward-recursive algorithm is developed to compute barrier option premia, with rigorous error bounds under Lipschitz assumptions. The approach is implemented and benchmarked against diffusion-bridge methods in Black-Scholes and a local-volatility (pseudo CEV) model, showing competitive speed especially for moderate discretization levels while highlighting trade-offs between grid size, accuracy, and computation time. These results indicate a fast, structure-preserving alternative to traditional path-simulation techniques for path-dependent options.

Abstract

This paper is devoted to the pricing of Barrier options by optimal quadratic quantization method. From a known useful representation of the premium of barrier options one deduces an algorithm similar to one used to estimate nonlinear filter using quadratic optimal functional quantization. Some numerical tests are fulfilled in the Black-Scholes model and in a local volatility model and a comparison to the so called Brownian Bridge method is also done.

Paper Structure

This paper contains 10 sections, 5 theorems, 103 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Euler Scheme
  3. Approximation of knock out option prices using diffusion bridge
  4. Estimation of the prices by marginal quantization
  5. The algorithm
  6. Error analysis
  7. Marginal functional quantization of the price process
  8. A brief overview on functional product quantization of gaussian processes
  9. Marginal functional quantization and transition probabilities
  10. Numerical illustration

Key Result

Proposition 3.1

We have and where $(U_k)_{k=0,\dots,n-1}$ are $i.i.d$ random variables uniformly distributed over the unit interval, $G^{-1}_{x,y}$ and $F^{-1}_{x,y}$ are the inverse functions of the conditional distribution functions $G_{x,y}$ and $F_{x,y}$ defined by and

Theorems & Definitions (11)

  • Definition
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.1
  • proof : $\textbf{Proof}$
  • proof : $\textbf{Proof of Proposition $\ref{prop_exp_max_double}$}$
  • Proposition 3.3
  • proof : $\textbf{Proof}$
  • Remark 4.1
  • Theorem 4.1
  • ...and 1 more