Modeling and Simulation of Financial Returns under Non-Gaussian Distributions

Paper: arXiv 2302.02769

Authors: Federica De Domenico, Giacomo Livan, Guido Montagna, Oreste Nicrosini

Abstract

It is well known that the probability distribution of high-frequency financial returns is characterized by a leptokurtic, heavy-tailed shape. This behavior undermines the typical assumption of Gaussian log-returns behind the standard approach to risk management and option pricing. Yet, there is no consensus on what class of probability distributions should be adopted to describe financial returns and different models used in the literature have demonstrated, to varying extent, an ability to reproduce empirically observed stylized facts. In order to provide some clarity, in this paper we perform a thorough study of the most popular models of return distributions as obtained in the empirical analyses of high-frequency financial data. We compare the statistical properties and simulate the dynamics of non-Gaussian financial fluctuations by means of Monte Carlo sampling from the different models in terms of realistic tail exponents. Our findings show a noticeable consistency between the considered return distributions in the modeling of the scaling properties of large price changes. We also discuss the convergence rate to the asymptotic distributions of the non-Gaussian stochastic processes and we study, as a first example of possible applications, the impact of our results on option pricing in comparison with the standard Black and Scholes approach.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a solid comparative study of non-Gaussian return distributions and their scaling behavior using Monte Carlo simulations. While it provides good mathematical coverage of heavy-tailed distributions and an application to option pricing, it functions primarily as a comparative survey rather than introducing groundbreaking new methodologies.

Research Flowchart

  flowchart TD
    A[Research Goal: Evaluate Non-Gaussian Models for Heavy-Tailed Financial Returns] --> B[Data Input: High-Frequency Financial Return Series]
    B --> C[Statistical Analysis & Parameter Fitting of Candidate Distributions]
    C --> D[Monte Carlo Simulation of Price Dynamics & Scaling Properties]
    D --> E[Comparative Evaluation vs. Standard Black-Scholes Model]
    E --> F[Findings: Consistency in Tail Scaling & Impact on Option Pricing]