Paper: arXiv 2404.05214

Authors: Nizar Riane, Claire David

Abstract

In this work, we give a generalized formulation of the Black-Scholes model. The novelty resides in considering the Black-Scholes model to be valid on ‘average’, but such that the pointwise option price dynamics depends on a measure representing the investors’ ‘uncertainty’. We make use of the theory of non-symmetric Dirichlet forms and the abstract theory of partial differential equations to establish well posedness of the problem. A detailed numerical analysis is given in the case of self-similar measures.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper relies heavily on advanced mathematical concepts such as non-symmetric Dirichlet forms and abstract PDE theory to generalize the Black-Scholes framework. However, it lacks empirical validation with real market data, focusing predominantly on theoretical well-posedness and idealized numerical simulations.

Research Flowchart

  flowchart TD
    A[Goal: Formulate Generalized Black-Scholes Model via Investor Uncertainty] --> B[Inputs: Classic BS Model & Non-Symmetric Dirichlet Forms]
    B --> C[Methodology: Define Pointwise Price Dynamics on Average]
    C --> D[Abstract PDE Theory: Prove Problem Well-Posedness]
    D --> E[Computational Analysis: Self-Similar Measures]
    E --> F[Outcomes: Option Self-Similar Pricing & Validated Numerical Framework]

Cited References (17)

  1. An inverse Black–Scholes problem (2021) — link
  2. The finite difference method for the heat equation on Sierpiński simplices (2018) — link
  3. Control of the Black-Scholes equation (2017) — link
  4. Method of lines (2007) — link
  5. Laplace operators related to self-similar measures on R d (2006) — link
  6. Computational Methods for Option Pricing (Frontiers in Applied Mathematics) (Frontiers in Applied Mathematics 30) (2005) — link
  7. Parametric pricing of higher order moments in S&P500 options (2005) — link
  8. Far Field Boundary Conditions for Black-Scholes Equations (2000) — link
and 9 more
  1. Anomalies in Option Pricing: The Black-Scholes Model Revisited (1996)
  2. Introduction to the theory of (non-symmetric) Dirichlet forms (1992) — link
  3. Nonlinear Functional Analysis and Its Applications: II/ A: Linear Monotone Operators (1989)
  4. Nonlinear functional analysis and its applications (1988) — link
  5. Analysis on Fractals (2009)
  6. The Black-Scholes Equation (2008) — link
  7. Fractals and Scaling in Finance (1997) — link
  8. Weighted Sobolev spaces and capacity. (1994)
  9. Problèmes aux limites non homogènes et applications (1968)

0 of 17 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.