A multifractional option pricing formula

Paper: arXiv 2303.16314

Authors: Axel A. Araneda

Abstract

Fractional Brownian motion has become a standard tool to address long-range dependence in financial time series. However, a constant memory parameter is too restrictive to address different market conditions. Here we model the price fluctuations using a multifractional Brownian motion assuming that the Hurst exponent is a time-deterministic function. Through the multifractional Ito calculus, both the related transition density function and the analytical European Call option pricing formula are obtained. The empirical performance of the multifractional Black-Scholes model is tested by calibration of option market quotes for the SPX index and offers best fit than its counterparts based on standard and fractional Brownian motions.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces a mathematically sophisticated option pricing model using multifractional Brownian motion and time-deterministic Hurst exponents. It extends fractional Itô calculus effectively and provides empirical validation via SPX index option calibration, demonstrating superior fit compared to standard and fractional Black-Scholes baselines.

Research Flowchart

  flowchart TD
    A[Research Goal: Address variable market memory in option pricing] --> B[Model Development: Replace constant Hurst exponent with time-deterministic Hurst function in mBm]
    B --> C[Mathematical Framework: Apply Multifractional Ito Calculus]
    C --> D[Analytical Derivation: Obtain Transition Density & European Call Pricing Formula]
    E[Market Data: SPX Index Option Quotes] --> F[Empirical Calibration: Test Multifractional BS Model vs Standard & Fractional BS Models]
    D --> F
    F --> G[Key Outcome: Multifractional BS Model provides superior fit to market data]