HJM Local Volatility Model

Paper: arXiv 2301.13595

Authors: V. M. Belyaev

Abstract

Local Volatility (LV) is a powerful tool for market modeling, enabling the generation of arbitrage-free scenarios calibrated to all European options. To implement LV, we need to interpolate and extrapolate option prices. This approach is significantly faster and more accurate than any parameterized model. The implementation is demonstrated specifically for interest rate swaptions and caplets. A key component of this method is the Small Volatility Approximation within the HJM interest rate model, which is used to calculate sensitivity of forward bond volatility. These calculations are deterministic and fast, with excellent calibration accuracy. A detailed description of the calibration procedure is provided.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 5.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a solid mathematical formulation adapting Local Volatility to the HJM interest rate framework via small volatility approximations. While the mathematical derivation and calibration steps are detailed and novel for fixed income, empirical validation and full reproducibility details remain moderate.

Research Flowchart

  flowchart TD
    A[Goal: Fast, Arbitrage-Free HJM Local Volatility Model] --> B[Inputs: European Swaptions & Caplets Market Data]
    B --> C[Interpolate & Extrapolate Option Prices]
    C --> D[Apply Small Volatility Approximation in HJM Model]
    D --> E[Compute Deterministic Forward Bond Volatility Sensitivities]
    E --> F[Execute Fast Calibration Procedure]
    F --> G[Outcome: Accurate, Highly Efficient Interest Rate LV Model]