Implied Probabilities and Volatility in Credit Risk: A Merton-Based Approach with Binomial Trees

ArXiv ID: 2506.12694 “View on arXiv”

Authors: Jagdish Gnawali, Abootaleb Shirvani, Svetlozar T. Rachev

Abstract

We explore credit risk pricing by modeling equity as a call option and debt as the difference between the firm’s asset value and a put option, following the structural framework of the Merton model. Our approach proceeds in two stages: first, we calibrate the asset volatility using the Black-Scholes-Merton (BSM) formula; second, we recover implied mean return and probability surfaces under the physical measure. To achieve this, we construct a recombining binomial tree under the real-world (natural) measure, assuming a fixed initial asset value. The volatility input is taken from a specific region of the implied volatility surface - based on moneyness and maturity - which then informs the calibration of drift and probability. A novel mapping is established between risk-neutral and physical parameters, enabling construction of implied surfaces that reflect the market’s credit expectations and offer practical tools for stress testing and credit risk analysis.

Keywords: Merton model, structural credit risk, implied volatility surface, binomial tree, real-world measure, Equity & Credit

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail
  • Why: The paper involves advanced continuous-time stochastic calculus (Merton model, GBM), risk-neutral/physical measure conversion, and volatility surface mapping, but it grounds the theory with specific calibration methods (BSM inversion, binomial tree calibration) and proposes practical applications for stress testing and credit analysis.
  flowchart TD
    A["Research Goal<br>Estimate implied credit risk probabilities & volatility under the physical measure"] --> B["Data Inputs<br>Market Prices (Equity/Debt) & Merton Model"]
    B --> C["Methodology Stage 1<br>Calibrate Asset Volatility via BSM"]
    C --> D["Methodology Stage 2<br>Construct Recombining Binomial Tree (Real-World Measure)"]
    D --> E["Computational Process<br>Map Risk-Neutral to Physical Parameters<br>Drift & Probability Calibration"]
    E --> F["Key Outcomes<br>Implied Mean Return & Probability Surfaces<br>Tools for Stress Testing & Credit Analysis"]