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

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  • 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.

Suggested Citation

  • Jagdish Gnawali & Abootaleb Shirvani & Svetlozar T. Rachev, 2025. "Implied Probabilities and Volatility in Credit Risk: A Merton-Based Approach with Binomial Trees," Papers 2506.12694, arXiv.org.
    • Handle: RePEc:arx:papers:2506.12694
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    References listed on IDEAS

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    1. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    2. Jagdish Gnawali & W. Brent Lindquist & Svetlozar T. Rachev, 2025. "Hedging via Perpetual Derivatives: Trinomial Option Pricing and Implied Parameter Surface Analysis," JRFM, MDPI, vol. 18(4), pages 1-32, April.
    3. Longstaff, Francis A & Schwartz, Eduardo S, 1995. "A Simple Approach to Valuing Risky Fixed and Floating Rate Debt," Journal of Finance, American Finance Association, vol. 50(3), pages 789-819, July.
    4. Hu, Yuan & Lindquist, W. Brent & Rachev, Svetlozar T. & Shirvani, Abootaleb & Fabozzi, Frank J., 2022. "Market complete option valuation using a Jarrow-Rudd pricing tree with skewness and kurtosis," Journal of Economic Dynamics and Control, Elsevier, vol. 137(C).
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