[Paper Review] A Merton-Like Approach to Pricing Debt based on a non-Gaussian Asset Model

Lisa Borland, Jeremy Evnine+1 more|arXiv (Cornell University)|Jan 1, 2008

Stochastic processes and financial applications18 references4 citations

TL;DR

This paper extends Merton's structural model for corporate debt pricing by replacing the log-normal asset return assumption with a non-Gaussian process that incorporates fat tails and skew, using a statistical feedback model. The approach maintains closed-form solutions and empirically fits credit spreads and option prices simultaneously, with parameters q ≈ 1.3 and α ≈ 0.3 providing strong fit across equity and credit markets.

ABSTRACT

This paper is a contribution to the Proceedings of the Workshop Complexity, Metastability and Nonextensivity held in Erice 20-26 July 2004, to be published by World Scientific. We propose a generalization to Merton’s model for evaluating credit spreads. In his original work, a company’s assets were assumed to follow a log-normal process. We introduce fat tails and skew into this model, along the same lines as in the option pricing model of Borland and Bouchaud (2004, Quantitative Finance 4) and illustrate the effects of each component. Preliminary empirical results indicate that this model fits well to empirically observed credit spreads with a parameterization that also matched observed stock return distributions and option prices. 1

Motivation & Objective

• To generalize Merton’s structural model for credit risk pricing by replacing the log-normal assumption with a non-Gaussian asset return process.
• To incorporate fat tails and skew in asset returns, which are empirically observed but neglected in standard Merton models.
• To maintain closed-form solutions and option pricing tractability while improving empirical fit to observed credit spreads and equity option prices.
• To test whether parameters calibrated on equity markets also accurately price credit spreads, supporting a unified modeling framework.

Proposed method

• Adopt a non-Gaussian stochastic process for asset returns based on a statistical feedback mechanism, generalizing the Black-Scholes-Merton framework.
• Use a power-law volatility function with parameters q (tail thickness) and α (skew) to model time-dependent volatility and non-Gaussian returns.
• Derive a closed-form solution for the default probability and credit spread using a transformed time variable ˆT and a quadratic payoff condition for default.
• Calibrate the model by matching theoretical credit spreads to observed market spreads across 54 firms using least squares.
• Back out implied volatilities under the standard Merton model to compare deviations caused by fat tails and skew.
• Map complex capital structures to a single-debt synthetic bond using a weighted average of outstanding debt.

Experimental results

Research questions

• RQ1 Can a non-Gaussian asset return model improve the fit of Merton’s structural model to observed credit spreads?
• RQ2 Do the same parameters (q, α) that fit equity option prices also accurately price credit spreads?
• RQ3 How do fat tails and skew affect the shape of the credit spread curve relative to the standard Merton model?
• RQ4 Is there a consistent parameter set (q, α) across equity and credit markets that supports a unified pricing framework?

Key findings

• The model successfully fits observed credit spreads across 54 companies using parameters q ≈ 1.3 and α ≈ 0.3, with minimal variation in α.
• The parameter values q ≈ 1.3 and α ≈ 0.3 are consistent with those that best fit equity option prices and stock return distributions in prior studies.
• The standard Merton model underestimates credit spreads for both highly leveraged (d > 1) and under-leveraged (d < 1) firms when fat tails and skew are present.
• The inclusion of fat tails (q > 1) increases implied volatility for extreme leverage ratios, while skew (α ≠ 1) causes asymmetric deviations in spread predictions.
• The model produces a 'volatility smile'-like pattern in implied volatilities when compared to the standard Merton model, reflecting the impact of non-Gaussian features.
• The empirical results suggest that a single set of parameters can simultaneously explain equity option prices and credit spreads, supporting a unified market model.