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Deriving Default Probability from a Lognormal Asset Model

Article Quant Q&A · Author: Carmen González
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Summary

The document derives the maturity default probability for a loan in a Vasicek-style credit risk setup. It models the borrower’s asset value as a geometric Brownian motion, solves for the logarithm of asset value at the horizon, and treats that log value as normally distributed. Default occurs when terminal assets fall below a specified liability threshold.

Taking logarithms of that threshold condition converts default into a standard normal tail probability, with a cutoff determined by initial assets, liabilities, drift, volatility, and horizon. The resulting cumulative normal expression gives the individual loan’s probability of default under the model. The derivation describes single-name default probability only; it does not address dependence across borrowers, portfolio loss distributions, or the additional assumptions used in Vasicek’s portfolio credit model. Its conclusions therefore depend on the geometric Brownian motion specification and chosen parameters.

Key ideas

  • Asset value is modeled as a geometric Brownian motion with constant drift and volatility.
  • The logarithm of terminal asset value is normally distributed under this process.
  • Default is defined as terminal assets falling below the loan’s liability threshold.
  • The default event becomes a standard normal probability after taking logarithms.
  • The excerpt derives an individual default probability but not portfolio dependence or loss risk.

Tags

Fixed incomeStatisticsRisk management
Full text
# Need to solve the stochastic differential equation of Vasicek Model


# Need to solve the stochastic differential equation of Vasicek Model












How to solve the stochastic differential equation of the Vasicek model for the analysis of credit risk? I search in the article "The Distribution of loan portfolio value" (Vasicek) but he doesn't solve the equation.

$$dA_i=\mu_i A_i dt+\sigma_i A_i dx_i$$

The solution is:

$$log A_i(T)=log A_i+\mu_i T-\frac{1}{2}\sigma_i^2T+\sigma_i\sqrt{T}X_i$$

The probability of default of the i-th loan is then

$$p_i=P[A_i(T)<B_i]=P[X_i<c_i]=N(c_i)$$

where

$$c_i=\frac{log B_i-log A_i-\mu_i T+\frac{1}{2}\sigma_i^2T}{\sigma_i\sqrt{T}}$$

and N is the cumulative normal distribution function.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.