This is a concise paper describing the Vasicek model. It serves the purpose of describing how banks should evaluate risk on loan assets, and provides a connection between the Black-Scholes option pricing model to pricing bonds or loans.

It starts with the geometric Brownian motion model, which we can describe the price movement of a stock with the following:

\begin{align} dS_t &= \mu S_t dt + \sigma S_t dW_t \\ S_t &= S_0 \exp((\mu-\frac12\sigma^2)t + \sigma W_t) \\ &= S_0 \exp((\mu-\frac12\sigma^2)t + \sigma \sqrt{t} Z) \end{align}

The Vasicek model describes the total asset value of an obligor as gBM as well, but the Brownian motion driving the model is a combination between two other Brownian motion:

$X_t = S\sqrt{\rho} + Z_i\sqrt{1-\rho}$

which $$S$$ is the systematic risk and $$Z_i$$ is the idiosyncratic risk, which only the former is shared across the same industry or market while the latter is specific to the obligor. The default probability of two obligors are related by the systematic risk, precisely, the covariance of their $$X_t$$ is $$\rho$$.

The convention of notation makes $$X_t\sim N(0,1)$$ and the asset price movement is described by

$A_t = A_0 \exp((\mu-\frac12\sigma^2)t + \sigma \sqrt{t} X_t)$

we assume the obligor defaults when its asset falls below some threshold $$B$$ at time $$T$$, i.e.

$\Pr[A_T < B] = \Pr[X_T < c] = \Phi(c) = p^*$

which $$c$$ is derived from the gBM $$A_t$$ and threshold $$B$$, and $$p^*$$ is the unconditional probability of default, a.k.a. through-the-cycle average loss. The quantity $$c<0$$ is the distance of current asset value from the default threshold, in terms of standard deviations. For a pool of loans, the correlation of defaults is due to the correlation between assets $$A_i$$ and $$A_j$$, i.e. the systematic factor $$S$$. If all obligors are correlated to each other in the same factor $$\rho$$, we call the pool equi-correlated.

Besides the unconditional probability of default, we can also have the probability of default conditional on $$S$$ if we fixed $$p^*$$:

\begin{align} \Pr[A_i(T)<B_i] &= \Pr[X_i<c_i]=\Pr[X_i<\Phi^{-1}(p^*)] \\ &= \Pr[S\sqrt{\rho}+Z_i\sqrt{1-\rho}<\Phi^{-1}(p^*)]\\ &=\Pr[Z_i<\frac{c_i-S\sqrt{\rho}}{\sqrt{1-\rho}}] \\ &=\Phi\left(\frac{\Phi^{-1}(p^*)-S\sqrt\rho}{\sqrt{1-\rho}}\right) \end{align}

This is the loss subject to credit condition $$S$$, and denoted by $$p(S)$$, which we can further find its pdf as:

\begin{align} \Pr[p(S)<x] &=\Pr[S>p^{-1}(x)] =\Phi(-p^{-1}(x))\\ &=\Phi\left(\frac{\sqrt{1-\rho}\Phi^{-1}(x)-\Phi^{-1}(p^*)}{\sqrt{\rho}}\right) \end{align}

Because of this, the KMV model suggested the relationship between a company’s liability (debts) and the equity (stock value). It considers that if the face value of the debt is $$D_t$$ to repaid at time $$t$$, and the asset is modeled with $$A_t$$, then:

• Equity can be considered as the call option on the asset $$A_t$$ with strike at $$D_t$$
• Debt can be considered as the risk-free debt of the same amount and shorting a put option on $$A_t$$ with strike $$D_t$$

so according to the Black-Schole formula, we can derive the value of equity $$E$$ and debt $$B$$:

\begin{align} E &=A_t\Phi(d_1)-D_te^{-rt}\Phi(d_1-\sigma_A t) \\ B &= De^{-rt}-[De^{-rt}\Phi(-d_1-\sigma_A t)-A_0\Phi(-d_1)] \\ &= De^{-rt}-P_t \end{align}

which the volatility of equity and asset are related as $$\sigma_E E = \frac{\partial E}{\partial A}\sigma_A A$$.

If we assume all those debts are bonds, then the bond’s yield to maturity is

$y = \frac{\ln(D_t/B)}{t}$

and the difference $$y-r$$ is the credit spread to compensate for the default risk.

## Bibliographic data

@techreport{
author = "Somnath Chatterjee",
title = "Modeling credit risk",
institution = "Bank of England",
}