Merton’s Structural Model and Extension

Pioneered by Merton (1974) and Black and Scholes (1973), structural (or asset value) model is one of the two primary classes of credit risk modeling approaches (The other one is the reduced form model.). It assumes that at time t a firm with risky assets A_{t} is financed by equity E_{t} and zero-coupon debt D_{t} of face value K maturing at time T>t: A_{t}=E_{t}+D_{t}.

When the firm’s asset is valued more than its debt A_{T}\geqslant K at time T, the debt holders will be paid the full amount K and the shareholders’ equity will be \left(A_{T}-K\right). On the other hand, when the firm fails to repay (therefore defaults on) the debt at T, the debt holders can only recover A_{T}<K and the sharehodlers will get nothing.[ref]Here is sidenote test page.[/ref] The equity value at time T can be represented as an European call option on asset A_{t} with strike price K maturing at T, E_{T}=\max\left(A_{T}-K,0\right). The asset value is assumed to follow a geometric Brownian motion process, with risk-neutral dynamics given

(1)   \begin{equation*} dA_{t}=rA_{t}dt+\sigma_{A}A_{t}dW_{t} \end{equation*}

where r denotes the risk-free interest rate, \sigma_{A} is the volatility of asset’s returns, and W_{t} is a Brownian motion under the risk-neutral measure. Applying Black-Scholes formula would give

    \[ E_{t}=A_{t}\Phi\left(d_{1}\right)-Ke^{-r\left(T-t\right)}\Phi\left(d_{2}\right) \]

where d_{1}=\frac{1}{\sigma_{A}\sqrt{T-t}}\left[\ln\left(\frac{A_{t}}{K}\right)+\left(r+\frac{\sigma_{A}}{2}\right)\left(T-t\right)\right], d_{2} = d_{1}-\sigma_{A}\sqrt{T-t}, and \Phi\left(\cdot\right) denotes the standard normal \textit{cdf}. The probability of default at time T is given by \textrm{P}\left(A_{T}<k\right)=\phi\left(-d_{2}\right). <=”" p=”">

A typical strategy of debt holders to protect themselves from the credit risk is to long a put option P_{t} on A_{t} with strike K maturing at T. The put option will be valued at \left(K-A_{T}\right) if A_{T}<k, and=”" worth=”" nothing=”" if=”" a_{t}="">K. Purchasing the put option guarantees that the credit risk of the loan is hedged completely as the debt holder’s payoff equals K at maturity no matter if the obligor defaults or not. It therefore forms a risk-free position

(2)   \begin{equation*} D_{t}+P_{t}=Ke^{-r\left(T-t\right)}. \end{equation*}

The price of put option P_{t} is determined by applying Black-Scholes formula as

(3)   \begin{equation*} P_{t}=Ke^{-r\left(T-t\right)}\Phi\left(-d_{2}\right)-A_{t}\Phi\left(-d_{1}\right). \end{equation*}

Taking account the credit risk spread (risk premium) s, the value of the risky bond is

(4)   \begin{equation*} D_{t}=Ke^{-\left(r+s\right)\left(T-t\right)}. \end{equation*}

Combining Eq.(2) — (4) gives a closed-form formula for the credit spread

    \[ s=-\frac{1}{T-t}\ln\left[\Phi\left(d_{2}\right)-\frac{A_{t}}{K}e^{r\left(T-t\right)}\Phi\left(-d_{1}\right)\right] \]

where \frac{A_{t}}{K} represents the firm’s leverage. Note that s depends only on A_{t} and \sigma_{A} which is in line with the economic intuition. Their nonlinear relationship can be observed from the below figures.

Many approaches have been proposed to improve the classical Merton’s model. The first passage model introduced by Black and Cox (1976) allows the firm may default at any time before the debt maturity. Jones et al. (1984) suggest to introduce stochastic interest rates to improve the model’s performance. Longsta and Schwartz (1995) employ a Vasicek process for the interest rate, dr_{t}=\left(a-br_{t}\right)dt+\sigma_{t}dW_{t}^{\left(r\right)}, while Kim et al. (1993) consider a CIR process, dr_{t}=\left(a-br_{t}\right)dt+\sigma_{t}\sqrt{r_{t}}dW_{t}^{\left(r\right)}, and Briys and De Varenne (1997) treat the interest rate following a generalized Vasicek process, dr_{t}=\left(a\left(t\right)-\left(t\right)r_{t}\right)dt +\sigma_{t}\left(t\right)dW_{t}^{\left(r\right)}. By comparing the Merton’s model and its four extensions Eom et al. (2004) find substantial spread predication errors that four models underestimate the spread observed from the market while the other one overestimate it.

References

  • Black, F., Cox, J. C., 1976. Valuing Corporate Securities: Some Effects of Bond Indenture Provisions. Journal of Finance 31, 351-367.
  • Black, F., Scholes, M., 1973. The Pricing of Option and Corporate Liabilities. Journal of Political Economy 81, 637-654.
  • Briys, E., De Varenne, F., 1997. Valuing Risky Fixed Rate Debt: An Extension. Journal of Financial and Quantitative Analysis 32 (2).
  • Eom, Y., Helwege, J., Huang, J., 2004. Structural Models of Corporate Bond Pricing: An Empirical Analysis. Review of Financial Studies 17 (2), 499-544.
  • Jones, E., Mason, S., Rosenfeld, E., 1984. Contingent Claims Analysis of Corporate Capital Structures. Journal of Finance 39 (3), 611-625.
  • Kim, I. J., Ramaswamy, K., Sundaresan, S., 1993. Does Default Risk in Coupons Affect the Valuation of Corporate Bonds?: A Contingent Claims Model. Financial Management, 117-131.
  • Longstaff, F. A., Schwartz, E. S., 1995. A Simple Approach to Valuing Risky Fixed and Floating Rate Debt. Journal of Finance 50, 789-819.
  • Merton, R. C., 1974. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates. Journal of Finance 2 (2), 449-470.

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