# Model the share price under the Merton Credit model

The project I'm working on requires me to model the share price of a firm through time using the Merton and Black-Cox credit models. The model is used here to induce the leverage effect in the share price.

I was initially simulating the share price path using a GBM for the value of assets, $A_t$, and simply equating the equity value, $E_t$ to $\max(0,A_t-D)$, where $D$ is the debt value. The rationale here is that this respects the accounting equation: $A_t = E_t + D$.

Now, I'm not so sure about this and was wondering if it is more correct to instead use $$E_t = BScall(V_t,D,...)$$ for $0\leq t \leq T$.

Would this be any different for the Black-Cox model (which allows for default prior to $T$).

Any references would be appreciated.

## 1 Answer

Your latter statement is correct. Under the Merton model, Firm Value (FV) = Value of Equity + Value of Debt. The percentage changes in FV are then assumed to be GBM. So, the value of equity will be the Black-Scholes call price. And the value of debt will be the face value of a zero coupon bond minus the Black-Scholes put price.

Black-Cox is an extension of the Merton model by allowing for default before maturity of the zero coupon bond. Whereas the Merton model only allows for default at maturity. So under this model the value of equity is (1) zero if default occurs at or before maturity, (2) a call on the firm's value multiplied by the probability of no default if there is no default.

• Thank you. I now understand the Merton credit model case. But why is the value of equity under the Black-Cox model equal to the firm's value multiplied by the probability of no default in case 2? I couldn't see any reasoning for that in the reference. – febstar Dec 27 '17 at 10:20
• This follows from the pricing of a defaultable zero coupon bond. Equity will be Firm Value - Value of Debt, and when simplified the probability remains from the Value of Debt portion. This is equation 4.13 in the reference, but as you note, it is not well explained. Value of Debt = $D_{T} = (F - max(F - V_{T}, 0)) * 1_{\tau_H > T} + V_{T} * 1_{\tau_H \le T}$ Value of Equity = $V_{T} - D_{T} = max(V_{T} - F, 0) * 1_{\tau_H > T}$ Where F is the face value of the zero coupon bond. And $\tau_H$ is the premature default time. – Forrest Dec 27 '17 at 19:31
• Thanks once again. I think I follow what you're saying, but not sure if I can agree with the statement that $E_t = call(V_t,F,t,T)\mathbb{P}(\text{no default})$. Would we not be pricing $E_t$ as a down-and-out call option (as mentioned in the reference just below eqn 4.13)? – febstar Dec 28 '17 at 11:09