Skip to content
All library documents

Valuing Equity as a Call in Merton and Black–Cox Credit Models

Article Quant Q&A · Author: febstar
Read the original

Summary

The document clarifies how to represent a firm’s equity when simulating its asset value under structural credit models. In the Merton model, firm assets follow a geometric Brownian motion and equity is a call option on those assets, with debt represented by the face value of a zero-coupon bond minus the value of a put. This means equity value is not simply the positive part of assets minus a fixed debt amount at every time before maturity; the option valuation reflects uncertainty about the firm’s value at the debt maturity.

Black–Cox extends the setup by allowing default before the debt matures. Equity is wiped out if the default barrier is reached; otherwise its value reflects the possibility of surviving to maturity. The response gives a verbal distinction between the models and points to a reference, but does not provide the full Black–Cox valuation formula, specify barrier assumptions, or discuss parameter calibration. These model-based valuations rely on structural assumptions and should not be confused with a direct accounting identity applied to simulated asset paths.

Key ideas

  • In the Merton model, equity can be valued as a call option on the firm’s assets.
  • Firm assets are modeled as following geometric Brownian motion in the described setup.
  • The debt value is linked to a risk-free zero-coupon bond and a put on firm assets.
  • Black–Cox allows default before debt maturity, unlike the maturity-only default feature described for Merton.
  • The explanation omits detailed barrier formulas and calibration considerations.

Tags

EquitiesFixed incomeOptionsDerivatives pricingRisk management
Full text
# Model the share price under the Merton Credit model


# 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.

## Answer by Forrest (score 1)

https://quant.stackexchange.com/a/37508

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.

Reference: https://www.fields.utoronto.ca/programs/scientific/09-10/finance/courses/hurdnotes2.pdf

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.