Deriving Equity Volatility in the Merton Structural Credit Model

Summary

The explanation derives the relationship between a company’s equity volatility and the volatility of its assets in the Merton structural credit model. It starts from the model’s interpretation of equity as a call option on firm assets, with liabilities acting as the strike. The option’s sensitivity to asset value is its delta, given by the standard normal cumulative probability term. Applying this sensitivity to a small change in asset value gives the change in equity value.

Dividing by equity value and expressing the asset change as a proportional return yields the equity return exposure: asset value divided by equity value, multiplied by the option delta. Multiplying that exposure by asset volatility gives the stated equity volatility formula. The derivation relies on the option-pricing setup and a diffusion-style asset variance assumption; it is a model relationship, not a direct empirical estimate of default probability. The document does not discuss calibration or departures from the model’s assumptions.

Key ideas

• The Merton model treats equity as a call option on firm assets with liabilities as the strike.
• The call option delta measures how equity value changes when asset value changes.
• Relative equity volatility scales asset volatility by asset value divided by equity value and by the option delta.
• The derivation assumes the structural model’s option-pricing and asset-process setup.

Tags

# Estimation of Default Probability using Merton's model


# Estimation of Default Probability using Merton's model












There is an explanation of `Risk Neutral Default Probability` using a Firm's Equity price here - https://www.mathworks.com/help/risk/default-probability-using-the-merton-model-for-structural-credit-risk.html.

However, there is an equation which states that

$\sigma_E = \frac{A}{E} N \left(d_1\right) \sigma_A$

How is this formula derived?

## Answer by Kermittfrog (score 4, accepted)

https://quant.stackexchange.com/a/59096

As you see in the third equation on that Mathworks page, the Merton model postulates that the value of equity equals the value on a residual claim on a company's assets after the creditor has been repaid. Economically speaking, equity is a call option on the asset value $A$ with strike price equal to the liability $L$, the formula for which is

$$ E=AN(d_1)-Le^{-rT}N(d_2) $$

We further note that the variance of the asset process is (with a bit of handwaverianism)

$$\sigma^2\left(\frac{dA_t}{A_t}\right)\equiv \sigma_a^2dt$$

Finally, we know for a call option that $ \frac{\partial E}{\partial A}=N(d_1)$ which is also colloquially called Delta. Thus

\begin{align} E&=AN(d_1)-Le^{-rT}N(d_2) \\ \Rightarrow dE&=N(d_1)dA\\ \Rightarrow \frac{dE}{E}&=\frac{1}{E}N(d_1)dA\\ \Rightarrow \frac{dE}{E}&=\frac{A}{E}N(d_1)\frac{dA}{A} \end{align}

and ultimately

$$ \sigma_E\equiv \sigma\left(\frac{dE}{E}\right)=\frac{A}{E}N(d_1)\sigma\left(\frac{dA}{A}\right)=\frac{A}{E}N(d_1)\sigma_A $$

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.