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In terms of Merton credit risk model need to find the initial value of counterparty's assets and the volatility of the assets. Both value are not directly observable thus we have to approximate them by solving the system of equations, one of which is
$$ \sigma_E E_0 = N(d_1) \sigma_V V_0 $$
I found the derivation of this formula but I couldn't find a good economic explanation behind it - or just what it means?
You could read it like this:
The typical change in equity value is equal to the typical change in asset value, adjusted for the probability of the assets surviving.
Note that the formula is not specific to Merton models, it's also true for regular options and their underlyings. It's just that volatility of option prices isn't typically a concern in "ordinary" cases.
Well, the main intuition of the Merton model is that a company's equity can be treated as a call option on its assets, thus allowing for the application of Black-Scholes option pricing methods. Let's consider a company that has assets $A_{t}$ financed by equity $E_{t}$ and a zero-coupon debt $B_{t}$ with face value K, and maturity T. At time of maturity T, we have:
$$ E_{T} = \begin{cases} A_{T} - K & \text{if } A_{T} > K \\ 0 & \text{if } A_{T} \leq K \end{cases} $$
The reason being is that, at maturity T, debtholders would be paid the full face value K if the company's assets at T are greater than K, leaving to shareholders an equity amount of $A_{T} - K$. However, if $A_{T} \leq K$, then the company would default on its debt payment. In that case, since the debtholders have the first claim on what's left of the company's assets, the equityholders end up with nothing.
Then: $$ E_{T} = max(A_{T} - K, 0)$$
This is exactly the payoff of a call option on $A_{T}$ with a strike price K and maturity T. Therefore, the Black-Scholes option pricing methods can be applied, (assuming the asset value follows a GBM). By assuming that the equity value $E_{T}$ also follows a GBM, and by applying Ito's lemma, you can show that:
$$ \sigma_E E_t = \frac{\partial E_t}{\partial A_t} \sigma_A A_t $$
By substituting B-S call option delta, we obtain: $$ \sigma_E E_t = N(d_1) \sigma_A A_t $$
In your own notation, at time t = 0, you set $A_0 = V_0$, then: $$ \sigma_E E_0 = N(d_1) \sigma_V V_0 $$
The equation stated in the question is not at the core of Merton's credit model, (Not saying you claimed it is) but is a simple device in helping to solve the system of linear equations.
The equation given simply establishes a relationship between the volatility of equity and the volatility of the assets and it follows from the application of Black Scholes that the equity delta equals N(d1).
Please see p5-6 here: http://www.ec.bgu.ac.il/monaster/admin/papers/1202.pdf
The derivation of $$ \sigma_E E_0 = N(d_1) \sigma_V V_0 $$ is actually in Merton (1974).
Equation 3.b is what you are looking for. The only difference is the notation. You can see F as E, and everything else is the same.
Reference:
Merton, R.C., 1974. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates. The Journal of Finance, 29(2), p.449.
