关于merton模型的求助

merton模型的变量与公司股票和债券价值关系：
                       value of firm   face value of debt    time to maturity   interest rate   volatility of the firm
value of debt             +                      +                        -                        -                      -
value ofequtiy           +                       -                        +                       +                     +
推导过程是什么？重点是对于看跌期权的空头罗马字母数值的正负号如何得出。
跪谢。

Under Merton's framework, the market value of the firm’s underlying assets follows the following stochastic process:

dV(A) =μ V(A)  dt+σ(A)  V(A)  dz

where
V(A)  ,dV(A)  are the firm’s asset value and change in asset value,
μ ,σ(A)  are the firm’s asset value drift rate and volatility, and
dz is a Wiener process.

The BS model allows only two types of liabilities, a single class of debt and a single class of equity, so

V(D) = V(A) - V(E)

If X is the book value of the debt which is due at time T then the market value of equity and the market value of assets are related by the
following expression:

V(E)=V(A)*N( d1) −exp(−rT)* X*N( d2)

V(E) is the market value of the firm’s equity,
d1 is similar to that in B-S,
d2=d1−σ(A)* T^(1/2) , and
r is the risk free interest rate.

From above equations, you can see the relationship between variables in your question.