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PUT option on the Assets Value + Loan = Risk-free bond. Then under no arbitrage condition:

$$ \large{ P_0 + B_0 = F e^{-r_fT} } $$

The PUT option price:

$$ \large{ P_0 = F e^{-r_fT} N(-d_2) - V_0 N(-d_1) } $$
$$ \kern 9pc \large{ d_1 = {ln(V_0 / F) + (r_f + {1 \over 2} \sigma_{V}) \over \sigma_{V} \sqrt{T}}, \kern 2pc d_2 = d_1 - \sigma_{V} \sqrt{T} } $$

The loan value:

Equilibrium interest rate on the Loan, \( r^* \):

Credit spread:

Probability of Default (using normal distribution of assets value assumption):