Continuous Time Merton
Overview
The continuous time merton model formulates the dynamics of the firms value using the language of
stochastic calculus.
Pricing Credit Risk
The dynamics of the value of the assets of the firm is stated as an
Ito Process.
{% d A(t) = \mu A(t) dt + \sigma A(t) dW(t) %}
An application of the
Girsanov Theorem
can transform the equation under the
risk neutral measure.
{% d A(t) = r A(t) dt + \sigma A(t) dW_Q(t) %}
Recalling the
Black Scholes Formula
for a call option
{% C(A_t, t) = N(d_1)A_t - N(d_2)Fe^{-r(T)} %}
Where F is the time T value of the debt.
{% D(0) = A_0 N(-d_1) + e^{-rT} F N(d_2) %}
Notice that this is not just the discounted value of the "future value" of the debt, F.
The difference represents the price of credit risk that the market demands.
