Let $$c(t,T):=-\frac{1}{T-t}[\mathrm{ln}(P_1(t,T))-\mathrm{ln}(P_0(t,T))]$$, with:

• $$c$$ measure of how a company is prone to fail;

• $$P_0(t,T):=e^{-r(T-t)}$$ price of no-defaultable bond.

• $$P_1(t,T):=\mathbb{I}_{(\tau>t)}e^{-\int_{t}^{T}R_sds}$$ price of defaultable bond, where $$R_s:=r_s+\gamma_s$$.
• $$\gamma_s:=\frac{f(t)}{\bar{F}(t)}:=\lim_{h\rightarrow 0}\frac{P(\tau\leq t+h|\tau>t)}{h}$$ hazard rate, where $$\tau$$ is a generical instant default of defaultable bond on $$(\Omega, F, {F_{t}}_{(t\geq 0)},\mathbb{P})$$ such that $${(\tau\leq t)}\in F_{t}$$.
• $$F(t):=P(\tau \leq t)$$ the CDF of a generical instant default.
• $$\bar{F}(t):=1-P(\tau>t):=1-F(t)=e^{-\int_{0}^{t}\gamma_sds}$$ the complement of $$F(t)$$.
• $$P(\tau>T|\mathfrak{H}_t):=\mathbb{I}_{(\tau\geq T)}e^{-\int_{t}^{T}\gamma_sds}$$ the survivale rate of defaultable bond over the maturity $$T$$.

Then let:

• $$V_t:=S_t+F_t$$ the portfolio's company, evaluated in $$t$$, that invests in stocks and bonds.
• $$F_t:=Fe^{-r(T-t)}\phi({d(t,2)})+V_t\phi({-d(t,1)})$$ the debt quote value of portfolio's company that acts like a put option with payoff $$F_T=F-(F-V_T)^{+}$$
• $$S_t:=V_t\phi({d(t,1)})-Fe^{-r(T-t)}\phi({d(t,1)})$$the equity quote value of portfolio's company that acts like a call option with payoff $$S_T=(V_T-F)^{+}$$.

I ask you: why it's possibile to observe that $$P_1(t,T)=\frac{F_t}{F}$$?