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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 $P_1(t,T)=\frac{F_t}{F}$?

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