# Put-call parity for equity share and debt share

Considering Merton's structural approach" for credit risk modeling, we arrive to prove that the pricing formules are $$S_t=V_t\phi(d_{T,1})-Fe^{-r(T-t)}\phi(d_{T,2})$$ for equity share and $$F_t=FP_0(t,T)-P^{BS}$$ for debt share, with

1. $$d_{T,1}=\frac{ln(\frac{F}{V_0})-(\mu_V-\frac{\sigma_V^{2}}{2})T}{\sigma_V\sqrt{T}}$$
2. $$d_{T,2}=d_{T,1}-\sigma_V\sqrt{T}$$
3. $$V_t:=S_t+F_t$$ the value of company's portfolio, with $$S_t$$ the share of financing on equity market for the company and $$F_t$$ the share of financing on debt market for the company.
4. $$P_0(t,T):=\mathbb{E}^{\mathbb{Q}}[e^{-\int_{t}^{T}r_sds}|F_t]$$ the price of no-defaultable bond under neutrality measure assuming the existence of $$\mathbb{Q}$$-equivalent martingale.
5. $$\frac{dV_t}{V_t}=\mu_Vdt+\sigma_VdW_t^{\mathbb{P}}$$ the MBG that describes the dynamics of portfolio with solution $$V_T=V_0e^{(\mu_V-\frac{\sigma_V^2}{2})T+\sigma_VW_T}$$
6. $$S_T:=max{(V_T,F)}:=\begin{Bmatrix} 0 & ifV_T=F \end{Bmatrix}$$ and $$F_T:=min{(V_T,F)}:=\begin{Bmatrix} F_T & ifV_T=F \end{Bmatrix}:=F-(F-V_T)^+$$, so equity and debt share present the same structure of a european call and put.

Now, professor says that applying put-call parity we obtain that $$F_t=Fe^{-r(T_t)}\phi(d_{T,2})+V_t\phi(-d_{T,1})$$. Look below, please: Really, I don't understand where he got that put-call parity, given that the put-call parity that I know is (for B&S) $$c+Ke^{-r(T-t)}=p+S_0$$.

I'm really hoping you can help me. Thanks in advance!

• Why not make an appointment and talk with your prof? The handwriting is really not something people can easily understand. – Gordon Jun 28 '19 at 13:20
• @Gordon Because the exam is next Monday. Can you help me to understand those passagges? – Marco Pittella Jun 28 '19 at 13:34
• I think you're more likely to attract a good answer if you reduce your question to the minimum needed to understand your question (so just the two or three lines that you don't get and then put those in as equations rather than as screenshot that probably nobody will care to decipher). What probably you're probably looking for is that the value of a position consisting of a long call and a short put (with the same strike) is the same as the value of a forward contract with the same strike (which you can easily derive from the one you mentioned in your post) – Bram Jun 28 '19 at 21:54