Put–call parity and the value of the firm

In short

• Firm value equals equity plus debt: $V_0 = E_0 + \text{debt value}$. At maturity T the debt's face value D is due.
• If $V_T > D$ (solvent), debtholders get D and shareholders get $V_T - D$. If $V_T < D$ (insolvent), debtholders get $V_T$ and shareholders get 0.
• Shareholder payoff $\max(0, V_T - D)$ is a call on firm value with exercise price D: unlimited upside, limited downside.
• Debtholder payoff $\min(V_T, D) = D - \max(0, D - V_T)$ is risk-free debt plus a sold put on firm value: upside capped at D.
• Parity gives $V_0 = c_0 + PV(D) - p_0$. The put's value reflects the credit spread; it rises as insolvency becomes more likely.