Debt and Equity Valuation using Option Pricing

Debt and Equity Valuation using Option Pricing:

Here, I will talk about how to value Debt and Equity of the firm using Option Pricing formula and the heart of Option Pricing Put-Call Parity.

Asset of the firm can be defined as the total value of its Debts and Equity.
Hence, Assets = Debt + Equity ---------------(1)

Using Put Call Parity from Black Scholes and Merton,
Call Option + PV (Strike) = Underlying + Put Option (Strike)  -----------(2)

Now, PV(Strike) can be thought of a Risk Free Bond that has the same face value as Strike price so you are confident that you have the money K at expiration. K is thought of the total DEBT value that firms owes to its debt holders at maturity. Underlying in this scenario can be treated as the assets of the firm itself. So, equation (2) can be re-written as:

Call + Risk Free Bond = Asset + Put Option --------------------(3)
From Equation (1), Debt = Asset - Equity --------(4)
Equity holders are nothing but Call holder with the strike as K as once the Debts are paid off, Equity holders own everything in the firm.
So, Debt = Asset - Call Option -------------------(5)

Using Call valuation from Equation 3, we can re-write, equation 5 as:
Debt = Asset - (Asset + Put Option - Risk Free Bond)

Debt = Risk Free Bond - Put Option ----------------------(6)

Which makes sense also, as Debt holders have their payoff maxed at K if firm value >= K else they own the assets which should be K - Put Pay off

Below image might make this intuitive sense clearer.

Similarly, Equity Holders can be valued as:
Equity = Asset + Put Option - Risk Free Bond --------------------------(7)

Intuitively, you can think of Equity Holders own the Asset and Put option to sell the assets after paying off the Debts (Risk Free Bond).

Nitin