Credit Analysis: Structural Models and Option Analogy

So I’ve done a lot of searching and looking for examples b/c I was really stuck on this concept of “investing in a company’s debt is similar to buying a zero coupon risk bond and selling a put option”. I have come up with my own example to interpret this mess of an analogy and I was wondering if someone would advise if this is correct:

Value of Debt = Min(A,K)

A = value of the asset

K= face value of the risky bond

Value of Mortgage/Debt = Min (House, Loan) Bank perspective: I lend money to homebuyer(company) to purchase this asset(home). The home is financed by 400k to buy a 400k dollar home. *recession hits, and home values plummet*. The home is now only worth 200k. The homebuyer chooses to stop making payments b/c in their perspective, “why should I continue making payments on a loan when the value of my house is less than that? if I sold my home today for 200k, I still won’t have enough to pay back the 400k that I still owe the bank. So eff it, I’mma just stop making payments and default on the loan”. The bank then realizes that their loan is not being paid and the value of that mortgage drops. To salvage whatever value is left, they will repossess the home as collateral, and sell it for 200k, which is basically what the value of debt is worth at this point.

To conclude, the Bank lost money on the mortgage (aka “risky bond”) and the put option in the analogy is simply there to simulate a loss when the value falls below face value (K).

That isn’t how I think of it. What is the difference between a corporate bond and a risk-free bond? You get a little extra yield each period in exchange for the small-ish risk that the whole thing could go south and you will take a beating. That is just like selling a put. You get a little premium each period in exchange for the small-ish risk that the underlying will go south and you will take a beating. Conceptually, I think the analogy makes sense.