Structural models of corporate credit risk - the option analogy

Hi folks,

Need some help on the option analogy for structural model of credit risk.

Here is the text:

Two questions:

  1. As per (1) in the image above, why are shareholders long a put option with an exercise price of K? And why is default equivalent to exercising the put option? If there is a default, value of assets is less than K, so shareholders will have no residual value - so why is default synonymous with shareholders exercising the put option and receiving a value of K?

  2. Under the put option analogy, why can investors in risky debt can be construed to have a
    long position in risk-free debt and a short position in that put option ?

Thanks so much folks.

Think of a compnay with no debt. and assets = A
As an equity holder you are exposed all the upside of assets and all the down side.

Now introduce debt a company with K debt and assets A
You exposure as an equity investor is positive to asset prices above K, and below K you receive zero.
So your payoff is max (0, A -K)
This is like a call option with strike K intrinsic value = max (0. S - X)
Legally as the equity holder you own all the assets (like owning a share)
But below K you have no exposure
You are not taking the exposure to dwonside here the debt holders have the exposre
Debt exposed to zero to K
Equity shareholders K to A
This is like a protective put
I own a share price 50. I buy a buy put strike 50
Above 50 I have all the positive exposure
Below 50 I exercise a put and the writer takes the hit.
Sp when I own equity in a compnay I know my exposure to is from K upwards, If the assets of company fal below K they are someone elses.

With the debt. The most I can get is “K” and I am exposred to any downside below K. If assets drop below K the debt holders are taking the hit.

You can think of interest above the risk free rate as the premium - equity holders pay a premium to debt holders to bear the risk below “K”

Drawign the diagrams will help here. Don’t look at someone elses diagram draw your own.
Split the page two halves side by side.
Draw a protective put, Draw what it looks like to be a shareholder
Similar for bond and wrtiing a put.

Hi @MikeyF ,

Thank you, this is very helpful. I fully understand (2) now.

Just to further clarify on (1) - in the diagram below, we are saying that for A < K, the payoff to shareholders is zero.

But if shareholders exercise the put option to get a payment of K from debt investors, how do we get a net payoff of zero for shareholders when A < K?

It is just of a way of framing what we are looking at.

Assume a company currently has “A” of Assets and 50 of debt. The value of the assets can vary.

Just thinking of the assets
Share holders own the assets and have exposure from 0 to A

But there is debt
Debt have a claim on the assts between 0 and 50

So the net shareholders have net exposure from 50 to A

You could think of this as the shareholders having a put on the assets at 50. Negative exposure of the assets below 50 is the concern of the debtholders. 50 to A is the shareholders upside.

Think about buying a share on the stock market at 50.
And buy a put with strike of 50.

You have no exposure to the downside below 50. The downside there is whoever wrote the put.

Above 50 you have positive exposure to the shareprice.

You need to step away a little from the precise derivatives definitions and jsut look at the types of payoff each hodler gets. No one is “exercising the put” it is more the shape of the payoffs you are looking at.

Thank you @MikeyF , I appreciate your help