# Why is the value of debt modeled as a short put option in Merton's model?

Can someone give me an intuitive understanding of why the Merton model models the value of the debt from the lender's point of view as a short put with a risk free bond?

I'm not well versed in this so I'd appreciate answers that are not heavy on math; I'm just looking for an intuitive understanding.

• I missed this on as this is off-topic since this site is dedicated to quant-finance professionals who would know the intuition behind it. Since it has been answered, I'll not delete/close it. – SRKX Aug 31 '15 at 10:46

If the company was risk free the lender would always get back the promised amount $L$ at maturity. So the lender would be holding a risk free bond.

But companies are not risk free, there is a chance that they won't be able to repay the full amount $L$. This can be modeled as a risk free bond plus a "thing" which will have negative value if the company defaults and zero value otherwise.

What is this "thing"? If the company value $V$ is less than $L$ at maturity, the company will default and the lenders will take over the company, which they can then sell to partially recover what they were owed.

So, if at maturity $V>L$ the lender loses 0 (no default) while if $V<=L$ the lender loses $L-V$.

If you are familiar with options you will see that the "thing" is identical to a short position in a put option. (You lose nothing if the underlying is above the strike at maturity and you lose (S-K) otherwise).

Here is a very to the point explanation of the Merton model.

• You should extract the answer from the paper and include it in the answer, otherwise this should be a comment. We're trying to have answers that contain as much information as possible in order not to depend on links as much as possible. – SRKX Aug 31 '15 at 10:40

Equity is the residual value of a company like a call on debt: $$\text{Debt}=\text{Assets}-\text{Equity}$$ According to call-put parity: $$S-C=Ke^{-rt}-P$$ The left side corresponds to the company's debt, while the right side is short put and long bond.

Here's an answer short on math, as requested.

First, understand the "risk-free bond" part:

Let's assume there's a magical company that always has a 100% chance of paying their debts, no matter what. Because our magical company has 0% chance of default, lending to them would be identical to a risk-free bond.

Of course, there's no such thing as a company that has no risk.

So to make this a realistic company, the Merton model adds a 'risk component' to that risk-free bond.

That's where the "short put" comes in.

In our "realistic" company, the bond holders have a zero-coupon bond with a par value equal to the company's debt. If this company's assets drop below the value of its debt, the bond holders obviously get less than par value. The most they can get is the total asset value of the company in that case.

Mathematically, for bond holders that's the same thing being short a put option. Why? Because the bond holders don't lose anything if the company's assets are valued higher than its debts. But the bond holders do lose if the assets fall below the value of its debt.