Merton model for calculating debt value

[sridhar]

Hi David,

I am watching your Credit A, Part 1 episode 7....I think I understand the Merton model for valuing the debt of a firm. I've also understood the mechanics of working the associated EditGrid that you've. However, I am missing the so-what aspect?

I know it is dancing in front of my eyes -- but how do I compute the PD using the results of computing the call price and the put price? In other words, how do I interpret the results in the context of computing the PD? To make things more concrete, I am referring to the results of the example on Slide 56.

In contrast, I think I do understand how you arrive at the PD (using the distance to default metric) using the KMV model.

--sridhar

 

[David Harper CFA FRM]

Hi sridhar,

It is a very *strong* observation. (I wish i had explained this better in the tut): Gets right to the heart of the difference between Merton model and KMV's adaptation of the same.

Slide 56 is using Merton model to infer equity value, which along with equity volatility is needed for the KMV approach.
Slide 56 employs risk neutral valuation principle; i.e., riskfree rate can be used to price the option.
But it will *fail* to produce a correct PD. The risk neutral idea does not extend to the PD! Trying to infer PD from slide 56, using a RF rate, will give an entirely too high implied PD (i.e., N(d2)).

If you take slide 56 and replace RF rate with expected return, a higher number, then N(d2) or N(-d2) will return a Merton model implied probability of default.

That's the difference which i think may resolve your observation: it is okay to use riskless rate for the option value (per risk neutral valuation), but if we are using Black-Scholes to compute a PD, then really we are using N(d2) and we must at least replace riskless rate with expected asset return
(note: N(d2) is the probability the Euro call option will be struck, so N(d2) in Merton is probability the firm value will breach the debt level)

I am glad you agree the KMV is intuitive...just to emphasize the point, note the KMV approach could have used Merton directly by replacing RF rate with expected asset return. Then N(d2) or N(-d2) gives the PD. Why don't they just cut to that; i.e., use N(d2) for the EDF? B/c they don't want to agree with the lognormal (parameteric) distribution, they just want to "use it" to compare DD to other firms and based the PD/EDF on historical (empirical) actual patterns. Hope that resolves!

David

append: just another way to look at this re: KMV approach. The first step in KMV is to use the merton model to infer firm value and volatility. In my view, that is the only option pricing involved. After that is done, the subsequent steps (i.e., compute DD and map DD to EDF) are not option pricing and, therefore, do not invoke the risk neutral valuation principle. Steps 2 and 3 in KMV are straightforward intuition: grow the firm value, compute DD, etc.

 

[David Harper CFA FRM]

.... I wanted to illustrate, as this is such a rich idea (I will include d2 in the cram).
Here is d2 in the Black-Scholes:

http://learn.bionicturtle.com/images/forum/d2inblacks.png

It is worth understanding d2, specifically:

if r = return on firm asset, then d2 is KMV distance to default.
In this case, (r=growth in firm asset), this is not option pricing. It can be intuitively seen.
ln(S/K) is the return, above default, already implicit in the capital structure, plus add the growth (r) which is eroded by volatility (per a "geometric expected return"). Convert this total growth (i.e., already in LN(S/K) plus the expected) into DD by dividing by scaled volatility. Per slides 50+, this is not option pricing, every explain of KMV/Merton diagrams this out, up with asset, down to the default...

Further, if r = return on asset, then N(d2) is the merton implied PD.

Now, change r to riskless rate: r = riskfree rate.
And, suddenly, we are in the parallel risk-neutral universe. N(d2) becomes the probability the option will strike only in this world. By convenience, we can use N(d2) here to value the option and transfer to real world. But N(d2) where r=riskfree is not useful to us
(except if r happens to be the expected return on the asset, which Hull often "cheats" with this assumption by assuming the asset has no systematic risk, but it is another matter).
okay, so it is maybe a long story, but rich with themes...David

 

[sridhar]

Great stuff David...I need to absorb your explanation...I am sure it will sink in, but as usual you've provided an authoritative response....I'll reach back to you, if the fog has not lifted....

On an unrelated note, on slide 46 of the Credit Risk A slides, you've....

Probability of default (from previous) = 97.25%

I think the red should be "Probability of repayment...." because in slide 44, the PD, i.e. 1 - p is 2.75%

--sridhar