de Servigny - Default Risk Quantitative Methodologies

[brian.field]

I am reading de Servigny's Default Risk Quantitative Methodologies chapter from the GARP printed textbooks.
Page 69 references the Black-Scholes pricing model for the value of a firm's equity under the Merton model.

Interestingly, the formula does not reference d1 yet the expression is immediately followed by a "where d1 = ..." statement.



Does any know if this is a careless GARP error or does the actual de Servigny text also have the same error?

Obviously, we recognize d1 from the Black-Scholes model.

Best,

Brian

 

[brian.field]

In fact, I can't seem to find from where the numbers in the d1 expression came! Perhaps they were presented in Chapter 1 or 2 of the de Servigny text.

 

[brian.field]

Another question here: Page 71 presents the following:

I do not see how the author is coming up with 2.8. Then, he uses the negative in the N() function. This is not presented very clearly.

Assume the following:

V = 3BB
X = 10 BB
mu = 5%
sigma = 9.6%

Do you come up with 2.8?

Also, why does the author change the sign of the 2.8?

 

[ShaktiRathore]

Yes Brian in your first post d1 is not explicity mention in merton version of the BSM. I think the author took d2 as k and mentioned d1 as k+sigma*sqrt(T-t) as relation d1-d2=sigma*sqrt(T-t) holds. Its a silly mistake in terms of using wrong notation.
Thanks

 

[ShaktiRathore]

Yrs Brian the answer do come to 2.8,
In merton model replace everything of equity in Bsm with asset, plug in the values and calculate[ log(12.511/10)+(.05-(.0196^2*.5))]/.0196*1
Vt is the current value of firm's assets=12.511b,X is debt of firm current=10b,mu=expected return of firm=.0,sigma(v) is volatility of firms assets which we assummed as equity'volatility itself because asset=equity+debt and Var(asset)=Var(equity)+Var(debt)+2*corr(debt&equity)*sigma(equity)*sigma(debt)*w(e)*w(debt) because sigma(debt)=0 which is assumed in the model therefore Var(asset)=Var(Equity) or sigma(asset)=sigma(rquity).
What we get in the end is N(-d2)(d2=2.8) which is nothing but the probability of asset price falling below the debt or firm defaulting.The N(d2) is the probability that asset price shall exceed debt thrrefore 1- N(d2) is probability of asset price falling below debt we can write 1-N(d2) as N(-d2) for normal cum probability distribution.Similar to Bsm where N(-d2) is the probability of stock price falling below exercise price X.
Thanks

 

[brian.field]

Thanks Shakti.....perhaps I am being careless myself.

Your expression does not equal 2.8 as far as I can tell. Rather, LOG(12.511/10) + (0.05 - (0.0196^2*0.5)) / 0.0196*1 = 2.6385.

Also, why would the author define V as 3 BB (market cap) and then give Ao = 12.511 BB without defining what Ao is? I assumed V = 3 BB.

Also, where are you getting the 0.0196? Sigma is defined as 0.096 no?

 

[ShaktiRathore]

Yeah sigma should be .096 instead of .0196, now check Brian 2.8 shall come,i have checked it. Vt is denoting A0 only Brian.
Thanks

 

[brian.field]

(Log(12.511/10) + (0.05-(0.096^2)*0.5))/0.096 = 1.49 whereas (LN(12.511/10) + (0.05-(0.096^2)*0.5))/0.096 = 2.8

Isn't that amazing?

The author reports Log but is actually using LN. Another example of outrageous carelessness that wasted a significant amount of time! He could have indicated that the base was e.

 

[ShaktiRathore]

Yes thats very careless in the author' side i had taken the base as e only. And in these merton and bsm we usually use e as bade for log terms.
Thanks

 

[brian.field]

Thanks for the help Shakti!

Brian

 

[brian.field]

Just realized after looking at the Study Notes for de Servigny.

The Distance to Default is d2 from the B-S model and the Merton PD is N(-d2) which explains why the 2.8's sign switches in N(-2.8) above.

 

[ShaktiRathore]

Brian you can get how sign really changes by visiting: https://forum.bionicturtle.com/threads/merton-model-a-summary-of-the-issues.5646/
Or watch video:

Thanks

 

[David Harper CFA FRM]

Hi @brianhfield

I am confused by the GARP books because here is the screenshot from the source De Servigny (kindle version):

Here is the sheet from my XLS, which can be opened here at @ https://www.dropbox.com/s/cxogqan42xkbrtj/0224-default-ch3-actual.xlsx?dl=0

Please notice:

• The first column is my (longstanding) assumption of firm value, V(0) = $12.75 in order to make the DD = 3.0 per the reading. Then PD = 0.13%
• The second column uses V(0) = $12.511 which returns PD = 0.25% (only because mine does not round the DD to 2.8).
  
• This shows that both use the basic Merton model correctly, FWIW