Merton formula

[shanlane]

Hello,

I am looking over the notes and in De Serv Ch 3. The k (or d) in the version of the Merton Model you show on slide 9 of video 6c does not appear at all in the chapter. I believe this is also the version that you have in the notes. I am looking at p 66 of the de Serv chapter. I believe the version that you use in the notes is from Stulz, but in problems, you seem to use a drift or average return term in the calculation for d, so I am nost sure if the Stulz version is ever really used. Is it?

Thanks!
Shannon

 

[David Harper CFA FRM]

Hi Shannon,

Yes, correct, I currently have the Stulz notation on p.9 (only because Stulz actually contains more material related to the AIM; the AIM actually more refers to Stulz than de Servign, GARP is loose that way; i.e., de Servigny Ch 3 does not show how to "calculate the value of a firm's debt and equity" ... it goes straight to Merton PD. The AIM is actually pointing to Stulz)

But, it doesn't really matter: It's all the same Black-Scholes to solve for value of the equity as call option (with risk free rate per risk neutral). I think i may, for the upcoming 2012, replace the page 9 formula with the Hull-type BSM.

Then the Merton PD (de Servigny page 69) is not BSM, but uses the drift/expected return. Thanks,

 

[shanlane]

If I use Black Scholes with two different "d" terms I will get two different answers. In spreadsheet 6c1, even whe you say you are using the "Stulz method" you are not using the "d" that you use in your notes.

My whole point is: when do we use the Stulz expression for "d"? You never seem to use it, but its the only version of it that you show in the notes.

Sorry to be a nag, but this is really starting to drive me crazy.

Thanks again!

Shannon

 

[David Harper CFA FRM]

Hi Shannon,

Stulz d is the same as d1 in BSM (although please keep i mind this refers to using option pricing to derive the value of equity as a call option, so it uses the risk free rate, instead of the expected return which is used in the non-option-pricing Merton PD; a forum search will show dozens of explains on this nuance. I just don't want you to let the detail around d1 or d confuse you into thinking this is the Merton PD in de Servigny, right? Merton PD is not using the risk free rate, is using expected return b/c it's not option pricing in PD, it's physical future tail estimation):

Stulz d = ln(V/P(T)F)/[sigma*SQRT(T)] + 1/2*sigma*SQRT(T)
But P(T)F is just the discounted debt face value, the discounted strike price = K*exp(-rt), so:
Stulz d = ln[V/K*exp(-rT)]/[sigma*SQRT(T)] + 1/2*sigma*SQRT(T) =
Stulz d = ln[V/K*exp(-rT)]/[sigma*SQRT(T)] + (1/2*sigma^2*T)/[sigma*SQRT(T)] =
Stulz d = ln[V/K*exp(-rT)] + (1/2*sigma^2*T) /[sigma*SQRT(T)], since LN[V/K*exp(-rT)] = LN(V/K) *rT,
Stulz d = ln(V/K) + rT + (1/2*sigma^2*T) /[sigma*SQRT(T)] = ln(V/K) + T*[r + (1/2*sigma^2)] /[sigma*SQRT(T)] = d1

I'm really really in rush between video productions, so feel free to check my parens (the above i don't even have time to check) but I know they are the same, so if i have a mistake an assumption that the final is correct (i.e., Stulz d = Hull BSM d1) can be used to fix anything in between. Thanks,

 

[shanlane]

That was awesome. I feel embarassed that I did not derive that myself.

Thanks!
Shannon

 

[David Harper CFA FRM]

I had to do it before but only b/c some previous customer forced me too (it was that or admit an error, i'm fine to admit an error, but i'd much rather get lucky). Some people don't understand why i spend so much time on the forum, but the thing is that all of the interesting stuff I learned from the forum in the Q&A the forum isn't as much for me to explain as is for me to learn, Without asking questions the retention is very shallow; I'll read something, think i understand, but i don't really understand until somebody talks about, is my experience

 

[shanlane]

Well, you are doing a great job. I always learn more from a conversation thatn I do just from reading. Sometimes, asking what I consider a dumb question leads to some really interesting results.

Thank you so much for helping me out!

Shannon

 

[Arnaudc]

Good morning all,

I am currently reviewing my notes on "Credit Counterparty risk" by Allan Malz and more specifically the AIM: "Describe the Merton model, and use it to calculate the value of a firm, the values of Debt / Equity.

Thanks to David, I could already calrify one misunderstanding I had: How do we know when to use Return on Asset or Risk Free rate?
I understood from the post above to use rf for pricing of options (value of Equity (call) and value of put (to be used to find debt value) )
and use Return on assets only to find the "physical" PD (risk-neutral PD is found using rf)

I have a remaining question mark: What is the Strike to be used in all these formulas:
- Call / Put options
- PD

Assume we have a bond par value 100 with one coupon remaining of 6. This is essentially a ZC bond with par value of 106.
Assuming risk free = 5%, the PV of this ZC = 106* exp (-5%) = 100.8303.
So, my question (to recap), Should we use 106 or 100.8303 as "K" in the various formula's?

Thank you very much to anyone who could clarify this for me

Kind regards,

 

[David Harper CFA FRM]

Hi @Arnaudc Please see my note (Merton model, a summary of the issues) at https://forum.bionicturtle.com/threads/merton-model-a-summary-of-the-issues.5646/ It goes into much detail.

With respect to your question about the strike price: the first step in Merton uses option pricing (the BSM, with its risk-free rate, as you noted) to infer the current market value and volatility of the firm's assets. It does this by assuming the equity is akin to a call option on the firm's assets: the assets are owned if the debt is retired. My model (therefore) uses the principal amount; in your example, that would be 100 rather than 106 or 100.83 for this step. Note that Malz also assumes the nominal value of the firm's debt, his variable (D). For the second step, which is not option pricing but a forward-looking (aka, "actuarial" in Malz' terms) estimation, here we have more judgement flexibility. Malz uses the final cash flow due (in your example, 106); my model, following most others that I am aware of, has always used the par (in your example, 100). In practice, per de Servigny, the critical issue here is, what's the correct default point? KMV (who's founders pioneered this procedure) estimated the default point somewhere between current and total liabilities. I hope that helps!

 

[Arnaudc]

@David Harper CFA FRM ,

Once again, many thanks for your explanation.
Regarding the par value, I would like to challenge you a bit...

Merton assumes a ZC bond in its model which is why bond's par value should be used as this is the terminal value of the debt and on debt's maturity, this is the value that have to be paid to bondholders (with remaining going to shareholders).
This is why I believe in case of a 1y Coupon bond, the terminal value of the debt should be par + coupon (if the coupon is not paid because firm's asset value is not enough, do we believe shareholders would get any value?)
I think Malz's value for the first step is D=106 (I performed the calculation and this is how I get the exact value of Equity)
Actually, this value of strike of 106 (par + coupon, which is in my opinion the future value of the debt, which needs to be retired before shareholders could have any claim on the firm's assets) seems to be consistently used in Malz paper. both for Firm's value estimation and risk-neutral or actuarial PD.

EDIT:
I see where my confusion came from for the 1st question.
In Stulz paper, he uses the PV of the debt par value as strike...
This is a bit confusing to have these divergences.. At the exam, which method should we use? Currently I see 3 ways of calculating the "strike":
- Par value (including coupon, imo)
- Debt present value
- KMV approach (with mix of ST and LT)

Thank you again for your time!

Kind regards,

 

[David Harper CFA FRM]

Hi @Arnaudc Okay, right, as I actually look at Malz currently, you are correct about his assumption in the first step (page 219). Sorry to misrepresent him on that, thank you for pushing back, I learned something here! (fwiw, my key influence is https://www.dropbox.com/s/t8ajnm8zlu3og4o/Crosbie-Peter-MoodysKMV.pdf?dl=0)

Malz page 219:
"Example 6.3 (Merton Model): We apply the model to a firm that has an asset value of $140. We’ll assume the firm’s sole debt issue is a bond, with one 6 percent coupon left, to be paid in one year along with the principal at the maturity of the bond. This is effectively a zero-coupon bond with a par value of 106, but looking at the debt this way conveniently centers current debt prices near 100 percent of par.