Merton Model & Distance to Default - Which formula to use

[trabala38]

Hi everyone, hi David,

I am reviewing the Merton Model and the Distance-to-Default concept. However, I am a little lost when it comes to formula. Especially, when reviewing BT T6 Notes, it is written on p.64:

S(V,F,T,t) = V * N(d) - Pt (T) * F * N(d- sigma*sqrt(T-t)).

where d=(ln(V/(Pt*F))/(sigma*(sqrt(T-t)) + 0,5 * sigma * sqrt (T-t).

On p 66, I can read:

PD=N(- (log(Vt/K) + (u-0,5*sigma^2)*(T-))/(sigma*sqrt(T-t))

On p.82, I can read:

d2* = (LN(V0/Fdpt) + (u - 0,5*sigma^2)*T)/(sigma * sqrt(T)

My 1st question is: why does the 1st formula (cf. o 64) does not use the drift rate (u ;mu) as an input? Is it a simplification or is there a specific reason? For standard equity option, the risk-free rate is used as an input (risk-neutral valuation), but not here, why?

My 2nd question is: I have been trying to re-price the equity value in Learning Spreadsheet 7.b.1, but I got slightly different results (4,02 instead of 3,98). I am wondering which is wrong in my inputs... Should I compute d1 and d2 with the drift rate? Or without the drift rate? Or with the risk-free rate?

Thanks,

trabala38

 

[David Harper CFA FRM]

Hi trabala,

I covered the difference in great detail here @ http://forum.bionicturtle.com/threads/merton-model-a-summary-of-the-issues.5646/
i.e., in pricing equity as a call option, present value pricing of the derivative (no arbitrage contingent claim) justifies risk free rate.
However, PD = N(-DD) is not pricing, it's future distribution estimation, requiring "real-world" asset drift.

Feel free to append/query on that thread, although I think you'll see why pricing the equity (in BSM) wants risk-free and PD wants expected return.

Thanks,

 

[trabala38]

Thanks David, it is exactly what I was looking for!

Cheers,

trabala38

 

[trabala38]

Hi David,

Well, in addition to your post, since I was still troubled by the absence of risk-free in the formula stated on p. 64 of BT notes (referring in fact to Stulz Chapt. 18).

In fact, there is a risk-free interest rate, but it is embedded in the discounted face value fo debt (cf p. 64 BT notes):

Initial formula => d=(ln(V/(Pt*F))/(sigma*(sqrt(T-t)) + 0,5 * sigma * sqrt (T-t).

=> d=(ln(V) - ln(F) - ln(Pt) + 0,5 * sigma^2 * (T-t)) / (sigma * sqrt (T-t))
=> d=(ln(V) - ln(F) - ln(e^-r*(T-t)) + 0,5 * sigma^2 * (T-t)) / (sigma * sqrt (T-t))
=> d=(ln(V) - ln(F) - (-r*(T-t)) + 0,5 * sigma^2 * (T-t)) / (sigma * sqrt (T-t))
=> d=(ln(V) - ln(F) + r*(T-t) + 0,5 * sigma^2 * (T-t)) / (sigma * sqrt (T-t))
=> d=(ln(V) - ln(F) + (r + 0,5 * sigma^2) * (T-t)) / (sigma * sqrt (T-t))

So, we can see that at the end, it is the classical Black & Scholes formula with risk-free interest rate appearing.

Hope that will help those who were wondering about the absence of a risk-free rate in Stulz Chapt. 18 reading).

Cheers,

trabala3

 

[David Harper CFA FRM]

Hi trabala, yes, thank you, agreed! I think our derivations (i.e., extract the riskfree rate from Stulz, so d in Stulz is reformatted to correspond to d1 in BSM) match,
see http://forum.bionicturtle.com/threads/merton-formula.5517/#post-15553