Merton Model

[ChadWOB]

Hi David,

A member of my study group pointed out the following question that I was hoping you could help with. Regarding the Merton Model formula used by Hull vs. de Servigny. Hull uses the formula that I am familiar with from FRM lev 1:

d1= [ln(V/D)+(r+0.5*sigma^2)*T]/[sigma*sqrt(T)] which I typically use (in accordance with N(d2)) to calculate the value of the firms equity (or call feature).

My group member pointed out the de Servigny uses the 'drift' term instead of using the 'risk free rate'. Is there any significance to this difference?

Thank you,
Chad

 

[David Harper CFA FRM]

Hi Chad

Yes, it's the key difference between Merton for PD and d2 in the Black-Scholes. I have an old video (below) on the topic. The riskfree rate in d2 for BSM is justified per a risk neutral idea: the option value is true per no arbitrage regardless of the asset drift (BSM doesn't need Rf rate, doesn't want it).

However, d2 as "distance to default" in Merton isn't option pricing at all, it can be grasped/visualized: the asset drifts up, and we compute the area in tail per a lognormal assumption (that returns are normal; the dubious assumption). In this way, there is a "striking resemblance" between d2 in BSM and the DD in Merton PD, but a big theory difference between the difference between riskfree rate (justifed in BSM) and actual (aka, physical) drift in DD.

It occurs to me there is an additional lens through which to view the difference: Jorion refers to a difference between derivatives valuation (i.e., uses a discounted, risk-neutral distribution) versus risk measurement (uses a future, physical distribution). This difference is manifest here, the DD in Merton is "simulating" a future distribution, so it wants the expected (physical) return of the asset; DD is not discounting to a price (mean) but retrieving a tail of the forward distribution.

While BSM is difficult to master, I do recommend understanding the DD in Merton because you can literally understand the whole formula, and it pays dividends.

I hope that helps, thanks!