Inference for an alternate D2 or Distance to Default Formula


New Member
I was reading the Stultz reading on Credit Derivatives and trying to replicate the same across excel.
One of the Stultz example has the following parameters:

Asset Value $ 120.00 S
Face Value of Debt $ 100.00 K
Time5 T
Risk Free Rate of Return10% r
Volatility20% Sigma
Return on Assets20% Mew

Although I was able to find the distance to default by the usual D2 formula of Black and Scholes,
1586884520192.png Distance to default by this formula works out to be 2.42015

I just saw that an alternative formula works as well:

d2 = ln((S*e^((R asset returm-sigma squared/2)*T))/K)*sqrt(T)

How i stumbled upon this formula,

1. I wanted to find out the value of Asset after 5 years.

Considering Return on Assets and Drift, The value of Asset after 5 years turns out to be $295.15
Value of V @ T=V*e^((Mew - Sigma Squared/2)*T))

2. I tried finding the log return of the Asset compared to the Face Value of Debt => ln(Value of S @ T/F) = 1.082322

3. I happened to multiply the log return is 2 with the square root of time {ln(Value of S @ T/F)*sqrt(T)}, this turned out to match the value of the distance to default which equals 2.42015

I tried the same exercise to find out the d2, by replacing the mew with the risk free rate and found that the same formula works and matches the d2. Hence, i hope that both these formula are matching.

However, I would like to ask you guys that is there any inference or meaning to the alternative formula that i stumbled upon.//d2 = ln((S*e^((R asset returm-sigma squared/2)*T))/K)*sqrt(T) //

Or is it just some mathematical transformation of the D2 Formula

//I have attached the excel for reference//


  • Black and Scholes D2 Calculation.xlsx
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