calculation of probability of default on debt interest and its repayment

Black – Scholes model assumes that underlying stock is distributed lognormally. I’ll try to illustrate:

S0 – current price, S – price in one year (lognormally distributed random variable). Assume that Rs is annual return on S. Then S/S0=exp(Rs), assuming continious compounding. Or ln(S/S0) = Rs – random variable. Volatility of Rs (which is actually normally distributed) is used in Black – Scholes formula.
Prediction of default
Assume that V – value of assets of the comany today, F – debt payable (in t years), if V distributed lognormally, using black-scholes formula you can calculate ‘value firms equity as european call option’ (from BPP study text):

E=N(d1)V-N(d2)F*exp(-rt)
d1=(ln(V/F)+(r+0,5*sigma^2)t/(sigma*sqrt(t))
d2=d1-sigma*sqrt(t), where sigma is volatility of ln(V) (in BPP study text sigma is volatility of V, which is inconsistent with Black-Scholes model assumptions)

1-N(d2) is the probability of default (when option is out of the money).

In Q3 from December 2008, for example, examiner in answers assumes that cashflow is distributed normally, then he calculates expected annual cashflow and its volatility, and finally probability that cash outflow is greater than cash in hand.

I did some simulations in Excel, and found that in 500 tries I had no default, I suspect there is a problem in examiner’s solution with calculation of annual deviation.

P.S. having read solutions to the exam questions, I would recommend just to do something (assume normality, for example), if you get stuck 🙂 .