Credit Risk Intro and Merton Default model
In short, credit risk is that borrower is facing some probability of financial distress.
Implicit put option
In reality, banks are lending money to mortgage borrower, or bond holders are lending money to company that issue debt.
For the mortgage borrower or company shareholders, they are actually get a put option.
Why is a put option?
They got money at the first place, it’s like the strike price in the put option. Max(0. K-S)
here the housing price or company asset value is like the stock price, that is the value you are going to lose if you choose to exercise the put option and default.
But this put option does not come with no price, on the company level, that means bankrupt, on the individual level, that person might not going to get a loan for a period of time because their trashed credit.
With this idea of implied put option, bond price can be valued.
bond price = e^(-rT)*[probability of default *At + (1-probability of default)*B]
T = time to maturity
At = asset value at maturity date
B = notional amount of debt
r = corresponding risk free rate
Now, let’s see how the probability of default determined.
Merton default model
Basic idea is to get the distance to default( measured by unit of standard of deviation) and assume the company asset return follow a normal distribution, then normdist that distance and get the probability.
1. asset value follow fa lognormal distribution and asset return follow a normal distribution
2. asset value is continuous, not jumps
3. no market friction
at time 0, distance to default is ln(S/K)
as time progress, mean return is mu, return volatility is SD.
return exception is (mu -0.5*SD^2)
d2 = distance to default = [ln(S/K) + (mu -0.5*SD^2)*t]/(SD*t^0.5)
true probability of using mu, risk neutral probability if using risk free rate.
N(d2) is the probability of default.
Expected loss(%) = probability of default * loss given default
credit spread = rho – r = (1/T)*ln[1/(1-probability of default*loss given default)