Financial Theory: Lecture 16 Transcript
October 29, 2009
<< back
Professor John Geanakoplos:
We're now at the stage where we're considering the implications of
uncertainty, so I hope that the subtlety, and surprise element of the class, will gradually pick up without
increasing the difficulty. The complexity will pick up a little bit, but not the difficulty. It's just you'll have to
keep a few more things in your head, but the mathematics isn't any harder.
So we ended last time talking about default and inferring default probabilities, and so I just want to finish off
that discussion. So suppose that at any stage of the tree, you know, lots of things can happen in the world.
We're always going to model the uncertainty in the future by a tree with different things happening, and at
each of these nodes people are going to have a discount rate.
So maybe it'll be r equals 20 percent, and here r could equal 15 percent, something like that, and we want to
add to this the possibility that there's default. So if we add the possibility of default, and these things keep
going and maybe there are payoffs at the end or payoffs along the way. If at any point in the tree like this one
we add a new possibility, which is the default possibility, so this happens--by the way, when do people
default?
They never default before they have to make a payment. So when do they default, exactly when they're
supposed to make a payment. So suppose that this guy is going to default here when he's going to make a
payment. At every possible scenario he would default there. So we've got a very simple model of default, so
not a very realistic one where the guy defaults in all of these following scenarios. So something's just bad.
Once he's gotten here you know that he's not going to make the payment the next period.
We further assume that not only does he default there, but he defaults on everything thereafter. So the payoff
is just going to be 0 here. So this is going to be--originally we had probabilities p
1
, p
2
, p
3
, let's say for the
probabilities, now we're going to have probabilities d for default and then 1 - d times all of these, right? So
essentially what have we done?
We've simply replaced in our calculation of payoffs and present values, we've simply replaced these
possibilities with probability p
1
, p
2
, p
3
. We added another possibility, but the payoffs are 0 here. Nothing's
going to happen from then on except 0, and we said that happened with probability d, which means
presumably, all of these have to be scaled down by that so they still add up to 1. So essentially the point I'm
trying to make is that default that leads to 0 payoffs thereafter is just like discounting more.
Why is that? Because whatever calculation you did for the value here of what the bond could possibly be
worth there is--it's all the same numbers as there were before except we've multiplied it by 1 - d, so it's the
same thing. So instead of going 1 over 1 + r times future payoffs, that's no default value, so that times future
payoffs. Now we've got default value under this special kind of default is going to be 1 - d times 1 over 1 + r