In this module we're going to discuss the cash account and the pricing of zero
coupon bonds in the context of the binomial model for the short rate.
The cash account and zero coupon bonds are extremely important securities and
derivatives pricing in general and so we're going to spend some time now
figuring out how to price them and understand the mechanics of these
securities. >> So let's get started.
If you recall this is our binomial lattice model for the short rate.
Ri is the Short- Rate that applies for lending between times i out to i plus 1.
In general it's a random variable because it can take on any of these values for
example at time 2. So time t equals 2 the short rate or 2
could take on this value at state 0, this value at state 1 or this value at state 2.
We also have our risk-neutral probabilities q u and q d, and of course,
q u plus q d must sum to 1. These are our risk-neutral probabilities,
so they're strictly positive, and what we will do is we price everything with these
risk-neutral probabilities. In particular, for example, if we want to
price a non coupon paying security, and I use the term, non coupon here loosely.
So, a coupon could refer to any cash flow. So if we want to price in non coupon
paying security at time i, state j. Well, we just do our usual discounted
risk-neutral pricing. So Zij, the value of the security at time
i state j, is one over one plus the short-rated time i state j, times the
expected value of the security one period ahead, where we take that expected value
with respect to the risk-neutral probabilities qu and qd.
Now as we said as well before, there can be no arbitrage when we price using three.
And the reason for that is as follows; If you recall our definition of a type a
arbitrage, for example. So a type a arbitrage was a security of
the form v 0 being less than 0, and its value at time 1, v 1 must be greater than
or equal to 0. So we said any security like this in the
one period model constituted type a arbitrage.
Well this is not possible over here if we price everything according to 3.
And the reason is, so Z will take the place or r v here.
We see that it is not possible for this to be greater than or equal to 0, and this to
be greater than or equal to 0, and yet, have this being less than or equal to 0.
This is not possible, because the q's and r are all strictly positive.
And if the zeds are greater than or equal to 0, then this must be greater than or
equal to 0 as well. So we actually cannot get a type a
arbitrage. That's not possible.
It's the same for a type b. If you recall, a type b arbitrage assumed
a security of the following form, V0 less than or equal to 0, V1 greater than or
equal to 0, but V1 not equal to 0. Which means that V1 is greater than or
equal to 0 in all states and is at least one state where it's actually strictly
greater than 0. Well, again, the exact same argument over
here would show that, that is not possible as well.
So there can be no arbitrage when we price according to 3.
I now want to talk a little bit about The Cash-Account.
The Cash-Account is a particular security that in each period earns interest at the
short rate. We're going to use bt to denote the value
of the cash account at time t and we would be assuming without loss of generality
that it starts off with a value of 1, so b 0 is equal to 1.
The Cash Account is not risk free. And the reason it's not risk free is
because interest rates are uncertain, they're stochastic.
In particular the value of the Cash Account at the time t plus s say, is not
known at time t for any value that's greater than 1.
However, it is locally risk-free because I do know the value of B t plus 1 at time t.
In fact, B t plus 1 will always be equal to B t times 1 plus the short rate.
And I'm going to know the short rate at time t and therefore I will know B t plus
1 at time t. So again think of your bank account
analogy. If I deposit money today for one month, I
know what rate will apply for that one month period and so I will know how much I
will get at the end of the month. But I will not know what interest rate
will prevail in one months' time and therefore will not know future values of
The Cash-Account beyond one month. So a quick thing to notice here, so Bt
therefore has this expression here, based on the argument I just gave you, I can
look at Bt plus 1 Bt and divide 1 by the other and see that I get one over 1 plus
rt. And the reason I want that expression is
down here, I want to derive equation four here.
So how do I derive equation four? Well, again, for a non-coupon paying
security, zt times z at time t, state j is equal to 1 over 1 plus rtj times the
expected value of the security one period from now.
So this is our familiar risk neutral pricing expression from the previous
slide. I can actually rewrite this expression as
the expected value under q, remember q is equal to q u and q d, the risk-neutral
probabilities. And I can replace my 1 over 1 plus rtj,
with this expression here bt over bt plus 1.
So therefore I can write the value of the non-coupon paying security at time t as
being the expected value at time t under q.
On Z t plus one, multiplied by B t over B t plus one.
So rewriting equation four, I can just bring the B t over to the left hand side
and I get this expression here. This is an important expression but I can
go a little bit further. I can actually iterate it to get the
following. So for example, I can write Zt over Bt is
equal to Et. Under q of, well we know it's zed t plus 1
over bt plus 1, but I can actually use this equation 5 again to write zed t plus
1 over bt plus 1 as the expected value under q.
Condition on time t plus 1 information of Zt plus 2 over Bt plus 2.
And using the law of iterated expectations, this is equal to the
expected value condition on time t information of Zt plus 2 over Bt plus 2.
And I could repeat the same trick again and again.
And so, it's easy to see that this condition is hold.
So this is our risk neutral pricing condition, for a non-coupon or
non-dividend paying security. In particular, it's the pricing equation
that we use for any security that does not pay any intermediate cash flows between
times t and t plus s. When we're doing risk-neutral pricing for
a coupon paying security, we use the exact same idea, so Ztj equals 1 over 1 plus Rtj
times the expected value under q of the value of the security plus the cash flow
at time t plus 1 that just gives us this expression here.
And for the same reason as before, we can see as long as we price any coupon-paying
security this way that cannot be an arbitrage.
There is now way that this quantity can be greater than or equal to 0 and yet to have
Ztj being less than 0. So you couldn't have a type A arbitrage
because if this is greater than or equal to 0 there is neutral probabilities we
know are strictly greater than 0, this is greater than 0 and so all of this
expectation must be greater than or equal to 0.
So in particular this is not possible. So you couldn't get a type A arbitrage,
and for the same reason, you couldn't get a type B arbitrage as well.
Alright, so we have seven. Well, it's easy to rewrite seven using the
same ideas we used in the previous slide. I can replace 1 over 1 plus RTJ with Bt
over Bt plus 1, bring the Bt over to the other side, and I get Expression eight.
Now I can iterate, I can substitute in for example, if I substitute in the following;
I know that Zt plus Bt plus 1 is equal to the expected value under q conditional on
time t plus one information. Ct plus 2 over Bt plus 2.
Plus Zt plus 2 over Bt plus 2. So if I substitute that in down here, I'm
going to get this expression here when s equals 2.
Now it's easy to see that this expression holds more generally for general values of
s, or for integer values of s greater than t.
So this is an extremely important condition.
We're going to use this throughout this section on term-structure models and
pricing fixed-income derivatives that tells us how to price fixed-income
derivatives using risk-neutral pricing and this ensures that there's no arbitrage.
In other words, it ensures that we're pricing fixed-income derivatives in a
manner which is consistent with no arbitrage.
Note also that equation six is actually a special case of nine, because we get
equation six From nine by just taking all of these cjs equal to 0.
So this is an extremely important result, it guarantees that we can price everything
with no arbitrage. Here is a sample short-rate lattice.
It starts off with the short rate r00 being equal to 6%.
And then the short rate will grow by a factor of u equals 1.25 or fall by a
factor of d equals 0.9 in each period. It's not very realistic.
These interest rates, as you can see, grow quite large here.
And given the current economy, global economy, where interest rates are very
low. This example wouldn't be very realistic.
But it is more than sufficient for our purposes.
In fact, it's good to have such a wide range of possible interest rates.
As it makes it easier to distinguish them in the examples that we'll see in the
future. At this point I should also mention.
That you should look at the spreadsheet that is associated with these modules.
The spreadsheet you'll see this particular example there.
And we're going to be using this example throughout this section to price various
types of fixed income derivatives. We're going to be looking at caps, floors,
swaptions. Options on zero coupon bonds and so on.
So we're going to use this particular short rate example as our model in all of
these pricing examples. So the first thing we're going to do is
we're going to see how to price a zero coupon bond that matures a time t equal to
4. So if we want to do that we're going to
use our risk neutral pricing, our risk neutral pricing result If you recall,
states that Zt over Bt is equal to the expected value conditional on time t
information of Zt plus 1 over BT plus 1, and this is a risk-neutral pricing
identity for securities that do not pay coupons or do not have intermediate cash
flows. And certainly that is true of a zero
coupon bond. If you recall, a zero coupon bond does not
have any intermediate cash flows. It only pays off its face value at
maturity. And this example matures at t equal to 4.
This face value is 100 and this indeed is if you like using our notation for 0 zero
coupon bonds Z44. So what we're going to do is we're going
to bond rate the price to zero coupon bond is to use this expression here by just
working backwards in the lattice 1 period at a time.
We know Z44, it's 100. At maturity the, the bond is worth 100, so
let's work back and compute its value at time t equals 3.
Well, to do that, we can just use this expression.
Another way of saying this, and we saw this as well before, this is equivalent to
saying that Zt Is equal to the expected value time t of Zt plus 1 over 1 plus or
t. And so in fact it's this version that
we're going to use. We're going to work backwards computing
the values t at every node by discounting and computing the expected value one
period ahead so that's all we're doing here.
So, for example, the 83.08 that we've highlighted here is equal to 1 over 1 plus
the short rate value at that node, and if we go back one slide we'll see the short
rate value at that node was 9.38%. So that's where the 0.0938 comes from
here. And then it's the expected value under q
of the value of the zero coupon bond one period ahead.
There are two possible values, 89.51, 92.22 and that's what we have here.
So we just work backwards in the lattice, one period at a time, until we get its
value here at time 0 and this is Z04. The time 0 value of the zero coupon bond
that matures the time 4. Having calculated the zero coupon bond
price at time0, we can now infer from that the actual interest rate that corresponds
to, to t equals 4. In particular if we assume part period
compounding and we let S4 denote the, the interest rate that applies to borrowing or
lending for four periods then we know that 77.22 times 1 plus S moved to the power of
4 must be equal to 100. So of course we can invert that to get
that S4. Is equal to 100 over 77.22, all to the
power of 1 quarter minus 1 and so that's how we get S4.
So there's always a one to one correspondence between seeing the zero
coupon bond prices, and seeing the corresponding interest rate.
Therefore it means that we can actually compute all of the zero coupon bond prices
for the four different maturities. So we can compute the zero coupon bond
price for maturity t equals 1, t equals 2, 3, and 4.
And from this we can actually back out, back out the actual interest rates that
apply to these periods. So, for example, we will get a term
structure of interest rates that looks like the following; We have t down here
and we've got the spot rate St here and maybe we'll see something like the
following or maybe it's an inverted curve but this point here, so for example t1,
that point corresponds to there and it corresponds to some spot rate st1 there.
So we can actually use this model to price all the zero coupon bonds.
And from all of these zero coupon bond prices, we can invert the message in the
previous slide to get the term structure. The term structure is the term structure
of interest rates. We can see what interest rate applies to
each time t. At time t equals 1, for example, we will
then compute a new set of zero coupon bond prices, and obtain a new term structure.
So for example at time t equals 0 were down here.
But at time equals 1 may be I'm up in this state of the world.
So if I'm up in this state of the world I could recompute the time structure of
interest rates, I could do that by a pricing all of the zero coupon bonds at
this point. I'm going to get a different set of prices
at the set of prices ahead at time t equals 0 and I can invert this new set of
prices to get the new time structure and may be that new time structure will look
different, may be will look something like this.
So I will get a new term structure times t equals 1, moreover the term structure I
see will depend on whether I'm up here or down here.
So what we've actually succeeded in doing is defining a stochastic model or a random
model for the term structure of interest rates by just focusing on the short rate.
So the short-rate or t is just a scalar random variable or scalar process by
focusing on modeling this short rate, as we've done, we've actually succeeded in
defining a stochastic or random model for the entire term structure.
And that's actually a very significant point to keep in mind when working with
these short-rate models.
