In this module we're going to discuss how to pass forwards on bonds.
We saw when we were discussing the binomial model for equities how to pass
forward in that case. We'll now we're going to extend that case
to passing forwards in bonds. In the next module will be after this one
will discuss passing futures, and then will be able to see an example of where
forward prices do not equal to futures prices.
So here is our familiar short rate lattice, we've seen this now a few times.
Short rate begins at 6%, grows by a factor of 1.25 or falls by a factor of 0.9 in
each period. This is the short rate that we're using
for pricing examples in these modules. So now we want to price the forward
contract on a coupon bearing bond. We're going to assume delivery takes place
at t equal to 4, and that the bond that's been delivered is a two year 10% coupon
bearing bond. We assume delivery takes place just after
a coupon has been paid. So how we going to price this?
How we going to compute the forward value? Well, we're going to use risk neutral
pricing and we're going to use what we know about risk neutral pricing.
So, we know the following. For a generic security, let's say it's got
price s at time t. We know the following.
We know that S 0 over B 0 is equal to E 0 undercue of Sn over Bn, and this is risk
neutral pricing, for any security that does not pay coupons or have an
intermediate cash flow between times zero and n.
So this is what we want to use to price our forward.
How we going to go from here down to here? That's the question.
Well, it's actually very simple. First of all remember we assume the B 0 is
equal to $1.00. If it wasn't, if B 0 was $100.00 then B n
would be B n times 100 and we could divide across by the 100 and get back to the case
where B 0 equals $1.00. So there's no difficulty seeing that B 0
equals $1.00, assuming B 0 equals $1.00. Sn, this is the value of the forward
contracted time n. Well what is the value of the forward
contracted time n? Well it's this quantity here, it's Z 4 6
minus G 0. This if the payoff of the forward
contract; it's the underlying security, which we're calling Z 4 6, minus the
forward price, G 0. So this is assuming that somebody is long
the forward contract. They're buying the security, they're
buying it for G $0, and they're getting a security that's worth Z 4 6.
So this is our Sn, B 4 is this quantity here, what about S 0 over B 0?
Well we know B 0 is equal to one, what is S 0?
Well if you recall how our forward contract works, it works as follows.
We choose the forward price G 0, so that the initial value of the contract is zero.
So actually S 0 is going to be equal to 0, and so we get this expression here.
And now the goal here is to figure out, what the value of G 0 is that makes this
true. Well, that's straight-forward.
This equation implies that, the expected value, at times 0, of G 0 over B 4, is
equal to the expected value at time 0 of Z 4 6 over B 4.
As a brief aside, I'll mention, I know I've mentioned this notation a couple of
modules ago where I said, we'd always use this notation to denote 0 coupon bond
prices, but here we're going to use it to denote an actual coupon bearing bond
price. So we get this expression here, but G 0 is
a constant at time 0. I know it's value at time 0, so I can take
this outside the Expecation, and I can then bring everything else over to the
other side, and I get equation ten here. So this is the forward price.
G 0 equals the expected value of Z 4 6 over B 4, all divided by the expected
value of 1 over B 4. Now if you recall the expected value of 1
over B 4, this is simply, this quantity down here, simply the time zero price of a
zero coupon bond that matures at time 4 and that has face value of $1.
This comes from this neutral pricing again.
So this is Z 0 4, and we actually calculated this in the last module.
It's actually 77.22 dollars if the face value is 100 dollars, if the face value is
1 dollar as we have here, then its price is 0.722.
So we actually know the denominator of this expression here.
So that means we only need to focus on evaluating the numerator, this.
And this is actually very easy to do using our backwards induction or working
backwards in the lattice. Let's see how we do that.
So to compute Z 4 6, what we will do is we'll work backwards in the lattice from
time t equal to 6. So at time t equal to 6, we know that the
value Z 6, 6 is 110. And it's 110 because that maturity, you
get the face value of 100 back, where you also get the 10% coupon, which corresponds
to $10. So the total payoff at t equals 6 at
maturity of the underlying bond is 110. And now we just work backwards at each
period, to try and find out the value at t equals 4.
So, for example, the 98.44 here is equal to 1 over 1 plus 0.1055, that is the short
rate that prevailed the time t equals 4, so we can see this if we drove back to
this period here, so 10.55 times the expected value of the bond one period
ahead. The bond one period ahead will either be
107.19[UNKNOWN] or 110.46[UNKNOWN], so that's how we get 98.44.
Note that we did not include the $10 coupon at this time, because we said that
the forward contract delivers the bond just after the coupon at this period has
been paid. So that's why we don't include the $10
coupon at this period. In contrast, when we're computing the
value of t equals 5, we actually would include the $10 that's paid in this
period, because that actually is a payment to the bond.
So we would say that the 102.98 equals $10 plus one over one plus the interest rate
of this notde times the expected value of the bond one period ahead which is 110.
So doing that, we find the value of the bond that t equals 4, and it's given to us
here. This is the underlying security of the
forward contract. So remember our goal is to compute the
value of this, this is the numerator in the expression for the fair value of the
forward price, so I need to compute this expectation here.
So we know this is the value at time t equals 4.
And I can just work backwards in the lattice and compute its value time t
equals 0, I get 79.83. So therefore, using the expression we have
for the fair value of the forward contract, which we see, it should have
been ten. So we see we're going to get 79.83 divided
by 0.722, which gives us a fair forward price, or an arbitrage-free forward price
of 103.38. We can go to the spreadsheet that you
have, and we can see how these prices were calculated.
So as before, these are the parameters for our short rate, so we have initial value
6% for the short rate and we see that over the next five periods it can grow up as
large as 18.31% or be as low as 3.54%. Given the short rate lattice, this enable
us to compute the 4 year 0 coupon bond price which we saw earlier at 77.22.
Remember we needed this in the denominator when we were calculating the fair value of
the forward price. So, the next step to do is to compute the
value of the 10% six years coupon bond. So recall, it pays off at maturity of
$110. This is the principle plus coupon of $10.
And then we work backwards to find its value at time t equals 5, including the
coupon, which is 10%. Work backwards all the way.
In fact, we can actually find that the value of the six year 10% coupon bond at
time zero is 24.14. So we do that here although we didn't need
to do this to compute the forward price. To compute the forward price we just
needed the x coupon price of the bond. So this is the price of the bond at time t
equals 4 but these prices include the $10 coupon that's paid at that period.
To get the x coupon price we just subtract $10.
And that's what we do down here. So we subtract the coupon, $10 in these
cells, and now we can actually work backwards in the lattice to compute that
quantity, 79.83 that we saw. And finally, we get our value of the
forward price which is 103.38.
