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In this module we're going to discuss how to pass forwards on bonds.

Â We saw when we were discussing the binomial model for equities how to pass

Â forward in that case. We'll now we're going to extend that case

Â to passing forwards in bonds. In the next module will be after this one

Â will discuss passing futures, and then will be able to see an example of where

Â forward prices do not equal to futures prices.

Â So here is our familiar short rate lattice, we've seen this now a few times.

Â Short rate begins at 6%, grows by a factor of 1.25 or falls by a factor of 0.9 in

Â each period. This is the short rate that we're using

Â for pricing examples in these modules. So now we want to price the forward

Â contract on a coupon bearing bond. We're going to assume delivery takes place

Â at t equal to 4, and that the bond that's been delivered is a two year 10% coupon

Â bearing bond. We assume delivery takes place just after

Â a coupon has been paid. So how we going to price this?

Â How we going to compute the forward value? Well, we're going to use risk neutral

Â pricing and we're going to use what we know about risk neutral pricing.

Â So, we know the following. For a generic security, let's say it's got

Â price s at time t. We know the following.

Â We know that S 0 over B 0 is equal to E 0 undercue of Sn over Bn, and this is risk

Â neutral pricing, for any security that does not pay coupons or have an

Â intermediate cash flow between times zero and n.

Â So this is what we want to use to price our forward.

Â How we going to go from here down to here? That's the question.

Â Well, it's actually very simple. First of all remember we assume the B 0 is

Â equal to $1.00. If it wasn't, if B 0 was $100.00 then B n

Â would be B n times 100 and we could divide across by the 100 and get back to the case

Â where B 0 equals $1.00. So there's no difficulty seeing that B 0

Â equals $1.00, assuming B 0 equals $1.00. Sn, this is the value of the forward

Â contracted time n. Well what is the value of the forward

Â contracted time n? Well it's this quantity here, it's Z 4 6

Â minus G 0. This if the payoff of the forward

Â contract; it's the underlying security, which we're calling Z 4 6, minus the

Â forward price, G 0. So this is assuming that somebody is long

Â the forward contract. They're buying the security, they're

Â buying it for G $0, and they're getting a security that's worth Z 4 6.

Â So this is our Sn, B 4 is this quantity here, what about S 0 over B 0?

Â Well we know B 0 is equal to one, what is S 0?

Â Well if you recall how our forward contract works, it works as follows.

Â We choose the forward price G 0, so that the initial value of the contract is zero.

Â So actually S 0 is going to be equal to 0, and so we get this expression here.

Â And now the goal here is to figure out, what the value of G 0 is that makes this

Â true. Well, that's straight-forward.

Â This equation implies that, the expected value, at times 0, of G 0 over B 4, is

Â equal to the expected value at time 0 of Z 4 6 over B 4.

Â As a brief aside, I'll mention, I know I've mentioned this notation a couple of

Â modules ago where I said, we'd always use this notation to denote 0 coupon bond

Â prices, but here we're going to use it to denote an actual coupon bearing bond

Â price. So we get this expression here, but G 0 is

Â a constant at time 0. I know it's value at time 0, so I can take

Â this outside the Expecation, and I can then bring everything else over to the

Â other side, and I get equation ten here. So this is the forward price.

Â G 0 equals the expected value of Z 4 6 over B 4, all divided by the expected

Â value of 1 over B 4. Now if you recall the expected value of 1

Â over B 4, this is simply, this quantity down here, simply the time zero price of a

Â zero coupon bond that matures at time 4 and that has face value of $1.

Â This comes from this neutral pricing again.

Â So this is Z 0 4, and we actually calculated this in the last module.

Â It's actually 77.22 dollars if the face value is 100 dollars, if the face value is

Â 1 dollar as we have here, then its price is 0.722.

Â So we actually know the denominator of this expression here.

Â So that means we only need to focus on evaluating the numerator, this.

Â And this is actually very easy to do using our backwards induction or working

Â backwards in the lattice. Let's see how we do that.

Â So to compute Z 4 6, what we will do is we'll work backwards in the lattice from

Â time t equal to 6. So at time t equal to 6, we know that the

Â value Z 6, 6 is 110. And it's 110 because that maturity, you

Â get the face value of 100 back, where you also get the 10% coupon, which corresponds

Â to $10. So the total payoff at t equals 6 at

Â maturity of the underlying bond is 110. And now we just work backwards at each

Â period, to try and find out the value at t equals 4.

Â So, for example, the 98.44 here is equal to 1 over 1 plus 0.1055, that is the short

Â rate that prevailed the time t equals 4, so we can see this if we drove back to

Â this period here, so 10.55 times the expected value of the bond one period

Â ahead. The bond one period ahead will either be

Â 107.19[UNKNOWN] or 110.46[UNKNOWN], so that's how we get 98.44.

Â Note that we did not include the $10 coupon at this time, because we said that

Â the forward contract delivers the bond just after the coupon at this period has

Â been paid. So that's why we don't include the $10

Â coupon at this period. In contrast, when we're computing the

Â value of t equals 5, we actually would include the $10 that's paid in this

Â period, because that actually is a payment to the bond.

Â So we would say that the 102.98 equals $10 plus one over one plus the interest rate

Â of this notde times the expected value of the bond one period ahead which is 110.

Â So doing that, we find the value of the bond that t equals 4, and it's given to us

Â here. This is the underlying security of the

Â forward contract. So remember our goal is to compute the

Â value of this, this is the numerator in the expression for the fair value of the

Â forward price, so I need to compute this expectation here.

Â So we know this is the value at time t equals 4.

Â And I can just work backwards in the lattice and compute its value time t

Â equals 0, I get 79.83. So therefore, using the expression we have

Â for the fair value of the forward contract, which we see, it should have

Â been ten. So we see we're going to get 79.83 divided

Â by 0.722, which gives us a fair forward price, or an arbitrage-free forward price

Â of 103.38. We can go to the spreadsheet that you

Â have, and we can see how these prices were calculated.

Â So as before, these are the parameters for our short rate, so we have initial value

Â 6% for the short rate and we see that over the next five periods it can grow up as

Â large as 18.31% or be as low as 3.54%. Given the short rate lattice, this enable

Â us to compute the 4 year 0 coupon bond price which we saw earlier at 77.22.

Â Remember we needed this in the denominator when we were calculating the fair value of

Â the forward price. So, the next step to do is to compute the

Â value of the 10% six years coupon bond. So recall, it pays off at maturity of

Â $110. This is the principle plus coupon of $10.

Â And then we work backwards to find its value at time t equals 5, including the

Â coupon, which is 10%. Work backwards all the way.

Â In fact, we can actually find that the value of the six year 10% coupon bond at

Â time zero is 24.14. So we do that here although we didn't need

Â to do this to compute the forward price. To compute the forward price we just

Â needed the x coupon price of the bond. So this is the price of the bond at time t

Â equals 4 but these prices include the $10 coupon that's paid at that period.

Â To get the x coupon price we just subtract $10.

Â And that's what we do down here. So we subtract the coupon, $10 in these

Â cells, and now we can actually work backwards in the lattice to compute that

Â quantity, 79.83 that we saw. And finally, we get our value of the

Â forward price which is 103.38.

Â