We can actually see that these elementary security prices satisfy the forward

equation. So these are the forward equations, we'll

explain where they come from in a moment. We note that P0,0 subscript e equals 1,

why is this? Well this quantity here is the time zero

value of $1 that is paid at time zero and state zero, i.e., today.

So $1 today is equal to $1 today. So certainly, this is true.

So, starting with this value, we can actually work forward in time to compute

the values of the Pij's at every note. For example, suppose this is our binomial

lattice. Well, we start off with knowing this value

it is one. And now we can use the forward equations

to get the values of these two nodes. So this value here of one is equal to

P0,0. I'll ignore the superscript e just to

avoid cluttering the, the slide. This value we want here is P1,0 and the

value we want up here is P1,1. Well, if you look at these we could see

how to get them. So we can get P1,0 from, this equation and

we can get P1,1 from this equation. In this case we would take k equal to 0.

And in this case we would take k even to 0 and that's how, sorry we would take yes k

equal to 0 and that is how we can calculate the state prices at time 1.

Given that we now know these three state prices we can go forward to calculate P2,1

or rather P2,0, P2,1, P2,2. Again using these two equations for the

P2,2 and P2,0 and we can use this equation here for p two one.

So actually that, this is why they're called a forward equations, we start off

with P0,0 equal to 1. And we actually use these equations to

work forwards in the binomial lattice to calculate the state prices for every node.

So, where do these forward equations come from?

We're going to answer that question on this slide and hopefully make clear.

Where these equations come form, so for example, let's consider this elementary

security. This is the elementary security that pays

$1 at time t equals 3, and stage two. And zero everywhere else, so by definition

the value of the security is P3,2. However, we can also compute the value of

the security another way. We can just treat this as a regular

security and use risk neutral pricing to compute its value.

So if we use risk neutral pricing, we will work backwards in the lattice in the usual

way to find its value. So let's do that, so if we work backwards,

we can come back to this node. This is node r2,2 up here.

Well, the value at node r2,2 is going to be 1 over 1 plus r2,2 times the expected

value of the security, 1 period ahead. With 1 period ahead the value of the

security is either 0 or it's 1. And so we get this quantity here which

simplifies down to this expression here. So this is the value of this elementary

security at time t equals 2 and state 2. It's value of node n2,0 is clearly 0

because at node. At time t equals 2 and state 0 you'll only

get 0 in the 2 successive states. So its expected discounted value must be

0. So therefore we have this, its no that

n2,1 which is here. Well that is given to us by 1 over 1 plus

r2,1 times the expected value of the security 1 period ahead and that is 1 with

probability a half. And zero with probability a half, and so

we get this quantity over here. So what we've done is the following: we've

seen, we've come to this elementary security, which is worth one of this

statement zero everywhere else. We know by definition the value of this

elementary security is P3,2, but by working backwards in the lattice we've

also computed its value at these three nodes.

It's equal to this at r2,2 is equal to this at r2,1 and it is equal to zero at

r2,0. So therefore, we can say that P3,2, over

here, must be equal to this quantity times the value of $1 at that node.

Well the value of $1 at that node is P2,2 plus this quantity times the value of 1

dollar at this node and 1 dollar at that node is actually P2,1 plus zero dollars

times the value of P2,0 which is the value of $1 at that node.

So in fact, all we're doing here is linear pricing.

We're actually breaking this security up into this many units of P2,2 plus this

many units of P2,1 plus 0 units of P2,0 and this is indeed, in this case equation

13. So this is equation 13.So this is the

argument that you get to show that the forward equations are true.

It's easy to see that this holds in general for any node any time k state s.

If we're at an extreme node at the bottom or at top.

Well there's only one predecessor known that's possible, so we would only get one

term in this equation, and that's why we would either get this equation here or

this equation here. So these are the forward equations we can

calculate the stay prices, or the elementary prices by working forwards in

time from t equal to zero. So lets go back through a familiar

short-rate lattice. This is the short-rate lattice we're

considering throughout these, these modules.

We start off with r equal to 6%, it grows by a factor of u equals 1.25 or falls by

factor of d equals point 9 in every period.

So we're going to actually compute the forward prices in this particular model by

starting with P0,0 equals $1, and working forwards to calculate the forward prices

at all future nodes. And this is the lattice with the

corresponding elementary prices, or state prices.

So the value at any node, nij is actually Pij.

The value of the elementary security that pays $1 time i state j.

So how do we get these values, well we know where the 1 comes from.

We can work forwards using the forward equations to get these values.

So for example, how do we get this value 0.3079, well we know from the calculation

we just did in the previous slide. The 0.3079 is going to be equal to 0.2194

time divided by twice 1 plus the short rate prevailing at this node plus 0.4432

divided by twice 1 plus the short rate prevailing at the node.

So this is just the same calculation that we did in the previous slide.

We find that P 0.3079 is equal to the sum of these two quantities here.

Now what can you do with these elementary prices or these state prices?

Well once you've calculated these state prices many other derivative securities

are very easy to calculate. For example, suppose we want to compute

Z04, the price of this zero coupon bonds. So this is the zero coupon bond, its value

at time 0, maturity 4. Well with face value a 100 we can compute

Z04 is just being 100 times. The state prices, the sum of the state

prices, in all of the states time t equals 4.

So what are those states a time t equals 4?

Well it is these quantities here. So remember a zero coupon bond is going to

pay $100. And this note a $100 here and $100 here

and so on. So, how much is that worth?

Well that must be worth 100 times this elementary security price plus 100 times

the elementary security price for this note plus 100 times the elementary

security price for this note and so on. Again that is just linear pricing in

action and so that's how we get Z04 equals 100 times.

The sum of these elementary security prices 0.0449 and so on, and actually that

summed to 77.22, which we've seen before, we saw this in one of the first modules in

this section where we computed zero coupon bond prices by working backwards in the

lattice. Well, we've done it in a different manner

here. We've done it here by first calculating

the forward prices by working forwards in time, and iterating those forward

equations. And, then given all of the elementary

prices it was absolutely trivial to compete the price of a 0 coupon [unknown].

It was simply the face value times the sum of all the elementary prices at that time.

We can also calculate other security prices using these elementary prices.

So, here's another example consider a forward starting swap.

That begins at t equals 1 and ends at t equals to 3.

The notional principle is $1,000,000. The fixed rate in the swap is 7% and the

payments are received at time t equals i for i equals 2 and 3.

So this is where the forward feature kicks in.

If it was a regular swap you would get a payment of time equals 1.

Here we're assuming that you only got a payment at time t equals 2 and time t

equals 3. And that payment is based as usual on the

fixed rate minus the floating rate that prevail at time t equals i minus 1.

So the first payment is at t equals 2, because payments are made in arrears.

So the question is, what is the value v0 of this forward swap today at time t equal

to zero? Well, how can we calculate that?

Actually it's very straightforward. So what we have here are the actual cash

flows. For this one, there are actually five cash

flows and we can go back to see these cash flows by looking at the short rate

lattice. So these cash flows are based on these

short rate, they occur in arrears, so we should be seeing these cash flows of 7.5

minus the fixed rate of 7%, 5.4 minus 7%, 9.38 minus 7%, and so on.

These are the [inaudible] of the underlying swap and so indeed here they

are. These are the fixed rate of 7% minus the

floating rate 9.38, 7.5, 5.4, 6.75 and 4.86%.

So they're the cash flows of the swap, but remember these cash flows are paid in

arrears, one payment ahead. So what we do is we take these cash flows

and discount them by the appropriate short rate.

So it's 9.38%, the same value as, as in here, 7.5%, same value as here and so on.

So now we've got the value of the cash flows that each of the nodes at which

these cash flows are determined. Given that we know the elementary security

prices for those nodes, it is just a simple matter of multiplying these values

by the corresponding elementary security prices.

And that is exactly what we have done here.

So we see, we got a value of $5,800 for a notional principal of 1 million dollars.

So again, we could have priced this forward starting swap if we liked, by

working backwards in the lattice using risk neutral pricing While instead we've

done something differently. We're still using risk-neutral pricing of

course because that's where the forward equations come from.

But what we've done instead is we've determined the elementary security prices

via the forward equations, and then used those elementary security prices to

compute the fair value or the albatrosary value of the cash flows associated with

this forward starting swap. All of these calculations are certainly

the calculations for the elementary security price available to us In the

spreadsheet, so we have our short rate lattice.

You can see how we actually calculate the elementary prices.

We start off with a value of 1 and then we just iterate the forward equation.

So we get 0.4717, 0.4717 and these two nodes are [inaudible] equal to 1, and we

can actually iterate forward the forward equations, by using if statements in here

to make sure that we're actually using the correct version of the forward equations.

There are three different versions that we saw.

We need to make sure that we're using the correct one.

And so that's how we do that? We just copy and drag these formulas

through the lattice and updating the elementary prices at each node or each sub

and the [inaudible]. Given these we can actually now compute

all of the zero coupon bomb, calculating the zero coupon bomb prices now absolutely

trivial, we just sum the corresponding, elementary prices, multiply them by a 100

and that's how we get the, the zero-coupon bond prices.

So down here, you can just see we're just summing the corresponding elementary

prices, multiplying by 100. And then of course, we can invert the

zero-coupon bond prices to get the spot interest rates for that maturity.

So for example, 6.68% we get by inverting the 77.22, assuming per period

compounding. And we did that calculation as well in an

earlier module.