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

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

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

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

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

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

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

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

Â lattice. Well, we start off with knowing this value

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â So, where do these forward equations come from?

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

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

Â security. This is the elementary security that pays

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

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

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

Â security and use risk neutral pricing to compute its value.

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

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

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

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

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

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

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

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

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

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

Â 0. So therefore we have this, its no that

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

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

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

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

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

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

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

Â also computed its value at these three nodes.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â considering throughout these, these modules.

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

Â factor of d equals point 9 in every period.

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

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

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

Â corresponding elementary prices, or state prices.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â So what are those states a time t equals 4?

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

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

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

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

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

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

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

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

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

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

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

Â here. We've done it here by first calculating

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

Â equations. And, then given all of the elementary

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

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

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

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

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

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

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

Â So this is where the forward feature kicks in.

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

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

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

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

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

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

Â to zero? Well, how can we calculate that?

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

Â flows. For this one, there are actually five cash

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

Â lattice. So these cash flows are based on these

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

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

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

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

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

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

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

Â and discount them by the appropriate short rate.

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

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

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

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

Â by the corresponding elementary security prices.

Â And that is exactly what we have done here.

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

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

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

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

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

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

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

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

Â this forward starting swap. All of these calculations are certainly

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

Â spreadsheet, so we have our short rate lattice.

Â You can see how we actually calculate the elementary prices.

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

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

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

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

Â There are three different versions that we saw.

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

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

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

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

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

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

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

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

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

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

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

Â compounding. And we did that calculation as well in an

Â earlier module.

Â