In this module, we're going to discuss the so-called forward equations. The forward equations enable us to compute the prices what are called elementary securities. An elementary security is a security that pays $1 at a particular time and in a particular state at that time. We'll see that by constructing the forward prices we'll be able to price many derivative securities very easily. So let's get started. We're going to let P subscript i comma J superscript E denote the time zero price of a security that pays $1 at time i, and state j, and zero at every other time and state. Such a security is called an elementary security. Actually, it is sometimes also referred to as an arrow Debreu security, for those of you with some background in economics. This type of security actually has a very important role to play in financial economics. So such a security is called an elementary security and we call P subscript i comma j superscript e its state price. 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.