In the last module, we saw to price forwards on bonds. In this module we're going to see how to price futures in bonds. You might recall that the mechanism behind a forward contract is very different that the mechanism behind a futures contract. So we're going to see in this module how we actually get a different price when we price Bond Futures to the price that we get when we price bond forwards. We are now going to price futures contracts on bonds. In particular, we are going to price a futures contact, contract on the same coupon bearing bond that we considered in the last module. In that module, we priced a forward contract on a coupon bearing bond. Here we're going to price a futures contract on that same bond. And I'm also going to compare the forward price with the futures price. In fact this example is interesting because we will see that the forward price is not equal to the futures price. So we're going to have to remind ourselves first about how futures prices are constructed. So let Fk be the date k price of a futures contract expires after n periods. Let Sk denote the time k price of the security underlying the futures contract, then Fn must be equal to Sn. After all, at maturity, the futures price and the underlying price must coincide. So we want to compute the futures price at t equals n minus 1, and we can do this by recalling that any time we enter a futures contract the initial value of the contract is 0. Now, what does that mean? Well, if you recall our risk mutual pricing, our risk mutual pricing states, that for any security, let's say security with price t, St over Bt Is equal to the expected value at time t using the risk neutral probabilities of St plus 1 over B t plus 1. This is our familiar risk neutral pricing for a security that does not pay dividends or coupons, or anti-intermediate cash flow between times t and t plus 1. We're going to use this as follows. So we're going to actually take t equal to n minus 1, and we're going to take s t to be the payoff of the futures contract, or the value of the futures contract. So we know St is equal to 0, because any time you enter into a futures contract you get 0. So therefore, we get 0 here, and then st plus 1, well, the value of the futures contract at that point is going to be Fn minus Fn minus 1. We could also actually interpret this using the more general form of this neutral pricing where we included a coupon plus the value of the security, divide the security would then be zero and the coupon would be this quantity here. So risk neutral pricing gives this to us, what does that mean? Well it means the following. It means that the expected value 1 to q at time n minus 1 of Fn over Bn. Well, that's Fn minus 1 over Bn is equal to the expected value at time n minus 1, with respect to q of Fn over Bn. Now, Fn minus 1 is known to us at time n. It's the futures price at time n minus 1. So that comes outside, this is also a very important characteristic of the cash account. It is known to us one period earlier. So the value of the cash account to time n is also known to us at time n minus 1. So therefore, this Bn can come outside as well, it will come outside here. This Bn will also come out, and they will cancel. And what we're left with Is that Fn minus 1 equals the expected value under q at time n minus 1 of Fn. And in fact, we've seen this before when we were discussing the binomial model for stocks and pricing futures in that binomial model. So we get this expression here, I can iterate this and actually I can get this expression more generally for any time k, I can get Fk equals the expected value of Fk plus 1, conditional time k information. And we can use law of iterated expectations to get this expression here or equivalently this expression here. Now, this holds regardless of whether or not the underlying security pays coupons. If you look at how we derived this, whether or not I paid coupons does not matter. In contrast, the corresponding forward price that we saw in the last module is given to us by this expression here. So this is the value of the forward price, this is the value of the futures price, both at time zero. If by the way, and you can see this immediately, if interest rates were deterministic In that case Bn would be a constant, so the 1 over Bn would come out over here, 1 over Bn would come out over here, they would cancel, and I would be just left with G equals E0 the value of Sn which is exactly what I have here. So certainly when interest are rates are deterministic G0 equals F0. In general, interest rates are not deterministic. I cannot take the one over Bn outside the expectation here, and in fact these two expressions then do not agree. So now let's compute the fair value of the futures contract on the same coupon-bearing bond that we considered in the last period. We know that F0 Is equal to E0 under q of Sn, where s n is the security underlying the futures contract. Well in this case, the security underlying the futures contract is that 10% coupon bearing bond. That is delivered at time t equals 4, just after the coupon has been paid at t equals 4. We saw in the last module that this vector here, vector here of prices, are the prices of that 10% coupon baring bond of t equals 4, so therefore we want to compute f0, which is the expected value of this pay off here. Well, we can do that easily by just working backwards in the lattice. Note, however, that when we work backwards in the lattice we do not discount by 1 over 1 plus the interest rate, because there is no discounting going on up here. This is a fair value of the futures contract at zero, and there is no discounting here, we're not dividing by Bn. So in particular, for example, you should see the 98.09 is equal to a half times 95.05 plus a half times 101.14. No discounting taking place when it work backwards in the lattice here. What do we see? We see a value of the futures contract be 103.22. This is different to the fair forward price that we computed in the last module, that was 1 or 3.38. Now, the numbers are close as you would expect them to be but in fact they are not the same. So here's an example where the forward price and the future's price are actually different. Let's go back and see why? So, if you stop and think about it you can see that F0 is a weighted average of s n at time n. And G0 is also a weighted average of Sn at time n, and if you think of how the rates work, the interest rates work It shouldn't be surprising that g 0 103.38 turns out to be a little bit larger then the futures prize 103.22. And that is because of the manner in which these quantities are computed. If we go to the spreadsheet again and you've seen this already, I hope you have the spreadsheet in front of you. You see here we have the short rate lattice. We have the four-year zero coupon bond lattice. We use that to compute the 77.22, which we used in turn to compute the forward price of the bond. These are all the calculations for the forward price. Over here we've the calculations for the futures price, so over here we've 91.66 down to 111.16. These are the same prices we used for the maturity of the forward and they're equal to actually the value of the coupon-bearing bond time four, but ignoring the coupon that's paid at that time. So we get these values, now they can be the future's price. We just work backwards in the lattice computing the price at each note. Note that there's no discounting going back, when we work backwards here and that's because Fk equals the expected value of Fk plus 1 under q. Not the expected value of Fk plus 1 discounted under q. So, you see we have our 103.22 here and we have our 103.38. So the forward price and the futures price are different. They're very close, but they are different. And this is an example showing you that forward prices and futures prices are not the same, theoretically, in an arbitrage free model.