0:00

In the last module, we saw to price forwards on bonds.

Â In this module we're going to see how to price futures in bonds.

Â You might recall that the mechanism behind a forward contract is very different that

Â the mechanism behind a futures contract. So we're going to see in this module how

Â we actually get a different price when we price Bond Futures to the price that we

Â get when we price bond forwards. We are now going to price futures

Â contracts on bonds. In particular, we are going to price a

Â futures contact, contract on the same coupon bearing bond that we considered in

Â the last module. In that module, we priced a forward

Â contract on a coupon bearing bond. Here we're going to price a futures

Â contract on that same bond. And I'm also going to compare the forward

Â price with the futures price. In fact this example is interesting

Â because we will see that the forward price is not equal to the futures price.

Â So we're going to have to remind ourselves first about how futures prices are

Â constructed. So let Fk be the date k price of a futures

Â contract expires after n periods. Let Sk denote the time k price of the

Â security underlying the futures contract, then Fn must be equal to Sn.

Â After all, at maturity, the futures price and the underlying price must coincide.

Â So we want to compute the futures price at t equals n minus 1, and we can do this by

Â recalling that any time we enter a futures contract the initial value of the contract

Â is 0. Now, what does that mean?

Â Well, if you recall our risk mutual pricing, our risk mutual pricing states,

Â that for any security, let's say security with price t, St over Bt Is equal to the

Â expected value at time t using the risk neutral probabilities of St plus 1 over B

Â t plus 1. This is our familiar risk neutral pricing

Â for a security that does not pay dividends or coupons, or anti-intermediate cash flow

Â between times t and t plus 1. We're going to use this as follows.

Â So we're going to actually take t equal to n minus 1, and we're going to take s t to

Â be the payoff of the futures contract, or the value of the futures contract.

Â So we know St is equal to 0, because any time you enter into a futures contract you

Â get 0. So therefore, we get 0 here, and then st

Â plus 1, well, the value of the futures contract at that point is going to be Fn

Â minus Fn minus 1. We could also actually interpret this

Â using the more general form of this neutral pricing where we included a coupon

Â plus the value of the security, divide the security would then be zero and the coupon

Â would be this quantity here. So risk neutral pricing gives this to us,

Â what does that mean? Well it means the following.

Â It means that the expected value 1 to q at time n minus 1 of Fn over Bn.

Â Well, that's Fn minus 1 over Bn is equal to the expected value at time n minus 1,

Â with respect to q of Fn over Bn. Now, Fn minus 1 is known to us at time n.

Â It's the futures price at time n minus 1. So that comes outside, this is also a very

Â important characteristic of the cash account.

Â It is known to us one period earlier. So the value of the cash account to time n

Â is also known to us at time n minus 1. So therefore, this Bn can come outside as

Â well, it will come outside here. This Bn will also come out, and they will

Â cancel. And what we're left with Is that Fn minus

Â 1 equals the expected value under q at time n minus 1 of Fn.

Â And in fact, we've seen this before when we were discussing the binomial model for

Â stocks and pricing futures in that binomial model.

Â So we get this expression here, I can iterate this and actually I can get this

Â expression more generally for any time k, I can get Fk equals the expected value of

Â Fk plus 1, conditional time k information. And we can use law of iterated

Â expectations to get this expression here or equivalently this expression here.

Â Now, this holds regardless of whether or not the underlying security pays coupons.

Â If you look at how we derived this, whether or not I paid coupons does not

Â matter. In contrast, the corresponding forward

Â price that we saw in the last module is given to us by this expression here.

Â So this is the value of the forward price, this is the value of the futures price,

Â both at time zero. If by the way, and you can see this

Â immediately, if interest rates were deterministic In that case Bn would be a

Â constant, so the 1 over Bn would come out over here, 1 over Bn would come out over

Â here, they would cancel, and I would be just left with G equals E0 the value of Sn

Â which is exactly what I have here. So certainly when interest are rates are

Â deterministic G0 equals F0. In general, interest rates are not

Â deterministic. I cannot take the one over Bn outside the

Â expectation here, and in fact these two expressions then do not agree.

Â So now let's compute the fair value of the futures contract on the same

Â coupon-bearing bond that we considered in the last period.

Â We know that F0 Is equal to E0 under q of Sn, where s n is the security underlying

Â the futures contract. Well in this case, the security underlying

Â the futures contract is that 10% coupon bearing bond.

Â That is delivered at time t equals 4, just after the coupon has been paid at t equals

Â 4. We saw in the last module that this vector

Â here, vector here of prices, are the prices of that 10% coupon baring bond of t

Â equals 4, so therefore we want to compute f0, which is the expected value of this

Â pay off here. Well, we can do that easily by just

Â working backwards in the lattice. Note, however, that when we work backwards

Â in the lattice we do not discount by 1 over 1 plus the interest rate, because

Â there is no discounting going on up here. This is a fair value of the futures

Â contract at zero, and there is no discounting here, we're not dividing by

Â Bn. So in particular, for example, you should

Â see the 98.09 is equal to a half times 95.05 plus a half times 101.14.

Â No discounting taking place when it work backwards in the lattice here.

Â What do we see? We see a value of the futures contract be

Â 103.22. This is different to the fair forward

Â price that we computed in the last module, that was 1 or 3.38.

Â Now, the numbers are close as you would expect them to be but in fact they are not

Â the same. So here's an example where the forward

Â price and the future's price are actually different.

Â Let's go back and see why? So, if you stop and think about it you can

Â see that F0 is a weighted average of s n at time n.

Â And G0 is also a weighted average of Sn at time n, and if you think of how the rates

Â work, the interest rates work It shouldn't be surprising that g 0 103.38 turns out to

Â be a little bit larger then the futures prize 103.22.

Â And that is because of the manner in which these quantities are computed.

Â If we go to the spreadsheet again and you've seen this already, I hope you have

Â the spreadsheet in front of you. You see here we have the short rate

Â lattice. We have the four-year zero coupon bond

Â lattice. We use that to compute the 77.22, which we

Â used in turn to compute the forward price of the bond.

Â These are all the calculations for the forward price.

Â Over here we've the calculations for the futures price, so over here we've 91.66

Â down to 111.16. These are the same prices we used for the

Â maturity of the forward and they're equal to actually the value of the

Â coupon-bearing bond time four, but ignoring the coupon that's paid at that

Â time. So we get these values, now they can be

Â the future's price. We just work backwards in the lattice

Â computing the price at each note. Note that there's no discounting going

Â back, when we work backwards here and that's because Fk equals the expected

Â value of Fk plus 1 under q. Not the expected value of Fk plus 1

Â discounted under q. So, you see we have our 103.22 here and we

Â have our 103.38. So the forward price and the futures price

Â are different. They're very close, but they are

Â different. And this is an example showing you that

Â forward prices and futures prices are not the same, theoretically, in an arbitrage

Â free model.

Â