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>> We're now going to see how to price forwards and futures in the binomial

Â model. We'll see that they're very

Â straightforward to price even though the mechanics of these securities are

Â different. We'll also ask the question, how do the

Â prices of forwards compare with the prices of futures?

Â We'll see that in the binomial model, they're actually identical, although we

Â will make the point that this is not true in general.

Â We're going to start with an n-period binomial model.

Â Although, in this case, we see n is just equal to 3, u equals 1 over d as usual.

Â And, of course, we also have our usual cash account which pays a gross risk-free

Â rate of r in each period. Okay.

Â So, consider now a forward contract on the stock that expires after n periods.

Â And we're going to let G0, denote the t, the date t equals 0 price of the contract.

Â Now, I have price, and in, in quotes here because there's often a lot of confusion

Â over the word price. Because people sometimes think that if

Â something is a price, then that is what you must pay to purchase the contract but,

Â of course, that is not true. G0 is chosen so that the contract is

Â initially worth zero. So therefore, when you buy a forward

Â contract, no money changes hands, in fact, the initial value of the forward contract

Â is zero. The so-called forward price, here, G0, is

Â just used to determine the payoff at the maturity of the forward contract.

Â There's also a similar situation with futures, which we'll discuss when we come

Â to the pricing of futures in the binomial model, in a few moments time.

Â Okay. So, back to the forward contract.

Â G0 is the price of the contract, but actually, the value of the contract, when

Â we enter into it, is 0. So, using risk-neutral pricing, the

Â initial value of the contract is zero. That's how much we must if we buy this

Â forward contract. We get nothing until time n, and at that

Â point, we get Sn minus G0, this is the payoff of the forward contract.

Â So therefore, risk-neutral pricing says that 0 is equal to the expected value

Â using the risk mutual probabilities of the payoff discounted and the discount factors

Â are to the power of n. So, this is simply, risk-neutral pricing.

Â Okay. Now, what we must do, is we need to figure

Â out, what is G0. That's the goal here, to figure out the

Â fair value of G0. We will do this by just looking at this

Â equation. We notice, first of all that Rn, is a

Â constant, okay, the gross risk-free rate is a constant so it comes outside the

Â expectation. And so, we actually just get 0 equals the

Â expected value of Sn minus G0. Remember G0 is also a constant, it's

Â chosen at time 0. So, it is not a random quantity, so we

Â don't have an expectation around it. So, of course, this implies G0 is equal to

Â this. And this is the forward price of the

Â contract. Okay.

Â And 10 holds whether or not the underlying security pays dividends.

Â We've now discussed dividends in the context of the binomial model.

Â We didn't mention dividends at all here. But, in fact, dividends can be president,

Â present in the model and this is still the correct price.

Â The only point where dividends will enter is in G, okay?

Â If you remember, the risk-neutral probability is Q, alright, given to us by,

Â it's going to be R minus d minus c divided by u minus d, okay?

Â And so, if the security, if the underlying security pays dividends, well, this is

Â going to enter into the risk-neutral probability.

Â It will lower the probability, the risk-neutral probability of up moves, and

Â make the forward contract a little cheaper than, would otherwise be the case, okay?

Â And we'll make the forward contract a little cheaper than would be the case if

Â dividends were not present. Okay.

Â Futures. Consider now a futures contract on the

Â stock that expires after n periods, okay? Let Ft denote the date t price of the

Â futures contract, and again, I put price in quotes because Ft isn't the value of

Â the futures contract, alright? If we enter into a futures contract at any

Â time, it actually costs nothing. The fair value of a futures contract at

Â any time is actually zero, okay? So, as was the case with the forward

Â contract, this futures price is really used to determine the payoffs of owning

Â the futures contract. So, we'll come to that in a moment when we

Â actually price the futures contract. Okay.

Â So, the futures contract expires after n periods.

Â So therefore, we know Fn equals Sn. This is almost by definition.

Â These would be the terms of the futures contract.

Â It expires at time n and according to the rules of the contract, Fn is equal to Sn.

Â So, this must be the case. Okay.

Â So, as I mentioned earlier, a common misconception is that Ft is how much you

Â must pay at time t to buy one contract or how much you receive if you sell one

Â contract. This is false.

Â A futures contract always costs nothing. The price Ft is only used to determine the

Â cash flow associated with holding the contract.

Â So, that plus or minus Ft minus Ft minus 1 is the payoff received at time t from a

Â long or short position of one contract held between times t minus 1 and t.

Â So, it will be plus Ft minus Ft minus 1, if we were long or we owned one futures

Â contract. And it would be minus if we were short or

Â we had sold one futures contract between times t minus 1 and t.

Â So, in fact, some people will often characterize a futures contract as

Â follows. They will say that a futures contract is a

Â security that is always worth 0, but that pays a dividend, I should put quotes here,

Â it's not a dividend like the dividend you get from a stock.

Â So, it pays a dividend of Ft minus Ft minus 1 at each time t.

Â And, of course, this quantity here can be greater than or equal to 0 like the

Â regular dividend, but it can also be less than 0, okay?

Â So, you can think of a futures contract, as being a security that's always worth 0.

Â After all, it never costs you anything to purchase or sell a futures contract, but

Â it does create a stream of payoffs afterwards, and these payoffs can be

Â thought as dividends, or generalized dividends, and these dividends are given

Â to us by this quantity here. Okay.

Â How do we price a futures contract in the binomial model?

Â Well, we're going to work backwards from, from time n, the maturity of the futures

Â contract. So, we know it costs nothing to enter into

Â a futures contract at time n minus 1. So, 0 is the initial value of the futures

Â contract. As I said in the previous slide, the

Â futures contract is always worth zero. So, using the one period risk-neutral

Â pricing, that's all we're using here, one period risk-neutral pricing says, 0 is

Â equal to the payoff of the futures contract at maturity, which is time n.

Â And that payoff is Fn minus Fn minus 1. We discount by R, and we see 0 equals this

Â using the risk-neutral probabilities. Okay.

Â From this, we get Fn minus 1 is equal to the expected value of Fn using the

Â risk-neutral probabilities and conditioning and time and minus one

Â information. This follows because R is a constant so it

Â comes outside and it disappears and Fn minus 1 is known to us at time n minus 1.

Â So, in fact, Fn minus 1 doesn't need an expectation around it at all.

Â So therefore, we have Fn minus 1, equals the expected value of the futures price,

Â one period ahead, using the risk mutual probabilities.

Â We can generalize this to any time t and t plus 1, and get the exact same

Â relationship, using the exact same argument to get this relationship here.

Â Okay. So, this is the relationship for general

Â t, okay? Now, we can also recognize the fact that

Â Ft plus 1, is equal to the expected value at time t plus 1 of Ft plus 2.

Â That's just using this relationship, but taking t equal to t plus 1 inside here, we

Â get this. So, we can substitute this in for t, Ft

Â plus 1, to get this quantity here, and we can keep doing the same thing.

Â We know Ft plus 2 is equal to the expected value at time t plus 2 of Ft plus 3, we

Â can substitute that in for Ft plus 2 and so on and we get to this point up here.

Â Then, we can use what's called the law of iterated expectations and the law of

Â iterated expectations just tells us that we can collapse all of these expectations

Â just into the expected value of time t under Q of Fn, okay?

Â So, that's what the law of iterated expectations tells us.

Â You should be familiar from this from some of your probability courses.

Â If not, you don't have to worry about it. We're not going to be using it too much

Â during this course and it certainly won't appear in any of the assignments.

Â Okay. So, the law of iterated expectations tell

Â us that Ft equals the expected value of Fn condition in time t information using the

Â risk-neutral probabilities Q. So, in fact, Ft is what's called a

Â Q-martingale. And indeed, we've recorded an additional

Â module, which introduces us to martingales, and that module can be found

Â on the course website, as well, if you're not familiar with the idea of a

Â martingale. Okay.

Â So, we can take t equal to 0, recognize the fact that Fn equals Sn by the

Â definition of the futures contract. So therefore, we find F0 equals the

Â expected value of Sn at time 0, using the risk-neutral probabilities.

Â And again, this holds, irrespective of whether or not the security pays

Â dividends, the dividends would only enter into the calculation of the risk-neutral

Â probabilities, Q, as I mentioned a few moments ago.

Â What's interesting to ask this point is, are the forwards and futures prices equal?

Â And yes, they are. You can see this expression in 11 is

Â identical to the expression we have in 10 as well.

Â So, even though they're different contracts, the futures marks to market

Â everyday, there's a payoff everyday, that, that dividend payoff we spoke about,

Â whereas, the forward contract pays nothing everyday until the maturity, they we

Â actually have the same price. F0 equals E0 of Sn, using the risk-neutral

Â probabilities, Q. This is not true in general.

Â It's only true in the binomial model and other certain types of models.

Â The reason it holds true here, in fact, is if you were to go back and look at these

Â slides, you'll see one of the reasons it's true is when interests rates are

Â deterministic. So, interest rates are deterministic, so

Â we have to take this R outside and go through the rest of the analysis and see

Â that we got the futures price equal to the forward price.

Â In general, interest rates are actually random.

Â They move about through time and as a result, you wouldn't be able to take this

Â Rn outside and so in models of models that have random interest rates, you would find

Â the futures prices and forward prices are not identical.

Â They would be very similar, but they wouldn't be identical.

Â