>> We're now going to see how to price forwards and futures in the binomial model. We'll see that they're very straightforward to price even though the mechanics of these securities are different. We'll also ask the question, how do the prices of forwards compare with the prices of futures? We'll see that in the binomial model, they're actually identical, although we will make the point that this is not true in general. We're going to start with an n-period binomial model. Although, in this case, we see n is just equal to 3, u equals 1 over d as usual. And, of course, we also have our usual cash account which pays a gross risk-free rate of r in each period. Okay. So, consider now a forward contract on the stock that expires after n periods. And we're going to let G0, denote the t, the date t equals 0 price of the contract. Now, I have price, and in, in quotes here because there's often a lot of confusion over the word price. Because people sometimes think that if something is a price, then that is what you must pay to purchase the contract but, of course, that is not true. G0 is chosen so that the contract is initially worth zero. So therefore, when you buy a forward contract, no money changes hands, in fact, the initial value of the forward contract is zero. The so-called forward price, here, G0, is just used to determine the payoff at the maturity of the forward contract. There's also a similar situation with futures, which we'll discuss when we come to the pricing of futures in the binomial model, in a few moments time. Okay. So, back to the forward contract. G0 is the price of the contract, but actually, the value of the contract, when we enter into it, is 0. So, using risk-neutral pricing, the initial value of the contract is zero. That's how much we must if we buy this forward contract. We get nothing until time n, and at that point, we get Sn minus G0, this is the payoff of the forward contract. So therefore, risk-neutral pricing says that 0 is equal to the expected value using the risk mutual probabilities of the payoff discounted and the discount factors are to the power of n. So, this is simply, risk-neutral pricing. Okay. Now, what we must do, is we need to figure out, what is G0. That's the goal here, to figure out the fair value of G0. We will do this by just looking at this equation. We notice, first of all that Rn, is a constant, okay, the gross risk-free rate is a constant so it comes outside the expectation. And so, we actually just get 0 equals the expected value of Sn minus G0. Remember G0 is also a constant, it's chosen at time 0. So, it is not a random quantity, so we don't have an expectation around it. So, of course, this implies G0 is equal to this. And this is the forward price of the contract. Okay. And 10 holds whether or not the underlying security pays dividends. We've now discussed dividends in the context of the binomial model. We didn't mention dividends at all here. But, in fact, dividends can be president, present in the model and this is still the correct price. The only point where dividends will enter is in G, okay? If you remember, the risk-neutral probability is Q, alright, given to us by, it's going to be R minus d minus c divided by u minus d, okay? And so, if the security, if the underlying security pays dividends, well, this is going to enter into the risk-neutral probability. It will lower the probability, the risk-neutral probability of up moves, and make the forward contract a little cheaper than, would otherwise be the case, okay? And we'll make the forward contract a little cheaper than would be the case if dividends were not present. Okay. Futures. Consider now a futures contract on the stock that expires after n periods, okay? Let Ft denote the date t price of the futures contract, and again, I put price in quotes because Ft isn't the value of the futures contract, alright? If we enter into a futures contract at any time, it actually costs nothing. The fair value of a futures contract at any time is actually zero, okay? So, as was the case with the forward contract, this futures price is really used to determine the payoffs of owning the futures contract. So, we'll come to that in a moment when we actually price the futures contract. Okay. So, the futures contract expires after n periods. So therefore, we know Fn equals Sn. This is almost by definition. These would be the terms of the futures contract. It expires at time n and according to the rules of the contract, Fn is equal to Sn. So, this must be the case. Okay. So, as I mentioned earlier, a common misconception is that Ft is how much you must pay at time t to buy one contract or how much you receive if you sell one contract. This is false. A futures contract always costs nothing. The price Ft is only used to determine the cash flow associated with holding the contract. So, that plus or minus Ft minus Ft minus 1 is the payoff received at time t from a long or short position of one contract held between times t minus 1 and t. So, it will be plus Ft minus Ft minus 1, if we were long or we owned one futures contract. And it would be minus if we were short or we had sold one futures contract between times t minus 1 and t. So, in fact, some people will often characterize a futures contract as follows. They will say that a futures contract is a security that is always worth 0, but that pays a dividend, I should put quotes here, it's not a dividend like the dividend you get from a stock. So, it pays a dividend of Ft minus Ft minus 1 at each time t. And, of course, this quantity here can be greater than or equal to 0 like the regular dividend, but it can also be less than 0, okay? So, you can think of a futures contract, as being a security that's always worth 0. After all, it never costs you anything to purchase or sell a futures contract, but it does create a stream of payoffs afterwards, and these payoffs can be thought as dividends, or generalized dividends, and these dividends are given to us by this quantity here. Okay. How do we price a futures contract in the binomial model? Well, we're going to work backwards from, from time n, the maturity of the futures contract. So, we know it costs nothing to enter into a futures contract at time n minus 1. So, 0 is the initial value of the futures contract. As I said in the previous slide, the futures contract is always worth zero. So, using the one period risk-neutral pricing, that's all we're using here, one period risk-neutral pricing says, 0 is equal to the payoff of the futures contract at maturity, which is time n. And that payoff is Fn minus Fn minus 1. We discount by R, and we see 0 equals this using the risk-neutral probabilities. Okay. From this, we get Fn minus 1 is equal to the expected value of Fn using the risk-neutral probabilities and conditioning and time and minus one information. This follows because R is a constant so it comes outside and it disappears and Fn minus 1 is known to us at time n minus 1. So, in fact, Fn minus 1 doesn't need an expectation around it at all. So therefore, we have Fn minus 1, equals the expected value of the futures price, one period ahead, using the risk mutual probabilities. We can generalize this to any time t and t plus 1, and get the exact same relationship, using the exact same argument to get this relationship here. Okay. So, this is the relationship for general t, okay? Now, we can also recognize the fact that Ft plus 1, is equal to the expected value at time t plus 1 of Ft plus 2. That's just using this relationship, but taking t equal to t plus 1 inside here, we get this. So, we can substitute this in for t, Ft plus 1, to get this quantity here, and we can keep doing the same thing. We know Ft plus 2 is equal to the expected value at time t plus 2 of Ft plus 3, we can substitute that in for Ft plus 2 and so on and we get to this point up here. Then, we can use what's called the law of iterated expectations and the law of iterated expectations just tells us that we can collapse all of these expectations just into the expected value of time t under Q of Fn, okay? So, that's what the law of iterated expectations tells us. You should be familiar from this from some of your probability courses. If not, you don't have to worry about it. We're not going to be using it too much during this course and it certainly won't appear in any of the assignments. Okay. So, the law of iterated expectations tell us that Ft equals the expected value of Fn condition in time t information using the risk-neutral probabilities Q. So, in fact, Ft is what's called a Q-martingale. And indeed, we've recorded an additional module, which introduces us to martingales, and that module can be found on the course website, as well, if you're not familiar with the idea of a martingale. Okay. So, we can take t equal to 0, recognize the fact that Fn equals Sn by the definition of the futures contract. So therefore, we find F0 equals the expected value of Sn at time 0, using the risk-neutral probabilities. And again, this holds, irrespective of whether or not the security pays dividends, the dividends would only enter into the calculation of the risk-neutral probabilities, Q, as I mentioned a few moments ago. What's interesting to ask this point is, are the forwards and futures prices equal? And yes, they are. You can see this expression in 11 is identical to the expression we have in 10 as well. So, even though they're different contracts, the futures marks to market everyday, there's a payoff everyday, that, that dividend payoff we spoke about, whereas, the forward contract pays nothing everyday until the maturity, they we actually have the same price. F0 equals E0 of Sn, using the risk-neutral probabilities, Q. This is not true in general. It's only true in the binomial model and other certain types of models. The reason it holds true here, in fact, is if you were to go back and look at these slides, you'll see one of the reasons it's true is when interests rates are deterministic. So, interest rates are deterministic, so we have to take this R outside and go through the rest of the analysis and see that we got the futures price equal to the forward price. In general, interest rates are actually random. They move about through time and as a result, you wouldn't be able to take this Rn outside and so in models of models that have random interest rates, you would find the futures prices and forward prices are not identical. They would be very similar, but they wouldn't be identical.
