Now, for our contracts have been in existence for hundreds of years as I mentioned before, but they are not really very practical. There is two main disadvantages relative to what will happen with the futures. One is if I want to know how much I will have to pay in the forward contract at maturity, then I have to remember what the forward price was three months before when I had when I entered the contract. We have to remember for everyone trading forwards when they entered the market at what price because the forward price is going to change every day, even for the same delivery. Let's say the delivery is on January 1st of 2020. Today, you entered the forward contract for that maturity, the price is going to be different tomorrow because the price, as we saw multiplies s times small t, but that's the small t moves from one t to another. With forward contracts, you will have to remember what the forward price was when you entered the contract to know how much it has to be paid at maturity. That's one thing. The other thing with forward contracts is that if one of the counterparties goes bankrupt or default and cannot deliver its promise, you have a problem. There is a default risk. There's a credit risk. Instead, the way forward contracts are mostly implemented in today's market is in a future exchange. Futures are our way to implement basically forward contracts in a way that avoids those difficulties first of all, because you don't really trade with a specific counterparty with trade with an exchange. You enter a futures contract with a futures exchange, so the exchange guarantees that you will have your promised payment at maturity if it's a payment. Otherwise, you may lose money. If the exchange doesn't go bankrupt, which is not very likely, you are okay, and the other thing is the futures will be revalued. You will get your profits and losses from the futures position every day, and it's called mark-to-market. It's going to be mark-to-market every day. We are going to split your profits and losses into small amounts every day instead of waiting for three months to have your total profit or your total loss. That way, you don't have to remember what happened the day before, or two days before, or three months before because it's already been accounted for. It's been mark-to-market, and there is no need to remember the past. It's a simple idea, and formally, let's see what's happening. Assume that you enter a futures contract at time, small t, and then the next day comes and the futures price changes to f of t plus 1. Let's just get a paint brush here, and then what happens with the margin account is that those two prices are compared to the yesterday's price, the initial price, and tomorrow's price, and this can be positive or negative. If it's positive, you get money into your margin account, and if this is negative, you have to pay from a margin account. That's what's called marking to market and you do it daily, and then you continue like that. The day after, you look at the difference from yesterday's price. Again, you either get money or pay money from your margin account until the end when finally, you get the difference between the last days futures price and the day before. Then assume for a second that we know that f of capital D has to be equal to s of capital T, that the futures price has to be equal to the spot price at the end, I'm going to argue that in a second. If that is true, here everything will cancel because here we have f of t plus 2 minus f of t plus 1 which is f of t plus 1 will cancel, then f of t plus 2 will cancel and so on, f of t minus 1 will cancel. The only two terms which will not cancel will be the final term, f of T which is s of T and the initial minus f of t term. You add up all these profits and losses of Romeo daily marking to market of the margin account and you get the same payoff as with a forward contract. It's just the forward contract which is split into daily parts but it's the same payoff now. Why is this final future price equal to the spot price? Well, what does it mean? F of T, it means the future's price on a future contract which immediately matures, which immediately ends at the same time it is entered, that's just buying the asset on the spot, has to be the spot price otherwise there would be arbitrage. The bottom line is the total payoff if you ignore the interest rates in the margin account, the total payoff is equal to the payoff of equivalent or an equivalent forward contract. Now the question is, is this is then the futures price which I still denote here as F of t should be equal to the corresponding for a price under perfect market conditions, no arbitrage. Well, we did have that law of one price which said that if the payments in the future are the same, then the price should be the same. But there was a comment that those payments should be the same at the same times. This is not the case here. In a forward contract, everything happens at the very end and maturity, that's when the whole payment is paid. Now, with the futures contract, the payment splits and you get it's across daily, either get it or have to pay it. The timing is not the same, so I cannot apply the law of one price directly. However, there is this results, this claim here that if you do know what the interest rate will be in the future, so if the interest rate is deterministic, then in a perfect market and also in arbitrage futures price has to be equal to the corresponding forward price. Let me argue that with the replication arguments. I'm going to take today to be zero just for notational convenience and I'm going to replicate the payoff of a forward contract by trading in futures. Then I will argue because I can replicate the forward contract by trading in futures and the price should be the same. Replication arguments, I can create the same payoffs at the same times by trading in futures the same payoff as for the forward contract therefore the price should be the same. Here's the argument, there is initial position, you go long into e to the minus r, t minus 1 futures. I'm going to assume that there is a continuously compounded constant interest rate just for simplicity, but the important thing is that the interest rate is known. Then at time one, I'm going to increase to e to the minus r, t minus 2 futures. T minus 2 is less with the minus sign is larger numbers. I'm increasing the position futures and so on. I keep increasing to e to the minus r, t minus 3. Finally, one day before maturity, I increase my payoff to the one future contract. I have exactly one future contract by slowly increasing the futures position. Let's see what the profits or losses payments are in each day. In particular, any arbitrary periods, k to k plus 1, if I held exactly one futures contract, my profit would be as in here, profit loss would be f of k plus 1 minus f of k. However, I don't have one, I have less than one. In fact, if you think about it in terms of notation, I have exactly from k to k plus 1, I'm holding e to the minus r, t minus k plus 1 futures contracts, that's by this strategy here. This is my profit loss f of k plus 1 minus f of k times the number of positions in futures which is the number of futures contracts which is this. But then I want to see if I keep this in my margin account which I do, then now I'm not ignoring the interest rate, I want to see how much this will be at maturity. This is the profit loss during the period k, k plus 1. The question is how much this will contribute to my final payoff at maturity T? What do I have to do? Well, I have to multiply this by the e to the r times the number of remaining days. But the number of remaining days is exactly t minus k plus 1. I'm multiplying this by e to the plus r of t minus k plus 1. Minus r, t minus k plus 1 will cancel with plus r, t minus k plus 1 and you add the exponents. You will get e to the zero which is one. The total contribution of this part or an integral k to k plus 1 in this period is exactly f of k plus 1 minus f of k. This is why we set up the strategy so that it will in the end increase to f of k plus 1 minus f of k. Then if I add all those contributions, I'm going to get exactly s of t minus f of zero, and now this is the payoff that I have at maturity. The timing is the same as the forward contract, this is in fact the payoff I will have accounting for the interest in my margin account. This is the payoff I will have at maturity. It's the same as the payoff of forward contract by the law of one price or by no arbitrage, the f of zero, the futures price at time zero has to be the same as the forward price of the corresponding contract. That's the argument. Now, it wouldn't work and doesn't have to hold if the interest rate is random and you cannot know what interest rate will be in the future. Then the forward price may be higher or lower than the futures price depending on the random properties of the interest rate in particular, how it is correlated with the underlying assets. Whether the interest rate tends to go high when the underlying asset goes up in price, or whether it tends to go down. That will affect whether when you are getting money into the futures, into the margin account, whether the interest rate is higher or lower or when you're losing money, is it higher or lower? That will depend on this correlation between the interest rate and the underlying asset. It can be either way. The futures price can be higher or lower than the forward price if the interest rate is random. But if it's not, at least until the very end of course for us the interest rate will be deterministic, then futures are basically equivalent to forward contracts. The price is the same, it's just a different way that the payments are paid. That's about the futures. Let's see what's next.