>> In this module we're going to introduce you to futures, why futures are needed, how one can hedge using futures, and in later modules we're going to introduce you to the mechanics of the margin account associated with the futures. We call that, we had introduced these contracts called forwards, a couple of modules ago. Forwards are contracts that gave, the buyer, the right and the obligation to purchase a certain amount of an underlying asset at a specified price at a specified time. The problem with these forward contracts were that, they are a multitude of prices here, and as a result of that, they cannot be organized through an exchange. So what do I mean by multitude of prices? So let's say that, there is an expiration time capital T. If you construct, a forward contract at time t equal to 0. Associated with that particular contract would be a forward price F0. If you construct it at some other time t k, t equal k, there would be a different forward price Fk. If you construct something at t equal to u there is be another price, Fu. So all of these contracts are expiring at the same time, capital T. And they have different prices, depending upon when they were construct. So in every other respect, these contracts are similar except for the price. And because of that, it's very difficult to exchi, to set these contracts up, through an exchange. Because they are not set up through an exchange, there is no price transparency. Recall that price transparency is very important, to make sure that supply and demand sets fair prices, or no arbitrage prices. If there is no price transparency, one would not be able to construct arbitrage prices. That is supply and demand would not equilibrate to an arbitrage-free prices. Because there is no, price transparency and these contracts are not organized through an exchange, there is something called double-coincidence-of-wants. When you construct these forward contract, you have to have somebody take the opposite side. In order to find, that counterparty, you have to go look for it. And that might, lead to problems where certain forward contracts cannot get written because their counterparty is not available. There's also default risk of the counterparty. We talked about this in the context of slops. If you take on, a forward contract with a counterparty and the counterparty is not willing, to make the payments when the time comes, or is bankrupt. Then you expose yourself to unnecessary risk. So one needs a contract that works like a forward contract, that's able to fix prices sometime in the future, but is able to solve these other problems associated with a forward contract. That is, it's organized through an exchange. There's price transparency. You can get in and get out of this contract very easily. There is no counterparty with which you are contracting it's, you're contracting through brokers who are organizing in exchange. [SOUND] One such contract, is called the futures contracts. It solves the problem of the multitude of prices for the same maturity by marking to market. It just gives the profits and losses at the end of the day. So, price, there's a single price for, a single maturity. There aren't any more prices, based on when the contract was set. The contracts, because of the fact that these multitude of prices don't exist, they can be organized through an exchange. They can be written on any underlying security which has a settlement price. You can write it on commodities, you can write it on broad-based indices, such as, the S & P 500, the Russel 2000, etc. You can even write it for the volatility of the market, for example, VIX futures. There are a whole number of different assets that are available, and I would encourage you to go to this website at the cmegroup to look at the various kinds of, commodities, indices, and other measurable quantities on which future contracts are written. So here are, how a future contract work. An individual opens a margin account with a broker. It enters into, a certain number of futures contracts with a certain price of 0. So this is the futures prize that is available at time t equals zero. And in order to enter into these futures contracts you have to set up. An initial margin, that depends on, [SOUND] whether you're a hedger, or a speculator, and so on. And it typically is around 5 to 10% of the total contract value. All the profits and losses are settled using a margin account. If there are profits, which means that the futures price goes up, and you are having a long position, then the profit that you make is credited to your margin account. If the price goes down, then the losses are also settle through the margin account. If the margin becomes below, the maintenance margin, than there's a margin call, that is a broker asks you to put more money in. And you have to make it back up to the initial margin. I've given a snapshot of a particular worksheet of a workbook that I'm going to be working through, in a later module, that gives you, how this margin account works, and how the mechanics of the futures work out. In this particular module we'll be looking at some of the more theoretical results associated with futures. In the later module on excel, I'll walk you through how, how the mechanics workout. [SOUND] So what are some pros and cons of futures? Well, the pros are, you can have high leverage. You only put 5 to 10% of the total notional amount of the contract in the margin account. As a result, you can control very large sums of money, by putting a very small amount of money upfront, so you can get yourself very high profit. Futures accounts are very liquid, so you can take exposure to very different, many different kinds of assets very easily. It can be written on a wide variety of underlying assets, so if you want to hedge or speculate, you can speculate on a very wide range of assets. The cons are, quite related to the pros. The high leverages means, that you expose yourself to high risk as well. Futures prices are approximately linear function of the underlying, so only linear payoffs can be hedged. So if you have cash flow that is normally linear function new of some underlying, asset price or some underlying market indicator. Then you cannot hedge them using future prices, because future prices are linear functions of the underlying. Futures may not be completely flexible. Futures are organized through [SOUND] exchanges, which means that they mature at a specified date, they're written for a specified quantity, they're only written for certain commodities, and so on. And so, if you want to hedge something which doesn't quite, fit. The specifications of futures contracts, you might have to construct a one-off forward contract. You might have to go back to the broker, and construct a one-off forward contract. Or take on something called basis risk, which is going to come later on in this module. What about pricing futures? So in order to price futures, we need something called a Martingale Pricing Formalism. And this, comes because in its full generality, the interest rates are random or stochastic, and when you have stochastic interest rates, we cannot use the simple Arbitrage arguments that we have constructed so far, to construct a price for a future. If on the other hand, the interest rates are deterministic, and we know that forward price is equal to the futures price. We know how to construct the forward price using our Arbitrage arguments. And therefore, we know how to price the futures price. One thing that we do know, is that at maturity, the futures price f capital T is equal to the price of the underlying S capital T. And that's, this is what we are going to be using in the next few slides, to make some hedging arguments. [SOUND] So hedging using futures, so long hedge. [SOUND] Suppose today is September 1st, and a baker need 500 bushels of wheat on December 1st. So the baker faces the risk of an uncertain price on December 1st. This baker can use, a futures contract to fix the price that he could be exposed to, on December 1st. Here's the hedging strategy. You buy 100 futures contracts maturing on December 1st, each for 5,000 bushels. So you now have, taken on futures contracts that are for 500,000 bushels. What happens to the cash flow on December 1st? The futures position at maturity is going to be F capital T. Whatever is the futures price at time capital T minus F0. Whatever is the current price for the futures contract. I know that at time capital T, F of T is equal to S of T. Therefore, the futures position at maturity is going to be S capital T minus F0. If you were to buy the bushels of wheat, in the spot market, you have to pay S capital T. So the effective cash flow that you get, that is you buy in the spot market, and you take the profits that you get from the futures position. The effect of cash flow that the baker gets is ST. Which is coming, which is the payment. [SOUND] This piece of the payment is coming from the futures position. [SOUND] And this is, [SOUND] the payment that the baker has to make at the spot market. So S of T cancels, and effectively, [SOUND] the price [SOUND] of the beaker is fixed at F0. Did this cost anything? If you look at the, initial position and the final position, it would appear as if nothing happened. You could get into the futures contract with putting any money upfront. You end up getting the difference between the, the spot price at the future time minus the current price, as the profit from the future's position, if you combine that by the cost in the spot market. It appears that by not putting up any money you were able to fix the price of wheat to be F0. Which is the current futures price. But in reality in order for all of this to workout, you had to put money into a margin account. And you have to keep adding money to the margin account in case there are margin costs. If at any point during the time from 0 to capital T, from September 1st to December 1st, there is a margin call and you're not able to provide the money necessary to keep your position going. The broker is going to cancel your position and all the benefits that you were thinking about getting from the future position are no longer available. So even though it's not transparent, one has to keep in mind that there are costs associated with making sure that there is enough cash there to put up the margin calls when necessary. In the previous example, we had a perfect hedge. We assume that the futures contract matures right at the time that we want the money, but perfect hedges are not always possible. The date capital T at which we have a cash flow may not be a futures expiration date. The cash flows associated with whatever the quantity that we are trying to hedge may not correspond to an integer number of futures contracts. It was very lucky that we were buying 500,000 bushels of wheat because the futures contracts were written on 5,000 bushels of wheat. A futures contract on the underlying may not be available. If I want to hedge kidney beans. I don't have a futures contract on it. The futures contract might not be liquid. I may not be able to get enough quantity of the futures contract that I need. The payoff P of T may be nonlinear in the underlying. And then, the futures contract, which only gives me a linear payoff, will not be sufficient to hedge. The difference between the spot price for the underlying, and the futures price is called the basis. Futures price here refers to whatever futures contracts that we are buying in order to hedge the underlying asset, whose stock price is stochastic or random. When there is a perfect hedge, then the basis is equal to 0. When there is no perfect hedge because one of the reasons listed above, the bases are not equal to 0 at time capital T. The spot price of the underline is not equal to the futures price of the contract that they are using to hedge the underline. This is what is called basis risk. A basis risk as arises because the futures contract is in a related, but different, asset, or expires at a different time. Here's an example, today is September 1st, and a taco company needs 500,000 bushels of kidney beans on December 1st. The story's the same except now instead of wheat this particular company needs to hedge the price of kidney beans. Taco company faces the risk of an uncertain price of kidney beans. The problem is, that there are no kidney beans futures available. So we have to hedge, this particular uncertainty using a futures contract written on some other underlying. And therefore, have to take on basis risk. So, I'm going to buy soybean futures to hedge kidney beans. And the reason I'm going to do this is because I think that the price of kidney beans is correlated with soybeans. And, as a result, because soybean futures price is going to be correlated with soybean spot price, perhaps, I can use soybeans to hedge kidney beans. I'm going to buy, an amount y of the soybean futures. Each of these futures contracts are for 5000 bushels of soybeans. So what are going to be my cash flows? The cash flow associated with the futures position at maturity is going to be F of capital T minus F0 times y. And then, I'm going to buy kidney beans in the spot market, so it's going to be some cash that I have to pay for that, which is P capital T. So the effective cash flow is going to be y of FT minus F0, plus PT, which is the cash flow associated with buying kidney beans in the spot market. PT is not equal to y times FT for any y, and therefore, a perfect hedge is impossible. So what happens? What can we do? So instead of trying to get a perfect hedge where, the effective cash flow is exactly equal to zero, I'm going to try and minimize the variance of the cash flow. Variance of CT, can be written as variance of PT plus the variance of y times Ft minus F0, plus two times the corea, correlation between Y, FT minus F0, and PT. [SOUND] Now F0 is a constant. [SOUND] Because a time, T equal to 0, this is known. So this expression can be written as the variance of PT which is the same term as before. Now I'm ignoring the constant. So I write this as variance of y times FT. I take the y outside, and whenever I pull a constant out of the variance, I get the square of the constant. So it become y squared times the variance of FT. I can ignore the constant here, when I'm calculating the covariance. I'm going to take the y outside. But when I take a y outside of the covariance, just the y comes out. So it's going to be 2 times y plus the covariance of FT and PT. I'm going to take the derivative, of this expression with respect to y. So y is unknown. I don't know how many of these contracts I want to buy. So I take the derivative with respect to y. You end up getting, the expression to be 2 times y variance of FT plus 2 times the corea, covariance of FT and PT must be equal to zero. This is, this is the amount of y that is going to give me the minimum variance hedge. And therefore, the optimal number of futures contract is simply given by the solution of that equation, which is minus the covariance of FTPT divided by the variance of FT. This tells me exactly, how many covar, contracts am I going to buy, in order to hedge the