Ses 10: Forward and Futures Contracts II & Options I

the following content is provided under a Creative Commons license your support will help MIT OpenCourseWare continue to offer high quality educational resources for free to make a donation or to view additional materials from hundreds of MIT courses visit MIT opencourseware at I hope you all had a good Columbus Day weekend the stock market certainly did any questions from last time no okay so what I want to do today is to continue talking about futures and forward contracts today we're going to finish up on these interesting instruments with a couple of examples and then with a specific method for pricing forward in futures contracts so let me refresh your memory it's been a week so I know so we're going to go back and look at a specific futures contract and I'm going to take this contract and then try to talk a bit about how you might use contracts like this in hedging your risks as well as in making certain kinds of bets if you will so remember that this contract is a contract that was issued on July 27th 2007 so the middle of the summer for oil to be delivered in December and there's a specific date in December where all oil futures contracts of this type will settle that is will come to maturity where the date is going to be specified in advance and everybody knows it and so in July when you buy this contract at a price of 70 506 per barrel and each contract is for a thousand barrels when you quote buy the contract what that means is that you are agreeing today July 27th you are agreeing today that come December you are going to buy a thousand barrels at a price of 70 506 per barrel okay so that's the agreement and the party who is selling you the contra the counterparty is agreeing to provide you with that oil at that price in December so the futures price is 75 oh six and as we said last time the current market price on July 27 2007 that's called the spot price the spot price may be higher or lower than the futures price depending on what expectations are for what's going to happen with oil over the subsequent six-month period okay now the initial margin as I mentioned last time was 4050 dollars the maintenance margin the margin that you need to maintain so that if the initial margin goes down in value you have to actually put money back into your brokerage account your margin account and if you fall below that 3000 dollar threshold you will get a phone call which is known as a margin call several weeks ago somebody once said that somebody told me that they've been getting lots of phone calls all from the same person the person called margin and you know that can happen when markets go awry now again no cash changes hands today because the value of the contract when it's struck is a zero NPV transaction and how do you know what zero NPV again because if it's positive for one party it's coming from the other party so which party would you like to be you'd like to be the party receiving that positive NPV nobody wants to be the party that is losing the NPV so the futures price will adjust in order to make it zero NPV in fact that's what we mean when we say that at zero and PV it is the futures price that makes it so so I'll give you an example if it turns out that somebody suggests a futures price a $60 a barrel on that day lots and lots of people are going to want to buy that contract because that's a good deal relative to where oil really should be and that means lots of people are going to want to buy but nobody's going to want to sell at that price of $60 so everybody wants to buy and nobody wants to sell what has to happen to the price exactly goes up and it keeps going up until the number of buyers equals the number of sellers that's the point at which it's a zero NPV transaction okay now let's take a look at what the payoff is of such a contract on date 0 in this case July 27 2007 the contract is worth nothing but if the if the futures price moves tomorrow then the contract could actually have value and a diagram of how that works is something like this if the futures price not the spot the spot obviously will move also but I'm talking about the futures price because the futures contract is specified so that every day's worth of gains or losses in the futures contract relative to its price you will have to either get paid or you will have to pay so this blue line shows you the payoff if you're holding a long position in one of these contracts if you bought a contract okay if you sold the contract then your payoff diagram is the dotted line all right now the blue line basically says that if the futures price goes above seventy five dollars and six cents then you make money if the futures price goes below seventy five dollars and six cents you lose money so when you buy a contract like this it is as if you actually bought the oil but you haven't really bought the oil all you've done is to buy the right and obligation to purchase the oil in the future okay now let me let me give you another example that will make this even more concrete all right the only way to understand this because this is not a natural security for most of us okay stocks and bonds you might find sort of natural futures contracts are weird in that you know they have zero investment today and so they're worthless today but they're not worthless after the initial date when you enter into that agreement so let's do an example yesterday yesterday you bought 10 December live cattle contracts on the Chicago Mercantile Exchange at a price of 74 55 per pound okay so you basically bought some cows and the contract size is 40,000 pounds of cow I don't know how much Cal that is but even if you're on the Atkins diet that's plenty of cow all right and so you what you have though in this contract is not the cows but rather you have the obligation to buy the cows in December for a price of 74 55 per pound and there's 40,000 pounds of it so the value of your position the value of your position is the size of the contract multiplied by the futures price multiplied by the number of contracts so it's two hundred and ninety eight thousand two hundred the size of your position or sometimes it's also called the notional size you've heard that term over the last few weeks notional well this is an example of a notional so you don't actually own two hundred and ninety eight thousand two hundred because of course we've said that the contract is zero NPV when you enter into it all you've done is to agree to buy 40,000 pounds of how in December at a particular price so the idea is that you control the notional amount of 298,000 201 you do specifically get is the profits and losses from that notional so let's do an example that was the position yesterday no money changed hands you've got some initial margin that you have to put down but that's really your money in a brokerage account you're not giving it to anybody it's safety money it's collateral right okay now today what happens let's suppose that today the futures price closes at seventy four thirty five all right it's just gone down by you know whatever two tenths of a set the value of cattle has gone down you're holding long this cattle contract maybe you're a restaurant you know some restaurant or you have a chain of steak houses and so you need to buy cattle on a regular basis so the price of cattle just went down did you make money or lose money yeah you lost if you're long on the other hand if you're a cattle farmer and you sold the contract you did a good thing because you locked in the price of 7455 and now the price went down so let's calculate what the value of the notional size of the position is it's twenty nine seven four hundred that yields a loss of eight hundred so you know what happens today today your broker will deduct eight hundred dollars from your account from your margin account and take that eight hundred dollars and put it into the cattle farmers account so now he has the eight hundred dollars now if the day after if tomorrow it turns out that the price of cattle goes up by two-tenths of a cent it goes back to 74 55 you know what happens you get eight hundred dollars back now the cattle farmer loses that e100 and gives it back to you you see this way you always settle up every day so if for some reason the cattle farmer ends up going bankrupt and isn't able to deliver any cattle to you then you're out at most one day's worth of movements and that's one of the reasons why futures markets are and futures brokers are so careful about closing down accounts that don't meet their margin requirements it's because they don't want to have credit risk lingering growing and you know unknown the first moment that you do not make a margin call you do not deposit the requisite margin the first time that that happens they have the right which they exercised always to close down your position you're out of the game and that's the end so it reduces dramatically the amount of credit risk that either counterparty has I don't have to trust you that three months from now you're going to actually have 40,000 pounds of cow for me all I need to do is to make sure that this contract settles every day and the uncertainty then gets resolved day by day but your credit risk is very well managed okay and mine is too so this is a very important innovation it's very different from a forward contract in the sense that forward contracts contain enormous amounts of credit risk right because once we enter into a forward that's just like a futures contract but the difference is that we don't exchange any money ever until the settlement date and by that point you could be so far out of the money you could be so far you know in in debt to me as well as to other creditors that you just can't afford to pay and so I'm stuck with this piece of paper that says you're going to give me 40 thousand pounds of cattle and you can't even afford to buy me a steak dinner a problem right so this futures exchange is a beautiful thing it reduces credit risk it also encourages liquidity it encourages trading why because at any point in time on any given day between now and December if you decide that you want to get out of the restaurant business and you don't want this contract anymore you can get out of it poof you just get out of it by doing an opposite transaction so if you bought a contract for December you know what you do when you wanna get out of it you sell a contract for December you literally sell so it's actually duplicated transaction but it's of the opposite sign and so they cancel out right because you're going to get delivery and you will provide delivery you know and those will cancel out yeah Justin what are we doing yeah so let's say I had a long position in oil and then I found out that I was with that money and I decided just for go mine report mine market what else would happen well you know you first of all you are you are liable for all of the losses not just the margin so the margin account is not meant to be a non-recourse loan they will go after your assets now you could declare bankruptcy personal bankruptcy and get protection under Chapter 7 but that will hurt your credit ratings and all other nasty things will happen to you if that occurs they close out your account when your margin yeah yeah like what if what if you so they close it out but your losses are higher than your marketing points anyway right so you're stupid you're still liable for those in addition to okay so you make a good point is it is it generally possible that your losses are greater than the amount of margin so in that case who gets left holding the bag you know who gets left holding the bag that blue box in the middle the futures Clearing Corporation but the reason that they establish a particular level of margin is to be able to ensure that that's a very unlikely event and it goes back to what are the likely daily swings in the futures price if you put enough margin in your account so that I can be sure that you can cover 99% of the time you can cover the daily swing then I don't have to worry about now of course if we had a day like last Friday or on Monday you know I mean it's that's pretty outrageous that's one of the reasons why a number of futures exchanges have increased their margin levels it's because the daily swings have gotten much bigger but as long as they can cover the one-day movement they don't have to go after your home or your other assets captain's contract oh no you certainly cannot sell it two dollars oh you gotta sell it that he says yeah really you are out of position because you don't have you don't have a claim or you don't have a commitment to enter into that trade in December that's what I mean when I say you're out of the position you also happen to be out of money in your example in other words you lost 50 cents that that's not coming back okay but what it means to sell is that you bought a contract that says in December you're going to buy 40,000 pounds of cattle okay you're committed to doing that now if on the other hand the next day you decide you want to get out of that commitment the way to get out of it is not to try to contact the counterparty say would you mind canceling my trade the way to do it is to simply sell a commitment at 40,000 pounds of cattle for December so your commitment to buy and your commitment to sell basically cancel each other out so on settlement date the futures Clearing Corporation will net out all of these buys and sells and the net amount will be transacted between the providers of the cattle and the buyers of the cattle that's right that's right so that's an example where if you bought and sold then you'll be netted out and you would not have physical delivery right yeah well their job is really not to make money but to create an exchange for its members so many exchanges are not for profit some of them are for profit but the objective of the futures Clearing Corporation is really not to make a lot of money what they're trying to do is just create a market and let people who want to trade with each other trade freely and efficiently they will charge a perhaps a small transaction fee that you have to pay in order to support the operations but they're not trying to make tons of profits off of that necessarily now they may be trying to make profits off of other activities but the objective of the Clearing Corporation itself is not to make tons of profits it's really just to provide a stable environment where people can transact and in some cases the members of the exchange own the Clearing Corporation so it's their own dollars that support the actual physical operations of the organization yeah I think the band in the papers was people are trying to sell it outside because otherwise it III recall a spike I certainly don't recall the logic about physical delivery I mean you know it could be that there are a number of people who are long the oil contracts that that that basically want it to be cash settled and the way that they have it cash settled at a particular point in time before settlement date is they close out their positions and so by closing out their positions they basically reverse the trade and that would actually put put put down push down the oil price so maybe the reverse argument a lot of short sellers were trying to argue that oil is going to go down and they wanted to cover their positions so they bought but you know the in any case you don't have to take physical delivery if you specify to your broker that you want all of this to be cash settled would you say that the credit risk involved in a forward contract is somewhat similar to the credit risk in credit default swaps and if so is there something analogous to a credit default swap that's similar to a futures contract if so the answer to your first question is yes because a credit default swap contract is basically a kind of a forward contract it does involve intermediate payments but if it turns out that the credit changes dramatically those intermediate payments either may be too much or not enough to cover the underlying value of the contract and after you strike a credit default swap it will take on value as for an exchange what a wonderful idea that exists exactly what's being proposed now it hasn't been done yet but there been a number of proposals to set up exactly a structure like this for credit default swaps in order to do that you have to standardize those contracts and you have to be able to do the paperwork in a relatively efficient manner and so that's actually being discussed debated and I think that there's a proposal by the Chicago Mercantile Exchange to start doing that if you do start doing that you will see that market growing even bigger than it is today and at at the same time the risks are actually going to decrease because with daily settlement of credit fault swaps just like with futures contracts all you need is one day margin in order to eliminate 99% of the problems so so let me that's a good question let me now talk about how to price futures and we'll take an interest rates explicitly okay so the question is what determines either a forward or a futures price you now know what a futures price is right it's the price at which you are willing to do a future transaction what determines that price we say supply and demand and the market but is there some logic that we can give to this process and the answer is yes we're going to use the exact same argument that we use for pricing everything else we're going to come up with two identical cash flows and two assets that have identical cash flows have to have the same what price value right okay so for now I'm going to actually ignore the difference between futures and forwards the only difference is the back and forth amount of money that we give to each other and therefore the accumulated interest or forgone interest that we pay when we put our money back and forth into each other's accounts so let me abstract from that and you know if you're interested you can actually see the derivation of that in in recitation I want to focus on the the bigger question of how these things are priced with respect to other prices so let me start with some notation I've got a particular contract let's say a futures or a forward contract and I've also got the spot price of the asset at a point in time so I'm going to let s T denote by spot price I'm going to let f of little t big t to determine the forward price and H of little t big t the futures price and for now I'm going to just assume that H and F are pretty close now notice that when I write down a futures price or a forward price I've got two sub-indexes right I've got little T and big T the reason – is that for every forward or futures contract there are two dates you need to worry about the date at which you are pricing the contract namely today and the settlement date right so you need to have those two indexes so right away we know that this contract is a little bit more complicated than say a stock where there is no settlement date right okay so I want to go back to a comment that was made by one of you when we first started talking about futures and forwards and the comment was why go to the trouble of using these contracts when you could just buy the asset itself and hold onto it if you if you need oil in December in order to make sure you have oil in December why don't you just buy it in October and hold it for two months then you have it in December and in fact that's exactly what we're going to do to figure out what the appropriate prices of the specific futures or forward contract so here we go I'm going to do my exact same analysis that I've done many times before when we tried to price bonds and stocks and other basic securities the left-hand column here is going to be the cash flows associated with a typical forward contract okay so a forward contract is one where you enter into the contract let's say a date zero and you pay nothing for the contract right there's zero NPV transaction and you are long the forward contract with the forward price F of zero comma T the only cash flow that occurs with a forward contract is on settlement date and on settlement date you've agreed to pay F of zero T for delivery of whatever it is that you bought the forward contract on right so the only cash flow that comes out of a forward contract is this F right here everybody see that right nothing up my sleeve it's very simple kind of calculation okay now I want you to look at the right-hand column which is going to be less simple the right-hand column I want you to imagine doing the following I want you to imagine buying the commodity at day zero however I don't want you to use any money I want you to buy it with no money down all right that's the the start of a scam it sounds like it but I promise you it's not so the way you're going to buy the commodities you have to pay the price the spot price and the spot price is s Sub Zero you don't have s sub zero so borrow it okay now I'm going to abstract from credit risk which I know is on everybody's minds today but let's suppose that you're all good credits so I'm not worried about loaning you the money at the risk-free rate so now you've borrowed S sub zero dollars and then you spent it right away buying the asset okay so as of date zero in the right hand column you own the asset now you have to wait t periods and while you wait you may have some costs for example if the asset that you bought is gasoline well you've got to store it in just the right way right you probably don't want to put it next to your furnace in the basement you probably want to put it in a cool place isolated and so on and so forth on the other hand if what you bought is uh you know pork bellies you probably want to put that you know in a freezer compartment as opposed to you know in your garage so you might have to pay costs for storing and at the end of that time T you have to pay interest on your loan so you borrowed S sub zero dollars you don't get that for free you got to pay interest on it this is the question about interest so you got to pay interest on that money and so you have to pay back at this point capital T you have to pay back the money you borrowed S sub 0 1 plus R to the capital T plus whatever your storage costs are but I'm going to allow that having the asset around might be kind of convenient there might be a benefit to having the asset around a convenience yield maybe if you need to use it sooner you have it there and and having it there saves you a little bit of trouble in order to be able to get whatever it is you need to get done with that underlying asset so I'm going to deduct from my cumulative storage costs any convenience yield that's a future speak for any kind of benefits that you get from holding on to the physical asset ok so your net storage costs are given here that's what you pay at the end of T periods I argue that these two cash flows give you the exact same value of the asset in other words in both cases you happen to have the asset at the time T so these two contracts have to have the same value because they offer the same set of cash flows in terms of the underlying commodity right you get the commodity in both cases so another way of thinking about it is if your objective is to have 40,000 pounds of cattle in December both of these will get you to the exact same point both of these cost you nothing on date zero and therefore if they cost you nothing and they give you the same outcome at the end they've got to sell for the same price so this has to equal this that's it that's the simple argument and the counter argument or proof that this has to be true is let's assume it's not let's assume that this is a lot bigger than this well if this is bigger than this then what should you do what right which one which one so forward contract and then buy this thing whatever it is do this okay now what if it's the reverse what if if this is bigger than this then by the forward and then do the opposite of this right flip it around short sell the asset if you can and then take the money and lend it out at interest rate R and dot dot dot you follow the logic right so that gives us a relationship between the forward price and other stuff and what is the other stuff the forward price has to be related to the spot price the interest rate the time to settlement and any other weird things about the commodity that may affect the value of it like the storage costs or the convenience yield you've got to factor that in okay so this is the relationship that tells you how to price a forward contract now a futures contract is almost like a forward the only difference is the interest differential on a daily basis where you actually are moving money back and forth into our accounts but the cumulative sum is going to end up being approximately the same so for the purposes of this class I'm going to assert that this is approximately the same in fact you can show that there's a another relationship that looks at the interest rate per period and it's a little bit more complicated but not much more complicated you can see that in your textbook okay if you're interested but for now I want to just focus on this relationship this relationship tells us how to price futures and forwards okay and now if I divide by 1 plus RF to the T then what I've got is that the forward price divided by the interest rate that calculates the the current value of that forward price has got to equal the spot price plus the present value of the net storage costs this is the relationship that we've been looking for and you guys have been struggling for for the last couple of lectures you've been asking well gee doesn't the interest rate belong in there and what about having the asset wouldn't it be nice to have it and so on and so forth all of those considerations are summed up in this one expression a very nice expression right very intuitive right what you pay at date t when you take the present value of it that has to be equal to what the thing is worth today plus any benefits for having the thing as opposed to not having the thing between now and settlement that's it now this is for the very beginning when you strike the contract what about at an arbitrary point in time between 0 and T well all of these arguments work exactly the same way when you're looking at two dates T and T as opposed to 0 and T so the relationship that I showed you it's a little bit more complicated now because you've got to take into account the fact that the time to settlement is not capital T its capital T – where you are today but that's the only change other than that everything is the same okay and you have to make sure that you accumulate the future value of all the net storage storage costs so that you actually move all of the costs to the end and then you bring it back to time T ok now let's take this out for a spin let's see how this works let's take a look at gold ok gold is easy to store there's no there's no storage cost really I mean gold is relatively compact a little heavy so you're gonna have to lift it and put it in your vault as some of you I'm sure are doing nowadays all right but but the bottom line is that the storage costs are negligible there are no dividends gold does not pay out dividends there are no real benefits either there's no convenience yield it's not like you know you need a little piece of gold every once in a while for your pleasure and so you want to scrape that off and enjoy it it just sort of sits there so if that's the case you factor that into that relationship that I showed you and that last term the pv of net storage costs is nothing and so the relationship is really simple the forward slash futures price today is just equal to what the current spot price is multiplied by one plus the risk-free rate of interest between today and settlement day if this relationship is violated when you look at gold futures and gold spot and you see that this relationship is violated that's a sign that there's an arbitrage you can make money off of that so that really is the way to make a million dollars with no money down you know is to try to find violations of this arbitrage relationship it's going to be hard because a lot of people that are looking at it all the time and so when there is an inequality of some sort it's probably not going to be very big and it probably won't last very long but to the person who found it first they might actually be able to make a little bit off of that discrepancy by either buying or selling gold and transacting these markets quickly let me do another example so this is gold what about gasoline gasoline it turns out is very different from gold first of all it's a pain in the neck to store safely so if you don't want to be blown up in the middle and this is really this is what I really mean by blowing up right gasoline you want to prevent that from happening you've got to pay a storage cost on the other hand there is a convenience yield if you've got the gasoline you can actually use it along the way you have to replenish it in order to get the same level of stock at the end but it's convenient that you have it instead of having to go get it because getting it involves trouble and costs so that's the convenience yield so if you factor that in then what you get is the futures or forward price is equal to 1 plus RF plus a convenience yield per period minus some kind of oh sorry up plus a storage cost per period minus a convenience yield per period and then raised to the t minus T power multiplied by the current spot price okay if this is violated then you're going to want to do one of two things either you're going to want to buy your own gasoline and store it or you're going to want to short it and do the opposite after Hurricane Katrina hit we had violations from this for for a period of time which suggested that it was actually worthwhile for you to go out and build your own storage facilities because the storage facilities were destroyed now it's only if you had the technology to build those storage facilities that you could actually profit from it but there are periods of time where market dislocation can occur and the discrepancy between futures prices and spot prices that gives you valuable information about what's happening in markets and in some cases in in non-financial context like commodities whether there's a shortage or whether there's a glut you know weather impacts these commodities and so by looking at this relationship you gather very valuable information ok here's another example another example is financials I'm going to take this example as the last one that I want to focus on because I want to now talk about how to use this for you know for your own purposes a financial future is a future futures contract on an index like the S&P 500 so they're all of those contracts are cash settle there is no physical delivery although you can easily imagine a situation where you could have physical delivery right somebody literally delivers 500 shares of stocks of 500 stocks with certain numbers of shares for each in order to get the S&P 500 but that's a pain and that defeats the purpose of the futures market which is to try to make things simple and to make it more efficient so a stock index future is really a pure bet on an underlying index and it gives you the investor or the hedger a way to get exposure or get out of exposure of that underlying in a very direct way now in this case they're there there's no real convenience yield but there is a dividend that gets paid by the particular set of securities so if you're holding the sp500 portfolio then you're going to be getting paid dividends for individual stocks in that portfolio and so you'd want to factor in in your futures arbitrage relationship the fact that you're getting a benefit like a convenience yield that you have to subtract off of this relationship right so you don't have a cost of storage because this is a financial futures but you do have a convenience yield in terms of a payment if you held the physical shares of the S&P 500 so that's the difference between a futures futures you don't get that dividend so you got to take that out in order to do the calculation that tells you what the futures price is relative to the spot so now if I give you an exam question that says today's spot price is such-and-such and the risk free interest rate over the next three month period is such-and-such you should be able to tell me what the no-arbitrage futures price should be today or vice versa if I told you what the futures price is and I told you what the interest rate is you should be able to infer from that what the spot price is going to be on October 19th 1987 the morning before the new york stock exchange opened there was a very big discrepancy between the spot price and the futures price that discrepancy caused arbitrage oars to rub their hands and say oh my god this is Christmas coming early I'm going to take advantage of this and so what they ended up doing it turned out because the relationship was violated in one specific way they ended up buying the futures and shorting the stocks that was the beginning of the October 19 1987 crash that within a day dropped the market by about 20 percent hey nowadays that's no big deal we're used to that but back then it was really it was really something okay so now that you have examples of how these prices are determined let me take this out for a different kind of a spin I want to show you how you use one of these things and the way I'm going to do that is with the S&P 500 what I skipped work more numerical examples that I would encourage you to go through on your own but this is an example that's important so I want to take you through it carefully make sure everybody understands suppose that you've got a million dollars to invest in the stock market and you've decided that you want to invest it in the sp500 you don't want to invest it in any other individual stocks you want a broadly diversified kind of an investment and the ESPY looks like a pretty good thing okay so there are several ways of doing this I'm going to focus just on two one of them is you can put your money in 500 different stocks and you have to spend a little bit of time figuring out what proportions are because if you want to replicate the SP the SP is evaluated indexed it's not equal weighted it's weighted by market capitalization so you can actually go through and figure out how big each company the SP is and then calculate those weights and then you've got to give this order to your broker and a million dollars isn't what it used to be so I suspect that that would generate some pretty tiny trades you got 500 securities and you've got a bunch of different odd lot trades good luck finding a broker that's willing to do it at a reasonable price so it's a pretty long list right or you can buy a futures contract in particular you can buy a contract on the S&P 500 futures so I want to go through and show you what that involves all right now let's take your million dollars and let's deposit it at the futures brokerage account okay so the money sitting there earning whatever interest they pay you on that account which is not much it's probably akin to you know a money market return okay so what you do is you want to buy futures contracts and you want to have the equivalent exposure of a million dollars invested in the sp500 now the way that the S&P 500 futures contract works is that the value of the contract the notional amount of the contract is 250 times the index whatever the index is worth they just make it make up a number like I don't know 250 and multiply that by the value of the index and they say that is what your exposure is okay for one contract so what is that let's suppose the contract let's suppose the S&P index is now at a thousand okay so the value of the futures contract is 250 times that and that's going to be a million dollars oh sorry 250,000 dollars in order for you to have the equivalent of a million dollars in the SP you need four of those contracts for times a notional of 250 is equal to a million now what does this say this says that you are essentially agreeing that you're going to buy the S&P 500 whenever it settles but you're not really buying this P 500 you're buying a pure bet that is equivalent to 250 times the S&P 500 okay so let's take a look at what that means suppose that the S&P index fluctuates bounces around then it turns out that you'll see that your cash portfolio the portfolio fluctuations if you had put a million dollars into the SMP directly it fluctuates in exactly the same way that your futures portfolio fluctuates if the SP goes down by nine to two 900 the notional value of your portfolio with four contracts is nine hundred thousand so you've actually lost a hundred thousand dollars and that's going to be deducted from your account if on the other hand the SP goes up by a hundred then your cash portfolio will be worth a million ten a million one hundred thousand rather and your futures portfolio will be worth the same you will now get a hundred thousand deposited into your account by holding this futures contract it's as if you were actually invested in the SP what you're getting is the daily fluctuations but you actually don't own the security you simply agreed to buy this so-called index on the maturity date and by doing so and because that contract value is so closely tied to the SP 500 index it moves in lockstep with the cash portfolio okay any questions about this yeah no no remain that we are going to wear them exactly right so you and I we're just gonna agree we're gonna bet okay we're going to bet and we're going to agree on a particular price for SP 500 three months from now and if it goes up and I bought the contract then I win if it goes down then you sold the contract then you win but it's a pure bet between you and me the futures Clearing Corporation sits in the middle to make sure that you and I don't run away why do they do this and why why do good a meeting smpm money well first of all in some cases they do so for example you could buy a contract on the number of degree days of a certain amount in Florida now why would you want to do that it turns out that the one of the largest crops of oranges are grown in Florida and it turns out that the output of orange groves is very closely tied to temperature so if it goes up to 39 degrees or below 32 degrees you can actually have very different kind of crops and so you can bet on that and and at some point you could actually trade on I don't know if you can trade on it now but there are markets for some of the wildest things and the reason that you have these markets is because when two mutually consenting adults have opposite views and they want to express them then you want to be able to let them do that and allow them to basically either hedge their risks or take on risks that they're able to do so this is an example of that you're an investor you want to buy stocks but you don't want to buy 500 little stocks one by one you want to get exposure right away now of course there's another way to do this you can put it in a mutual fund but the problem the mutual fund is that it only gets priced once a day whereas this thing gets priced every second of the day when the futures exchange is open of course nowadays you can buy an ETF an exchange-traded fund so that's another way of getting exposure but the SMP futures was around long before ETFs and allowed people to do all sorts of hedging transactions now I'm gonna give you a second example that I think will make it a little bit more clear and actually will answer a question that was asked uh I think two lectures ago when I first started this lecture I said that maybe a company would only want ahead twenty five percent of its risk and somebody asked well what does that mean twenty five percent and I said I'll answer that question well so I'm going to answer that question now so suppose now as a different example you have a diversified portfolio of large cap stocks worth five million dollars so you already own the stocks and it's currently worth five million but you don't have any confidence that the market is going to stay where it is you think it's going to go down and so you want to hedge some of that risk you don't want to hedge all of it because you do have faith that over time markets will do well but you just want to be able to dampen a little bit of the downward spiral if it does occur so you might consider selling twenty five percent of your portfolio getting rid of twenty five percent of it and putting that in cash that's one way to do it but the problem is you know is that it's not that easy to sell 25 percent of 500 stocks because you have to again slice the portfolio stock by stock you can have a trade list of 25 stocks a sorry 5 500 stocks which comprise 25 percent of your portfolio so it's a pain but here's an easier way to do it you can short sell 5sp contracts and I'm arguing that that will do the exact same as if you just liquidated 25 percent of your portfolio now let's see let's see if that's right so let's go through the exact same table the – portfolio let's see what happened to the cash portfolio if the SP goes up or down by a hundred points if it goes up by a hundred points then you've made money you've got five and a half million if it goes down by a hundred points you've lost money you've lost two four and a half million right now let's see what happens if you don't do anything with the cash portfolio but you simply short sell five S&P futures contracts if you do that then obviously if the SP doesn't change then nothing happens to your portfolio but if the SP goes up then you're going to make some money sorry let's say yeah your if the SP goes up you're going to lose money in the sense that what's going to happen is that your short positions are going to cost you a hundred and twenty-five thousand dollars how did I get a hundred and twenty-five thousand dollars anybody works through the math for me the S&P 500 goes up by hundred points the futures price goes up by 250 times 100 points my position I've got five of these contracts I've just lost $25,000 per contract I've got five of these contracts I've lost 125 thousand dollars now what about the downside the downside if SMB goes down by 100 then the price goes down by 25,000 I'm short so I make 25,000 per contract I've got five contracts I've made 125,000 so look what happens in this case when the SP goes up I don't make as much because my hedge works against me on the other hand when the SP goes down I don't lose as much because the hedge is working for me because I've only taken out 25 percent of my portfolio with this hedge its dampening but not eliminating that kind of fluctuation yeah represented an interest and so you still be okay well that's the argument that I gave earlier which is that you have to sell 25% of your portfolio this is a way of doing it and not only that if you did it this way it would be a lot cheaper to implement in the sense that you basically don't have to you don't have to do 500 transactions you do one transaction right so the transactions cost is a lot cheaper and it's also easier to keep track of you don't have to figure out what the price of 500 securities are you've got the price of just one security to worry about yeah I think you're also you're not losing out on what you could have had in cash in terms of interest because that interest is factored into the futures that's right remember we use that interest equation so all the forth on interest is in there right okay so the meaning of I want to hedge 25 percent means I'm going to use the futures contract so that the notional exposure is 25 percent of the current value of my portfolio so if you're merck pharmaceutical company that has a certain percentage of their revenues in foreign denominated currencies you can hedge half of the risk of those exchange rate fluctuations by taking half of the revenue stream let's say it's you know ten billion dollars and buying or selling depending on which way you're going the amount of futures or forwards to get rid of that exposure yep example we put the our million young margin right yeah we only put as much as them that's right that's right yeah you don't have to put a million in the margin account because typically the margin is going to be something like you know in this case seven or eight percent of the notional exposure so you can take the rest of that money and go to Las Vegas if you like although some would say this is better than Las Vegas yeah that's right that's right with an ETF if you want a million dollars of exposure you got to put a million dollars into the ETF with the futures contract if you want to put a million dollars of exposure on you need 7% and the reason is obvious is because that daily mark-to-market okay so ETFs have not killed the futures market but it does provide another vehicle for retail investors who may not want the leverage who may not need the leverage to not have to worry about the leverage okay this leverage leverage is a scary thing as I said before this is the chainsaw that you don't want to be giving your 8-year old as a toy right because when when prices move quickly you're going to have very big swings in the underlying value of your margin account so you know if you've got only seven percent margin in an account think about it that means that if the prices go down by seven percent you are wiped out your entire margin account is gone when futures brokers take your money they assume that you know what you're doing and so they assume that you know the margin that you're putting down is margin that you can afford to lose and that you understand that what you're getting is much bigger exposure that presumably is either for speculative purposes in which case you won't over leverage or for hedging purposes in which case you've got some other assets that are counterbalancing these swings like in this case you know obviously when you look at the fluctuations in your positions they are extraordinarily big relative to the margin right let's do a quick back of the envelope calculation let me tell you what I mean suppose that you put 5% margin down okay you buy a contract put 5% margin down and let's suppose that the price of the futures contract drops by two and a half percent okay what is the rate of return on the amount of money you put down as margin if that's your initial investment you can think about it as an investment because that's the only way a futures broker will let you buy a contract if you put down $100,000 and the futures price goes down by $50,000 what's the rate of return on your investment – 80 % right that's a big move that's a huge move on a day so when you deal with margin you have to be extraordinarily careful you have to have very very tight risk controls you have to understand what the swings can be and you have to manage that risk very very carefully on an intraday ly basis in some cases because these futures prices can swing a lot even within a day okay any any other questions well that's it for futures and forwards you now know how to price them you now know how to use them for hedging purposes and there are all sorts of other kinds of futures and forwards interest rate bond currency single-stock futures now exists in fact there are even futures contracts on the VIX there's futures contracts on electricity usage there are futures futures contracts on the presidential election if you go to the Iowa electronic markets the University of Iowa they created a futures exchange that has two contracts one that pays a dollar if McCain gets elected and the other that pays a dollar if Obama gets elected and by looking at the prices you can actually see what the folks that are trading the futures contracts are thinking in terms of who's got the edge so the futures prices contain an enormous amount of information but keep in mind the information is only as good as you are by you I mean the market if the market is comprised of knuckleheads the prices you get will be knucklehead prices okay if the market contains really smart sharp sophisticated individuals you'll get extremely informative prices so prices while they are the best thing that we have as a guide for the future they're clearly not perfect and there are periods of time when the market prices are less perfect than others and as I told you before you know for the next three weeks finance theory is going to be on vacation in the US stock market because all the uncertainty that has been building up over the last several years are now focused on the next three weeks markets will be swinging back and forth pretty wildly and it's because people are reacting emotionally not necessarily with their full logical capabilities okay that's it for futures and forwards and now what I'm going to turn to is options these are the last set of securities that I want to go through with you that are not like the securities that we've done before and let me just pull up the lecture notes for options I want to start with a little bit of an introduction for how to motivate options I think most of you know what options are their name is quite apropos because they do give you options futures and forwards require you to engage in a transaction but options don't they give you the right but not the obligation so you have the option of not entering into that final transaction at settlement date I'm going to start with some motivation then go through some payoff diagrams go through options strategies and then I'm going to talk very briefly about valuation of options I have to talk to you guys about black Scholes weak you can't leave them IT without hearing about black Scholes so I've got to do a little bit of that but really the derivation is quite a bit more sophisticated and that's why you might want to take 15 for 37 options and futures where the entire course is devoted to these instruments they are that comply and and that important so let me first describe exactly what an option is an option actually is a specific example of something that you now know of more generally as a derivative a derivative security gets its name because the value of the security is derived from yet another security okay it's it's derivative as opposed to I guess fundamental or primary and examples of derivatives are warrants versus options options are securities that you can think of as pure bets between two parties warrants are options that are issued by a company on its own shares so the the net supply of options is zero but the net supply of warrants is not zero all right it's issued by companies and there are two different kinds of options calls and puts a call option is a piece of paper that says the holder of this piece of paper is allowed to buy a security on or possibly before a particular date usually called the exercise date okay or maturity date and the difference between being able to exercise early versus exercising at the maturity only is the difference between an American and a European option an American option is one way you can exercise it early and a European option is one where you can only exercise it on a specific date the maturity date or the expiration date and puts are the opposite of calls instead of giving you the right to buy it gives you the right to sell or to put the stock to somebody else okay and the prices at which you get to either buy in the case of calls or sell in the case of puts is called the strike price or the exercise price alright so I'm going to define a limit notation stock prices SMT strike prices K notice that K does not have a time subscript because it's fixed at the time the options are issued and it doesn't change throughout the life of that option it's part of the contract terms and then the call price is CT put prices PT and the value of these contracts at maturity is actually pretty simple if today a particular stock is trading at $50 a share and you purchase an option to buy that stock at $70 a share in three months does that piece of paper have any value the current price of 60 this piece of paper gives you the right to buy it at seventy in three months is that worthless why not the price is at sixty you can get it at sixty today so why would you want it at seventy three month from now exactly the price may go up the lease in the piece of paper is not worth zero today is that there is a chance no matter how small you might think it is there is still a chance that something wonderful might happen in the next three months and then the price will go up to 80 and if it goes up to 80 you'll be very happy that you have the right to buy it at 70 how happy will you be you'll be ten dollars per share happy right that's what that expression says on the expiration date you get to buy the shares if you're holding a call option you get to buy it for K dollars but in fact the market has determined that the price at that time is really s T dollars so if you're holding this piece of paper this is your profit s t minus K per share now if it turns out that you get to buy it for 60 and it ends up trading at 40 well then you're not going to exercise that right you're going to let the option expire and when it expires it'll be worth if this number is negative it can be negative of course but you're not obligated to buy it on the other hand if this were a futures contract you certainly are obligated to buy it and then you'd get a negative return but an option is a wonderful thing in that the payoff is never negative it's either zero or it's s t minus K that's for a call now put option it's exactly the reverse if you get to sell the stock then your profit is what you get the salad at vs.

What it's really trading at and so you actually hope that it's really trading at a very low price because if you get the sell it at a high price but it's trading at a low price you profit the difference okay so the payoff for a put option is exactly the reverse maximum of zero and K minus s now it's very important that you understand this asymmetry because that asymmetry is going to lead to all sorts of interesting things about these instruments and before we go and talk about that kind of asymmetry I want to give you another way of looking at options which is to look at options as a kind of insurance contract because actually all insurance contracts are a form of an option so let me let me give you an example I suppose that you want to insure the value of a particular stock that you're holding you're holding General Electric and it's trading at 20 bucks a share and you'd like to make sure that it never goes below 18 dollars a share you want to buy insurance that if it goes below 18 dollars a share you will get paid 18 dollars a share okay well the way you do that is you buy a put option a put option on General Electric where the strike price is $18 chair because if it goes below 18 dollars a share you get to sell General Electric for that 18 so you'll get the $18 regardless of whether it goes to ten or five or who knows what it turns out that the put option is exactly like insurance and let's take a look and see why these are the typical terms of an insurance contract what's the asset that you're insuring General Electric what's the current asset value twenty dollars a share what's the term of the policy how long do you have the policy for it's the time to maturity right what's the maximum coverage what you covered for $18.00 a share that's right that's what you bought the coverage for that's what you're gonna get if it goes below that what's the deductible how much could you lose before the insurance kicks in $2 a share exactly that's the deductible and finally what does it cost you to buy this insurance what's the insurance premium exactly the price of the put that's it beautiful thing a put option is nothing more than an insurance contract on the value of a stock and it's going to turn out that a call option will be intimately tied to what a put option is so every call option can be converted into a portfolio that includes a put so all options you can think of as insurance contracts but there are a few differences the difference between an option is that you can exercise it early so for example for whatever reason if you decide that you want to buy General Electric at $18 a share when it's trading at 1750 and you still have one month to go but you want to get paid that 18 now you can do that you can't do that with your car car insurance right I guess you could you could you know Ram it into a post and I want to get paid now so let's but that's not really considered a proper thing to do so early exercises one different second difference is Mark ability if at some point you don't want the insurance anymore you can't get rid of it and give it to somebody else you can't transfer your auto insurance to your friend if you decide you don't need it anymore right but you can transfer the insurance policy here you can sell the option you can sell it and also there are dividends that are being paid on the stock that you have to worry about with options whereas with a typical insurance contract you know a car doesn't necessarily pay dividends and the reason that's important is when it pays dividends the value goes down and so you have to make adjustments for that with an option you have to protect an option for dividend payments you don't need to do that for insurance because typically you don't assume that the insurance value goes down that much over time yeah well no that's actually not true with a put option it gives you the right to sell the stock right if you buy the stock and you hold on to it and you also buy a put that protects the downside but the upside that's all yours right because the as the stock goes up what happens to the value of the put exactly it stops at zero so the stock goes up the put doesn't have any value any more it becomes worthless worth less and less and on the date of expiration if the stock is way above the value of the strike then it expires worthless it doesn't go negative if it went negative if you had a futures position then you'd be right you've actually capped your gains but this doesn't you see this you get the best of both or so it seems you get the upside but it protects the downside and as you all probably know insurance is not cheap so it sounds good but you got to pay for this and so you bet that the price of a call option or put option is not zero when you strike it when when you set that cut and unlike a futures contract that's worthless an option is not worthless on day zero it's worth a lot for example right now what's really expensive and if you want to check this you could take a look for fun if you want to buy insurance on the S&P 500 now you know we've had a great rally on Monday the S&P was up a thousand points okay if you want to buy insurance on the S&P 500 index you can do that there are options on the index so you might say okay let's say that the SP is at a thousand today I would like to buy protection that over the next month it doesn't go down by more than a hundred points ten percent so what do you do you buy a put option on the SP with the strike price of what nine hundred right okay for a month that's what you want to buy go out and calculate that price you're going to be shocked at how expensive it is to get that insurance for four weeks four weeks that's all it's really expensive today it's approximately I think it's approximately ten times more expensive today than it was a year ago the implied volatility is up by at least an order of magnitude so if you want that insurance it's available but you have to pay for it so the question in all of these things is is it worth it in order to decide whether it's worth it you've got to do two things first look into the innermost workings of your own soul and ask how frightened you truly are and the second you got to do is look at the market and is the market functioning reasonably well or is the market reflecting all of these kinds of crazy fears in order for us to be able to talk about it intelligently we need a way to price it we need the kind of logic that I showed you with futures contracts and we're going to get that logic I'm going to show you how to price these things using a very very simple model that is incredibly powerful but we're not there yet before we do that I want to make sure you understand what these contracts can do for you in terms of changing your payoff profiles of your portfolio yeah well no that's not what makes it similar to a futures because while you cannot exercise it early you never have to exercise it at all so a European option gives you only one date where you are able to exercise but even on that date you never have to exercise it with the futures contract you have to exercise it on that date you've made a commitment no no it won't because still on that date you have a positive amount of protection like the example I gave you let's suppose that I bought a European SP option for the day after election day you know Wednesday November 3rd is it that will have positive value today in other words I'm going to have to pay money in order for you guys to sell it to me because you're going to be providing me with some protection that if the wrong thing happens on Tuesday the world is not going to blow up on Wednesday okay I'm not telling you what the wrong thing is I I'm neutral in all of this but but that's an example where that insurance really has value so you're not going to give it to me for free and I'm willing to pay for it okay all right since we're out of time I'm going to just leave you with this diagram that shows you the difference between a call option and a futures contract remember the futures contract what that look like that was a straight line right exactly this is not a straight line this is kinked very kinky security and so we're going to talk next time about a how to price kinky securities and how to combine them and engage in even more kinky kinds of payoffs

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