Math 176. Math of Finance. Lecture 01.

course can't read that can you math of Finance I like to joke and say that the course really is math of getting rich or math of getting a good job and in fact in the past students have gotten good jobs and I still hear from several of them all verification I heard from one who has heard not who he was telling me about her new condominium she's only was out two years three years she has her new condominium and getting a new car and she just got a promotion another one up in San Francisco is also doing well couple of others are finishing up in graduate school so if you take this course seriously you will have an advantage in fact that is my goal my goal is to make sure that each and every one of you if you're interested in learning a material each and every one of you will learn the material and what happens is you will have the opportunity of getting a good position now to do that you're gonna have to work hard I'm going to have to work hard I will do whatever it takes to try to get you through on the other hand because I will do whatever it takes to get you through you will discover that I have very low patience level if somebody is busily working on their homework for the next class or something else you will be asked to get out and so what happens is I will be here to try to make sure that you learn the material I want you to be able to reciprocate you really work to try to get this through and as I've stated students coming out of this class have done well they've done quite well in the past now come on in come on we waiting for you now how do I run the class well you see the material we're talking about the math of Finance we're going to be learning about such things called puts what's a put put yeah in other words you're not really sure are you what's a call so we're gonna learn about these mysterious things called puts and calls put-call parity we're gonna learn about the black Scholes equation how to derive it Auto modify it we're gonna learn a lot of material and learning this material what I want you to do is to develop your intuition I'm gonna hear our math majors okay how many Recon okay about half and half now when you go into a job and they ask you something from economics or from mathematics whichever area you're from you really don't have time to go look up that old book and to derive and to derive the formula or something else you need to have some intuition about what's going on and why the models and economics are what their models that are used to try to develop instinct intuition about what should and shouldn't happen and the mathematical tools that we use from the mathematicians folks they're limited they're not good enough so you have to know why and where and so what we're going to be doing throughout this course is worrying about assumptions we're going to be worrying about what it takes for you to be able to develop a sense of what's going on no that means you got to learn the material and as I said I'm gonna do whatever I can to make sure you learn it that means I will entertain questions at any time you have a question about something ask it and let's not worry about it let's let's get it out right away I'm gonna have class management at times I may not be able to answer all questions right away but you will get an answer I don't believe in that and you will learn in your next course nonsense that all of you have heard in fact the most important question you can ask is why gap talk about who's gonna ask you all know what walk that means what vlog at me all righty who gives and I'll let you finish it who gives a and you can finish the rest of the word why dad is a question that we can ask at all times because what we're doing is we're asking hey this is a pragmatic course where is the pragmatism where can I use this to get myself another job to better understand reading the business pages various other type things so these are the type of questions you can ask me and next time I'm going to get better markers but that's the type of questions you can ask me and folks you will discover starting today it's the type of question I will be asking you and I say it's a dialogue I mean it's a dialogue I will be asking you type questions now let's get to the logistics and then let's get to the course any questions up now okay logistics in terms of grades how our grades determined etc well you see tests and quizzes I believe are effective tools and getting you to learn the material what does that mean it means that you can expect quizzes unannounced I do not collect homework homework tells me who you know quizzes tell me what you know and so I will give you quizzes on an ounce sometimes two in a class just depends on where I think I need to give a quiz to find out what it is why do I give quizzes number one so that you can understand what you know oh ho you can see I'm right here he's already saying oh I know this material then he goes and takes a quiz ensues whoops I guess I didn't better to know it now than on an exam so he can know more about me you all like to play that game of trying to psych out these instructor don't you I did when I was a student what kind of questions is he going to ask and so what happens on quizzes you see the type of issues that I'm going to emphasize I learned more about me boy did I give a brilliant lecture oh that was so good that was wonderful chippy on YouTube it's wonderful the lecture and then I give a quiz and find out nobody understood a word I said and so I have to go back and recover the material and finally it tells me about you very rarely in these in the last several years but occasionally in the past I would run into a student who does wonderfully on the quizzes not because who they're sitting next to Bo does wonderfully on the quizzes but freezes on the exams all right with this we can identify who that person is and we can try to help finally what happens is what about these quizzes well if you can do the homework because remember I said the homework I see everybody out there working as I walk by working with each other and homework and by the way I proof of that but if you can do the homework and understand the homework you should get the quiz perfectly because the quizzes are essentially homework problems are different numbers etc that's all that's all now there are no makeups for quizzes here or in the discussion sessions you miss a quiz you get a zero there on a 10-point basis you go to zero on the other hand what do I do is I will drop your two lowest grades so if you've missed two quizzes those are the two if you missed three you're missing too much class okay the quizzes are in a 10-point basis as I said and they will amount to approximately one test problem but you will have no more than a hundred points on quizzes there are more if I get more quizzes than that I'll just drop more quizzes your lowest quizzes our exams you will have two of them I will give you one week advance notice calendar week advance notice before an exam I am NOT announcing an exam today so you know Thursday and Tuesday you don't have an over exam so I give you one calendar week advance notice and I also make it very clear on the class period preceding an exam what to worry about and then you have your exam now what is the policy look a lot of students in this class somebody's going to be sick somebody's going to have an auto accident on their way to class these are things that have happened in the past somebody is caught in Chicago where they use sub-zero weather and can't get out the planes aren't getting out alright if you miss the exam that's fine don't even bother it's telling me I believe you have a legitimate excuse are we happy to give you a make up in fact not only will I be happy to give you a make up I'll be happy to give you a copy of the exam and the answers oh I can see all the economists in here going through the incentive Theory saying ho ho ho how are we going to get a take advantage of this the reason I do this is the following it's that the makeup is that much harder you need to copy of the exam and the answers to prepare for the makeup when I'm writing up an exam I write down a large number of questions and then I say oh no no this one you really you really need to know a little bit more material you really got to understand these questions here before you can really tackle this question so those are throwing off to the side for the makeup exam and so the makeup exam is ready right when the regular exam is but it is going to be another level hard it's not impossible it's not unfair it's very fair I've had people come in and really do quite well but it's going to be harder and that's only fair because you're ill and I know you're ill you would never never believe otherwise so she's not there but she's it knows what's on the exam and that hurts you so really what we will do is you will have make you know the makeup will be more difficult there is no makeup for a makeup and the final is whenever the final is I haven't checked yet when is the final scheduled nobody knows Thursday on that exam week oh yeah okay so that's when it is the same rules apply to that how should you prepare for this course remember I said I will do anything I can to make sure you understand that includes extra sessions extra description discussion anything how can you prepare what you can do is you can get together in study groups I strongly strongly strongly recommend study groups in the past the students who have by the way please do that again you won't be here if you are in the study students that have been in serious study groups have always done better almost always have done better now a study group of one is useless a study group of one you get to this problem is a little just a little wording you can't understand it and you're stuck all night and you get nowhere study group of one is useless study group of two it's better optimal is a study group of three and four beyond four be honest put away the book bringing the keg have your party okay so groups of about three or four very good and what is wise is if you get started soon look around now and see who you want to get into your study group be wise if you start soon you start learning the material going over the homework together going over make sure there's no freeriders but you're going over the homework together and you're going over the exams and quizzes and everything else okay any questions any questions about the course any part of the course okay and next time I will introduce our your TAS what you some of you will meet him beforehand okay now I don't know about you but I was glued to the TV last night I was busy watching a couple of football teams do you know that all Byrne was practicing over here preparing for their a game here they were practicing up here in the field right were Feltes inand what's the name of the street over here well anyway over here on the field getting ready for the game and so I guess maybe it was that reason that I was busy hoping for Auburn to have an upset and boy it was close right that was one of the best football games I've seen how many how many watched it last night okay now there's some good games coming up this is a time of the year that I really like for example that's Sunday we got San Diego Chargers over in Denver and who do you favor you favor San Diego why California and they did beat giver before - what kind of hunt you gonna give me Peyton Manning spoken odds two to one do - what your first name Quinn alright and I promise I'll get better markers next time but Quinn is taking San Diego and he's giving two-to-one odds what does that mean - the one odds means that if I've bet five dollars with him and if eight and Manning comes through like I expect you will what happens is he owes me two dollars for each dollar I'd bet with him but if San Diego pulls an upset which they can they'd beat 10 dar they beat Denver on the road that what happens is he keeps the money I've got rhythm who here likes Denver yeah odds you're gonna give me three and a half no no I want points I want odds yeah and your first name Corey I'm gonna use K because it looks too close to Q so Corey and he's going to give me one point two five two one and he is taking Denver so you see what happens here see what happens here is that there's going to be this game on Sunday in the afternoons on CBS I don't pay attention CVS against San Diego and Denver Denver wins he keeps all the he pays each person that bet with him two bucks for each book that they bet he keeps all the money he has what happens is his San Diego wins did I get that screwed up no San Diego wins he keeps the money he pays a dollar 25 for each dollar that was bet rhythm here you're a poor struggling student you got a hundred dollars for books that's it who you gonna bet on so maybe San Diego you can put all hundred dollars on San Diego what half-and-half what's gonna happen what about you we've been a bad time what about you I [Music] mean don't forget there's a risk involved here isn't there do you know much about football now the risk is even bigger isn't it I thought you didn't because when I asked to watch the game you were looking down so what happens is that there's a risk involved here and so at this risk that's involved what we want to do is why should I have bet a hundred bucks darn let me take a look and see what else they had I'll hear it to try this oh look at this you got a hundred dollars if you pick your hundred dollars on San Diego and Denver wins you have no money and all those books you can't buy if you bet the other way around and they lose no money but you know what there's a way to get rid of the risk and this is going to be the economic term called hedging hedging you bet on both so let's take a look and see what we're going to do here Brinn Quinn what we have is he takes San Diego and he's giving two-to-one odds so you bet X dollars with Quinn I don't know what X is that's what we have to figure out and then what we do is we bet the rest of the money so I'm betting X dollars on Denver and here I'm going to have the odds are one point two five to one and I'm going to bet a hundred minus X dollars and I'm going to bet on San Diego and the goal here is to find out is it possible to cover my bets on both ways hedging so that I'm going to do okay all right so let's take a look and figure out what what we should do number one uh-huh that's good enough that's good enough number one is suppose San Diego wins if San Diego wins then I lose 100 minus X dollars Oh No women excuse me excuse me if San Diego wins then Denver loses and if Denver loses I bet on Denver right here so I'm going to lose that money so I'm going to lose X dollars but I'm gonna go over here and say hey hey hey you owe me some money you owe me a buck and a quarter for each Buckeye or I bet right here so I'm going to earn that's how much money I'm going to earn and I want to find the x value so that I know I'm going to do okay so we just collect these terms that's going to be minus five-fourths 9/4 correct me on my arithmetic if I'm wrong 5/4 and 4/4 is nine fourths plus 25 four goes into 25 25 and so what happens here is I want to ensure that this is greater than zero so I want to be sure that 125 is greater than nine fourths X or X is less than four five hundred over nine which is about fifty five dollars or somewhere in that neighborhood so if I bet if my X satisfies this right here and I'm gonna win make some money if San Diego wins okay everybody see what I did there all I did is figure out what my losses are and what my profits are just added them together now let's suppose Denver wins that means San Diego loses all right so that means I'm gonna go see Quinn and say hey I bet X dollars with you i watt twice that amount of money to X dollars so my profit is going to be two EPs and my losses it's going to be 100 - excellent which is equal to 3x minus 100 and so therefore if X is greater than 100 over 3 I'm doing ok so if we take these two boards together we have that then if I choose X somewhere in here somewhere in these values right here I'm free I will win no matter what right no matter what happens I will win ya know what I did is I bet X dollars on Denver here if the inverter wins I'm going to get 2x dollars back okay and then here I'm losing this money right here because San Diego lost so remember no money changes we're doing this for friends no money exchanges hands till the game is over and so therefore I get here this is what I won with Denver won I bet on Denver so I'm going to get 2x dollars back that's the amount of money I make and here I'm going to your because I bet on because San Diego lost and I bet the hundred minus X dollars on San Diego I lose all of that money right there and so minus a minus is a plus so that's 3x minus 100 and I do it like this no no how did my total of a hundred yeah no no no no money exchanged hand in fact what happens here and that's going to tell you that I'll come back to that in a few seconds so therefore if X lies between here if X lies between this amount right here I am guaranteed to earn money notice not a penny came out of my pocket not a penny came out of my pocket I'm guaranteed without using any of my money because what happens is if I bet this amount of money right here let's say 50 right in the middle if I bet 50 what happens is if den San Diego wins then what I'll do is I'll take my winnings and pay off by losing and I'll still make that amount of money right there okay because this right here is my losses here's my winning so if I choose 50 here that fifty inside there that's 50 divide through multiply through subtract off with X is 50 and that will give you exactly the amount of money that I'm putting in my pocket so I have enough money from him to pay off him okay and I didn't take any money out of my pocket and the other way it goes that way - all right and so on this fashion right here we got it yeah so so if we do that if to throw $50 here suppose we put $50 on Denver Denver loses lost 50 but I'm going to get back I'm gonna have $12.50 no no no no no let's take a look I'm betting $50 here what happens is I'm getting a dollar 25 for each one so I'm getting sixty two dollars and fifty cents yeah money no money changes hands we're not going we're not talking about a racetrack where you put your money down at a racetrack you put your money down because they don't believe you're gonna come there if you lost we're just doing it on the exchange yes got it got it got it and so that's what we have here okay now what we have then is we have two curves we have two curves and they're going to be the profit curves you can draw the profit curves for these two situations for the Denver curve the Denver curve if x equals zero I'm up to 125 when x equals whatever this number is right here some number right here I got serum and so my profit is always here here's my profit you choose that amount that's my profit choose that amount for X that's my profit it's all I'm doing is taking this curve right here and graphing it this is the curve y equals minus 9/4 x plus 125 okay so that's the profit curve yes oh no money be change his hand until the game's over no no yeah so we're not doing it uh you see you're asking good question so let me let me clarify it here right now before the game we're doing an among friends or people to have heavy baseball bats I used to be a live in Chicago okay so what happens is no money changes hands until after the game now if you go to a bookie or go to a racetrack or is anything else of that type are you going to show up if you lose oh by the way I owe you $100 probably you probably be five seats away okay if I got enough okay so they're they want the money down in advance here we're not putting money down in advance here we're just talking in terms of friends sitting down having a coca-cola watching the game watching the football game and so at the end of the game that's when money exchanges hands we're getting to that next very good we'll get to that next now what happens is while you draw this curve right here 3x minus 100 and this curve right here goes up like this this is the curve 3x minus 100 it's this curve right here and so what happens then is this interval right here is this interval right here if I'm anywhere in here then no matter what happens I either get this profit or I get that profit depending on who wins the game this is just a profit curve okay question got it you see what I'm doing here okay now there is an issue here the issue is we're not in this game in this bedding she doesn't care who wins the game and she doesn't find any excitement from watching the game the next Sunday at 2 o'clock what she wants to do is she has a hundred bucks for books and she wants to be able to buy more books so she wants to maximise she wants to maximize her outcome independent of who wins where is that the intersection of the two curves it's the intersection of the two curves if she bets that amount of money right there then no matter who wins she's going to earn that amount of money if she bet less if this team wins she's going to win the little bit the other team she'll win more likewise here but she doesn't know what's going on she just wants to be guaranteed a certain amount of money coming out okay everybody with me on that how do I find that intersection point set the two equations equal to each other exactly and so let's do that that'll give me the value of x and then I can figure out the profit so if I find the set the two equations equal to each other I'm going to have 3x minus 100 and I'm going to that's 3x minus 100 that's this curve right here this live profit line right here which is going uphill and I'm going to set it equal to minus 9/4 X plus 125 and so now all I do is solve and I'm going to have three and 9/4 x equals 225 well 12 21 21 fourths x equals 225 so if I let X be 225 divided by 21 x 4 I know exactly how much what is this about 42 dollars or 43 dollars I know exactly how much to bet and what happens is not only do I know how much to bet but when I take this value of X and I substitute it into one equation or the other it doesn't matter because they both cross I will find what is my winning whatever are my winnings okay everybody with me on that okay so what is this called hedging is where you bet on both sides hedging is a form of insurance we will discover later on as we go along a hedging is in a very important form of insurance it covers you if the market is all of a sudden going to drop a friend of mine in the state of Michigan is a very wealthy lawyer what he does is if you lose money on the market what he will do is he will take on a suit for you and he will check whether they hedged properly henching is insurance in case the market goes this way or that way it's exactly the problem that we had with her she doesn't know who to bet on so hedging is to cut your losses so hedging is a form of insurance and we're going to make this very precise as we go along and it is a way to cut your losses as I said my friend in Michigan the lawyer what he'll do is if you indeed you lost too much money he will check to see if they if the finance firm did appropriate hedging to find out to protect you if you invest money in a company quite often they will ask do you want to take risk or do you want to be very conservative what is what are your objectives in your investment those terms right there are partially involved with hedging so I'm using this football example simply because it's easier to understand the graphs the idea of hedging yet on both sides there is another concept arbitrage arbitrage AR bi t RR AG e arbitrage is where you will make money or guarantee to make money you're guaranteed to make money does arbitrage exist oh yeah it does another one of my friends took a two-year leave from her academic position and because she recognized a little time gap when things were and she found a way to have arbitrage and became quite quite wealthy so arbitrage is where you are guaranteed to make money on here no let's go to your comment what was your comment about the gambling that's right what happens now if the two of them got together and they found out that didn't matter what happened and they were sitting around we're sitting around in the same room and they're finding out no matter who wins I'm going to make money they're going to start adjusting their odds aren't they they realize I'm taking advantage of them that is an important role a very important role of arbitrage what arbitrage does is when people realize that there's an opportunity for money a lot of people will jump in on it when a lot of people try to take advantage of it the people that are losing the money are going to start making adjustments and that is why we have a certain amount of smoothness etc when we write down equations later on when we get to the black-scholes equations and several others we're going to be suing that certain equations you can differentiate them whoa I can see where you know I can differentiate that from physics but why can I differentiate an equation that comes from your opinion your opinion your opinion and we're putting them together how are you going to do that and where that comes from the ability to be if you assume that things are continuous and differentiable etc a lot of it comes from arbitrage people adjusting so that if it isn't there you're going to get it down so it is there and we're going to see exactly how to do that so what are some of the things that we're gonna be looking at you now know what hedging is and I hope you wrote it down in your notes you now know what arbitrage happens to be and these are two terms that we're going to be using throughout the theory and our term our course they're going to be central when we start putting together portfolios we're going to be putting forth portfolios that go both ways and I'll give you several examples examples that came from when I've taught this class before and I created this course when I was at Northwestern University and a couple of the students looked at thee because I always encourage the students to read the business page I've never read the business page before go ahead read it find out see what's there and what happens is they came in and they said we think we see an arbitrage opportunity so we talked about it class some of them wrote it down in their notes some of them reached out their wallet on Friday and made money by Monday so we're going to be looking at situations of this type and we're going to see how that affects our intuition about what happens how that affects how the market actually behaves so arbitrage and hedging okay now I was going to write something else down but I have to get better better markers and what I have here so what I'll do is I'll get into something else what we really want to do is we want to have some idea remember what I stated at the very beginning the goal is for you to develop some intuition about when you could make money when the mathematics needs changing when whatever happens your goal is to develop some instinct some sense about that so to do that we're going to look at this problem right here and say what is there about this particular problem where I should have guessed in advance that it is to my advantage or what types of problems are going to be put forth to me and I'm going to say ooh stay away from that to do that we're going to introduce a couple of things called a fair bet but to do that we have to introduce or remind some of you reminding I hope a reasonable portion of you but not all of you I am going to introduce a little probability in statistics we're gonna need a lot of that in this course you'll get what you need if you don't remember it through you haven't had it don't worry we'll get it now what is the probability probabilities are really in some sense the frequency or the likelihood that something's going to happen so p1 is a number between 0 and 1 and it is a likelihood that event 1 is going to occur if I flip a coin what's the likelihood of heads 1/2 so therefore I have that the probability of heads is 1/2 the probability of tails is 1/2 so what we want is if I have distinct events distinct events already there's no overlap and we're talking about a finite probability space right now then what I have is the probability of each event the probability of each event is P I and the summation of them has to equal what has to equal unity unity is the likelihood that something's going to happen it's the likelihood that something is going to happen so what we want to do now is let's take a look at a couple of problems we're going to use this to invent something which is called expected values expected value is exactly what the words say what do you expect so it's expected value is an expected value of a function well the trouble with a function is you know if you have a function like f of x equals x squared you take the X point and you stick it in there and you find that u square can you get the value in probability what I'm going to do is I'm going to put in the event of maybe getting heads or tails and so what happens is it's not a function where I put in the value it's where I put in one of these events that can occur and so we call that a random variable don't worry about this random variable just think of it as a function that says if this event comes up here's what you can earn if this event drops up here's what you're going to earn example random variable random variable X there are two events the events is that the temperature outside is going to be 68 and the temperature outside is going to be 75 that's the only two temperatures and so what is random variable here the random variable at the temperature is 68 it is go to class and if the random very offended variable might be skipped class go to the beach so that's the kind of random variable that you have been using all along anyway you're looking at an event and you say if it's going to do this I'll do that if it's going to do that I'll do that that's all and the events are not necessarily deterministic if they're not deterministic you get a random variable so let's try to use this random variable and it will work out a couple of problems first one random variable of heads is $10 what if tails comes up it is minus $20 and what you want to do is how much can I expect how much can I expect and so what happens is what we can expect is the following the expected value of the random variable X the expected value of the random variable X it's going to be the obvious it's going to be how likely is it to get heads and if I get heads what's my reward plus how likely is it to get tails plus what's my reward okay and so we just now substitute in the values 1/2 the probability of heads is 1/2 where we wrong here somewhere way over there and I'm going to get $10 if that's true the probability of tails is 1/2 and I'm going to lose 20 bucks and so what I'm going to have is 5 minus 10 or a minus 5 dollars so in that bet right there you can expect to lose and so what you will have throughout this course is a lot of expected values what can I expect to happen let's try another one let's write down the definition and then try out another one that really the expected value of a random variable it's going to be how likely is the IP vent going to happen x times what is going to be my reward if that happens and let's sum over everything that can happen okay what would be the expected value if I change this to $20 pardon Cyril that's a fair bet a fair bet fair bet is where the expected value is equal to zero okay now suppose I take a penny put it on a table and spin it on the edge what's the probability of heads what happened nope not at all I haven't tried this on a penny for the last Oh at least decade but some of the older pennies and I found this out in my class probability and statistics where I told them I fix pected the answer is going to be one half but I said let's go try it so each student in the class of 60 had to go out and spin the Capanna penny 100 times and report the results in class to the next day first students as i pointed I I think I screw it up hi I've got twenty nine heads and the other one said I got thirty one it went through the class did the statistical analysis and to my surprise with all the tools of Statistics at least pennies then and I said I haven't done it for a while probability of heads is about 0.3 tails about 0.7 why if you look at a penny particularly the older pins now they're made out of I don't know what but the head is a little heavier the face is a little heavier than the backside so when a spinning it changes that axe serve rotation so that the head will go down first and the tail will crop up more often okay so what happens is after I discovered that I happened to be in Paris and so a friend of mine and I we were sitting on the floor maybe we had too much wine but we're sitting on the floor do you in French francs that was before it was euros and we were spinning the hose to find the probability and on and on and it turns out that indeed a lot of coins if you spin them because they're not quite symmetric you will get a different answer to finish the story about my class this was a class at Northwestern what happens is that night and I know none of you would ever do this but that night a large number of them went back to the dorms I said pizza call it now what do you say if someone guys through the call everybody says heads okay call it heads Oh tails I guess the students cleaned up they also cleaned up on stories like the football game or using it on politics and anything already so now what we do is for spinning the probability of heads is equal to 0.3 and the probability of tails is equal to 0.7 for this one right here find the expected value okay the expected value is going to be the probability of heads times what I'm going to get if head strolls up the probability of tails times what I'm going to win if tails show up already so we just need to substitute the probability of heads is 0.3 this right here is $20 plus 0.7 and this right here is minus $20 which is equal to 0.6 or $6 minus 14 dollars minus 8 so you can see how the probabilities strongly influence expected value would get expected value of 0 if you're flipping the coin see if you flip the coin that slight lack of symmetry isn't going to affect it that much okay because it's spinning in the air but you do it on the edge now it's changing that axis of rotation all right so that's what we have here now I'm going to give you a problem remember what a fair bath is take San Diego let's find out what he thinks there's the likelihood of San Diego winning we're assuming it's a fair bet what does it mean for a fair bet means the expected value is zero what does it mean means you'll take one side or the other if it's a fair bet at those odds you'll take San Diego or he'll take Denver Y doesn't really matter it's a bet because you don't have an advantage when we were the other so if he thought it was a fair bet from this we should be able to find out what he thinks is the probability of San Diego winning so computed got an answer mm-hmm okay so let's write down what it should be in this situation he's stating that the probability it's a fair bet so zero is equal to the expected value of x so what does this mean this means that this is equal to the probability of San Diego winning and if San Diego wins he's going to get $1 plus the probability of Denver winning and here he's going to lose two dollars correct you're prettier with me on that now what is the probability of all I have to find out what's the probability these are our unknowns I've got two unknowns one equation fortunately I've got another equation that the sum of the probabilities has to equal one somebody has to win the game so this can be written as the probability of San Diego winning minus two times one minus the probability of San Diego winning equals zero and what that gives me when you carry out the arithmetic that gives me two is equal three times the probability of San Diego winnings so therefore his opinion is that the probability of San Diego winning is 2/3 yeah everybody with me on that okay now what we have is one person's probabilities and so therefore for me the probability for me that San Francisco is going to win San Francisco wrong game Denver is going to win is one third okay now what I want you to do is we go over here and we find what is he gave the odds of one point two five oh great here we go one point two five and what he was doing is he was betting on Denver so find what was the probability of Denver for him so if he thinks it's a fair bet zero is equal to the probability of Denver winning and he'll get a dollar Denver wins plus one minus the probability of Denver winning and he will lose 5/4 which is OAS payoff and so what we have then when we go through the arithmetic it's that 5/4 is equal to 9/4 X probability of dead verb excuse me let's see that will go on that side that's 4/4 and that's 5/4 so it's nine fourths and so therefore the probability of Denver winning is equal to 5/9 and so for me the probability of San Diego the odds are it's four ninths okay everybody with me on that so you have a simple way now this is important because one of the things which we do in our classes is we give you all kinds of answers and so therefore you're having trouble when you get to something new that's not in the textbook what you want to do is you want to get into the habit of looking at these equations and saying what can i what can I extract out of there you have a little bit of Vantage over him you're gonna make more money by trying to figure out what extra information you can get out and here we have the information about what they have now let's take a look remember that some of the probabilities has to equal one right talk get so let's take a look at my probabilities for me I have that the probability of Denver Denver winning a San Diego winning plus the probability for me of Denver winning should equal one let's see so that means that 4/9 what was the other number one third or three nights that's two equal one does it nah it's smaller than one the fact that the probabilities for my for me that they sum up to less than one means there's another little nudge inside there that the probability of my Gamble's of what's going to happen it ends up less than everything means there's another little nugget inside and that's what allows me to take advantage of the situation so whenever the sum of the probabilities in my Gamble's adds up to less than one you got an advantage folks you've got an arbitrage opportunity what happens if the summation is the greater than one you're being taken to the cleaners now go over to the odds-on who's going to be in the NBA Finals they're Lowe's they're posted assume each of them is a fair bet and find out what's your probability if you bet add them up and the number will be not less than 1 not equal to one not more than greater than one it'll be about two or three means that's stacked against you it's really stacked against you okay so try it go home and check some of these things check some of the things where the likelihood of such-and-such happening is listed your odds are such-and-such some of them up figure out what the for the fair bet figure out what it is some find out how badly you're being taken fairly badly on there okay so let me quickly summarize and tell you about it because we got three minutes four minutes number one where's my book where's my book I will be sending you chapter 1 through the internet what I decided to do is that rather than having you go buy something I'll send you chapters of the book as I sharpen them up free in other words number 2 course policy I sent you an email where can I find it you can find on the webpage course webpage that gives you more the policy of the course than what I have stated here be sure you read it so what we will do is number three homework probably about six or seven tonight I'll send you out your homework problems I got a class until five and then I do have to have dinner so maybe about seven so so tonight what I do is rather than rather than giving homework in class I give it by email after class why I'm not interested in presenting material I'm interested in changing the course as I did today changing my lectures see what I do is they prepare about two three different lectures notice no notes you have notes than a presenting material okay so I present prepare actually about two to three different lectures and then depending on time depending on questions depending on something like that by going one direction or the other so therefore I want the homework to reflect what we covered and that's why I do it after class in the evening after I've reflected on what we've covered so you're gonna get the notes you're gonna get for the chapter one you're and I'll give you a homework assignment you have the policy page and everything else and I guess for you're out with one minute to spare okay see you Thursday all right excuse me what discussion sessions start tomorrow how's that discussion session start tomorrow there's one of the TAS over there

Channel: UCI Open

Views: 165,356

Rating: 4.9136071 out of 5

Keywords: UCI, OCW, OpenCourseWare, UCIrvine, Saari, Math, Finance, Expected Value, Variance, Random Variable, Probability, 176, Mathematics (Field Of Study)

Id: JZn0GhwjjD4

Length: 74min 16sec (4456 seconds)

Published: Fri Jan 31 2014