Math 176. Math of Finance. Lecture 10.

all right here we have the x is normally distributed zero bearing slot y is equal to x to the fourth find p d f for y go ahead by the way we're now moving into material that's a little bit more technical a little bit more difficult expect many wishes expect many quizzes based on homework from yesterday on okay all right this one right here is actually worked out our first workout with notes and it's done exactly the same way that's what i'm erasing that is i want to find i want to find the probability that y is less than or equal to t this is equal to i'm putting in y's other name this is the probability that 0 is less than x to the 4 is less than or equal to t now when i take i have to solve for x because that's the only thing i really know it is a normal distribution with mean zero variance one and so therefore solving for x i have the probability of t to the minus one fourth final e to the one fourth and a minus in front is less than x is less than t to the one fourth any questions on that okay and this right here is equal to 1 over root 2 pi the integral from minus t to the one fourth power to t to the one fourth power e to the minus x squared over two dx yes um if we separated the probability with the two inequalities sure that's fine that's fine and then you add them together that's fine that's perfectly fine as long as you get the right answer yeah okay good so you see his confidence he already knows he's got the right answer okay and so now i want to find the derivative with respect to t of the probability that y is left y is less than or equal to t and this is going to be just differentiating this integral so it's going to be 1 over root 2 pi e the upper limit this is chain rule upper limit of e to the minus t to the 1 4 uh quantity squared over 2 times the derivative of this which will be 1 4 t to the minus 3 4. and then now i take the lower limit that's going to be minus e to the minus minus t to the 1 4 power squared over 2 times the derivative of this which is going to be minus 1 4 t to the minus three fourths you left it in that form that's fine if you collect if you collected terms you probably made arithmetic here did you so but if you left if you got that far it would be said yeah you got that okay any questions yeah um as long as you got the right answer i mean there's probably only about 50 60 ways to do this problem and uh so we're mainly interested i mean as you know from uh from the test what we're interested in is what you know not if you can add or subtract and so it'll be graded in terms of does he know what he's doing okay in your case his in her case because she know what she's doing oh you didn't okay all right so that's eight questions on this okay now got the new notes and i'm going to be uh expanding them a little bit when i have a chance this weekend but i want you to pay attention to the following thing we started off we started off the taylor series early on so if i have f of s t equals f of s plus a change in s f of s plus the change in s t plus the change in t we know from taylor series that this right here is going to equal f of s t plus the first partial of f with respect to s times the change in s plus the first partial with respect to t times the change in t plus other terms that just comes from taylor series all right so what we have is a new pen in a second so what we have it's this right here well i thought i had again ah i got a broken pen now oh boy so we gotta find a different pen um let's try this one so what we have is when i take this on this side we have the change in f is equal to the partial of f with respect to s the change in s plus the partial of f with respect to t the change in t okay everybody see that what does it say always okay what does it say all the way on the left on the top first turn first term the first term is yeah if i start with f of st and i'm going to change s5 delta s and i'm going to change t by delta t okay that's going to equal f of sp by the taylor series these are the first terms and so what we have then is this relationship right here and this is an important relationship which is used a lot considerably everywhere now what do we do is often we just simply say let's let let's let the first term the partial f with respect to s hey maybe that's a constant for y so that's where we get the constant and so what we have to expect is i'm going to try to find another darker pen now i don't like that i've gone through just about every color pen i can find so what we have we have then it's quite frankly it's not this is good the change in s is equal to mu s the change in t plus mu s the change yeah mu sigma s the change in x but this right here is very very natural it really comes from taylor series this is the first approximation for what's going on it's a first approximation for what's going on therefore you have to expect this equation to pop up almost everywhere you have to expect this equation right here delta s is equal to mu s delta t plus s sigma s delta x to crop up where we found out now that delta x to make sense really has to be normally distributed delta t variance we have to expect this equation to crop up in physics medical problems in uh economics finance everywhere because as such a fundamental it's just the first partial part of a taylor series expansion of a more complicated type problem all right that means that we want to know how does s change now this problem right here is called scale 3d it's called scale free really what i'm putting down here you know the first part of the lecture today he's really directed to point out that what we did last time do you have a question you have a question yeah okay that what happens is that the material that we felt last time carries over to almost all disciplines that we're doing it in economics finance but it carries over to almost everything else so suppose so let's look at that equation and let's let u be equal to 10 to the sixth s whoa this really expands everything very very large so s might be this big and u is huge talking about something very very different scale and so what is the change in you the change in u is equal to 10 to the sixth the change in s so let's try to find what happens here in this equation right here when i put in u and here i'm going to have the change in u and what is going to be uh s well u is going to be 10 to the 6 s and so when i find the equation for delta u i get the same equation i get the same equation because the 10 to the 6 cancels out on both sides the 10 to the 6 cancels out on both sides and so scale three that i get this equation for looking at really micro type things and i get the same equation looking for huge huge scale things and in any problem any area where this is a reasonable approximation where surface area of something like here is changing depending on what the current surface area is alpha d plus indeed a random type term oh i've got i got a mole right here and the mole is going to be increasing that's the size of the mole increase well i don't know but that is increasing depending on what on each little increment so it depends on the current size of the mole on all the sides and the random effect so just about anywhere you are you're going to run into this equation all right that means it is crucial that you review the notes to understand what happens to find what is the probability distribution for s of t we need to know this we need to know how to derive it in you need to know all of that what we did is we use ethos lemma where we chose f of s is equal to the natural log of s i was asked that the extra session we had on friday yesterday yesterday meaning tuesday that where in the world did you get the natural log of s well what happens if i try to simplify this by dividing by s the way we follow the value of money if i try to simplify this by dividing by s i get delta s over s equals mu delta t plus sigma delta x and this sure looks like it came from the natural log of s this sure looks like it came from the natural log of s so that's why we said let f of s equal to this then we use ito's lemon to find delta natural log of s so natural delta natural log of s will not equal this when i use equals lemma it will equal another term so one half sigma squared s squared delta s squared and so uh what we do is we're going to find this right here we add them all up this is all in your notes from last time your notes are not complete what you do is you just take a look at the notes that i sent out yesterday and it's all listed there you just add all of the little increments so what is that that's going to be here is t what is delta of natural log of s in this interval it is natural log of s of t 1 minus natural log of s of t 0 that's going to be the change in the natural log and then what we do is we write what's on the other side from e to slimmer we write that down we then say what's going to be the change in the delta s right here you add that in that's going to be the natural log of s of t2 minus the natural log of s of t1 equal whatever the right hand side is and we go all the way down and it turns out to be a magical a magical magical cancellation so what we end up with then is what we end up with then is the magical type statement but natural log of s of t by just going through that summation here it's in your notes and notes which have been passed out is going to equal the natural log of s of p 0 plus u minus 1 hat sigma squared t minus t plus w w is normally distributed means zero and uh variance sigma squared t minus t so we've got that elusive or at least a form of that elusive what is the likelihood of speed you're in medical school you want to know how is that cancer changing what's the likelihood the cancer is going to do such and such here's the type of equation you probably would use okay and so what we want to do is we want to find what is the likelihood of s of t so let's define the probability of s of t less than or equal to s that means i have to write this as the probability of the natural log of s of t is less than or equal to the natural log of s and you know what the probability distribution is for the natural log of s it's a normal distribution repeat mean given by the probability distribution the mean given by this and variants given by this so this gives you a wonderful way of being able to give predictions just think just just suppose suppose what happens is um you got an ipad and what happens is you're going to well it's going to be the price of an ipad on march 14th okay you know the current price of an ipad on the market than the free market or something like that but what you do is you go through this right here to find what is going to be the price of the ipad you can't find the precise price but you find the probability that it's going to range between on the free market between 350 and 450 high probability you see or low probability i'll sell my buy one later on so that gives you information about what to do so therefore what we have is by knowing what this thing right here is by not finding the pdf for this you can solve a lot of problems this is a homework problem i'm not going to solve for you i'm going to expect you to solve it i think that's a strong hint of what to expect but i will give you the answer i will give you the answer if y equals if the natural log of y is distributed that'll be the ugly term over there and sigma squared variance that's going to be the second term right here you will find when you go through this computation that the ddf is well let's let's first write down what would be the media for this the pdf for this would be 1 over root 2 pi e e to the minus x minus mu squared over sigma squared one-half right that's the mean for the normal distribution we carry that off the class what it will be here it'll be one over root two five each set e each stack to the minus one hand place of x we write down the natural log of s minus mu over sigma squared times one over s or s greater than zero that will be your answer so in other words all i'm doing is replacing replacing the x with the natural log of s and then putting the derivative of the natural log of s in the front are we missing a yes we surely are i just did that uh make sure that you caught okay and she says hey you left on the signal you gotta have a signal here and i have to have a signal here thank you that gives you a better answer okay now let me show you how this is a question you got a question i usually when people have a question that's something that five other people want to know okay so here i have if the natural log of y is normally distributed mu sigma then the pdf for y probability distribution function for y is one over s one over root two pi sigma got my hands left i forgot that first e to the minus one half natural log of s minus mu over sigma quantity squared or sigma greater than zero and he wanted to know the signal outside the roof and it surely is so let's double sure let me know this by putting on another parenthesis i'll decide something good now we're going to show you how i'm going to show you another relationship see what we're going to what this does is it says that this log normal distribution is closely related to the normal distribution this log normal distribution is closely related to the normal distribution so it's going to look when i look at the curve you know the bell shaped curve it's going to look a little bit warped and squashed and everything else we're going to talk about that in a second but i want to show you the close relationship between the normal distribution and the logic normal distribution and this is where that q i had earlier comes in for the normal distribution i know that one is equal to 1 over root 2 pi sigma the integral from minus infinity to infinity of e to the minus u minus mu over sigma squared to the one half pu okay that just says you sum all the unit all of the little probabilities you have to have probability one okay certainty now you see that u in there u stands for ugly okay all the substitutions you make in calculus you always use u for the substitution and that's because you let you the stand for the ugly part of what you're trying to get rid of so what i'm going to do now is i'm going to ask you to use the following change of variables ugly is equal to the natural log of s go ahead find the change of variables and see what what happens here so okay enough of you are looking up that i figure you have that so first everywhere i see a u i'm going to write in the natural log of s minus 1 natural log of s minus mu over sigma squared now i have a d u so i have to find what is d u d u is equal to one over s d s and so therefore i'm going to have a 1 over s ds with the change of variables there's two other places where x crops up one of the places where x crops up is in the lower upper limit here this is saying that u is approaching infinity if u is approaching infinity what happens to s it also approaches infinity so the upper limit is infinity the lower limit here is my u is approaching minus infinity how do i get u to go to minus infinity with s so therefore the lower limit here is serum and what we do is we have then that this whole thing is equal to one here is a probability distribution function if you see it's only a change of variable from here to here if you take a look it is precisely what we i stated earlier will be your pdf of the log normal distribution so this is an important distribution it's important does not does not come from the fact it's a change of variable a lot of books say it is a change of variable so it's important nonsense total nonsense it is important because of this expression right here which crops up almost everywhere and when we wanted to find what is s of t we know that s of t is going to be the probability distribution is going to be this thing right here natural log of s minus mu and you know what that mu is is that natural log of s of t 0 plus all that other terms and you know what sigma is going to be and you plug that in here that's why it's important so the log normal distribution is important because it is the natural natural distribution associated with delta s type terms and that's why you will see the log normal distribution everywhere okay any questions we've got one more question and then we're going to get on to something else one more question is what does it look like what does it look like and i'm going to leave some of this for your homework but i'm going to give you an idea the normal distribution the normal distribution you know the old bell-shaped curve here is mu the normal distribution looks like that no if you choose a smaller sigma the normal distribution is well it's got to get closer and closer to mu and you see why i'm not an artist uh shoots up like that this area under the curve has to equal one so it's gonna have a stronger heat now for the log normal for the log normal i have to expect that some of this is going to be squashed in i have to expect for the log normal let's see if this one works gonna log normal i know i can't go to the left of zero i cannot go to the left of zero because what happens is when we went through this distribution we had natural log of s s is only defined when s is greater than zero when we did this right here we see that s goes between zero and infinity so s has to be to the right of zero so what we're going to do is we're going to squeeze in all of this into a certain part of the region and all of this into another part of the region take a look where do we think things are going to hang around on the normal distribution x is hanging around both sides of mu here i'm expecting that the natural log of s is going to hang around mu the natural log of s is going to hang around mu because this is symmetric one side or the other all right and so i expected natural log so let's look at a special case of when u is equal to zero when mu is equal to zero we expect s to hang around one not quite not quite and this is going to be your homework exercise that i'll be sending out today the homework exercise is to find what is the peak of this what is the peak of this probability distribution see what is the peak of this one right here that's easy when i differentiate that's the concept i differentiate that i'm going to get x minus mu set that equal to zero x is right hanging around view or mu is equal to zero x is hanging around zero as we have from that drawing here you're going to find that the peak is hanging around something like something like about one over e to the sigma squared so sigma is getting closer and closer to zero in other words we're getting really really tight it's hanging right around one if sigma is larger that's a little bit to the left and so the probability distributions with the log normal look a little bit like this if sigma equals one and then here i and mu equals zero it's going to look a little bit like like that okay distorted it's a distorted uh bell shaped curve where i'm trying to get all of the information i have to the left over there squeezed in here type thing and then it's moving off there if sigma is equal to approximately 0.25 then what happens is it looks like something like that okay so the pete do you have a question nope in front of you right there do you have a question yeah okay uh what happens is that as sigma is uh closer and closer to zero that peak point that we had here is getting closer and closer to one that's if mu is equal to zero you can make the adjustments when you with something else when sigma is getting larger like one it has to peak over here it comes down like this you should have some sense of what the log normal distribution looks like because it props up so many places okay any questions yep that isn't very clear is it so let me rewrite that green isn't very clear one over the summer it should be something of the order of e to the sigma squared okay and so when sigma is getting closer and closer to 0 e to the zero is one and so therefore that peak is going to be closer and closer to one when sigma is equal to five that's e to the twenty-fifth to zero and so what you're able to do then is you have a sense and i want you this is one of your homework problems is to see how far off am i on this right here and to find what's going to be the peak of a log normal distribution and it'll be to try to sketch the log normal distribution i want you to know that because it's going to crop up again and again and again and finance just about everywhere any questions all right do you have a question oh yeah just moving around okay um let's quickly review what we did so far today what we did is we started off by reviewing and having a quiz on how to find pds probability distribution functions that's important you will see this again and again be sure you know how to do it secondly what we did is we used ethos lemon or i didn't use equilima i reminded you from your notes last time that if i use ethos lemon worth something like this that i'm going to get a probability distribution function for what will be the price of the acid what would be the price of the spread cancer in your soil that will be the spread of radium or radioactive material or anything else because they all come from that kind of equation so then what we did is i pointed out the importance of the log normal distribution which is something really truly have to have a sense like we have a pretty good sense for the bell-shaped curve you want to have a sense for the log normal distribution okay now what we're going to do is get into a question that was asked at the extra session and that was by the way i'm going to try oh yeah announcement thursday march 6th next exam thursday march 6 next exam that's right more than a week's advance notice uh second um i'm going to try between now and then time permitting to have extra sessions every week and i'll try to schedule a change today now what you did is you had a calculus problems like the quality the rate of change of f of t dt equals what's f what does he have where does head come from where does that come from we're trying to slide away where does f come from in calculus we have derivative of f with respect it's just there it happens to be a problem that is a sign of whatever you're trying to look at an issue that you're trying to look at maybe it is speed maybe it is how fast is uh water spraying across the floor whatever the issue you're happy to be interested in analyzing and so therefore when we get to ethos lemma i was asked where does f come from where does f come from you remember what ethos lemma happens to be this lemma happens to give you an and here and another obvious quiz problem is finding ethos lemma for the different delta s's that i had and that's on your homework i'm sure you do know how to do those but we know that this is equal to the partial of f with respect to s delta s the partial of f with respect to t delta t plus one half of sigma squared s squared the second partial of f with respect to s delta t and so the question is where does f come from it comes from wherever you happen to be interested so let me give you one right now and this was actually already on your homework but let's work out a little bit more generally suppose my portfolio my portfolio is equal to v of s of t minus alpha of s what's v you're going to ask that weren't you why portfolio is like a function ah why is proposing such a function excellent question what is what we want to do is we want a hedge and so i'm going to have b is going to be either a put or call it's going to be either a put or call or some combination i don't care you decide what it's going to be so he's asking uh he's asking the important vlog question who gives a iron what in the world would i have a portfolio like that all right excellent question so let's take a look at it let's take a look at a call if you have a call what are you thinking about the future yeah it's probably value's going to rise so how would i hit i could use a put but i could also go short i also could go short on commodity because if i'm going to go short on the commodity if i'm going to go short on the commodity what am i thinking that meant i borrowed the stock i think the price is going to plummet and what i can do then is i'll buy it at the lower price and then replace it so what is this this is a hedge it's a hedge it's a hedge between forward thinking in terms of the future that's the call negative thinking that the price is going to drop that's going short the put would be the other way around wouldn't it and so therefore this right here this portfolio right here is a simple portfolio you can make more complicated ones whatever you want to put for b but a special case is where i just have a call minus alpha s or put minus alpha s here alpha would probably be positive or negative so i get a positive value okay does that answer your question okay what i'm asking you to do right now is i'm asking you to compute compute delta pi of s that's right so so okay straight forward this is equal to this term partial of f with respect to t partial of pi with respect to t is going to be the partial of v with respect to s minus alpha delta s so we get from here from this term right here this term right here there's no t's in here explicitly implicitly but not explicit in here so this is going to be the partial of b with respect to t delta t that's all that survives and here i'm going to have one half sigma squared s squared i've got to take the partial of this one more time with respect to v and i'm going to have the second partial of v with respect to s and to make it a little bit more explicit going to have a partial of v with respect to s minus alpha times sigma s delta x plus partial of d with respect to s minus alpha mu s delta t plus the partial of b with respect to t delta t plus one half sigma squared s squared the second partial of v with respect to s okay everybody with me on that where's your last facebook um let's see i've got this term right here is multiply times this and this right here i brought that off but we know what delta s is delta s is equal to sigma s delta x plus mu s delta t so i multiply this coefficient i'm going to have a sigma s delta x times that and it's that times this right here this right here is going to stay the same and that's going to be right here did i answer your question or i lost where you are now do that any questions for your formula you have a very large data tea right yeah i have the opportunity at the end oh i lost it oh oh my that's crucial because i need that delta t thank you i need that now what's the purpose of hedging from the first day of football or games or basketball games what's the purpose of hedging yeah to eliminate risk in particular to eliminate risk an important part of hedging is to eliminate risk i told you about my friend it's quite wealthy as a lawyer you do have me in the market and you'll go and check your portfolio to make sure that they eliminated risk that they hedged appropriately so hedging is remember insurance typing eliminate risk where is the risk in this equation where would you say the risk is in that equation that last equation we got something excellent she's saying risk risk risk random behavior that's random that is risk notice i didn't tell you what alpha equals so how can i get rid of risk that's right so what happens is we get rid of risk i get this coefficient to equal zero and so this is a very important term on the market set alpha equal to the partial of v with respect to s in the finance books in the finance books this is called delta i can't use delta because i'm using deltas all over the place but in the finance books that's called delta that is a variable that the people on a market managing portfolio use constantly because what they do is they can compute by looking at the market they can compute what is the rate of change of say a call with respect to s and that will tell them how much to have for alpha i'll actually go long or short or if v is a foot the partial of p with respect to s tells me how much to go long or short so this is one of the tools that you use on the market to try to manage risk any questions on that now is this the only way to hedge nah i could just take what what are you going to do to hitch yeah oh if i just gonna okay well you know another way to hedge another way to hedge is to say hey i'm going to take i'm going to go short on all of this remember we talked about going short on things go put the money in the bank we've already talked about that a couple of times another way to hedge is to say that let's take the money out let's take the money out put it in the bank does it have to be the bank nah nah it can be whatever you want but you're putting it in somewhere else take the money out go invest somewhere else maybe i'll invest it in bonds maybe i'll invest it in in uh trinkets uh ant eater trinkets who knows what i'm going to invest in but i'm going to take the money out and there's what's going to be the rate of change of the money if i put it in the bank yeah interest rates are and he's asking what's the interest rate and the question is that our claim now you're a banker the rest of you didn't realize he's a banker and he wants people to invest in this bank so if not many people are investing in your bank what are you going to do where's your interest rate yeah you're going to raise it up a little bit suppose a lot of people are buying uh putting money in your bank you're going to lower it so we come to our good friend arbitrage our good friend arbitrage says that the money i can make on the market which is this is equal to the money i get if i take it out of the market and put it in the bank that interest rate will change and so what do we have here this is zero this is zero so we lost our drift term so that's all i have here so what we end up with is the equation his equation partial of b with respect to partial of v with respect to t delta t that's that equation there plus one half one half sigma squared s squared second partial of v with respect to s is equal to r here's a delta change in my money here's changing my money by going into the interest rate and i know they're equal because of arbitrage you know lower and raise the interest rate is going to equal r i forgot my delta t again is equal to r so now let's put down what that is that's b v minus what's alpha alpha is partial of b with respect to s delta t okay you see what i did here all i did was use all i did was use ito's lemon or a natural portfolio of going long or short to rip the item and with a putnam call that's all i did that's the first equation second equation is he said he's going to take the money out of the market put it in the bank he doesn't want any more risk if enough people do this the interest rates in the bank are going to adjust so both sides are going to equal that's good old arbitrage and so what we end up with is this equation here anyone see any cancellations delta t i can factor of delta t i can factor out that if i factor out delta t i'm going to have one half sigma squared s squared second partial of b with respect to s that's this term right here then i'm going to have plus r s partial of v with respect to s so i brought that out to these premiums on this side right here minus r v e uh plus the partial of b with respect to t equals zero did i do that right that looks right that looks right looks like a nice partial differential equation and if you had discovered that you would have got a nobel prize that's the black shultz equation that is the black scholes equation that's all there is to it see how easy it is to drive the derivation of the blockchain's equation is to simply state that my portfolio that i want to have is going to be b minus alpha s by my equals what is that get rid of risk as you get rid of risk that gets rid of this term in this term and saying that well wait a minute i can also take the money out of the market and put it in the bank set the two equal because arbitrage is going to allow you to set them equal by delta t and wamel the equation you have is the black it's that famous black scholes equation that you've heard about for a long time that's all it takes we're going to learn how to derive all sorts of variants of intellectuals equation if you throw all sorts of things in got a long weekend coming up have a good time you

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Keywords: UCI, UCIrvine, OpenCourseWare, OCW, Saari, Math, Probability, Expected value, Scale Free, exponential, ln, mean, standard deviation, substitution, derivative, delta, Finance

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Length: 61min 56sec (3716 seconds)

Published: Thu Feb 20 2014