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Mathematics and Finance

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thank you very much it's true pleasure to speak here on mathematics and finance so I'm on the mathematical and of this broader interdisciplinary endeavor here and wealth might talk of course will again be quite different from the from the previous starts With his secular the effect show it will be different because I will not deliver a speech on the paper of Miandoab book of mine and I have decided to give it rather non-technical pork and which ones to elaborate what was the influence of mathematics on finances what did good will be possibly do some harm but also in the other the other way around a little bit of Hoggard financing influence of mathematics and well so how can we liberate on these things and after some considerations I came to the conclusion it's a good idea to have a close look Howell models were developed in which context because it is usual see people have very precise questions to answer and if they provided good answers then maybe the small became successful and where applied possibly to similar
situations but
possibly also 2 quite different situations the which happened here sometimes to the good sometimes to the not so
good in order to get a better understanding of this I decided to have a somewhat historic you look back at how things developed in a in mathematical finance so the 1st part will be all the more historic nature where I have a closer look how things developed and then in the 2nd part I will come to more recent developments were will touch to some of my papers and I will try to draw some conclusions on their financial reality so let us start without our hero Beshear who in 1900 defended his thesis in Paris with a beautiful title cheerleader just because you're so at this time things were still named by the proper names and that act in over his an interesting character he was already 30 years old when he defended the history says he never went to 2 1 of the elite schools in in France it was an outsider to the system he was poor who was working as a kind of Secretariat that the ghost bodies so that the stock exchange and you can feel from every line of trees pieces that he was fascinated by this what's happening on the stock exchange my personal theory but this is speculation is this guy was a gambler there because I mean he was he came from a quite well-off family of wine merchants and larval when his parents passed away at the age of 19 he took over the company and the nothing's known and a few years later he's at the stock exchange in Paris at his poor what happened and an OK but maybe there are some more professional historians who can turn this speculation into some more solid knowledge little is known about him that's the only soda apparently as a way seen with a young guy so what you want to do he I was assassinated in particular by the option trading and he wanted to have a rational of of pricing and to understand his approach it is important to have a closer look at the concrete situation which affects their cell there was heavy trading and ended in particular there was 1 very interesting assets the so-called rounds of aren't so these a perpetual bonds and I have generously 1 Holland wanted to do a little history I can tell the story which goes back to the French Revolution and after that many of their noble man emigrated and when they came back in the restoration wanted to have the property back but after 25 years of revolution Noboru Matsuda this was difficult and so the Government of Louie 18 and they had this wonderful idea that gave them a perpetual bonds which would pay a quarterly interest and but the capital was never paid and the during the whole 19th century they were passed on the families to provide them with them than with an appropriate income and that was penetrating and also there was a futures trading in this which was fact typical trading and they where Delaware but options on these things which came up in a natural way also a elaborate too much but the these uh home to these perpetual bonds they had a nominal value of 100 Louie and the Patriot percent interest per year in quarterly coupons of 75 something and have but of course they were traded novel was at the at a price of 100 but it will go up and down but it would be particularly not deviate too much from its nominal value there would be as low volatility and the options he conceded that they were very short leaving options be to 3 weeks something in this order of magnitude was just that it was a futures contract and you combine it with the kind of secure which with a kind of insurance so instead of doing the forward contract you could also pay a sum of higher-priced but all Vermont you wear when prices went down you at risk was limited to 5 something and something with 20 something they the prices of the options they were standardized in strike price would vary in and they resulted in approximately at-the-money options OK so this was his situation and what does he do it well this picture I think everybody this from knows that this is the payoff function of such a call option disappears in this this is exactly as it well with different letters but otherwise it's exactly the same thing I don't liberate about and OK yeah now he becomes the 1st big step I mean his approach is probably I mean we're so used to this but I want to give a uh a reconsideration how does come probability here into into the game because after all no no .period at the stock market and he elaborated a very nice way on this let me quote from his thesis he distinguishes 2 kinds of probabilities that result with the 2nd 1 the 2nd 1 is what we have just seen that their personal at all you call personal probabilities or individual probabilities the age in Europe every the subjective subjected yet I mean an infected he even uses almost the same words this probability it's dependent on future events as he puts it consequently impossible to predict that the mathematical manner this last is the probability that the speculate that tries to predict the end he contrasts this With the probability which might be called mathematical which can be determined aptly Audi and which is studied in games of chance it's a little bit of a mystical this passage in contrast to the rest of the teachers but in hindsight it was some goodwill you can see here their personal probabilities of here their risk neutral probability of pricing probability as we call it today and have 10 yeah and he uses the word personal that the 2nd 1 I think directions of the speculators are absolutely personal since his counterpart in the transaction necessarily Harris the opposite opinion OK and now we terms To beautiful passage which recalled today efficient market hypothesis and what about this is it seems that the market the aggregate of speculators at a given instance Kimberly should meet their market rise nor market fall since each quoted prices there as many buyers and sellers where in 1988 and now comes the culmination of the home whole thing dispalcement enmity to speak to the news so the mathematical expectation of the speculate use the euro and wills the reason is exactly this paragraph and he says the consideration of troop
prices but true prices he means properly discounted prices we also corrects for coupon payments etc so all this he does perfectly right into it's easy for him to do the right thing discounting because they were doing things and forward to obtain house there has also been bit premium for the for the option was only paid an exercise so there was no problem here he does everything perfectly well and he says the consideration of troop prices permitted the statement of this fundamental principle the mathematical expectation of speculative 0 OK so and if you admired already here immigration will probe the next paragraph at least if you appreciate the mathematics here let me then we start with the example for example I buy a with the intention so the bonds I hear the riskiest sets that I buy a bond with the intention of selling it when it will have appreciated by 50 centimes the expectation of this complex transaction is 0 exactly is if I intended to sell my bond on the liquidation played wide any time or whatever so whatever you dove in average you that looms nor gain and this paragraph so much In mathematics in Inwood said that 1st of all this time when it hit will have appreciated by 50 something this is the perfect example for stopping time and when you stop to Martingale the optional sampling theorem of Tutu tells us that a medal expectations is always 0 so and yet you have been away you have the that idea of the Martingale here and in the 1st in the 1st paragraph which is not really a is describes whatever you do the operations which we they say trading on a Martingale you just don't change that at the expectation which always remains the erupt OK How can now the next day in the model which especially chooses and how he reasons for it I think most people know with which he has ended up well 1st of all he speaks about this transition probabilities of annotation is if we go ahead From today we go by into the future so safe in 1 month and 2 days the stock prices X then and we want to know what is the distribution of the stock price in wouldn't 1 month from now and the transition probabilities while should be homogeneous in time which means between now and in months should be the same from in the weeks to a month plus a week end should be homogeneous in time and also in space so whether his company is at 100 0 at 101 it should not matter OK and if she does these 2 things he immediately comes up because he knew the central limit theorem immediately canes of what we call today the Brownian motion has In the usual edited sense while I put here it's the unique solutions he was not deliberating on uniqueness and I'm in the mathematics when a different style than today also is only the unique some new solutions for the mathematicians would find at variance otherwise you also have this also stable processes it that but that's not what is of interest here he gets the solutions he analyzes it very well he makes the connection with the heat equation which is also a wonderful thing it's at the end he has of course this crucial parameter Sigma here which today we call the volatility and which he of course called much much nicer the coefficient of nervousness of the market much more beautiful than with the but things develop their way so Our pride is of course the Badgley it was 5 years earlier than and stand and independently more costly in physics and as you although it's called after this bottom is browned with 18 20 6 with 27 looked into a microscope and there were these particles that had this erratic movement but I was very happy at some stage when somebody pointed out to me that this is already much older and gather observations in the 18th century but much much older and here I cannot refrain from giving you this beauty here look dates from 99 to 55 before Christ so he was a contemporary of Tito and he road they did on of . 2 would say about the nature of things with 5 books so this is from the 2nd book so if he was in this maybe Korean philosophy of it be sticking reasoning and and of course his arguments I'm beautiful Lehtinen examined the whole community young like this he can among the other parties while I don't go on they're here is the English translation for those who have not so fluent in and a case so he describes a situation which does not correspond no not quite correspond to the room here you are in a completely knocked Roma but outside there is a bright sun and there is some Sunbeam coming in and that what happened observe what happens when somebody inside made it into a building and shed light on the shadowy places you will see a multitude of tiny particles mingling in a multitude of ways that dancing is an actual indication of underlying movements of matter that I hidden from our side it or it originates with the at times and at times it's interesting because they the Latin word for instance is pinned CPI so at the same time as the original things and it's the at times in this inter-Korean period it originates with the add-ons which move off themselves then those small compound body is such that I least removed from the impetus of the atoms of set in motion by the impact of the invisible blows and into cannon against slightly larger body so the movement mounts up from Apple's and regulator marriages to the level of our senses so that those bodies in motion that we seen some and move by blows that remain invisible this is really I mean that of course it corresponds exactly to what what engine and company did 2 thousand years later an OK so this is the background for Brownian motion OK no His fix them all along and he's already in perfectly good shape because his recipe is you prize everything pie expectations .period being by isn't immediate consequences from this fundamental principle the mathematical expectation of the speculators 0 you just have to take expectations and here it's really easy it's easier even than in the usual Black-Scholes analysis which we have to do for our students and
so for example would you have to do you have their doubts and distribution and you have to integrate such a function would really elementary and I just picked you a couple of nice results which is not yet the interesting thing he did not end up with a formal you can easily do the formula because our usual way is we fix the strike price and we calculate that the premium for them it was the other way round and apparently there is no way known closed form how to write to the theory that the former lover in the in the other direction but he gives very good recipes how you have to calculate its effect to that end he has beauties so and at-the-money the money call option at the money means that the strike price is exactly the actual price or to be precise that forward price OK for an at-the-money money call options 1 pertains a very nice form so it's proportional to statement to this small city in the edit its sense of that she is proportional to the square root of the time too maturity and for mathematicians of course particularly nice comes up and OK In another beauty which is not matched in the air in the Black-Scholes formal of course as simple particularly these things also hold true for them for the Black-Scholes formulation to the lean years station but not in this but the East the time to exploration so you have so it is a told it was maximum 2 months that it his time he was wondering what is the probability To make a net gain when buying an at-the-money option because I mean you have to pay the premium and your pay of should be in the east of the what have paid for the premium and it's it's easy to see you just have to plug in this into the density of the normal distribution but this with the segments where the 15 but also the normal distribution has to be properly read normalized and if you do a very easy calculation the segment the square root of the cancels out and what you get is a quantity which does not depend on Sigma 14 it's something and this is this is about probability is so you have probability of one-third of making again you should probably give 2 hoots of losing but this of course does not contradict that the fundamental principle because the losses abounded and their gains can be high and this is reflected by the but of course is a beautiful formulas it's a better OK so I made it shouldn't overdo it yes next step there comes up the notion of arbitrage and we know and this this will be my next step arbitrage probability related in any other trash comes out in but the I'm not under this name and that of course is again he just writes very nicely these for him these operations in which 1 of the traders would profit regardless of the eventual prices perfect definition of arbitrage OK and what he what he can he he mentions we will see that such spreads and never found in practice no arbitrage principle in just common sense OK not a big deal for him he used it for several things so for example when you have the option price is that this is a complex function of something like this he proved was that these arguments but otherwise that the basic relations between the probability theory in 1 hand and the lobby try from the other there he had not while he did not do and in my opinion he did already so much and about to do such a thing he would have I had to make quite a number of additional steps in his slowly and over again but back to them back to the basic power comes probably into the getting and there it's there it's nice to have again and look at his pieces where at the end of the day of the season does some empirical work he looks at actual option prices of course old soul determines what we call the political activity that looks to which volatility or nervousness of the market the absurd prices correspondent and how they they are consistent among them and and then he writes in a somewhat pompous way if with respect to several questions treated in this study have compared the results of observation with those of the it was not to verify formally established by mathematical methods but only to show that the market unwittingly obeys the law which governs in loft probably OK so histories is was he was an outsider he just gave in his thesis but they made a jury consisting of 3 persons and 1 of them was on the point of In .period very close the parameters had nothing to do before this but he understood immediately perfectly with this guy was doing and a very positive and insightful wrote a report but I just quoted this passage where he does this and he's less enthusiastic so .period today what should not be expected varied sectors protection the principle of the mathematical expectation holes in the sense that if it were violated there would always be people who would act so as to re-establish its and they would eventually notices but they would only notice it if the deviations were considerable the verification then can only be crossed so if this holds an expectation it should only be being noticed when the deviations of so that there is something wrong the ouster the author of diseases gives statistics where this happens in a very satisfactory manner so but this is not the only passages read he altogether broke vary positive report it was the only report but still he appointed was not given that he was outside of the system to get the most your although Labrador and this 100 years ago exactly yesterday means thank you very much goodbye there and because the French system unit emotional testimony related to it would have to make an academic career so he had a hard time and only after World War One with so many mathematicians have died he got a chance imbecile sold soldiers to get a professorship but altogether was that not so a lucky life as well let me go on with their with the questions are whether it is reasonable to use probability well as it was cold in there at the time in their shells were fired today we would say the social sciences and pointed heralded very interesting books Johnson book in some years later where he dedicates a whole chapter 2 chance and probably did and 1 of the reasons he got I should tell a little bit would us there so he elaborated on what we call jobs
and it is very much in this period of dynamical systems he says that the obvious example being flipping a coin or shuffling cards of something like this the situation must be a dynamical system for him where the whole situation is complex where are the people changes in the initial conditions may have a large effect on the outcome a and well what else you will see the moment and but I this kind of reasoning and then he asked candies b translated tool social sciences and that after some reason he has a very clear statement the loss of chance do not apply these questions and 1 of the reasons this is perhaps unfortunate since if he did call the city's method would protect us against miscarriages of 1 word about this conversation who was a match he but he was an active revolutionary in the French Revolution and he was he was already studying voting behavior it said that so he was the 1st to observed that when you when you have the choice of 3 people more than a can be preferred to be valued all I mean this was the beginning of the way too arrows impossibility theorem and among other things he also elaborated on how big a jury should be in order to have a good chance to get a good result he didn't extremely simple model I'm each church a probability p e to do the right thing and this should be bigger than one-half and then he concludes the more judges them more likely is that the uh corresponding to the good result comes out and this is what the point here is referring to that it's unfortunate but it's simply if it doesn't work for this .period Kerry argues in future years is his argument is so beautiful have we attempted to add tribute facts of this nature and where he says the applications in see also widened remember 8 years earlier he wrote the report on the butterfly's pieces of this nature to chance because the causes of obscure this was 1 of the things which is that this is how much of the causes of pure but this is not true job the causes are unknown to us it is true and that they even complexes where constituents for him where we can operate with each other but they are not sufficiently complex since they preserve something OK this was his argument before there should be nothing preserved this this was in this thinking we have all 4 of sportsmen with the with the ergodic conjecture that the matter that should be sufficient meets this was the the idea at this time he gives examples where in part shuffling through the simplified cut shuffling you just change the position of 2 cards whenever it .period comes up with heads of state and this is of course a very quickly nothing is preserved of the original structure and said that he says you can also do it with a bias .period no problem casing thing after some time you will have lost all the structure but he the coin is such that it always falls on hand that's different than there is the situation is too simple something is pursued and this is what it says here when it comes to a human being something is preserved men that comes a very poetic passages which is often quoted we have seen a young 1st of all that the preservation and this is something is a distinguishing mark of 2 simple cause it's OK when manner brought together they no longer site by chance and independently of each other but they react upon 1 another many causes come into action they travel demand and draw them this way and that but there's 1 thing they cannot destroy the habits they have off Panucci so the sheep of convergence uh French tale of a belief in the 16th century and they end up jumping into the sea business that so the obvious herding behavior and he said this herding behavior this is the thing which is preserved this it's 1 thing which we don't get their way in which we find all the time and I think it's a very nice and beautiful warning that at least 2 extreme events like all jumping into the water no 1 should be very careful to apply probability of at least .period Kennedy had a very outspoken opinion so Lalas go to the to the more recent developments as we all know then but she is model was slightly modified by Paul Samuelson who knows because he get postcards by Jim Savage who knows the work of specially tutelage was looking for a book of question did not for his thesis and Paul Samuelson looked at MIT and elaborate find went to Harvard looked to the library and there was this thesis and took it in with his high school French somehow deciphered what this guy was doing which was completely forgotten at least among economists is that no influence and immediately was fascinated and among other things he wrote out there the multiplicative pollution of it while you may allow of 4 adrift but as we all know that provincially it's is independent so this was natural because they feared that there was not any more than natural scale of 100 francs but it's it's not a big deal that they did the multiplicative the edited version With Yusuf Pathan we once calculated for the original date of passion you what is the difference when you calculate option prices with the with the additive model or with the with the Black-Scholes model of we wrote a small paper mathematical finance about this and it is of the order of 10 to the minus 8 of the underlying what 10 to the minus 6 of the option pricing so so this 1 is tempted to call this completely negligible up he would be as a specially called it because he got he addresses even his thesis that that in his mobile that stock prices may become negative but he dismissed with completed negligible aptly would be the same thing is for for the for the height of a person of a decisive issue so we have no problem of taking a normal distribution in statistics also a negative side of a human being is at least as absurd as negative while you for a stock OK but let us come to the heart of the matter no appetite this is that important to a new ingredients and Black-Scholes and then in a famous full load of the power their paper they attribute this insight to Robert Merton uh OK this is now their relations with their no arbitrage principle that in the model In the amyloid but is model this is a bachelor is method is the only prices which does not give arbitrage possibilities and let me just
mention this fundamental theorem of a surprising this is in the in the years after the the paper by the blackened shells and by Merton these 3 also said in various combinations in 3 important papers around this time made it clear that no arbitrage is essentially the same as the existence of a Martingale measure for the price process as in effect has mentioned with about a week a general mathematical context we get rid of the would essentially have made there a precise mathematical theorem others there there the I should mention here this is really the influence of the other way round is also important that many of these these questions coming from finance the triggered quiet their development within mathematics and made some progress in that the instant casting processes which which got to applications also outside of Of the applications in China and say
OK but let us come back to to this dynamic trading in no arbitrage principle I as I mentioned this was beyond the scope of year and this led to the uh notions of complete markets we have perfected replication then there was this serious dynamic portfolio insurance I mean to ensure it to protect your portfolio you can buy put options of course is always was possible but not people told you you don't have to buy put option you can replicate it adheres to the trading strategy and put it on the field it said that in this was 1 of the 1 of the main reasons of the crash in 1987 when the Dow Jones went down by 3 . 2 per cent of something like this within 1 day when implicit volatility shot up to 150 % which was even double as much as it in 2008 and this was already the perfected illustration of what point had predicted when people with the sheep somehow In a joint movement than the laws of probability do not apply very well anymore but this is not what Bush did or origin wanted I mean he was not doing his theory 4 for these extreme events but I'll come back to this in a moment let me yet at this stage I should this Council little bit her on the role of of course as we know from Black-Scholes and exactly the same way with but everything hinges on the Sigma which is assumed as a constant and especially already notices of course it's not constant that isn't something indicates that it is not a constant but changes according to the nervousness of the market and how do we deal with the case now the 1st thing was noted already in 1 of the really paper signed by Robert Martin it's that Sigma's constant he's not the essential thing they are potential thing is that it's deterministic OK or the only thing which really matters is the accumulated volatility from time 0 2 penalties if this is the real number and all random in exactly the same theory applies you really have nothing to change you just make a change of coordinates and Betsy so of mathematical speaking this is rather obvious but conceptually it's it's it's interesting so what you really have to to predict the is not there volatility but how much volatility will be accumulated from today until the expiration of the of volatility and now we go 1 step further there was again mathematical it's not that a very difficult infection this volatility Sigma it may not only depend on the it can just as well depend on as long as this is in the deterministic ways nothing changes you can OK you don't cannot write up a formula but can easily calculate that conceptually nothing new is happening so but while it well does this make sense economically to to model that the volatility is deterministic function of time let's start here as well as on the domestic front of time doesn't it make sense to more of the volatility will be 20 per cent in a week you will be the position of the not make much sense but they're down here this makes even less sense to say well if the surgery there since vegetable activities the 25 percent it's at that hour so but why was this invented because it was noted that very soon this at the volatility and it's not even consistent uh it at the same moment of time and with the same expiration you have to say most of you will know you have to smile and skew effects so you have to use their these strike price is far away From the actual price then the implicit volatilities higher this proof is like this this is sometimes called the Smiley effect and also so how to cope with it conceptually it is quite clear because there when you when you're away from today's prices than the extreme events play a stronger role and of course the extreme events are grossly underestimated by their normal distribution so by putting here for In a which is away from the present price and higher-value you could up at home accommodate for this and what do hear know what he what he designed is well you look at it all the prices of so-called plain vanilla options such as European options have been but with different terms of exploration and strike prices these you take a given and then you calculate the implicit volatility OK this gives you a curved and with this local volatility you calculate not the original offer prices because they had the input anyhow you don't have to recalculate them again that you can use it to calculate exotic options like a very options or you just think pocket payment options which are more complicated not this is this is a nice way of handling with this but we're losing all economic inside mean this is an abstract functions which should get out of Kelly breaking things too uh given the option prices without any good economic interpretation and of course it's wonderful it's from an American point of view this is a nice exercise from given this universe problematique that and it's quite hard to implement this on the computer and then you get out some numbers for exotic option everybody's happy it reminds me a little bit on the story also too costly and to cost you once said that the problem with the politicians is uh at nite they sit together with the journalism when they're talking to them the unlikely to them and where they read next the newspapers they stopped believing in themselves that are in a way and you you stuff to to involve all going very well and that there are the reasons why such a thing should exist center or why should you and me in this heavily results on the hypothesis that we have diffusion so uh process with continuous parts and and but then you you turn the computer chip that some numbers come out and of course they must be correct OK this is the point where financial engineering starts and where we did somewhat dangerously From the situation of Black-Scholes about where there is a clear inside into what is going
on so volatility in their mathematically satisfying way to cope with locally he used to say of course we will itself is forecast so it may go up and make a validated the let's put it that more worried as their stochastic process but the problem is all the duties of a of the Black-Scholes theory almost all the duties are gone and the this illusion of replication misses out this is political said uh dispersed we don't have complete markets anymore what I'm a little bit oversimplifying but essentially once you pass the stochastic volatility you have an incomplete markets so you may still calculated the balls of arbitrage free prices super-sub replication but you don't get a unique prices anymore the nicest thing was that this was independent of any economic arguments you just had to prefer a more to less but all of a sudden we factories along lyrics we have to use a utility functions of pricing by marginal utility it stipulated that but practitioners don't like this so very much and let me go yet 1 that 1st answered this is the meeting on robust finances there has been recently acquired after some development on robust financing by robust finance I mean in this context he model free finance so you don't assume any model the only thing which you assume no other Try this is quite harmless to assume no arbitrage yeah In from no arbitrage what we know is that the forecasting model stock prices they should be be and not under the pricing measures but nothing more and can we make a non-trivial conclusions from this and the answer is yes but only in some very special cases well anchored almost all conclusions but that there elements in some special cases but there's nice mathematics involved and it is of some economic relevance
and I want to show this to you this is not a little mathematics which put in but it's elementary so this is in a recent paper With some colleagues from Indiana so I don't think any quality uh let me know if it's need is there the Martingale then we compare the terminal value off the cause of the Martingale with their terminal about also the so-called maximal function or we take that supremo and at the time that we give financial interpretation so Frank wouldn't your favorite stock the I think so that is OK Damon never again and again so I promise to a pay you this is the bar with pieces a year from now and we look from today until the new year we take the highest-priced where that was in a pay you the square of the attack and so I'm afraid and this situation which we do here we know however that European options we have options on the uh stock Daimler In a year from now now because of the foxes but we can combine them and essentially every faction we can we can take cell we have on the market that we have a uh uh uh derivative securities which pays us the price of Daimler it's time the and the square of the but of course I am shimmering to my bones then we will have fewer 50 years 0 this is the area that we will go out and then it will go down the St and you little bit this is this is what I'm really afraid and I will have to pay you a lot and I can hardly protect myself with these things because why do I want to protect myself and there is still CD-quality which tells you that's when you compare the payoff of the maximal function square with the payoff of terminal while you you have an inequality with 14 so buying for European option should Everage like in the in the thinking of going to hell in a show should protect me against what I have to pay to OK but this is just an expectation now comes robust finances and we what we observed here is I can protect myself not an average but in a possible 1 way and here is there has been there that the statement we may find functions which we call trading strategies the IHT In the interpretation is how many of the Daimler stock I hold that tied the search that if I follow this aged and added that trading on the Daimler stock with the entity I combined it with 4 times their square off the terminal while you off as he hopes this is terrible mission here is a part this thing what we have here this high-value is tablets but now for every parts and I mean it now for every this is very important itself almost all because we don't have a problem with the disease every year the past bypassed I by the way this is not very well is such an elementary statement must have an element proved it is easy to prove this in fact this is a nice way 2 have approved for this to be the quality which is 1 of the pillars of stochastic analysis because when you take expectations here and remember you should have address square that the very definition of a Martingale tells you the expectation of this should cancel out so when taking expectations you come from year to year OK so the story about a robust finances and this with this I wanted to give you 1 example is you don't assume any mobile the only thing you assume is that you have to Martingale or even here you don't explicitly see anymore the Martingale but you can transform a Martingale inequality into at pathways inequality in any tells me to translate it into mathematical finance that I cared perfectly cover myself with this dangerous conflict which have sold to France by find for such the European derivatives and following this trading strategy OK now yes let me mention where these this is of course a nice example of what these these power options infected exceeds the reality but of course they have somewhat restricted to importance but there are some situations where this is all practical relevance let me mention the work of Peter Carlin wrote to me there is volatility derivatives volatility swap it that they had this theory works very nice let me there are nice results for hedging of burial options and yet I want to show you the law contract was realized volatility which is related to this paper here effective use of based on the inside of my bag and yes I want to show you that the idea in mind that's the last mathematics and doing by illustrating it with it with this simple model so our model is the yesterday the usual Black-Scholes model but the volatility was a we or anything it's just any predictable process we will we make no assumptions on it and we look on their logarithms well let's go go down here the logarithm at the time to maturity the knowledge it's a strange thing to look at the Morrison not so strange because Black-Scholes spirit you have everything in exponent and you take the long view get it out so if you this is just the Dow's formula uh if you do it then you can just write it in this way the integral way this is the law he has here uh term with the and of term with the half and now if you take expectations just as before you get this line here because the expectation of the integral on a Martingale this goes away so we know what is the price all this contract here
namely it's given by the accumulated while not quite that volatility but it's more natural to sum up the square of the volatility so this is the pricing formula but even more interesting is this formula here which is a path last fall this fall will makes sense for every trajectory that and now you can interpret this as a hedging to this is you always invest into the stock 1 over St this means the same as always In this 1 roll into the spot and in this case this thing here which is the prize together with this strategy exactly replicates the log of independently of what the statement he was so this is a nice inside and I wanted to yeah I should jump to conclusions so at there but I have to give some reference to robust finances yeah now this was all quite successful so far but now we come to a different range of applications let me recall this famous 15 report that is whether Stone who was the CEO of cheaper Morgan at this time in the aftermath of the of the 1987 crash he wanted to have 1 number in at 415 how much can we lose on our and this is an important word he had trading portfolio but were stalled because we had all this uh these models here and what people came out the fame of value at risk you chose to calculate the Kwan Tyler and I think this was a wonderful answer but it was this was internal analytical each instrument this is what it was invented for it was not number which is reported to the authorities and where is referred today where then there is the optimization of how was this of all of their risk weighting in its leader of course if you collaborate on this you can bring it down or as was noticed by the banner ads that debate he it has it has no good properties for their they're the troopers is also expected to that performed for a time research forest management and so this is a adding a very good example of that there is a tool which worked very well in this consideration that it was so successful and people now use it for things which is not so successful let me here here briefly some more applications beyond option pricing while along stories most for the term structure of interest rates well this is this is a very nice way also giving a coherent structure 2 the world all forms which has different kinds of maturity it's a tie which still builds on good inside and had to but starting from this there were will flow of mathematics was involved in credit refuse we heard today the most sophisticated mathematics was in this Watergate ship security station but all this is not based on on sound economic ideas and be dead the famous example is this doused popular which was decided for the CEO's so to to model the dependence also doubts in the 2002 credits and the gas popular the only advantage of as it is not easy to do calculate with but is so obviously wrong to that model something in particular this credit risk where you don't have the same situation of state that mean stock prices it is that you have a huge data you understand very well how market prices malls this was designed for things which do you did not have the spaces so I have to jumped to conclusions lessons to be drawn when applying mathematical methods and make sure you understand them all this is why I am I'm I'm not so much I'm not blaming Black-Scholes wasn't much else of course it is the to plug in the day that the correct the volatility and but people have a feeling they understand what it is it there good they also know you should normally applied to extreme events would not apply to risk management take .period Kerry seriously if this was very nicely described by CFO of large Austrian bank public display shows this wonderful mean it works so accurately in a 99 per cent of the 99 . 5 per cent of the cases except for that 1 or 2 days per year which really meant there it but it's clear they did is the obvious mistakes that it models their details in the wrong way or whether in principle you can make good sense of security behavior by probabilistic methods so this should lead to some modest and in particular when it comes to legal and accounting issues keep it simple keep the mathematics implement them in this what what we were told in modern his story this optimization of risk I think this is of course an obsolete and I and I am very much favoring have easy rules like uh just taking all their sense another starting with the risk risk-weighting or to have a conservative approaches for their accounting issues for how to how to count the assets and I want to make sure the comparison with the insurance industry that in life insurance they there and in the 19th century lots of insurance companies went bankrupt for a for a for lots of reasons but among among other things there is the necessity to use some mathematics in life-insurance because you have to calculate his research and you have to have put these reserves on right side of the balance sheet but there they have made the step it's really simple I mean such as locality pay which makes the probabilities to die this is something that every everybody in the building initiated camp a somehow appreciating can say whether this is really not that interest rate which apply admitted that these are things which is simple enough to be understood maybe not the senior management of at least by the by the people who do it and let me come to the conclusion mathematics is
a wonderful tool in the analysis among others of finance but remember good have when a measure becomes a target it ceases to be a good measure and love this phrase very much when you chose to try 2 maker of the Benetton headaches that they produce you some result then you should not expect the mathematics to do something good for you thank you very much and yes yes the well I've time to some questions the government so you think you were rented
Mathematik
Gruppenoperation
Abgeschlossene Menge
Vorlesung/Konferenz
Numerisches Modell
Eins
Natürliche Zahl
Besprechung/Interview
Familie <Mathematik>
Unrundheit
Term
Statistische Hypothese
Physikalische Theorie
Richtung
Topologie
Erwartungswert
Normalform
Spieltheorie
Rotationsfläche
Kontrast <Statistik>
Kontraktion <Mathematik>
Gerade
Kategorie <Mathematik>
Physikalisches System
Frequenz
Ereignishorizont
Erwartungswert
Mereologie
Größenordnung
Ordnung <Mathematik>
Eigentliche Abbildung
Finanzmathematik
Resultante
Impuls
Distributionstheorie
TVD-Verfahren
Einfügungsdämpfung
Prozess <Physik>
Nabel <Mathematik>
Natürliche Zahl
Formale Potenzreihe
Gesetz <Physik>
Statistische Hypothese
Richtung
Übergang
Einheit <Mathematik>
Perfekte Gruppe
Wahrscheinlichkeitsrechnung
Theorem
Minimum
Translation <Mathematik>
Vorlesung/Konferenz
Wurzel <Mathematik>
Regulator <Mathematik>
Lineares Funktional
Addition
Nichtlinearer Operator
Parametersystem
Multifunktion
Statistik
Gebäude <Mathematik>
Flüssiger Zustand
Rechnen
Frequenz
Dichte <Physik>
Normalverteilung
Menge
Erwartungswert
Koeffizient
Mathematiker
Explosion <Stochastik>
Standardabweichung
Wärmeleitungsgleichung
Subtraktion
Gruppenoperation
Zahlenbereich
Unrundheit
Ordinalzahl
Bilinearform
Sigma-Algebra
Physikalische Theorie
Ausdruck <Logik>
Zentraler Grenzwertsatz
Erwartungswert
Mittelwert
Stichprobenumfang
Pi <Zahl>
Indexberechnung
Varianz
Leistung <Physik>
Analysis
Beobachtungsstudie
Einfach zusammenhängender Raum
Markov-Kette
Mathematik
Relativitätstheorie
Komplexe Funktion
Eindeutigkeit
Gibbs-Verteilung
Physikalisches System
Kreisbogen
Brownsche Bewegung
Quadratzahl
Lateinisches Quadrat
Numerisches Modell
Resultante
Subtraktion
Abstimmung <Frequenz>
Einfügungsdämpfung
Prozess <Physik>
Punkt
Nabel <Mathematik>
Ortsoperator
Wasserdampftafel
Natürliche Zahl
Gruppenoperation
Unrundheit
Anfangswertproblem
Kartesische Koordinaten
Komplex <Algebra>
Statistische Hypothese
Dynamisches System
Algebraische Struktur
Existenzsatz
Theorem
Rotationsfläche
Zeitrichtung
Vorlesung/Konferenz
Schnitt <Graphentheorie>
Einflussgröße
Auswahlaxiom
Leistung <Physik>
Zentrische Streckung
Parametersystem
Statistik
Grothendieck-Topologie
Matching <Graphentheorie>
Mathematik
Physikalischer Effekt
Relativitätstheorie
Kombinator
Frequenz
Ereignishorizont
Sinusfunktion
Normalverteilung
Last
Mereologie
Ordnung <Mathematik>
Explosion <Stochastik>
Aggregatzustand
Finanzmathematik
Numerisches Modell
Punktprozess
Randverteilung
Resultante
Subtraktion
Prozess <Physik>
Punkt
Ortsoperator
Momentenproblem
Extrempunkt
Gruppenoperation
Zahlenbereich
Element <Mathematik>
Sigma-Algebra
Term
Statistische Hypothese
Physikalische Theorie
Ausdruck <Logik>
Dynamisches System
Wahrscheinlichkeitsrechnung
Reelle Zahl
Freie Gruppe
Vorlesung/Konferenz
Analytische Fortsetzung
Grundraum
Einflussgröße
Parametersystem
Lineares Funktional
Vervollständigung <Mathematik>
Mathematik
Güte der Anpassung
Gleitendes Mittel
Ereignishorizont
Deterministischer Prozess
Konstante
Arithmetisches Mittel
Chirurgie <Mathematik>
Normalverteilung
Verbandstheorie
Beweistheorie
Mereologie
Strategisches Spiel
Körper <Physik>
Faktor <Algebra>
Numerisches Modell
Resultante
Prozess <Physik>
Extrempunkt
Minimierung
Kartesische Koordinaten
Element <Mathematik>
Gesetz <Physik>
Raum-Zeit
Radikal <Mathematik>
Vorlesung/Konferenz
Kontraktion <Mathematik>
Gleitendes Mittel
Gerade
Lineares Funktional
Addition
Exponent
Physikalischer Effekt
Kategorie <Mathematik>
Güte der Anpassung
Gebäude <Mathematik>
Stellenring
Ereignishorizont
Arithmetisches Mittel
Rechter Winkel
Strategisches Spiel
Explosion <Stochastik>
Aggregatzustand
Subtraktion
Gewicht <Mathematik>
Sterbeziffer
Zahlenbereich
Derivation <Algebra>
Bilinearform
Trajektorie <Mathematik>
Term
Physikalische Theorie
Ausdruck <Logik>
Spannweite <Stochastik>
Erwartungswert
Algebraische Struktur
Logarithmus
Ungleichung
Stochastische Analysis
Leistung <Physik>
Wald <Graphentheorie>
Mathematik
Schlussregel
Paarvergleich
Neunzehn
Summengleichung
Modallogik
Quadratzahl
Flächeninhalt
Mereologie
Numerisches Modell
Finanzmathematik
Resultante
Mathematik
Güte der Anpassung
Vorlesung/Konferenz
Einflussgröße
Analysis

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Titel Mathematics and Finance
Autor Schachermayer, Walter
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/19501
Herausgeber Zentrum für interdisziplinäre Forschung (ZiF)
Erscheinungsjahr 2015
Sprache Englisch

Technische Metadaten

Dauer 1:03:59

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Fachgebiet Mathematik

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