THE INFLUENCE OF AMATINO MANUCCI

AND LUCA PACIOLI

 

Dr Fenny Smith

 

Right, that's what [Neime] looked like in those days, and we had various partners in this company.  There was [Giovanni] himself, there was [Panio], who was second in command, there was Amatino, sometimes known as Matino, who kept the books, and there was this Baldovino chap, who was the Chief Buyer.  There were some other partners: there were - I don't know whether they were brothers - Borrino and Vitale Marsoppi, Ughetto Buonaguida, and another couple of chaps.  It's not clear whether they're partners, but he's obviously in the family.

The major customers were the Archbishop and a chap called Bertrand des Baux, who wasn't actually a customer, but he's often mentioned in the accounts and obviously very friendly with the Archbishop, and obviously had a lot of power, and so, Lee thinks at any rate, he was good PR for the company.  We have some other customers: this village Lambesc - Lambesc was a village a few miles to the east; we have a draper in Salon; we have a chap Gaetano, who came from near Florence; I'm not sure whether a misprint has crept into this chap's name over the years because it looks like a woman, but anyway, he is the son of La Fiorentina, and this rather slippery looking chap, because sometimes he's know as Giovanni Giuserani and sometimes he's known by his French name of Jean Jusserand and you wonder what his close family called him.

Suppliers of this company? There's the Archbishop, there's Tano, there's this chap Giuserani, there's Tommasino Farolfi, who's obviously in the family of Mondragon, which is a bit further north, and there's an interesting chap called Astruc, who formed part of a thriving Jewish community, which had found sanctuary there in the South of France under the Holy Roman Emperors, and I like to think of these people as a group of merchants and traders who were all getting on, trading with each other, and not making any further judgements about each other than whether they're good to trade with.

So what did they trade in?  Wheat, barley, oats, wine, wool, cloth and yarn; they also dealt with exchange of money and the lending of money, which was all tied up into it; and we often here that there was an official ban on usury from the church in those days, and then we see that the Archbishop himself is happily borrowing money from the company at 15%!

So here's a sample page of Farolfi's accounts.  It's very neat.  It accords to the same paragraph format that we saw before, only there's only one column in the page.  There's a wide margin, and they're working in Roman numerals here, and they're doing their sums in the age-old way that people always used to do them from classical times onwards, and it was very comfortable and very easy, with counters on a board.  I'll have a bit more to say about that in a bit.

Okay, so now you can read this?  Here he says, "He gave us, on the day 27^th," so that's XXVII, "of February in the year novantanove," which is 99, 1299, and it has been posted where he has given - he's given it, so where he...the complementary entry is that he needs to have it in the ledger, al Quaderno de le spese, on the 8^th carte, on the 8^th leaf, £26 17s 2d, so there you see we've already got our complementary entries, with the positive on one side and the negative on the other.

So this document, it's in Florence, it consists of 56 leaves. There's a lot missing - there's a whole chunk from the beginning, and by the cross-references, you can see there's a whole chunk missing from the end.  Some of the pages have been stuck together, so poor old Castellani had to use a mirror and try and sort out what was the original writing and what had been stuck on top.

The first half of the ledger contains the accounts for the debtors, for the expenditure, and the debit side of the head office accounts, so that's the balance account; and the second half contained the accounts for the creditors and the profit and the credit side of the balance, so you've got this algebraic opposition.

With all the cross-referencing, he mentions all sorts of other ledgers that other parts of the accounts have been entered in, which we haven't got, but we could lead managers to build up quite a good picture of them.  

So we have the White Ledger, which is divided into two halves, and he thinks that the General Ledger was a continuation of it, because there's no closing balance to be seen on this ledger and the General Ledger has no opening one, and it's the earlier references that come from this.  So they filled up the White Ledger and they copied the balances over - they continued over into the ledger that we have.

There's the Red Book, which contained the main merchandise accounts, so this is your cereal, your wine, your oil, your cloth and whatever, and that's similarly divided into your two halves.  You've got the purchases at the front and the sales at the back, and the closing balances were then transferred to the debit of the balance account in the General Ledger.

You've got a couple of other ledgers.  I shouldn't have mentioned cloth because the cloth had a ledger to itself; and then there was the Expenses Ledger, so there was the current expenses and the things relative to the fixtures - that's the furniture and the warehouses.

There are a couple of other ledgers. There's what Lee translates as the Plush Book, and this was for the one partner, Borrino Marsoppi, for his business expenses.  They totalled it up from time to time and they debited to the Expenses account and credited it to his account, so you can see again you've got this algebraic opposition, and his account being kept separate from anything else.  Then there was another ledger, which seems to be the same thing but for everybody else put together.

Now, Manucci has been quite sophisticated in the matter of rent.  So he's got 4 accounts which concern pre-paid rent on various premises.  So he's got £16 paid to this chap on 17^th May 1299 for four years' rent in advance for a warehouse, and the sum gets transferred from the White Ledger to the debit of the rent account.  Then one year later, exactly, on 17^th May 1300, you get one year's rent, which is £4, and that's credited to the rent account and debited to the current expenses, and you assume that every year from there on, he's going to transfer £4 in the same way.

And he does the thing with the other rents.  So he's got advance rent of £12 for a shop for four years, and after a year, he transfers £3.  He's got £1 10s paid in advance for one year for another warehouse, and that's all written off at the end of the year.  Then he's got £12 paid in advance three years ago in 1297, for a second shop, and what we see in these accounts is the last balance of £4 transferred in the same way.

So in some, we've got, from Manucci, it's very sophisticated.  We've got extensive cross-referencing; we've got different sorts of accounts: we've got personal accounts; we've got real accounts, as I say, for goods and for rent; and we've got nominal accounts, for fixtures, for current expenses, and for expenses of eating and drinking.

Now, we have evidence that he balanced the accounts, but we can't check because we've only got the debit side, and all the references to the credit side are in a chunk that's missing.  But Lee did a lot of detective work and chased up various accounts or various transactions that the other side was missing because we don't have the right...we don't have the manuscript with them on.  He managed to make it balance, but it was clear that Manucci's system had the means of checking his balance, so he's got all these components that Lee put forward as being on your way to double-entry bookkeeping, and the only reason that we're not sure whether he is is that we don't have all the evidence, but we know he's there.

So now we're going to talk about Luca Pacioli, who included the first account, printed account, of double-entry bookkeeping, and in contrast to Manucci, we know quite a lot about him.

He was born around 1445, in a tiny small place called Borgo San Sepolcro near Arezzo in Tuscany, the same place Piero was born.  This is how Piero drew it.  There's a little sketch in a detail of one of his paintings, and that's what it looks like now. 

This is how his compatriots remembered him in 1878.  They say he invented it, but we know he didn't, but it was in pretty sophisticated state by the time he got there, and he answered a need for instruction in it.

Now, he does give us some biographical information, and again, this looks a bit like the last transcript.  This is about a quarter of a page, and the 600 pages all look more or less like that, with a few diagrams.

He began his career as a tutor to the three sons of a rich Venetian merchant called Antonio [Rompiansi] in one of the islands in the [?] in Venice.  When he was there, he attended the maths classes of [Dominico Bragadeno] at the Rialto, and he showed an early start in his career in 1470 by writing a book on arithmetic and algebra for his students, the sons of [Rompiansi].  We don't have that anymore, unfortunately. 

He travelled to Rome in the same year, where he stayed as a guest of [Leon Batiste Alberti]. Some time between 1470 and 1477, he was ordained as a friar, Franciscan friar.  Now this is a mendicant order, an itinerant order, so that's quite consistent with him spending all his career as an itinerant teacher of mathematics, and almost every town you can think of Italy, Pacioli was there at some point, teaching mathematics, and even spent some time in what is now Zadar on the Croatian coast. 

He spent some time at the court of...[Montefeltro in Molbino] and Piero was there as well.  This is one of Piero...s paintings.  You can see the Duke himself kneeling in the foreground at the Madonna and Child, and you can see Pacioli on the...if I had my cursor, I could show it to you.  He's on the right, between St Peter and I think it's St Francis, and he looks much like the other pictures we have of him, including his woodcut, which you've probably seen.

That's his woodcut, capital L. Every time you need a big capital L, that's Luca Pacioli himself giving a geometry lesson.

This is the frontispiece of the Summa, printed in 1494 in Venice. He supervised it, and then after that, he went to Milan, where he was at the court of the Milan and worked closely with da Vinci on this, on De divina proportione, on the Divine Proportion.  Pacioli wrote the words, da Vinci drew the pictures, and I'm sure you're all familiar with these wonderful illustrations he made of the regular solids.  There are a couple of manuscripts of that, and the book itself was published, in black and white, in 1509.

So now I'm going to give you a bit of a diversion, because one thing that I"ve found"I may have hinted at it, in the literature, these guys, the early guys, the 1300s, they're all using Roman numerals, and [I'm one of the] people who think, gosh, how on earth did they do their sums with Roman numerals, you know, how did they manage - they were all waiting for the Arabic numerals to be introduced so that they could do their sums properly!  It wasn't like that at all.  They were perfectly happy with it.  Italy seems to have been rather further forward than everywhere else, and as I've mentioned earlier, Leonardo Fibonacci wrote his "Liber Abaci" in 1202, and the word seems to be the same as an abacus, which we think of as a bead frame, but it was nothing to do with that.  It's how you use the Arabic numerals, how you do all your commercial sums with them, and the Italian merchants adopted it and they embarked on an intensive campaign to educate their children in how to use these Arabic numerals and do all your calculations with them.  They sponsored schools, so sometimes they were private schools, sometimes the commune actually paid for the school, where most male children would go and learn how to do their sums with the Arabic numerals.  These maestri, the teachers, wrote their own books, the merchants kept their own manuals, and they form a corpus called the libri d'abaco, all about the Arabic numerals.  The first one, apart from Fibonacci, comes from the late 13^th Century, there are 15 from the 14^th, and there are over 100 of these from the 15^th, and there are over 200 printed by 1600. 

I couldn't find any examples of the manuscripts, but there you see, the one on the top left, that was the first printed abaco, from Treveso in 1478.  This is [Philippo Calandri], bottom left, in 1491, and here you have a page of Pacioli's own manuscript that he wrote in 1478 for his students in [Peruja]. It's about the most illegible handwriting I've ever come across.

Now, these libri d'abaco, they're all slightly different, they all cover the same material, so there's lots of overlaps and lots of different emphases: some would mention some things other people would miss out; some would give more detail, some less.  They often used the same numerical examples, so you'd have the same problems that would go all the way through, and they would be common property.  Everybody would use them.  Sometimes they'd be all exactly the same; sometimes their solutions would be different; sometimes people would use arithmetic solutions; sometimes people would show, oh look, we can use algebra for this - algebra is the great thing of the future, let's show you can do it through algebra; some had more explanation and some had less explanation, and some corrected each other.

But in Northern Europe, we weren't the same.  We rather liked still using our counters that we'd always used since ancient times, and we did.

Here's an Ancient Roman, 1^st Century AD.  In the top picture, on the left, you can see, he's on his deathbed, he's writing his will, dictating his will, and behind him, on the left, and there's a close-up down below, there's his steward, totting up all the things he's giving to make sure that, you know, it all adds up properly.

We've got a vase, 5^th Century BC.  It's known as the [Darius] vase.  It's full of pictures, and one of them is Darius' steward, taking in all the tributes, totting it all up, and he has a table with various denominations marked on it, and he's got his counters and he's shoving his counters around, casting his accounts, adding it all up.  Doing your sums with counters, it's intuitive, you don't need a lot of education, it's very easy, people liked it, people trusted it.  The only thing is you can't check your sums.  We, in Northern Europe, we were very, very loathe to give it up; we hung onto it.  We were still using this method in the 18^th Century.

The table you can see there dates from France, from the 16^th Century, and it's marked out for doing your sums.

The first arithmetic books, printed in England and Germany, on the left, you've got Robert Recorde's.  They were hedging their bets a bit - they showed you how to do sums both with the Arabic numerals and with the counters.  This was from Adam Riese, who published...possibly not the first, but the most influential, in Germany, and again, he offered both sorts.  His first edition I think was 1522, and I've shown an edition from 1576. Obviously, there was a demand, and that's the way people liked to do it.

Here's a blatant piece of propaganda, from 1508, from Germany.  This is the Margarita Philosophica, published by Gregor Reisch, the pearl of wisdom, and here she is presiding over two chaps doing their sums.  We've got poor old Pythagoras on the right, still using his counters.  He's looking a bit shabby and care-worn, a bit worried.  In contrast to him, on the left, we've got this posh young chap, looking very confident and pleased with himself, with his fancy hat and his frilly shirt and his dainty fashionable boots, and he's going far, and he's using the new numbers!  It didn't work.  We were still using them well into the 18^th Century.

But let's get back to Pacioli, in Italy, where education was more forward, and where he's cashing in on the fact that there is a need for instruction in double-entry bookkeeping.  That's the first section of double-entry bookkeeping. It's dedicated to the Duke of [Urbino], as is the whole Summa, with the intention of giving his subjects all necessary guidance in the successful conduct of business, and he recommends double-entry bookkeeping, in particular the Venetian method, which is what he's grown up with, which surpasses all the others, and if you're familiar with this, then you'll be able to cope with any other style.  It's in the same style of the rest of the Summa.  It's full of exhortations and prayers and homilies and little advice, and the first piece of advice he gives you is that you need to be a good accountant and quick at computation and you need to arrange all your affairs systematically.

Now, I'm going to give you just a very quick summary of what's in it, because there's loads of translations, loads of people have written on it, so I'm just going to give you a flavour of it.

So he tells you how to make an inventory.  You've got your three ledger books - the memorandum, the journal, and the General Ledger.  He goes through all your aspects of commercial life and tells you how to enter various transactions into the books and for various different accounts you might have, and for the all-important closing or balancing.

So, the inventory does have an example, which is dated November 1493.  He says the important thing about the inventory, you've got to do it all in one day, everything that you own, all your stock, you need to have a record of what that is on that one day, otherwise you might find that you're in trouble.  You've got to be as detailed as possible, and it helps you because you've really got to be vigilant in all your business transactions, and then he says: "There are more bridges to cross to make a good businessman than to make a doctor of laws," and the big I like is that, "The law helps those who are awake, not those that sleep."

Then he tells you how to maintain your 3 important books, and again, there's a warning: have your books officially examined, stamped by the commerce officer, and watch out, because there are these people who have two sets of books and they show one to the buyer and one to the seller!

Then he shows you how to enter your transactions.  So you've got different ways of purchasing goods, as we saw.  You can pay cash, you can pay with time, you can exchange goods - that's barter, or you can write out a draft, and various combinations of these.  He shows you how to enter these different combinations using your double-entry in your various ledgers. 

Then he tells you how to deal with various public offices, and again, keep careful accounts with banks, especially with banks, as the clerks can sometimes mix them up.

Now, with the Office of Exchange, as we saw, you had to pay a brokerage fee.  This was normally split between the buyer and the seller.  He gives you a worked example using 4% as the fee.  So what you're doing is you're getting 98% of the price, the buyer is paying 102%, that's the 4% difference for the fee, and he shows you how to enter all these various things in your ledgers.  It is a bit complicated, so he's quite comforting here and he says, "If you don't anything, well, you don't make any mistakes, and if you don't make any mistakes, you're not going to learn anything.?

There's more guidance: how to enter all the aspects of your business trips; you might have separate stores with different accounts; how to pay by draft or through the bank or both, and this is how you enter it in your various books. But you can detect a certain theme all through: don't, you know, don't trust anybody any further than you can throw them, especially a bank!  Then he shows you how to enter exchange of goods in your ledgers, and bills of exchange. 

You record all the details of everything in the Memorandum...when you're dealing with partnerships.  So, you've got all the details of your partnership there: the objective, you know, what it's for; how long it's going to last; who is employed; who's invested what; what the assets and liabilities are; and you keep this partnership account separate in the ledger from all your other dealings.

Then you have different expense accounts: the ordinary, the household; the business; the wages.  It's quite interesting because the Extraordinary Expenses, these are things that were lost.  So you might have lost things through sea, or lost money, or it might be stolen, or you might have just lost it at the gambling tables.  Actually, the example he gives is archery, so it's a bit like people betting on horses - you might bet on the outcome of an archery competition.  That's what occupied them, in those days.  But okay, so you've lost it, well, you know, put it down in the accounts.

Then he shows you how to do"this is how to enter, how to enter in your ledgers your income and expense; profit and loss; how to carry the books forward; how to correct mistakes, because the mistake's there and you've got to leave it there, so you've to make sort of complementary entries to put it right; and the all-important closing and balancing of the accounts, and getting an assistant to go through one ledger while you do all the totals in the other, and you have your trial balance just to make sure everything's okay, and you're still buying and selling while you're doing this, so what to do with all the things, transactions, that are happing as you do it.

This is almost the last essentials.  Your ledger's got to be closed every year - that's the way we do it, and that means, you know, you always know where you are, and "frequent accounting makes for lasting friendship"!  And then a little bit of an addenda: how to keep your correspondence in order; and then a summary of the rules for keeping a ledger, a summary of what's all gone before; and a summary of all the important kinds of transactions - where you entered in the Memorandum, you record it in the Journal, and you post it to the Ledger.

Okay, so it's full of inconsistencies and it's certainly not an ideal first text for a learner.  There's no sample set of accounts.  There are individual examples, but no sample set.  It's a bit like...I don't know if you've ever read a technical manual for, you know, your video or your machine - it's perfectly clear if you know what you're doing, but if you don't know what you're doing, you don't know where you are!  But it was popular, and essentially it was for merchants who knew what they were doing - they wanted something to refer to for the details.  It was popular.  There was a second edition in 1523, which was very influential, particularly among 16^th Century mathematicians, and the two great figures, Cardano and Tartaglia, they both based their big arithmetical, algebraic works on Pacioli's Summa, and they both included a small section on double-entry booking, because that's what he did, so that's what we do as well.

So what was his influence?  Well, [Basil Yame] says that the important people, all the best books on accountancy that came up in the 15^th Century were all based, to a large extent, on Pacioli's Summa.  There were some others, but they never seemed to be so popular. 

So you've got Manzoni in 1540, and it's based, to a large extent, on De Scripturis.  He tidied it up - he probably got rid of all those homilies.  He showed you, it's interesting, how to correct a badly kept set of account books, and most importantly, he has a sample set of accounts that you can look at and see, oh yes, this is what you put in there, that's what that bit is for.

We have Jan Ympyn in Antwerp, and again, he kept closely to it, he improved it, he had an illustrative set of account books, and this was very, very popular.  It was published - there was a French edition in 1543, in the same year, and there was an English edition by Grafton in 1547.

Then we have Hugh Oldcastle, in England, a Profitable Treatyce, and he borrows very closely from Pacioli, even up to not containing a model set of accounts.

We have Schweicker in Germany, and that's very derivative of Manzoni and indirectly on the Scripturis.  Interestingly enough, for all the charges that have been levelled against Pacioli for plagiarism, neither Manzoni nor Oldcastle actually mention Pacioli.  They don't say we've pinched this from him; they just copy it, so everybody was doing it!

Ympyn, there were many editions of his books, and some of them mention him and some of them don't.

What happened later?  Well, things just took off and burgeoned, so after that, enormous numbers of treatises in double-entry bookkeeping, a great variety of subject matter and treatment, but there's one thing that runs through all of them: Pacioli's name is continued to be remembered and he's still called the "Father of Double-Entry Bookkeeping".   The first time I ever came across Pacioli, outside my own little world, was in the 1990s, this great run-up to the year 2000, and here was Pacioli 2000, this wonderful new software for keeping your accounts, with a really grotty picture of him on the front!

So here we have two important figures in the history of double-entry bookkeeping.  Neither of them invented it, but they both had their part to play.  Manucci was one...he was a steady practitioner - he just did it - and he was one of a body of chaps that they were all using whatever tools and skills they had available, probably swapping notes, "Oh, I found this a good way to do this", "Oh, I didn't write this down and I wish I had, so let's make sure we write this aspect down in the future", and they're all going forward, and all kind of crystallising their way towards double-entry bookkeeping.  It was already there and established by the time Pacioli came along, but he was a flamboyant teacher.  His main aim in life was to get it out, get it disseminated, get it promoted, and he promoted double-entry bookkeeping, along with all the other good things, in the Summa.  So they both deserve their place in the history.

I think I've managed to finish in time, so if you have any questions, then if I can answer them, I will!

[Audience applause]

Audience Member

Is there any evidence in the works of these two gentlemen of marking-to-market or revaluation of stocks?

Fenny Smith

I'm not exactly sure what that means?!  So market-to-market...

Audience Member

If you had some stocks in your...in your boat or something, revaluing them to a different market...

Fenny Smith

To a different market value of what they were already...  I'm not sure that they mention that; I don't know.  The only thing I can say is that, in his inventory, Pacioli gives the advice that you don't know what the value is of a lot of your stuff, so mark it up rather than down, so give it a higher value in your inventory than you think it might do, but I don't know.  You'd have to ask the experts who've actually delved very hard in this as to whether that happened.

 

 ©Dr Fenny Smith, Gresham College, 25 April 2008

 

 

LOUIS BACHELIER AND HIS THEORY

OF SPECUALATION

 

Mark Davis

Well, thank you for that kind introduction and for the invitation to speak here.  You'll note, on my biography, the word "history" appears nowhere, and I've no pretensions whatever to being a historian, but in the course of the last three-quarters of an hour, I've acquired a whole lot more respect for people who do pretend to be historians, for the sheer tenacity they display in reading 7^th Century old accounts.  I find it hard enough to read my own accounts, let alone wade through anyone else's!  I'm also totally amazed that we all managed to survive so long without a spreadsheet!

This is kind of an abrupt change of channel from the point of view of topic because what I want to do is talk about something which definitely relates to the 20^th Century and really has no precursor before that, and that's the work which followed on from Louis Bachelier.

Actually, Louis Bachelier was a...started in life, well, he started in life before that, but at one point, he was a PhD student in Paris, in the late-19^th Century - he presented his thesis in the year 1900 on...the title of the thesis is "Theory of Speculation".  He was a PhD student with Henri Poincaré, who was certainly the most distinguished figure in Paris at the time I suppose.  In traditional French style, he presented the thesis, and then there's this handwritten evaluation by the jury of the thesis, and there's the comment that the material studied in the thesis is "far away from the topics usually considered by our students", you know, a slightly [?] thing, but nonetheless, they, quite rightly, gave him a lot of credit for originality, which you could hardly deny that he had.

So the point about Bachelier's thesis is sort of two-fold really.  On the mathematical side, he kind of independently came up with a whole lot of things which were precursors of a whole lot of probability theory in the 20^th Century, so he was a sort of manifesto writer in a sense, although I don't think he would have thought of himself that way. 

So you find in there things which are generally attributed to other people, because other people did it better than he did, frankly, but nonetheless, the ideas are there, and among them are Brownian motion, which is, you know, an object of study for a lot of the 20^th Century right up to the present day.  That's generally thought to have been sort of systematised by Einstein in his paper on Brownian motion in 1906, but this is studied by Bachelier, introduced and lots of properties produced.  He uses the idea of Markov process - I mean, none of these things are defined, so Markov defined the Markov process in 1909, but lots of properties of it are used, implicitly, by Bachelier in his thesis in 1900, and specifically, you know, specific equations related to Markov process, like the Chapman-Kolmogorov equation, which has sort of acquired its name from works dating from the 1930s.  He got the connection between Brownian motion and analytic objects like partial differential equations and the heat equation, which is a sort of theme in modern analysis for decades ahead, and he obtained results, technical results, in Brownian motion which appear in modern textbooks, like the so-called Reflection Principle, which he...you know, he got the right formula for Reflection Principle,.  An actual...if you want, a rigorous proof of that thing, covering, you know, in complete detail, that really dates from about 1956.  So he was way ahead of his time in terms of getting results. 

So that was the mathematical side of what he did, and then the...I'll go through these.  Maybe I'll just sort of summarise what the idea is first, and then we can cover these slides.

Then there's the economic side.  So the purpose of his thesis was not directly to introduce all these objects in probability theory, but he used them to study pricing of financial options.  That's something where I think the history is a bit sort of fuzzy, because if you look at?  So, a financial option is, you know, the right to buy something.  So I do a trade with...with William, and I say, "Okay, William, you can buy from me, you know, one share of IBM stock at $100 three months from now," and that's a right, but not an obligation.  So then, if William pays me some money for that right, then if the price is above $100, he can buy from me at essentially below market price, and so he can walk away with a profit; or if the current price is below the strike price, then he can just tear up this contract and walk away.  So he's paying me something now in order to acquire a right later on, which has either zero or positive value, but never negative value, and the question is how much should he pay me now for this.  That's something which nobody had really considered in any scientific way beforehand. 

Actually, these things were traded centuries before that.  When I was looking at the history in connection with this book - I guess I should hold up the book at least once in this presentation! - there's another little book, also published by Princeton University Press, called "The First Crash", I think it's called, by Richard Dale, and he delves into the history and points out that, in the 17^th Century, you could trade options in London, not more than a mile from here I suppose, and all kinds of quite sophisticated financial contracts.  Of course, this is just gambling.  I mean, no one had the slightest idea of what these things were really worth, and they probably didn't really care very much either.  So that wasn't - there was a big history of trading, but no real history of analysis, and I suppose that Bachelier was more or less the first person to have a go at this.  It turns out that the formula he came up with is remarkably close to the Black Scholes, which is the sort of gold standard of option valuation, which was produced in 1973.  So, you know, from the sort of formulaic point of view, he was very, very close to the right answer, but he didn't have the right idea.  It was a concept, but he didn't really have a theory that worked, but at the same time, the theory he did have was not improved on by anybody for 65 years beyond that.  So the - there's a kind of twin track kind of history with this.  So if you think of Bachelier as here, in 1900, and the mathematical side of what he did, which is these...you know, where he independently introduced a whole lot of things which turned out to be important later on, were sort of, in fairly linear fashion, carried on by lots of other people, some of whom knew about Bachelier and some of whom didn't, but lots of them did.  He certainly wasn't forgotten.  What he did was too sloppy to be really much use in the end, but he had...he had the kernel of the right idea.

So that was the mathematics side, a sort of fairly linear progression.  The economic side was a kind of step function, where he was completely ignored.  I mean, he had no impact whatever on economics for 65...well, 60 years I think.  Absolutely zero.  I mean, no one in economics had ever heard of him, and they had only heard of him when told, you know, by actually statisticians - I'll come to this later on.  Then, in the 1960s, it turned out that...the sort of...he kind of ignited a spark somehow, and a whole lot of development in financial economics then took place in a very rapid fashion, and within, you know, 10 or 15 years, we had a huge industry of option trading, essentially using formulas which are not so different from Bachelier's 1900 formulas.  So that's a sort of curious story.

The other sort of curious part of the story is that the probability side of this was - I mean, certainly no one on the probability side had the slightest interest in financial economics.  If they had any applications in mind at all, they were probably in physics.  So, there was certainly no intention of sort of developing some kind of calculus which would be useful for people in financial markets.  No one knew anything about it, they had no interest, it was completely unknown, and yet, when somehow this all became relevant, you know, when people in financial economics were sort of alerted to this sort of circle of ideas, it turned out that the sort of mathematics that had been produced was exactly and precisely, you know, could not have been better tailor-made for the job than what had been produced.  So somehow, there was this completely independent activity which, when sort of reconnected with the other half of Bachelier, was just exactly the right tool for the job.  That's kind of remarkable, so that's really what this story is all about.

So, if we just look at what he did, you see?  So, the key thing in Bachelier's thesis was Brownian motion.  So Brownian motion is a model for random, you know - you can think of it as the model for a stock price if you want.  That's the way Bachelier thought of it.  So it's a random motion and the idea is that it has a continuous path, so you think of this in continuous time, so you'd have a continuous function, and if you look at the increment at any time, so the gain or loss over some fixed period, then those gains and losses are independent over independent periods, and they have a normal distribution.  So the gain over one time has a normal distribution with mean zero and variants equal to the length of the interval.  So that's the standard definition for Brownian motion.

It has all these sort of properties that the [probability-ists] love to study.  It's a Martingale, it's a mark of process.  So Martingale is the - I mean, this also sort of comes from gambling.  Martingale is the model for a fair game.  You know, if you simply repeatedly toss coins, and you win or lose a pound on heads or tails respectively, and you just look at the evolution of your fortune, then that's a Martingale.  So you expect to have tomorrow what you have today, so the probability of gain or loss is equal at every time, and you're sitting with this pile of coins and your expected fortune at any future time is just equal to the size of the pile of coins you have today.  So that's...that's a stochastic process, but that property is called Martingale.  That hadn't been introduced in any formal way either by the time Bachelier came around, but he sort of used it because his idea in the thesis was that markets display some kind of balance, so that every trade in the market has a buyer and a seller, and so there can't be any kind of consistent bias in favour of one or the other, so otherwise, the market would not be in equilibrium and everybody would be piling in on the same side.  So the only way to have an equilibrium in the market, according to Bachelier, is to have a situation where the expected profits on both sides are zero, otherwise, somebody is getting consistent advantage.  So had this wonderful way of putting it: ["??"], that's what he said - so the expected gain of the speculator is nothing.  So what that is saying is that if you hold a contract, then it's a Martingale. You know, so whatever its value is now, you don't expect, you know, on average, to win or lose anything in the future, otherwise the market could not be in any state of balance. 

So that was the sort of basic philosophy behind what he did.  That's sort of...kind of true and not true, but anyway...what he then did was to go ahead with and assumption - and this is never really explicitly stated in any very clear way, but this is what's going on - that he assumed that this price process would be Markovian, so its values in the future only depend on its value now, not on sort of how it got there.  He introduced this transition density, which says, you know, that if you start at some point, y, then at time, s, then your probability of being at some point, x, at time, t, is given by some density function.  He derived what's now known as Chapman-Kolmogorov equation, which says, you know, that if you think of two, three times like this, and you start here and you end at the third time, then you must go through some point, an intermediate time, and so you can break down the probability of getting from a point here to a point over here by just looking at wherever you happen to be at the intermediate time and kind of integrating out that dependence, so that gives you a relation which has to be satisfied by these transition functions.  That's equation one there, called the Chapman-Kolmogorov equation.  You can find this in every textbook on Markov processes will have this in.  He noted that the Brownian motion, the way I just described it, you know - so that's saying that, if you start at a point, then the value at some future time is just normally distributed with mean equal to the level you have started at and the variant is equal to the length of the time interval.  So if you take that as your transition function, then it does satisfy the Chapman-Kolmogorov equation like this.  He didn't sort of think to point out that all kinds of other things might satisfy this as well, which they certainly do, so there's a kind of element of rough and readiness about all this, but that was sort of one of two routes he took to getting the sort of analytic properties of Brownian motion organised.  So one of them was get the Chapman-Kolmogorov equation, do some sort of little analysis - I mean, just look at what happens over short times, and you find that this transition function, Q, which I've put there as equation 2, satisfies the partial differential equation at the bottom of the page, which is the standard heat equation, the heat equation describing the...you know, the flow of heat from some fixed initial temperature distribution.  It's the evolution of temperature distribution, say, on a plate or something like that.  If you start at a given distribution, then time flows on, so that the heat kind of diffuses around, and it obeys this second order partial differential equation there, and then the same equation is satisfied by the transition function for Brownian motion.  That's a sort of key result in probability, because you start with some sort of description of random motion, you want to work out something about what's the probability of things, and it turns out the way to do that is to solve the partial differential equation, and doing that is a well-studied problem in analysis and numerical analysis, so that turns the problem from something, you know, which is potentially kind of tricky to something which is just a standard piece of applied mathematics if you like.

So just, you know - this is sort of for the experts within.  There's this little result about the distribution of the maximum of Brownian motion, which you fine - again, you'd find any modern textbook on the subject would include this result, and it was given by Bachelier in the thesis - but as I pointed out, it actually depends on something called the Strong Markov Property of Brownian motion, and that wasn't introduced or properly studied until Hunt in 1956, a paper by Hunt, and it wasn't really until then that you could actually say that that result had been proved.  So it was a long, long way ahead of its time.

Okay, if we turn to the economics side of this, this is - I mean, his objective in the thesis was to use these stochastic methods to study the valuation of options, and so, as I explained before, the standard call option is just the right to buy something at a fixed price at a specified time, some time in the future, say, [?] t, and you have the right to buy a price k, and if you do that, then either this will be worth nothing - so if the spot price of your asset at the exercised time is less than the strike price, there's no value in exercising the option, so it has value zero.  If the spot price is higher, then you can buy for the lower price and sell for the higher price, and you get a profit which is equal to the difference, so you get this hockey stick function as the exercised value for a call option as a function of the spot price of the asset that it's written on at the time when you exercise it.  So that's the formula at the bottom of the page is just a formula that goes with the picture.  So that's the standard thing.

Now, it turns out that the sort of options that Bachelier was concerned with - okay, so the first thing is, of course, this thing has non-negative value.  The value is either - the exercise value is going to be either zero or something positive; it's never anything negative.  So if you want to acquire that right, you should certainly pay some premium for it, and the standard way in which that's done nowadays is you pay the premium at time zero.  So if I want that contract, I pay now a premium and then I exercise later if I want...if the conditions are such that I should exercise.  But that was not the way that things worked out in Bachelier's day.  The way that the French markets worked at the time was that you didn't pay anything when you - so you entered an option contract with no payment on either side, and then at exercise, you paid the fee, or the premium, but only if you didn't exercise it.  So either you exercised it and you just got the exercise value, or you paid - or if you didn't exercise, then you had to pay a premium.  So this hockey stick diagram then gets kind of modified to take account of that. 

So this is the exercise picture if you pay up front - except you have to take into account that you have paid up front, so there is a sort of negative bias to the whole thing.

But in the old French system, it looked like this, so it turns out that if that's the gain, then the best thing to do is exercise when your spot price is above the strike price minus the premium.  So that's the optimal exercise then and then you creep into profit at time K, so the value is zero at K, but when you move off to the left of the strike price, then of course you're losing something, and if you don't exercise, then you just lose the value of the premium, which is P.  So you've got this sort of...the whole hockey stick has been kind of shifted downwards like this, and so you get a loss on one side and a profit on the other side, and that's the right thing because you've paid nothing for this contract at time zero, so there must be some profit and loss, otherwise you just get something for nothing.

This goes very well with his idea that the speculator's net, you know, expected process is zero, so his idea for pricing this thing was the expected value of that, the pay-off from that function, should be equal to zero.  His model for the evolution of the price is Brownian motion.  It's an easy calculation.  You can set your students this problem, if they're not historians, which would be to figure out what the value of that contract would be, you see, and all you have to do is just look at the...you just look at the expected exercise value, you know, minus the premium, and it's Brownian motion, so the expectation is just the integration of the usual normal density function, so you can write down what that is, and then you just have to solve it for the one unknown thing, which is the premium, and you come up with a formula, and that's Bachelier's option pricing formula.

So it's...and it's not bad, I mean, it's not bad, this.  So that isn't really a theory, because why should this [be], have the expectation zero - you know, what exactly is the reasoning here?  That wasn't really clear.  It was clear in some sort of general sense.  I mean, it's obvious there must be some kind of balance in the market, but does that really translate into something about probability when you and I may not agree on what this probability is?  You know, I mean, I may have some view of what the future of the market is, you may have some different view, so how can we come up with one formula for this thing?  You know, we have to agree on one price if we're going to trade, but we may not have the same view on the market, so how can this possibly be the correct thing, you see?  So there's all kinds of objections to this.  It's not a theory; it's a formula, but it's a pretty good one, it turns out, so that's the good news then.

So, as I pointed out, if you look at the economics, absolutely nothing happened in this area for 65 years.  Why not? 

There are obvious reasons for this.  I mean, the main reason is simply that the time wasn't right for it.  You know, there was no real serious market in options.  You know, options was a gamble, there wasn't a whole lot of trading, it was a very peripheral part of financial economics, no one was really interested, so there wasn't really any sort of great need, perceived need, to get into this area at all.

Another point, which should never be forgotten, is that it would be completely impossible to handle option trading without computers.  It's too complicated.  As you'll see later on, you know, a big part of this is hedging, which is, you know, doing some sort of offsetting trades to manage the risk, and you just can't do that.  I mean, certainly, you couldn't do it with Roman numerals, but I don't think you could do it even with these wonderful newfangled gadgets of Arabic numerals.  I don't think you could do that without a computer, you know.  So it would be impossible, even with the best system of quintuple entry bookkeeping, to manage an option book in any sensible way, so it's just not feasible, you know, and you have to bear that in mind.  I think it's very important to bear in mind that the whole thing goes along with the technology, you know, and unless you have the technology, it's simply impossible, however good your theory happens to be.  So that's important.

And then, you know, people's minds were just elsewhere.  There were a couple of things going on in the 20^th Century - something called World War One and something called World War Two - and there was there was the Great Depression, which certainly wasn't any great incentive to indulge in sort of financial engineering on any great scale, and the Great Depression, you know, so all of this stuff was a reason why, you know, financial markets were just directed in a different area.  After the Second World War, there were various agreements, like the Bretton Woods Agreement in the late-'40s, which regulated exchange rates, so that really removed the opportunity for kind of the serious financial engineering in the foreign exchange markets, because the rates were, in theory, fixed.  So it wasn't until the 1970s, when this whole exchange rate mechanism broke down, that there was, you know, suddenly a huge opportunity to invent new ways of managing foreign exchange risk.

So, for all those reasons, nothing at all happened. 

So let me just...what I want to do I think is just sort of go through the mathematical development, which I said was sort of linear, from Bachelier onwards, and I'll just mention the sort of key people and the key events in this, and then we'll go back to the economics and say, alright, well, how did this thing sort of get into...back into currency, if you like, and then you'll see...kind of see the whole picture.

So if we look at the story of probability, and I think, you know, if - so if you look at this list of characters here, I would say everybody except...Einstein had probably heard of Bachelier.  I mean Einstein definitely would not have done, but I think everybody else on this list probably had heard of Bachelier.  [Wiener], I'm not sure - I mean, at the time, in 1923, [Wiener] may not have heard of Bachelier, but he certainly would have know about him later on.  So in no way was Bachelier sort of forgotten by this cast of characters.

So, as I say, the usual sort of...I mean, Brownian motion is called Brownian motion because it was an observation by a botanist called Robert Brown in 1826, that if you look at particles of pollen or something suspended in fluid, through a microscope, you see that they undergo very sort of irregular motion, and that was an observation by Brown.  I think what he did was - or at least either Brown or some later scientist, looked at these motions and they observed how far these particles, these little pollen particles, moved in a certain time, and they got something like a square root law, so the average displacement was proportional to the square root of the time. [Of course] the root mean square displacement was a proportion of the square root of the time.  That was an empirical fact which had to be explained somehow, and that's what Einstein did.  So Einstein formulated this whole thing in the context of, you know, the molecular theory of gases, and he figured out, you know, what the mutual bombarding of a collection of particles would do, and he got the same transition function, this thing that we saw before.  He got...yeah, so he got all this list of things down here.  You know, he got this transition function two, and he got the heat equation three, satisfied by it.  So that was...a sort of major achievement. In fact, there's a whole experimental side going along with this, which I won't go into, but I mean, after this theory, you know, experimental physicists were then trying to establish whether this theory was correct, and it gave an estimate for Avogadro's Number, which was subsequently verified, you know, so it's a big part of sort of early 20^th Century physics, this story. 

But it wasn't - as it stood, it wasn't really a big part of 20^th Century mathematics, because it was, again, you know, so now you want to have a theory, a mathematical theory, of Brownian motion, you know, and so that was a kind of analytic theory because it tells you what the transition function has to be, but is there a real, you know, is there a real, is there a sort of mathematical object that corresponds to this, and that's not obvious at all.  What Einstein pointed out in his paper was that this model, this Brownian motion model that he produced, could not be valid down to arbitrary small timescales because it's - he produced, you know - I mean, it's just a fact of the theory that the mean, root mean square displacement is proportional to the square root of the time.  So if you look at - for the average velocity of the particles, it's like one over the square root of the time.  So if you look at a very short time, the average velocity is blowing up, so you have this very weird thing.  So this is saying, so okay, so this must be something which is valid down to some minimum timescale, and below that, we'd have to do something else, because it's simply not a physical model if you allow infinite velocity, you see.  But from the point of view of mathematics, I mean, do you have to, you know, do you have to sort of re-scale everything at low, you know, for small timescales or don't you?  Norbert Wiener, in this extraordinary paper in 1923, showed that you didn't, you know, that you could have, you know, a perfectly consistent mathematical set-up, where you have a probability distribution not on - you know, this is an achievement, actually, it was the first example of a distribution of probability, of a random variable, which was not [finite] dimensional.  So, you know, if you think of an ordinary one-dimensional random variable, you just have a density function on the real line; if you have a vector, you know, something in n-dimensional space, you have a multi-dimensional, multi-variant distribution function; but Brownian motion is a path, it's a continuous function, and that's not a finite dimensional set, and so can you have a probability distribution on something that's not a finite dimensional set.  That was, you know, that was an obvious question that was kind of left over from integration theory, as developed by Lebesgue and others in the early first decade of the century.  I think this was the first concrete example of something that was a probability distribution on an infinite dimensional sense, in this case, the space continuous functions.  So the property was you have a probability of distribution which describes a motion of a particle, and that motion is continuous, and the distributions are exactly the ones of the Brownian transition function as derived by Bachelier and Einstein.  So that was a major achievement in mathematics, and it put Brownian motion sort of firmly on the map as a sort of well-defined mathematical object, which...it really wasn't as a result of what either of the previous contributors had done.

So then there's a whole list of kind of distinguished contributors that come after this.  I mean, Kolmogorov is, you know...actually, I was arguing with somebody yesterday whether you'd have to put Kolmogorov among the 20^th Century's best five mathematicians.  You could argue that I suppose either way, but he's certainly up there somewhere.  I mean, he was an extraordinary character, who worked in many areas, but a major one was probability, and the sort of standard framework for probability as we see it today, and based on measured theory, is the result of Kolmogorov's work in the early-'30s.

Now, Paul Levy is an interesting guy because he, in a way, was slightly more in the Bachelier tradition.  He was a French mathematician working in the '30s and '40s, and I think he was responsible for a kind of shift of perspective, which is maybe not so easy to describe, but is a key thing I think, which is, you know, probability, there's always two sides to it.  One side is kind of analytics.  You can look at a probability distribution as a distribution function; you know, it satisfies some equation like the Chapman-Kolmogorov equation, or like the heat equation, so you just look at the analytic objects which describe the probability distribution of something.  Probability, essentially, did that right up until the 1930s, and didn't really do very much else. 

Actually, there's - I always enjoy it - there's a little...there's a book by Leo Bryman on probability - it's one of the standard textbooks which is sort of first year graduate courses on probability.  He says, in the preface to that book, that there's two sides to probability, you know, it has two sort of...it's rooted in two different places.  One of them is analysis, you know, so that's the study of distribution functions; and the other is gambling.  Of course this is true.  No one can deny the connection between probability and gambling, and a lot of [intuition] comes from, you know, the results of random experiments.  So in a sense, there's sort of two ways you want to - actually, then he credits - Bryman credits Michel Loeve, who wrote this big tome on probability for...treating him [on the] serious side, and David Blackwell, who's a statistician at Berkeley, for teaching in the real side.

So there have always been those two ways of looking at things, so somehow, the mathematicians had sort of settled on this analytical side because it was sort of more comfortable to them, but they somewhat neglected the other side, from which you get a whole lot of intuition, and, in the end, a whole lot of mathematical technique.  I think probably the contribution of Paul Levy was to sort of shift the attention to the other side, so you use methods which are definitely based on sort of looking at the sample paths of some process as opposed to just looking at the probability distribution of that process.  You know, you get things like stopping times, where stopping time is, you know, the first time some process hits the level or something like that, and that's something which is essentially stochastic.  You know, you have to think in terms of the sample paths to do that, and results like this...this reflection principle which I mentioned, which Bachelier produced, I mean, if you want a proof of that, this is a sort of sample path idea, because you look at some level, you look at the first time your process hits that level, and you think about what happens after that, you know.  So that's...it will never sort of pop out of a kind of analytic treatment of the problem; you have to think in terms of how the process actually behaves, so you have to think in sort of pictorial terms to do this.  Bachelier did this, and I think he was...he was, in a way, his big contribution was to re-cast probability in that kind of spirit.  You know, it's a little difficult to sort of pin down, but...and then Paul Levy, who's simply just a much more professional guy, proceeded with the same kind of agenda, so he cleaned up a lot of this and introduced, you know, sort of much more formal methods and produced a lot of results, but in a way, it was a sort of development of Bachelier's thing.  Actually, they were sort of antagonists, but that's another story.

In the sort of - in the post-'30s period, there was, you know, we already mentioned the idea of Martingales, and sort of, from Bachelier's perspective, the Martingale was kind of the key thing, although he didn't have that name, and the name Martingale was introduced in the late-'30s. but he was using what essentially was a Martingale property in his kind of expectation zero sort of hypothesis.  So Martingales were introduced in the late-'30s.  The big book is Joseph Doob, published this book in 1953, which is the big, you know...I mean, everyone loves this book.  It's a great book on...which just gives a very, very nice treatment of Martingale theory, much of it originally due to Doob himself.

The other thing that turned out to be a key factor in this whole story is the theory of stochastic differential equations or stochastic processes, and this is due to Kiyoshi Ito, who amazingly enough was doing it in the middle of the Second World War in Japan.  The first paper was published in 1944 under the proceedings of the Imperial Academy of Sciences in Japan - truly amazing.  So his idea was how to get the connection between - you know, we already saw, right at the front of this talk, there's a connection between Brownian motion and the heat equation, so what is that connection in some more general context?  If you just move away from just Brownian motion but you look at other sorts of random, you know, some kind of transformation of Brownian motion, you get other sorts of equations, how - what is the connection between partial differential equations and stochastic processes?  In order to solve that, he invented this thing called Ito stochastic calculus, which has been tremendously influential over the last 50 years and is an essential component of this analysis that we're using in finance.

Where it comes from is just looking at the sample path - again, it's a sample path thing.  You look at Brownian motion, and you compute it's so-called quadratic variation, so you look at these increments and you sum up squares of increments, and you do that over a partition of some finite time interval, and then you refine that partition, so you get it smaller and smaller, you know, larger number of points and smaller and smaller distance between the points, and as you take that limit, it turns out the limit is simply equal with probability one to the length of the interval.  So you get a random motion, but for every part of that motion, you have a deterministic quantity, and that's its quadratic variation.  It's a totally amazing sort of property of Brownian motion, but this is the property that distinguishes, you know, Ito calculus from ordinary calculus.  So what Ito was pointing out, or the idea behind this, is that when you do [a kind of tailor series of expansions] involving Brownian motion, you know, you get the usual two terms, but then you get a third term which involves, you know, the square of the increment and that's not a second order term anymore.  That's a first order term, because of this - the property at the top.  That's an increment - that square is of order DT, not of order DT squared, so you have to keep it in the list of terms.  You can't throw that last term away; you have to include it.  That gives you the Ito formula.  So Ito developed this.  His calculus is not the same thing as ordinary calculus because you're dealing with functions which are rougher than ordinary functions.  You know, any function which has a bounded variation, this...that correction term at the end would disappear, but Brownian motion has exactly the right, you know, the right properties, and you can replace that DB squared by DT and that is the Ito formula.  So that's Ito's great result, and you find this everywhere.  Part of the Black Scholes formula is the Ito formula.

In the 1960s, the 1960s was a sort of big consolidation period, in a way, for all of these sort of ideas, because we had...the next...character in our cast list is Paul-Andre Meyer, who ran a school of probability in Strasburg, which became very, very famous, and he proved the so-called super-martingale decomposition theorem, which opened the way to extending Ito's stochastic intervals, much to...to sort of general martingales.  So instead of just integrating [?] Brownian motion, you could integrate with respect to any kind of martingale, and so that, from the sort of practical point of view, enlarges the modelling framework to, you know, encompass a whole lot more things.

This connection between the heat equation and stochastic processes was really sort of fixed once and for all by Strook and Varadhan in 1969, the so-called martingale problem formulation diffusion, that somehow, you know, martingales came in in a very nice way, and even in the sort of analytic side of probability.

Then I said, at the end, you know, there's this famous result by Bichteler and Dellacherie in 1979 was a...a sort of...a termination point for the whole theory of stochastic integration, because they - I mean, Meyer and co invented something called a semimartingale, which is a sort of sum of a martingale plus some bounded variation term, and the whole theory is based on this, and the question is why - you know, it seemed rather artificial at first.  But then, Bichteler and Dellacherie had a very beautiful theorem in 1979 which showed that if you wanted any kind of reasonable theory of integration for any kind of process, it would have to be a semimartingale, otherwise there's no reasonable theory. So they were saying, yes, you know, the other guys got it right, that is the correct class of process to look at; you can't really look at anything else, and so it just sort of drew a line over the whole thing.  The theory was there for semimartingales, and they showed you cannot do - that's it, you know, there is no other theory, and that was...sort of set the whole thing very nicely in context.

So the bottom line here is, you know, this whole sort of area started out as an area for pure mathematical specialists.  I mean, I should think if you looked at, you know, 1960...anything up to 1963/4 or something, and you said how many people in the world really understood Ito's stochastic calculus, then the answer was probably in the hundreds. There was no textbooks, no one had any sort of applications in mind...it was a very, very narrow speciality, and you know, if you look at the number of people who understand it now, then I supposed it must be...I don't know what it is, but it's everybody, you know, two miles from here and one mile from here down there understands it, okay!  So, you know, I mean, so now, you get textbooks, you know, this is a piece of pure - this is a piece of applied mathematics essentially.  There are well-developed rules, you know, it's widely understood, there are courses, you know, taught by people like William and myself, you know, on this area, and it's just simply part of the picture.

So if we then move on to the economic side of this, there's a story behind this, which is, you know, the key figure here is Paul Samuelson.  Samuelson is an economist who - well, I actually talked to him in 2003 in connection with this Bachelier book, when he was a young man of 89 - and he is a young man of 89, just an extraordinary guy, so he's now a young man of 93 presumably.  So Samuelson was - he's best known I suppose as the author of a textbook on economics.  The first edition came out in 1947 I think, and there have been successive editions, so you know, every economics student in America sort of learns economics from Paul Samuelson and his team of updaters.  He got interested in option pricing for reasons I'm not entirely sure what they were, but somehow, in the '50s, it started to be sort of a germane question again, you know, how to manage financial risk, and so options were things to consider in a more serious way.  So Samuelson started doing this in the '50s, and then he received a postcard from somebody called Jimmy Savage.  Jimmy Savage was the sort of leading mathematical statistician of the early sort of...late-'40s, early-'50s period I suppose, a very famous guy in mathematical statistics.  He of course knew about Bachelier, because mathematicians did, and he thought it's about time the economists woke up, you know, so he wrote postcards to his economist friends saying if you haven't read Bachelier, do, you see.  As far as we're aware, the only person that actually did was Samuelson, and when he did, he realised that there was a whole cornucopia of technique there which he needed to know.  So that was a...you know, so he was really sort of inspired by this. When I went to talk to him, I said to him, "What was it, you know - I mean, this was like 65 years before you were doing this, so what on earth had Bachelier got to teach you?" you know.  He said, well, it was sort of the methods, you know, so the whole idea of treating things as stochastic processes, you know, with probability distributions and the sort of stochastic analysis side, you know - it seemed to be just the right way to handle this sort of problem, and that's why they wanted to do it.  It turns out that there was a sort of big debate. It's very hard in these days I think - it's hard, nowadays, to kind of appreciate what this debate was all about at the time, but I think the idea was that, you know, if you think of what economists thought they were there for, in the 1950s, or maybe, you know, at any time, if you're...what's a financial economist trying to do?  Well, they're trying to explain the process of price formation, so you have a whole lot of interacting agents who are trading things with each other, in the way that we've heard described, and how do they arrive at prices, you know.  That's what financial economists thought they were there to do, you know, so look at the structure of a market, and you'd say, well, how do prices arise from the activities of these agents. 

There were beautiful theories about this in the 1950s, by Kenneth Arrow and Gerard Debreu in particular, where they show that, you know, you have a bunch of people who are, you know, who have some goods but they can trade goods with other people, and they do that and so - and they do that in a way which is sort of in their own best self-interest, then the whole system can be put together in such a way that you get a unique set of prices for trading these commodities between the agents.  So this was a beautiful piece of work.

Then, along comes Paul Samuelson and says, okay, we-re going to throw away all that and we-re just going to say our price is Brownian motion, okay.  So the economists say, well, you-re crazy, you know, you-re throwing away the whole - this isn't economics, this is just playing around, you know, because it has no connection with how these prices are formed.  So that was the sort of debate that was going on, and this is why Samuel ran into some opposition and why there was some sort of debate about this, but in fact, he was on the right track, and the reason he was on the right track was because, when you start looking at option pricing, you are not really interested in why the price of your IBM stock is what it is.  What you're interested in is why is it that the price of an option on IBM stock is related to IBM stock the way it is, you know, and that's a completely different question, and one that's best answered by a stochastic process approach, and this is what Samuelson realised.  So that was why he was interested.

So he looked at Bechelier and he said, well, one thing that was no use, no good about Bachelier is the fact that the price is Brownian motion, so that has a normal distribution, so it can be positive - in fact it will be positive - it will take negative values [to the] positive probability, whereas prices must be positive things.  So he suggested using geometric Brownian motion, so you just look at [?] of the Brownian motion instead of Brownian motion, and then obviously that's something which is always positive.  This is called geometric Brownian motion, and that's basically Samuelson's price model.  As soon as you do that, you realise you've taken a non-linear function of Brownian motion, so if you want to do anything, you have to use Ito calculus.  That's why Ito calculus became, you know, a definite part of this particular picture.

So, the equation I've written down there is the standard sort of Black Scholes, Merton, Samuelson model for prices, where you have a stochastic differential equation - that's equation five - which just looks like sort of a geometric increase, like constant interest rate [new] is some sort of noise perturbation, and if you work out the explicit solution for that, it's given - you know, this involves Ito calculus to get from five to the next equation down - you get an exquisite solution for price in terms of a Brownian motion, so the price turns out to be a non-linear function of Brownian motion and time.  So that's... that's good.

So, it wasn't long after that - in fact, Samuelson himself didn't get the Black Scholes formula.  You know, he had - there's a lovely paper called Rational Theory of Warrant Pricing by Samuelson, which has all kinds of wonderful ideas, but it doesn't have the idea.  The idea was produced by Fischer Black and Myron Scholes, with sort of...Robert Merton sort of sitting on the sidelines, but the actual paper is by Black and Scholes.  The quote there is from the abstract of their paper and it says: "If options are correctly priced in the market, it should not be possible to make sure profits by creating positions of long and short - or taking long or short positions and their underlying stocks.  Using this principle, theoretical variation formula is derived."  That was the key - that actually is the key idea.  So the idea is that you should look at what trading, you know, so there should be no inconsistency in holding an option and trading the asset.  So if you could have a situation where you could trade the underlying asset and produce something which has the same value as an option, then the amount of money you'd need to start trading with must be the value of the option, otherwise there's riskless profit - you could just buy one and sell the other, and you'd walk away with money for nothing, okay.  So it's a no arbitrage - so the idea is no arbitrage, that you can - the price is something that should be fixed by fixing the initial capital you need to replicate its payoff by trading in the market.  That's their idea. 

It turns out that works.  So if you, if you believe in the [?] price, model five, then there is a value, there is a uniquely specified value, based on that principle, for a call option, and that value is the Black Scholes formula, for which two out of three of these people won the Nobel Prize in Economics, if you think the Economics Nobel Prize is a Nobel Prize, which not everybody does, okay!  The one who didn't was Fischer Black, who unfortunately died before receiving the Prize, but I think everyone agrees that he was the smartest.  Maybe not everybody agrees on that - I don't know?!

Okay, so now there's - I'm sort of running out of my time, but it's...there's a sort of...after that, there was a sort of big cleaning up operation.  I mean, Black and Scholes had this paper.  It was, again, it was a little bit rough, they missed a few tricks, and there's still a sort of...the underlying principle was sort of clear up to a point, but not completely clear, but there was a sort of cleaning-up operation over the next six or seven years, which put the whole thing in good shape. 

So, just to mention a few things?  There's work by Cox, Ross and Rubenstein in the late-'70s, and they introduced this thing called a Binomial Tree, which is just like a sort of discrete, geometric, random walk, approximation to Brownian motion is what it is, and if you have that sort of structure, it's very, very easy to do all the calculations, much easier than the Black Scholes formula itself.  It starts out, you know, so you just start out with a...something that has its price S0 today, and then the assumption is that it just moves to one of two prices tomorrow, and then tomorrow, it will just move to one of two prices the day after that, and so on.  So you get a sort of a tree structure for prices, and then everything is finite probability, you can work out everything, extremely easy.  Then what you find is that you don't have to specify - the really nice thing about this is when you have a tree structure like that, in order to find out what the value of an option is, you do not need to know what the probabilities are for moving up and down.  There's a set of implied probabilities which give you the pricing formula, so it turns out that the pricing formula for options in that binomial tree are just given by taking expectation, or average value of the option pay-off, with respect to some implicitly defined set of probabilities, and that's called the risk neutral measure or martingale measure.  Those prices, those probabilities are the probabilities that make the underlying asset a martingale.  So the martingales sort of pop up in a very natural way, and you get this list of five properties which is, you know, that it says that you can only have - you know, this little simple model - there'll be no arbitrage, so there's no opportunity for riskless profit, if and only if you can find some set of probabilities which make the prices martingales.  Then, any contingent [claims] or any option has a unique value, which is consistent with absence of arbitrage, and the unique value is the replication value that I already pointed out, and you can express it in terms of just an expectation.  So that is saying that, you know, Bachelier's idea is correct, if you use this equivalent martingale measure thing.  So it's not the real probabilities; it's some probabilities that are implied by the structure of the market.  That works for the binomial tree, and then the question is how far does this go, you know, how far can you push this list of properties, you know, how far does it extend?  Well, it certainly extends to the Black Scholes theory because that works in exactly the same way. So it's not obvious that it does, but it does, and you get this list of properties.  The question is how much further can you go. 

This was sort of gradually answered over the next little while.  There's a famous paper by...or two papers by Harrison & Kreps and Harrison & Pliska in the late-'70s, where they sort of brought in general martingale theory, you know, modern theories of stochastic integration, and they formulated those requirements in a general context.  I mean, that list is essentially the list from the Harrison-Pliska paper.  You know, by that time, you'd got the whole - you'd fundamentally got the whole picture, so you know, that reduced...it turned, if you like, financial economics into mathematical finance, in the sense that everything you see is a well-defined mathematical object.  There?s no economic principles which don't have a clear mathematical formulation, you know, so the whole thing has been mathematised, okay.

There are still some - you know, the definitive answer to this question about how far the relationship between no arbitrage and martingale measures would go was still not answered, and it was answered much later in 1994 by Delbaen and Schachermayer.

Now, if we look at the - this is I think my last slide...yes, it is.  If you look at the way that that impinged on the outside world, everything happened very, very quickly actually, because - so 1973 was the Black Scholes formula, and that was the same year that a traded market in options started in Chicago, which is the first sort of real...the first traded options market, as opposed to over-the-counter deals in options.  So that started right away.  I don't think the two things are related directly.  I mean, I think that, you know, it takes longer than a few months to plan an options market, so they must have got going before Black Scholes was invented, but the two things did happen in the same year.

1979 is a sort of key year, in a way, because this process of sort of mathematisation was really done by Harrison, Kreps and Pliska in those three years.  The [basis of Bichteler and Dellacherie], as I mentioned, sort of shows that, you know, semimartingale is the only way to go, so you know, the theory was correct because it was impossible to have any other theory. 

There was trading in the fixed income world, in interest rates, really started taking off in 1980, so around that time, you know.  So there was a whole, a huge expansion of the range of assets which were treated by option-like methods. 

And I think, you know, again coming back to this point about computation, I mean, this was the beginning of the cheap memory era in terms of computation.  You know, you could - you can't run an options business without cheap memory and good computers.  So 1979 was the first IBM computer, so it sort of opened up the whole era - you know, the technology was in place, as well as the mathematics being in place, and that combination sort of wins the day.

Then I should sort of just mention, you know, that if we look at what's happening now, we're all in a state of slight uncertainty because there's certain little problems in the markets, which you may be aware of, right now, and this is relating more to credit risk trading.  So trading in credit as a sort of option-like endeavour began much later, in the late-1990s, sort of between 1996 and 1998 I suppose, and now that's, again, like the swaps market earlier, that's a huge part of the market, but it's one which is still far less well understood than the ones before.  You sort of have a feeling that maybe the original economists got it right, and that you had to look a little back, a little further into the process at sort of price formation and so on to really understand this credit risk market, and so maybe this whole thing has been just a big detour in the wrong direction, who knows. 

Okay, thank you very much.

 

 

©Mark Davis, Gresham College, 25 April 2008