A historian Joel Kaye has written widely on the development of science in medieval Europe. In Economy and Nature in the Fourteenth Century he presents a thesis that
In broad terms, the conceptual landscape that emerged in the fourteenth century resulted from a striking shift in the models derived to represent order and activity in the natural world: from a static world of numbered points and perfections to a dynamic world of ever-changing values conceived as continua in expansion and contraction; from a mathematics of arithmetical addition to a mathematics of geometrical multiplication, newly accepting of the approximate and the probable; from a world of fixed and absolute values to a shifting, relational world in which values were understood to be determined relative to changing perspectives and conditions; and from a philosophy focused on essences and perfections to one dominated by questions of quantification and measurement in respect to motion and change. Each of these new directions proved to be of great importance to the future of scientific thought.
Monday, 7 December 2009
Mathematics, Economics and Decision Making
Labels:
using mathematics
The 2009 Alan Tayler Lecture was delivered by Prof Lord Desai of the London School of Economics and Political Science at the University of Oxford on 30 November.
I did not attend the lecture, but I understand that Lord Desai called for economists to use more sophisticated modeling techniques - i.e. not to rely on single-factor linear models when trying to understand the economy. Makes sense to me.
I shall try and get a copy of the talk.
I also recently attended a talk by Prof Lord Skidelsky who was talking about Keynes, and publicising his book Keynes: The Return of the Master. A key point made by Skidelsky was Keynes belief in "irreducible uncertainty". From the perspective of maths, there is nothing controversial in Keynes is position, the Fundamental Theorem of Asset Pricing, in noting that there are infintely many martingale measure in an incomplet market, makes the same point. As a Russian mathematician has pointed out to me, when economists talk about "rational expectations", ask them under which measure they are calculating the expectation.
I did not attend the lecture, but I understand that Lord Desai called for economists to use more sophisticated modeling techniques - i.e. not to rely on single-factor linear models when trying to understand the economy. Makes sense to me.
I shall try and get a copy of the talk.
I also recently attended a talk by Prof Lord Skidelsky who was talking about Keynes, and publicising his book Keynes: The Return of the Master. A key point made by Skidelsky was Keynes belief in "irreducible uncertainty". From the perspective of maths, there is nothing controversial in Keynes is position, the Fundamental Theorem of Asset Pricing, in noting that there are infintely many martingale measure in an incomplet market, makes the same point. As a Russian mathematician has pointed out to me, when economists talk about "rational expectations", ask them under which measure they are calculating the expectation.
Sunday, 30 August 2009
Culture clash?
Labels:
using mathematics
The FSSC report Graduate Skills and Recruitment in The City has the following quotes (p 45) from a graduate recruitment manager at an investment bank
Perhaps a solution is to separate financial institutions into "traditional banks", undertaking lending and M&A activities relying on sales and marketing skills and "speculative banks" specialising in trading, derivatives and risk management. However, this is not the only model. Many manufacturing firms, particularly pharmaceutical and oil companies, are based on core expertise in science and engineering but where the majority of employees are involved in sales and marketing. So it appears the cultures can co-exist.
“Sales managers prefer graduates who are academic, with some interest in finance but more social skills with extra-curricular activities e.g. being the Head of the rugby/cricket club. They’re less fussy on the degree subject, but quite fussy in terms of the university they have attended – Oxford, Cambridge, LSE, Imperial.This quote resonated by an interview with a former Lehman's executive published in the Sunday Times on 23 August, Lehman Brothers failures exposed:
The [derivatives] trading/research side is much more focussed on the technical. Many of our senior managers are from French universities, with experience of engineering schools and écoles polytechniques in France. They look for expertise in engineering, maths, qualitative [quantitative?] finance. They like Oxford, Cambridge, Warwick”.
"Instead of reviewing the bank’s risk management systems, the executive directors would have ndless meetings discussing the corporate dress code as Fuld was a stickler for appearances. A angerous lack of awareness about the technical financial products the bank was playing with didn’t elp matters."Most people employed in finance are involved in sales and marketing activity, and so the soft skills needed for these roles are frequently highlighted by employers to the likes of the FSSC and Universities. However, the core of the banks' business is is highly quantitative, to quote from p 14 of the FSSC report:
“…[banks] need high level maths skills because that’s how the bank makes money – vanilla roducts have very little margin.”The financial innovation (or more accurately, the re-introduction of derivative products into financial markets) since the 1970s has changed the nature of banking, has the culture of banking kept up?
Perhaps a solution is to separate financial institutions into "traditional banks", undertaking lending and M&A activities relying on sales and marketing skills and "speculative banks" specialising in trading, derivatives and risk management. However, this is not the only model. Many manufacturing firms, particularly pharmaceutical and oil companies, are based on core expertise in science and engineering but where the majority of employees are involved in sales and marketing. So it appears the cultures can co-exist.
Wednesday, 26 August 2009
It's academic
Labels:
using mathematics
On of my favourite quotes from the credit crisis comes from the 2/02/08 edition of The Economist, the article "No Defense" on fraud issues at Societe General, states
"In common with other French banks, SocGen was also thought by many to take an overly mathematical approach to risk. “‘It may work in practice but does it work in theory?' is the stereotype of a French bank,” says one industry consultant."
In response to "Of couples and copulas", Prof Paul Embrects sent the following letter to the FT
Dear Sir
The article "Of couples and copulas", published on 24 April 2009,
suggests that David Li's formula is to blame for the current financial
crisis. For me, this is akin to blaming Einstein's E=mc² formula for
the destruction wreaked by the atomic bomb.
Feeling like a risk manager whose protestations of imminent danger
were ignored, I wish to make clear that many well-respected
academics have pointed out the limitations of the mathematical tools
used in the finance industry, including Li's formula. However, these
warnings were either ignored or dismissed with a desultory
response: "It's academic".
We hope that we are listened to in the future, rather than being
made a convenient scapegoat.
"In common with other French banks, SocGen was also thought by many to take an overly mathematical approach to risk. “‘It may work in practice but does it work in theory?' is the stereotype of a French bank,” says one industry consultant."
In response to "Of couples and copulas", Prof Paul Embrects sent the following letter to the FT
Dear Sir
The article "Of couples and copulas", published on 24 April 2009,
suggests that David Li's formula is to blame for the current financial
crisis. For me, this is akin to blaming Einstein's E=mc² formula for
the destruction wreaked by the atomic bomb.
Feeling like a risk manager whose protestations of imminent danger
were ignored, I wish to make clear that many well-respected
academics have pointed out the limitations of the mathematical tools
used in the finance industry, including Li's formula. However, these
warnings were either ignored or dismissed with a desultory
response: "It's academic".
We hope that we are listened to in the future, rather than being
made a convenient scapegoat.
The Royal Society calls for more funding of financial maths
The Royal Society published a report on Science, Technology, Engineering and Maths impact on the service sector. They devote a whole chapter (3) to financial innovation, see Hidden Wealth. The report was covered in the FT.
I was contact by the Royal Society in mid-July, this is (their) summary of what I said:
1. The financial crisis was not a homogeneous events; some institutions did (much) better than thers.
2. The root of the problem is in banks mis-pricing assets. When pricing assets the banks were relying on mathematical models. Some banks had "engineered" their own models, others "bought in" models (either by buying "off the shelf" or by hiring individuals from the innovating banks). Banks who treated models as "black boxes" have done far worse than those that had a reputation or developing their own.
3. Many quants (the majority ?) have a background in physics and engineering (there is a factoid that the majority of engineering graduates from top UK universities go into finance rather than engineering - you could check this. Similarly The City is the largest employer of physics PhDs). They understand deterministic systems but only have a rudimentary understanding of modern probability theory (the majority of maths graduates are in a very similar position, I spoke to a maths teacher who had a degree from Glasgow who told me he had not done probability since he was 16).
4. Financial economics developed in the late twentieth century using relatively straightforward maths. The models it produced are "too simple" (see pp 4-12 in "An Engine, Not a Camera"), but are "elegant".
The physicists / engineers were given a simple framework in which deterministic approaches appeared to give definite results. This approach is related to the "Crash of 87" and "When genius failed" in '98.
5. Throughout the industry a belief emerged that maths, "rocket science" removed risk. In the 1990s, mathematicians began to investigate financial models and re-evaluate them. A more rigorous approach to financial economics revealed the naivety of the assumptions that led to the simple models.
6. Because the financial mathematics community is small and peripheral in British science, it lacks authority , and the theory (from the mathematicians) became disconnected from the practice in industry.
7. In Europe, because they approach probability as a branch of analysis rather than from the perspective of statistics, there seems to have been a better appreciation that simple models that fitted data were inadequate. This is a subtle point. The issue is whether the "quants"
really understand stochastic systems. Does the UK education system (schools to universities) produce the volume of people with (basic to advanced) skills in probability and statistics. There is a view that France & Germany are better at this, as demonstrated by the large number of continental scientists and engineers employed in London banks.
8. The credit crisis provides a tangible example and introduction to a wider problem about our poor understandings of complex and stochastic systems. The concern is that many of the global challenges we face involve these sorts of systems. Is our science up to the task?
9. There is a basic competency issue but also a need to develop new mathematics able to describe what Lord May describes as "ephemeral" systems, but in other areas relating to fundamental maths.
I was contact by the Royal Society in mid-July, this is (their) summary of what I said:
1. The financial crisis was not a homogeneous events; some institutions did (much) better than thers.
2. The root of the problem is in banks mis-pricing assets. When pricing assets the banks were relying on mathematical models. Some banks had "engineered" their own models, others "bought in" models (either by buying "off the shelf" or by hiring individuals from the innovating banks). Banks who treated models as "black boxes" have done far worse than those that had a reputation or developing their own.
3. Many quants (the majority ?) have a background in physics and engineering (there is a factoid that the majority of engineering graduates from top UK universities go into finance rather than engineering - you could check this. Similarly The City is the largest employer of physics PhDs). They understand deterministic systems but only have a rudimentary understanding of modern probability theory (the majority of maths graduates are in a very similar position, I spoke to a maths teacher who had a degree from Glasgow who told me he had not done probability since he was 16).
4. Financial economics developed in the late twentieth century using relatively straightforward maths. The models it produced are "too simple" (see pp 4-12 in "An Engine, Not a Camera"), but are "elegant".
The physicists / engineers were given a simple framework in which deterministic approaches appeared to give definite results. This approach is related to the "Crash of 87" and "When genius failed" in '98.
5. Throughout the industry a belief emerged that maths, "rocket science" removed risk. In the 1990s, mathematicians began to investigate financial models and re-evaluate them. A more rigorous approach to financial economics revealed the naivety of the assumptions that led to the simple models.
6. Because the financial mathematics community is small and peripheral in British science, it lacks authority , and the theory (from the mathematicians) became disconnected from the practice in industry.
7. In Europe, because they approach probability as a branch of analysis rather than from the perspective of statistics, there seems to have been a better appreciation that simple models that fitted data were inadequate. This is a subtle point. The issue is whether the "quants"
really understand stochastic systems. Does the UK education system (schools to universities) produce the volume of people with (basic to advanced) skills in probability and statistics. There is a view that France & Germany are better at this, as demonstrated by the large number of continental scientists and engineers employed in London banks.
8. The credit crisis provides a tangible example and introduction to a wider problem about our poor understandings of complex and stochastic systems. The concern is that many of the global challenges we face involve these sorts of systems. Is our science up to the task?
9. There is a basic competency issue but also a need to develop new mathematics able to describe what Lord May describes as "ephemeral" systems, but in other areas relating to fundamental maths.
Monday, 6 July 2009
Driving by using the rear view mirror
Last October the EPSRC, my funders, asked me to contact the Science Media Centre about media coverage of the credit crisis. This was because there was a risk, which materialized, that mathematics would come into criticism in light of the credit crisis. A the time the SMC took the view, on the basis I was the only person making the point, that the credit crisis had nothing to do with science. Following my "Was the Credit Crisis a Science Story" piece and an intervention by Lord Drayson’s office, the Science Media Centre arranged for a session on the credit crisis to be included at the World Conference of Science Journalists, which I attended.
At the conference I explained the links between modern financial markets and science, gave a “scientific” perspective of what happened and why, discussed the role of mathematics in relation to finance and in relation to science in general, emphasizing the fact that maths is looking to describe complex and stochastic systems. An issue critical to the future development of all of science.
Three other people were on the panel. During the Q&A an Italian print journalist (I believe from the Italian equivalent of New Scientist) reported that his magazine had devoted a whole issue to the crisis and it had generated the most positive feedback the magazine had had all year. At the end of the session, the Chair asked me what the next big story in finance would be, I said the big story is that successful finance is based on good science, the Chair informed me this was not a story that could be sold to her editor.
Following the session a stream of European science journalists came up to me and told me that in their countries, science journalists had been covering the story and generally there had been a very positive response from the public.
In particular I had a lengthy discussion with two German journalists. One reported that the main German TV science magazine show had, also, devoted a whole program to the crisis, and again it had been very well received. What was disturbing was that he was scathing about the level of understanding of the crisis his British counterparts had displayed in the session. The fact that one of them did not recognize the importance of basing finance on good science “completely misses the point”.
We talked about why British science journalists take this view, and the conclusion was it came down to the British tradition of empiricism; science is about gathering data to build models that explain the world. There is a blind spot to Cartesian rationalism, that science also involves abstract reasoning. In relation to the credit crisis this explains the British (and American) dismissal of French Risk managers with the saying that worry “That’s all very well in practice but what about the theory”. The point is, Anglo Saxon risk management techniques were all very well in practice, up until the point they went wrong. French mathematicians (Artzner) had a theoretical basis for dismissing Value at Risk long before British financiers found out that in practice, it was flawed.
The empiricist-rationalist divide also explains why Continental Europeans have a more catholic view of science. If science is about rational thought it encapsulates mathematics, sociology, anthropology and even history. If it is about the interpretation of numerical data, it does not. Ironically, it looks as if to understand one aspect of the credit crisis, the fact that bankers made decisions based on what they saw in their rear view mirrors, seems to be a sociological question, not a question that can be addressed by the physical sciences.
P.S. Gillian Tett (17/7/09) has discussed the sociological interpretation of "herd behaviour".
At the conference I explained the links between modern financial markets and science, gave a “scientific” perspective of what happened and why, discussed the role of mathematics in relation to finance and in relation to science in general, emphasizing the fact that maths is looking to describe complex and stochastic systems. An issue critical to the future development of all of science.
Three other people were on the panel. During the Q&A an Italian print journalist (I believe from the Italian equivalent of New Scientist) reported that his magazine had devoted a whole issue to the crisis and it had generated the most positive feedback the magazine had had all year. At the end of the session, the Chair asked me what the next big story in finance would be, I said the big story is that successful finance is based on good science, the Chair informed me this was not a story that could be sold to her editor.
Following the session a stream of European science journalists came up to me and told me that in their countries, science journalists had been covering the story and generally there had been a very positive response from the public.
In particular I had a lengthy discussion with two German journalists. One reported that the main German TV science magazine show had, also, devoted a whole program to the crisis, and again it had been very well received. What was disturbing was that he was scathing about the level of understanding of the crisis his British counterparts had displayed in the session. The fact that one of them did not recognize the importance of basing finance on good science “completely misses the point”.
We talked about why British science journalists take this view, and the conclusion was it came down to the British tradition of empiricism; science is about gathering data to build models that explain the world. There is a blind spot to Cartesian rationalism, that science also involves abstract reasoning. In relation to the credit crisis this explains the British (and American) dismissal of French Risk managers with the saying that worry “That’s all very well in practice but what about the theory”. The point is, Anglo Saxon risk management techniques were all very well in practice, up until the point they went wrong. French mathematicians (Artzner) had a theoretical basis for dismissing Value at Risk long before British financiers found out that in practice, it was flawed.
The empiricist-rationalist divide also explains why Continental Europeans have a more catholic view of science. If science is about rational thought it encapsulates mathematics, sociology, anthropology and even history. If it is about the interpretation of numerical data, it does not. Ironically, it looks as if to understand one aspect of the credit crisis, the fact that bankers made decisions based on what they saw in their rear view mirrors, seems to be a sociological question, not a question that can be addressed by the physical sciences.
P.S. Gillian Tett (17/7/09) has discussed the sociological interpretation of "herd behaviour".
Wednesday, 10 June 2009
Maths and the Markets (II)
Following my last post I drafted a letter on behalf of the leading researchers in financial maths in the UK. After some edits, Sir David Wallace, Chair of the Council of Mathematical Sciences, sent this on our behalf to Lord Turner.
The FSA have responded positively to this, as discussed by Clive Cookson in the FT today.
Its a good result for both the FSA, who need the support of mathematicians that is out of their price range to employ and tghe research community will get some sort of support.
There is an angle that is not so positive. The government is keen that academics do more public engagement, and I think this is a good thing. If financial mathematicians do something that is not pointless, they should be able to inform the FSA on good regulatory practice. My concern is that the institutions between the government and academics are not efficient at enabling academics to engage. Typical examples are the RAE/ REF process, which focus academics on learning more and more about less and less until they know everything about nothing. Similarly the Research Councils are keen to facilitate public engagement, but at the end of the day most grant applications are assessed by a panel who are only (in maths at least) in world clas smathematics research - and pure mathematicians don't like being associated with anything as messy as the real world.
These are general comments, there is a specific example. The Technology Strategy Board recently conducted a call to establish a Knowledge Transfer Network in Financial Services. Some of the signatories to Lord Turner's letter submitted a proposal, which was unsuccessful. The question is, did the successful group make any submission to the FSA regarding the Turner Review? My point is that sometime government funding of research activity results in activites that "tick the boxes" but that is about all. These comments are informed by a private conversations with people from a number of consotiums who submitted proposals to the TSB.
Is there a better way? When I explore issues like the shortcomings of the RAE and research councils, members of the government counter that "that is what the (elite) universities want". It was Universities UK that lobbied government to keep the RAE as was/is. The high and mighty like the research councils panels, because it enables them to control science funding. However, people tell me that the US system, managed by the National Science Foundation, is more researcher focussed and less buraucracy focussed.
The FSA have responded positively to this, as discussed by Clive Cookson in the FT today.
Its a good result for both the FSA, who need the support of mathematicians that is out of their price range to employ and tghe research community will get some sort of support.
There is an angle that is not so positive. The government is keen that academics do more public engagement, and I think this is a good thing. If financial mathematicians do something that is not pointless, they should be able to inform the FSA on good regulatory practice. My concern is that the institutions between the government and academics are not efficient at enabling academics to engage. Typical examples are the RAE/ REF process, which focus academics on learning more and more about less and less until they know everything about nothing. Similarly the Research Councils are keen to facilitate public engagement, but at the end of the day most grant applications are assessed by a panel who are only (in maths at least) in world clas smathematics research - and pure mathematicians don't like being associated with anything as messy as the real world.
These are general comments, there is a specific example. The Technology Strategy Board recently conducted a call to establish a Knowledge Transfer Network in Financial Services. Some of the signatories to Lord Turner's letter submitted a proposal, which was unsuccessful. The question is, did the successful group make any submission to the FSA regarding the Turner Review? My point is that sometime government funding of research activity results in activites that "tick the boxes" but that is about all. These comments are informed by a private conversations with people from a number of consotiums who submitted proposals to the TSB.
Is there a better way? When I explore issues like the shortcomings of the RAE and research councils, members of the government counter that "that is what the (elite) universities want". It was Universities UK that lobbied government to keep the RAE as was/is. The high and mighty like the research councils panels, because it enables them to control science funding. However, people tell me that the US system, managed by the National Science Foundation, is more researcher focussed and less buraucracy focussed.
Thursday, 26 March 2009
Maths and the Markets
The Turner Review of Financial Services Regulation identifies one of the causes of the financial crisis as a "misplaced reliance on sophisticated maths". The FT was quick to criticise this naive and ill-informed view.
The banks that failed seem to have abused maths, financial firms that succeed use maths.
Take for example BNP Paribas, Bank of the Year in 2008. The lead quant is Prof Marek Musiela, a well regarded research mathematician who would be snapped up by the maths department of any prestigious university. In employing Prof Musiela, BNP is setting the culture of the company.
Much of financial innovation is coming out of hedge funds, and while these "shadow banks" come in for some criticism, a number of hedge funds were created by the frustration innovative scientists felt working in the blue chips banks (which are failing).
Last year's top earner was Jim Simons, who runs Renaissance Technologies. Jim is another research mathematician, having won the American Mathematical Society’s Oswald Veblin
Prize in Geometry. A third of Renaissance's 200 staff in its main offices have PhDs and one senior manager is quoted as saying “I’ve always said Renaissance’s secret is that it didn’t hire MBAs,” - i.e. it does not hire "finance" experts, it hires scientists.
In the UK, while the MAN Group has posted a halving of profits (on a falling market), their share price rose, the reason being they did better than expected. The MAN Group takes maths so seriously they fund Oxford University mathematicians, drawing world-leading researchers to Oxford and into dialogue with the hedge fund. Not only does this give MAN and edge, but it gives the UK as a whole an edge by being a "brain sink" rather than a "brain drain".
Have these leading firms really been getting their use of maths and science wrong? Anecdote is not evidence, but while RBS was investing millions in "sports ambassadors", BNP, Renaissance Technologies and the MAN Group were all investing in mathematics and science. The impact of these two strategies appears obvious, and investors should bear this in mind in the future.
The banks that failed seem to have abused maths, financial firms that succeed use maths.
Take for example BNP Paribas, Bank of the Year in 2008. The lead quant is Prof Marek Musiela, a well regarded research mathematician who would be snapped up by the maths department of any prestigious university. In employing Prof Musiela, BNP is setting the culture of the company.
Much of financial innovation is coming out of hedge funds, and while these "shadow banks" come in for some criticism, a number of hedge funds were created by the frustration innovative scientists felt working in the blue chips banks (which are failing).
Last year's top earner was Jim Simons, who runs Renaissance Technologies. Jim is another research mathematician, having won the American Mathematical Society’s Oswald Veblin
Prize in Geometry. A third of Renaissance's 200 staff in its main offices have PhDs and one senior manager is quoted as saying “I’ve always said Renaissance’s secret is that it didn’t hire MBAs,” - i.e. it does not hire "finance" experts, it hires scientists.
In the UK, while the MAN Group has posted a halving of profits (on a falling market), their share price rose, the reason being they did better than expected. The MAN Group takes maths so seriously they fund Oxford University mathematicians, drawing world-leading researchers to Oxford and into dialogue with the hedge fund. Not only does this give MAN and edge, but it gives the UK as a whole an edge by being a "brain sink" rather than a "brain drain".
Have these leading firms really been getting their use of maths and science wrong? Anecdote is not evidence, but while RBS was investing millions in "sports ambassadors", BNP, Renaissance Technologies and the MAN Group were all investing in mathematics and science. The impact of these two strategies appears obvious, and investors should bear this in mind in the future.
Wednesday, 25 February 2009
A Brief History of Financial Engineering
Labels:
history of financial engineering
c 550 BCE Thales, the father of western science makes money by using his scientific knowledge to become rich through olive speculation (Aristotle, Politics Book 1 part 11).
1202 Leonardo the Pisan (Fibonacci) introduces Hindu-Arabic numbers to Europe to aid in commercial calculations.
c 1250 Pope Innocent IV, in a commentary on cannon law, justifies the charging of risk premium for assets (Murray Rothbard, Economic Thought Before Adam Smith , Edward Elgar,1996)
c 1260 St Thomas Aquinas endorses insider trading (making profits based on information not know to the buyer) (Summa Theologica, Second part of the second part, 77, 3)
c 1564 Girolamo Cardano identifies the concept of mathematical probability.
1610 Galileo becomes the first “quant”. Having used a telescope to observe Jupiter, Galileo publishes his results, leaves Padua, and becomes “First and Extraordinary Mathematician of the University of Pisa and Mathematician to his Serenest Highness Cosimo II de Medici”. At Cosimo’s request, he investigates a gambling problem and publishes Sopra le Scoperte dei Dadi (Upon the Discoveries of Dice).
1654 The first derivative pricing formula is developed by Pascal and Fermat in answering the Problem of Points. The solution to the problem of points is essentially the same as the Cox-Ross-Rubenstein model. A special case of the discrete time CRR model converges to the continuous time Black-Scholes option pricing model.
1696 Newton becomes a quant, moving from Cambridge to the Royal Mint. In 1717 Newton moves England from the silver standard to the gold standard, fixing the price of silver to gold.
1738 Daniel Bernoulli publishes “An Exposition on a New Theory of the Measurement of Risk” in Russia. The manuscript appears to have been lost, reprinted in Germany in the late 19th century and then in English in 1954, 10 years after von Neuman and Morgenstern had introduced Expected Utility in the Theory of Games and Economic Behavour.
1860-1926 Maxwell develops the kinetic theory of gases, investigated by Einstein who discusses Brownian motion, which is defined mathematically by Weiner in 1926. In the late 1960s Robert Merton develops the theory of continuous time finance based on the Weiner process.
1933 Kolmogorov identifies probability with measure, enabling financial mathematicians to use the ideas of conditional probability and equivalent measures, which are the fundamental tools of derivative pricing.
1202 Leonardo the Pisan (Fibonacci) introduces Hindu-Arabic numbers to Europe to aid in commercial calculations.
c 1250 Pope Innocent IV, in a commentary on cannon law, justifies the charging of risk premium for assets (Murray Rothbard, Economic Thought Before Adam Smith , Edward Elgar,1996)
c 1260 St Thomas Aquinas endorses insider trading (making profits based on information not know to the buyer) (Summa Theologica, Second part of the second part, 77, 3)
c 1564 Girolamo Cardano identifies the concept of mathematical probability.
1610 Galileo becomes the first “quant”. Having used a telescope to observe Jupiter, Galileo publishes his results, leaves Padua, and becomes “First and Extraordinary Mathematician of the University of Pisa and Mathematician to his Serenest Highness Cosimo II de Medici”. At Cosimo’s request, he investigates a gambling problem and publishes Sopra le Scoperte dei Dadi (Upon the Discoveries of Dice).
1654 The first derivative pricing formula is developed by Pascal and Fermat in answering the Problem of Points. The solution to the problem of points is essentially the same as the Cox-Ross-Rubenstein model. A special case of the discrete time CRR model converges to the continuous time Black-Scholes option pricing model.
1696 Newton becomes a quant, moving from Cambridge to the Royal Mint. In 1717 Newton moves England from the silver standard to the gold standard, fixing the price of silver to gold.
1738 Daniel Bernoulli publishes “An Exposition on a New Theory of the Measurement of Risk” in Russia. The manuscript appears to have been lost, reprinted in Germany in the late 19th century and then in English in 1954, 10 years after von Neuman and Morgenstern had introduced Expected Utility in the Theory of Games and Economic Behavour.
1860-1926 Maxwell develops the kinetic theory of gases, investigated by Einstein who discusses Brownian motion, which is defined mathematically by Weiner in 1926. In the late 1960s Robert Merton develops the theory of continuous time finance based on the Weiner process.
1933 Kolmogorov identifies probability with measure, enabling financial mathematicians to use the ideas of conditional probability and equivalent measures, which are the fundamental tools of derivative pricing.
What mathematics can offer finance
Labels:
using mathematics
I emphasise to my undergraduate students that Kolmogorov’s 1933 identification of probability with measure theory is, in my opinion, the most important scientific development of the twentieth century. To paraphrase Bertrand Russell, probability is important because all science is inductive, and prior to Kolmogorov there were significant problems, philosophical, theoretical and practical, with identifying probability with the frequency of events.
The context is as follows. Students learn that if a market is arbitrage free then asset prices today are simply expectations of future asset prices calculated using, special, risk neutral probabilities, rather than any “natural” probabilities. The idea of a probability as an abstract measure is introduced along with the idea of equivalent measures, so the students understand that there is nothing special about, so called, natural probabilities. They are then shown that if a market is complete (idealised) there is a unique risk neutral probability, while if the market is incomplete (for example, it includes transaction costs) then there are an infinite number of possible choices of risk neutral probabilities. Finally, in a complete market, all risk can be removed in pricing derivatives; the same is not true for incomplete markets, which are the reality.
The next question is, how do we choose what risk neutral probabilities we should use to price assets in a market if there are infinitely many choices? A number of solutions have been presented, the most widely accepted is to use utility-based methods, an approach initiated by Prof Mark Davis in his paper Option Pricing in Incomplete Markets (1998). Essentially, given an agent’s utility function, we can identify the correct risk neutral probabilities to use in asset pricing.
Why is this significant in economics? Von Neumann-Morgenstern Expected Utility Theory, published in 1944, states that given a set of natural probabilities associated with lotteries (generating a probability distribution function) an agent’s utility function can be identified (preferences over distribution functions precede preferences over outcomes). However, von Neumann-Morgenstern Expected Utility assumes probabilities are objective and exogenously defined by nature (the Kolmogorov formulation of probability was not accepted in the US until around 1948). Derivative pricing tells us to ignore these probabilities and price assets using equivalent risk neutral probabilities. However, in realistic markets, there are infinite choices of the correct probabilities to use and we should use utility functions to identify the right one. Pricing assets starts with utility functions. The implication is, we need to understand utility to understand asset pricing; finance needs to be interdisciplinary.
Within the banks, with probability 1, there are scientists who are comfortable with probability as measure. I would be willing to wager that there are few people managing banks who are familiar with the idea that financial mathematics tells us that there is no unique probability measure under which to calculate expectations, rational or not.
Another topic I raise with students is why Pythagoras was responsible for the Black Monday in 1987. The context here is that the ’87 crash was associated with portfolio insurance, based on hedging investments in portfolios by taking out positin in index futures. I have to teach the students (because it is part of the Actuarial syllabus) a the optimal hedge ratio. This is derived on the assumption that investors choose between expected returns and variance of returns. This idea was introduced by Markowitz (mean variance portfolio selection) and developed by Sharpe, Litner (the capital asset pricing model) and yielded a handful of Noble Prizes. Why did Markowitz suggest this approach? There is no evidence to suggest that investors actually do choose between expected returns and variance of returns, Markowitz suggested it because, thanks to Pythagoras’s theorem, comparing expectations and variances is mathematically tractable.
Markowitz is a nice example of mathematics driving financial theory, rather than, as happens in the natural sciences, the observations driving the development of mathematics. Newton developed calculus to build his models. I tell my students that we are in an exciting time in financial maths and compare the state of the science to that of aviation around 1904. We are using glue and tying things together with string, but we are on the right lines. Much of finance theory was developed in the 30 or so years following the Second World War based on assumptions of probability that date from the seventeenth and eighteenth centuries. Using measure theoretic probability, scientists are able to look at financial problems without the constraints of nineteenth century mathematics and as a result, are able to build more accurate, though still simple, economic models.
The following is a personal selection of papers that I feel are significant contributions to finance theory from mathematics
Fernholtz, R., Karatzas, I, Stochastic Portfolio Theory: an Overview
This addresses issues relating to Markowitz style portfolio selection
Musiela, M., Zariphopulou, T., Portfolio choice under space-time monotone performance criteria
This addresses a significant issue in classical finance. At time t=0 the agent sets their objectives at some time t=T>0. In the interval ]0,t[ the agent locks themselves in a room, closes their eyes and sticks their fingers in their ears. The reality is that in ]0,T[ the investor is affected by the economy. Energy managers (a PhD in maths and a PhD in theoretical physics) at Scottish Power recently raised this issue with me, they need to manage portfolios dynamically, not statically.
Hugonnier, J., Kramkov, D. Schachermayer, W., On Utility-Based Pricing of Contingent Claims in Incomplete Markets, Mathematical Finance, Vol. 15, No. 2, pp. 203-212, April 2005
A refinement of Prof Davis's work.
H. Jin and X. Zhou, Behavioural portfolio selection in continuous time Mathematical Finance, Vol. 18 (2008), pp. 385-426.
Provides a mathematical basis for behavioural finance, as introduced by Kahneman and Tversky.
Friday, 13 February 2009
Don't blame the quants
Labels:
financial maths in the news
Wednesday, 11 February 2009
Getting serious about banking
We do not think banking is a serious occupation. If we take a job seriously, like the law, medicine, accountancy, engineering, teaching or plumbing, we expect the workers to have passed exams and to keep up with developments in the profession throughout their careers. The Financial Services Authority, rightly, does not allow anyone to sell an endowment mortgage to a private individual unless they have passed a series of exams. If someone fails their FSA exams, not to worry, they could get a job in an investment bank and sell Mortgage Backed Securities to other investment banks. It is up to the banks to make the judgement as to whether or not their employees are competent. The consequence of the chairmen and chief executives of RBS and HBOS not being qualified, is that they were not competent to judge if their employees were capable.
Hector Sants, the Chief Executive of the FSA, said in October that he had over thirty years experience in banking. That is nice to know, but if I owned a jet aeroplane in 1950 I might prefer a mechanic who has three years experience of jet engines to one with thirty years in aviation. This is the root of the problem; finance is developing too fast for people with thirty years experience. As a financial mathematician, researching and teaching how you make good, scientific, decisions in the uncertain world of finance, I am not overawed by the complexity of the financial products the banks are trading. I am stunned by the simplicity of the mathematics that many bankers were using to manage these products. Nuclear power plants are complex, and so we expect the engineers who operate them to use cutting edge technology. The human body is complex, we expect doctors to use up to date research in treating us. In investment banking, Collateralised Debt Obligations are complex but the reaction of the "quants" designing them was to dumb down the maths so that the managers, who did not have the professional qualifications of an engineer or a doctor, could keep up.
What happened in the lead up to the credit crisis is not difficult to explain. When I was in my late teens, l embarked on a short-lived experiment in producing ginger beer. Having brewed the beer in a large bucket, I bottled it in twenty-four bottles and stored them in my wardrobe. If I had presented a newly sealed bottle to an independent expert and asked them to asses the probability of the bottle exploding, they may have given the chance as one in twenty four, and so I could expect one of my twenty four bottles to explode. If I was concerned with losing many bottles, I could do some basic maths and conclude that there was a 99.99% chance that I would not lose more than five bottles.
A few weeks later one bottle exploded, followed by the remaining 23 over the course of a few days. When I had done my sums, I had treated each bottle as being independent of the others. Of course, since each bottle contained the same product, if one exploded, they probably all would. This was an object lesson in what are called dependent risks, essentially there was a one in twenty four chance of losing everything rather than a near certainty of only losing a little.
Investment banks had bought beer, sub-prime mortgages, and sold crates of bottles to investors on a sort of "money back guarantee" deal. Like some of Sir Alan's naive apprentices, they focused on turnover, since salesmen are paid on commission, rather than worrying about profitability.
Bankers, not just in the UK but all around the world, blame "nature" for mis-behaving when all the bottles exploded. This is a smokescreen to protect their reputation. Since 2000, there have been a series of concerns raised by mathematicians as to whether the equations bankers were using to manage the risk of their investments were adequate. In particular, that the key equation they use would underestimate the number of bottles that would blow up, and another, that valued their finished products would over re-act to a bottle being lost and the value of all the beer, not just the ginger beer, they were selling would collapse. The problem is, not only did bank management not understand this, but also, it is exactly what happened.
Hector Sants, the Chief Executive of the FSA, said in October that he had over thirty years experience in banking. That is nice to know, but if I owned a jet aeroplane in 1950 I might prefer a mechanic who has three years experience of jet engines to one with thirty years in aviation. This is the root of the problem; finance is developing too fast for people with thirty years experience. As a financial mathematician, researching and teaching how you make good, scientific, decisions in the uncertain world of finance, I am not overawed by the complexity of the financial products the banks are trading. I am stunned by the simplicity of the mathematics that many bankers were using to manage these products. Nuclear power plants are complex, and so we expect the engineers who operate them to use cutting edge technology. The human body is complex, we expect doctors to use up to date research in treating us. In investment banking, Collateralised Debt Obligations are complex but the reaction of the "quants" designing them was to dumb down the maths so that the managers, who did not have the professional qualifications of an engineer or a doctor, could keep up.
What happened in the lead up to the credit crisis is not difficult to explain. When I was in my late teens, l embarked on a short-lived experiment in producing ginger beer. Having brewed the beer in a large bucket, I bottled it in twenty-four bottles and stored them in my wardrobe. If I had presented a newly sealed bottle to an independent expert and asked them to asses the probability of the bottle exploding, they may have given the chance as one in twenty four, and so I could expect one of my twenty four bottles to explode. If I was concerned with losing many bottles, I could do some basic maths and conclude that there was a 99.99% chance that I would not lose more than five bottles.
A few weeks later one bottle exploded, followed by the remaining 23 over the course of a few days. When I had done my sums, I had treated each bottle as being independent of the others. Of course, since each bottle contained the same product, if one exploded, they probably all would. This was an object lesson in what are called dependent risks, essentially there was a one in twenty four chance of losing everything rather than a near certainty of only losing a little.
Investment banks had bought beer, sub-prime mortgages, and sold crates of bottles to investors on a sort of "money back guarantee" deal. Like some of Sir Alan's naive apprentices, they focused on turnover, since salesmen are paid on commission, rather than worrying about profitability.
Bankers, not just in the UK but all around the world, blame "nature" for mis-behaving when all the bottles exploded. This is a smokescreen to protect their reputation. Since 2000, there have been a series of concerns raised by mathematicians as to whether the equations bankers were using to manage the risk of their investments were adequate. In particular, that the key equation they use would underestimate the number of bottles that would blow up, and another, that valued their finished products would over re-act to a bottle being lost and the value of all the beer, not just the ginger beer, they were selling would collapse. The problem is, not only did bank management not understand this, but also, it is exactly what happened.
Thursday, 5 February 2009
Was the credit crunch a science story?
A recent report from the LSE has raised some important questions concerning financial journalism. Specifically, in relation to the credit crisis it reports that Robert Peston, the BBC’s Business Editor, had found it difficult to explain some of the complex derivatives being constructed by banks since “even the bankers creating the CDOs were unable to explain them in terms that make sense to non-specialists”. The role of these complex derivatives is central to understanding the credit crisis, and key to understanding the derivatives is mathematics. This raises the question, could science journalists have contributed to reporting the credit crisis.
The storm clouds of the credit crisis broke following BNP Paribas’s announcement that it was unable to value assets on 9 August 2007. The following day, Goldman Sachs reported that one of its funds had lost 30% of its value in the week, equiring a, now paltry, $3 bn bailout. In explanation the Chief Financial Officer of Goldman Sachs said
It is unlikely that a science journalist would have accepted the pseudo-scientific explanation Goldman’s offered. Would they have done a better job understanding the crisis as a whole?
Since the 1960s investment banks have been using increasingly complex mathematics, literally rocket science, to price and manage the risks of the assets they trade. The current credit crisis is unique in that for the first time, quants, physicists and engineers, using mathematical formulae rather than economists using their knowledge of markets, have been managing the assets at the heart of the crisis. Since experienced managers in banks have been at a loss to grasp the mathematics, it is hard to see how a financial journalist should be expected to do so. However, a journalist who is accustomed to investigating safety critical systems in technology is well capable of coming to terms with risk management systems in an investment bank. While a journalist who is able to explain the latest research coming from a physicist in a research lab, would be able to describe the pricing algorithm developed by a physicist in a bank.
If the types of journalists investigating science turn their attention to banks, would the risk of future crises be reduced? There is a strong argument that it would. A key role that science-reporting plays is in ensuring that technology is not misused and that it is properly regulated. Medicine is better off for the reporting of Thalidomide and the environment benefits by journalists scrutinising the energy industry.
In the 1990s, central bankers became concerned that the rules for defining how much capital a bank held in reserve were quickly becoming redundant, given the explosive growth of sophisticated derivative products. In 1996, the Bank for International Settlements, the gnomes of Basel who regulate the banks in relation to the credit crisis, began a process of re-writing banking regulation, called Basel II. Realising that they would never be able to keep up with the activities of the banks, the regulators specified a framework for regulation rather than a set of hard and fast rules. The framework, which is only now beginning to be implemented, is based on three pillars; the minimum standards for calculating how much money a bank needs to keep in reserve; the supervisory review process in the bank overseeing those calculations and market discipline ensuring the reviews and calculations are adequate.
In 2000, a mathematician working at the US Federal Reserve, checked the formulae used in Basel II to calculate capital reserves, but raised concerns
--> about whether it would be adequate if finacial products became too sophisticated (in the Discussion). The Basel committee had selected a formula that did not model the connections in the economy in a sophisticated manner and so it was inadequate if loans were inter-connected. The root cause of the credit-crisis has been that loans were connected, by the housing market, banks ran out of capital, precipitating the liquidity crisis that led to the collapse of the retail banks and the credit crunch.
The Basel II accord still uses the simplified formula, because the accord is clear; under the supervisory review process, a bank needs to match the calculation of capital reserves to the omplexity of their operations. The accord expects that banks will adequately execute the supervisory review process because of market scrutiny. Consequently, informed journalism plays a fundamental role in the new framework for banking regulation. Science journalists should take up this role given their knowledge and skills.
The Public Relations machines of banks are happy that discussion around regulation focuses on issues such as bonuses and banning complex derivatives, because the banks know that governments will not set such regulations. Turning the public’s attention to the science underpinning the bank’s business will not only put the focus on where the problem is, but is already part of the regulatory framework. In addition, putting banking managers under pressure to explain the details of the technology they are using will force better communication, debate nd understanding within the banks bridging the gap in banks between the quants and conventional financiers.
Finally, if the failure, because of poor use of science, of a section of the energy or pharmaceutical businesses had lead to a trillion-dollar rescue package, it is inconceivable that the managers of he industry would still be in place. Why have so few bankers been fired by their shareholders? Is it because the bankers have successfully blinded the shareholders with pseudo-science?
The storm clouds of the credit crisis broke following BNP Paribas’s announcement that it was unable to value assets on 9 August 2007. The following day, Goldman Sachs reported that one of its funds had lost 30% of its value in the week, equiring a, now paltry, $3 bn bailout. In explanation the Chief Financial Officer of Goldman Sachs said
“We were seeing things that were 25-standard deviation moves, several days in a row. There have been issues in some of the other quantitative spaces. But nothing like what we saw last week.”Essentially, his point was that the economic environment had been so perverse that the losses, incurred by the most respected investment bank in the world, were inevitable. The Financial Times2, along with much of the established financial press, took this as a reasoned explanation but, almost immediately, commentators in the blogosphere revealed it to be what it was, rubbish. Goldman’s models were wrong, not nature.
Financial Times, August 13 2007, Goldman pays the price of being big
It is unlikely that a science journalist would have accepted the pseudo-scientific explanation Goldman’s offered. Would they have done a better job understanding the crisis as a whole?
Since the 1960s investment banks have been using increasingly complex mathematics, literally rocket science, to price and manage the risks of the assets they trade. The current credit crisis is unique in that for the first time, quants, physicists and engineers, using mathematical formulae rather than economists using their knowledge of markets, have been managing the assets at the heart of the crisis. Since experienced managers in banks have been at a loss to grasp the mathematics, it is hard to see how a financial journalist should be expected to do so. However, a journalist who is accustomed to investigating safety critical systems in technology is well capable of coming to terms with risk management systems in an investment bank. While a journalist who is able to explain the latest research coming from a physicist in a research lab, would be able to describe the pricing algorithm developed by a physicist in a bank.
If the types of journalists investigating science turn their attention to banks, would the risk of future crises be reduced? There is a strong argument that it would. A key role that science-reporting plays is in ensuring that technology is not misused and that it is properly regulated. Medicine is better off for the reporting of Thalidomide and the environment benefits by journalists scrutinising the energy industry.
In the 1990s, central bankers became concerned that the rules for defining how much capital a bank held in reserve were quickly becoming redundant, given the explosive growth of sophisticated derivative products. In 1996, the Bank for International Settlements, the gnomes of Basel who regulate the banks in relation to the credit crisis, began a process of re-writing banking regulation, called Basel II. Realising that they would never be able to keep up with the activities of the banks, the regulators specified a framework for regulation rather than a set of hard and fast rules. The framework, which is only now beginning to be implemented, is based on three pillars; the minimum standards for calculating how much money a bank needs to keep in reserve; the supervisory review process in the bank overseeing those calculations and market discipline ensuring the reviews and calculations are adequate.
In 2000, a mathematician working at the US Federal Reserve, checked the formulae used in Basel II to calculate capital reserves, but raised concerns
--> about whether it would be adequate if finacial products became too sophisticated (in the Discussion). The Basel committee had selected a formula that did not model the connections in the economy in a sophisticated manner and so it was inadequate if loans were inter-connected. The root cause of the credit-crisis has been that loans were connected, by the housing market, banks ran out of capital, precipitating the liquidity crisis that led to the collapse of the retail banks and the credit crunch.
The Basel II accord still uses the simplified formula, because the accord is clear; under the supervisory review process, a bank needs to match the calculation of capital reserves to the omplexity of their operations. The accord expects that banks will adequately execute the supervisory review process because of market scrutiny. Consequently, informed journalism plays a fundamental role in the new framework for banking regulation. Science journalists should take up this role given their knowledge and skills.
The Public Relations machines of banks are happy that discussion around regulation focuses on issues such as bonuses and banning complex derivatives, because the banks know that governments will not set such regulations. Turning the public’s attention to the science underpinning the bank’s business will not only put the focus on where the problem is, but is already part of the regulatory framework. In addition, putting banking managers under pressure to explain the details of the technology they are using will force better communication, debate nd understanding within the banks bridging the gap in banks between the quants and conventional financiers.
Finally, if the failure, because of poor use of science, of a section of the energy or pharmaceutical businesses had lead to a trillion-dollar rescue package, it is inconceivable that the managers of he industry would still be in place. Why have so few bankers been fired by their shareholders? Is it because the bankers have successfully blinded the shareholders with pseudo-science?
Do the maths
In a week that has seen the disappearance from the financial landscape of such venerable firms as Lehman Brothers, Merrill Lynch and the Bank of Scotland it is inevitable that people question the competence of bankers. Along with many financial mathematicians, in both the financial industry and academia, I take the view that if you give people who have only ever driven bullock carts, Ferraris, they are likely to end up in a ditch. The people running the banking industry have not had a firm grasp of the technology underpinning their business.
Financial mathematics supports the practical disciple of constructing financial products in the same way that mechanics and thermodynamics supports motor engineering. Derivatives, the products that financial mathematicians construct, transfer risk in financial markets just as engines transfer energy in the physical world. Fixed rate mortgages are a popular product of financial mathematics, the borrower passes the uncertainty in the future interest rate to the lender, who takes it on for a fee. Just as in any industry, most people working in the financial markets are involved with the sales and servicing of products and only a few people actually design, and so have a deep understanding of, the products themselves. Jeremy Clarkson does not need to understand automotive engineering to be considered an expert on cars.
It is not surprising that many bankers did not understand the science they work with, because financial mathematics is literally rocket science; it comes directly from the mathematics developed in the fifties and sixties to control rockets. The international investment banks are well aware of this and they are becoming less interested in recruiting graduates with a degree in economics from the LSE, who understand theory, and are more interested in engineers, physicists or mathematicians, who can do the complex calculations underpinning derivatives. This is nothing new, for around a decade the most popular destination of physicists on completing their PhDs has been The City and more graduate engineers in the UK now head for careers in finance than in engineering.
Although young British scientists can contemplate fat salaries with investment banks, they are at a disadvantage. If you visit the sharp end of a City firm, you will notice that there are a large number of continental graduates working there. This is no accident; continentals have a much better education in probability than UK graduates. The reason for this is cultural. The UK was at the heart of the development of statistics in the nineteenth century but at that time most mathematicians thought statistics heresy, because it is inductive rather than deductive, and today most UK universities still have separate maths and statistics departments. Seventy-five years ago this summer, a young Russian mathematician Andrey Kolmogorov defined probability in terms of rigorous mathematics placing probability firmly in the mathematics syllabus taught at school and through university in the rest of Europe.
There is a conceptual gulf between British and French graduates. When using probability to think about the future, English speakers will talk about an expectation, the French talk about an esperance, which is associated with 'hope'. The British and American bankers expect something, the French hope for it. The bankers creating asset-backed securities believed that they had removed risk because they lacked a real understanding of probability. If we want to maintain our position in the lead of finance, we need to train more people in probability so that they are better able to deal with an uncertain future and look after our money.
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