I'm taking a break from all economics/finance/accounting-related activities (blogging included) tonight to attend the opening ceremony for the inaugural Fields Medal Symposium, which celebrates the work of a recent Fields medalist each year, right here at the Fields Institute.

This year's symposium is dedicated to Ngo Bao Chau and you can watch his public lecture from 7:00pm onwards by following the link provided in the website above (his lecture will actually start a bit later, after the ceremonial speeches by a string of dignitaries, but if you tune in at 7:00 you can catch a glimpse of yours truly, sitting beside Ngo on the second row of the theatre and looking totally starstruck).

## Monday, October 15, 2012

## Saturday, October 13, 2012

### Of course it's a model, duh! A final post on income, expenditure, and endogenous money

Many comments on the different threads related to Ramanan's critique of my paper with Steve Keen amount to saying that if we were trying to write down some kind of model for a perceived phenomenon (in this case the role of private debt in macroeconomics), then it would be ok, but because we violated an accounting identity (or more) in the process, oh boy, we have been very very naughty indeed.

The thing is, we never claimed to be doing any accounting, let alone violating it. Accounting is about recording stuff during a given period (a year, a month, a day, but NOT an instant, since you need to wait for stuff to happen to record it) and in the only part in the paper where we mention any recording (Appendix, page 24, last paragraph of the paper) we say that "recorded expenditure and income over a finite period (t2 − t1 ), such as those found in NIPA tables, necessarily agree".

So I'll say this again in a separate line and in capital for emphasis (with some superlatives in bracket, as commenters like):

RECORDED EXPENDITURE AND INCOME OVER A FINITE PERIOD NECESSARILY AGREE (*always, toujour, siempre*) !!!

Now suppose you read income statements for an economy months after months, year after year, and wonder why recorded spending (= recorded income !!) for the different periods happen to be different. You might think it has something to do with the Mayan calendar, or with the incidence of flu during that period, or maybe that it's completely random. If you are an economist you might want to explain it with a DSGE model that ignore private debt. Heck you might even write down a regression model that includes the change in private debt in one period as an explanatory variable for the spending (= income !!) to be recorded over the next period, as one commenter suggests. Or if you are Steve Keen you write down a model using differential equations, because they happen to be tractable and cool and predict many properties that sort of look like what goes on in real life. But none of that is accounting - all of it is modelling.

Everything else we wrote in the paper was with the view of explaining why the heck recorded spending (= recorded income !!) changes from year to year. If along the way we wrote stuff down that looked like a violation of an accounting identity, then I profusely apologize for it (in fact I already bought a whip to punish myself) and pinky-promise never to do it again. So will the accounting police chill out and move on? Unless you actually care about the model, in which case please read on.

As far as the model goes, what we are trying to capture is Minsky's assertion that "for real aggregate demand to be increasing, . . . it is necessary that current spending plans, summed over all sectors, be greater than current received income and that some market technique exist by which aggregate spending in excess of aggregate anticipated income can be financed."

So our Y_E represent "current spending plans" (per unit of time) and our Y_I represent "current received income" (per unit of time). Equation 1.5 in the paper is the key behavioural assumption that links investment to change of debt, and is a schematic representation of the mechanism that both Steve and I have in the back of our minds, what I call the "Keen model" described in this paper, where investment (the rate of change in capital) is a function of current net profits, but can exceed profits in times of boom and therefore be financed by debt.

All of this is pure modelling: in reality nobody looks at a differential equation before spending. The true test of the model is to see if it predicts the right behaviour for the key variables (employment rate, wage share, output, level of private debt, etc) over time, once the parameters of the several structural equations are calibrated using historical data (which includes income and flow of funds statements over many periods).

As a final word, notice that neither Y_E nor Y_I are meant to represent recorded expenditure or income over a period anywhere in the paper (which again, are necessarily equal !!). Both are modelling abstractions of what goes on in the economy and could include stuff like the Mayan calendar and incidence of flu, but happen to depend on the level of private debt.

The thing is, we never claimed to be doing any accounting, let alone violating it. Accounting is about recording stuff during a given period (a year, a month, a day, but NOT an instant, since you need to wait for stuff to happen to record it) and in the only part in the paper where we mention any recording (Appendix, page 24, last paragraph of the paper) we say that "recorded expenditure and income over a finite period (t2 − t1 ), such as those found in NIPA tables, necessarily agree".

So I'll say this again in a separate line and in capital for emphasis (with some superlatives in bracket, as commenters like):

RECORDED EXPENDITURE AND INCOME OVER A FINITE PERIOD NECESSARILY AGREE (*always, toujour, siempre*) !!!

Now suppose you read income statements for an economy months after months, year after year, and wonder why recorded spending (= recorded income !!) for the different periods happen to be different. You might think it has something to do with the Mayan calendar, or with the incidence of flu during that period, or maybe that it's completely random. If you are an economist you might want to explain it with a DSGE model that ignore private debt. Heck you might even write down a regression model that includes the change in private debt in one period as an explanatory variable for the spending (= income !!) to be recorded over the next period, as one commenter suggests. Or if you are Steve Keen you write down a model using differential equations, because they happen to be tractable and cool and predict many properties that sort of look like what goes on in real life. But none of that is accounting - all of it is modelling.

Everything else we wrote in the paper was with the view of explaining why the heck recorded spending (= recorded income !!) changes from year to year. If along the way we wrote stuff down that looked like a violation of an accounting identity, then I profusely apologize for it (in fact I already bought a whip to punish myself) and pinky-promise never to do it again. So will the accounting police chill out and move on? Unless you actually care about the model, in which case please read on.

As far as the model goes, what we are trying to capture is Minsky's assertion that "for real aggregate demand to be increasing, . . . it is necessary that current spending plans, summed over all sectors, be greater than current received income and that some market technique exist by which aggregate spending in excess of aggregate anticipated income can be financed."

So our Y_E represent "current spending plans" (per unit of time) and our Y_I represent "current received income" (per unit of time). Equation 1.5 in the paper is the key behavioural assumption that links investment to change of debt, and is a schematic representation of the mechanism that both Steve and I have in the back of our minds, what I call the "Keen model" described in this paper, where investment (the rate of change in capital) is a function of current net profits, but can exceed profits in times of boom and therefore be financed by debt.

All of this is pure modelling: in reality nobody looks at a differential equation before spending. The true test of the model is to see if it predicts the right behaviour for the key variables (employment rate, wage share, output, level of private debt, etc) over time, once the parameters of the several structural equations are calibrated using historical data (which includes income and flow of funds statements over many periods).

As a final word, notice that neither Y_E nor Y_I are meant to represent recorded expenditure or income over a period anywhere in the paper (which again, are necessarily equal !!). Both are modelling abstractions of what goes on in the economy and could include stuff like the Mayan calendar and incidence of flu, but happen to depend on the level of private debt.

## Friday, October 12, 2012

### More on income, expenditure, and endogenous money - a non-vacuous response

A reader of Mike Norman's very useful blog calls my last post a vacuous response to Ramanan. Of course it was not a response at all, merely a commentary. Notice how I said that "all the criticisms can be defended in two words: endogenous money!", not that they were defended... Plus I thought it implicit that whenever one says something like "I got n words for you: word_1 ... word_n", one's tongue is firmly in one's cheek. But I'm quickly learning that there is no such thing in the econo-blogosphere.

In any event, after the avalanche of comments on Mike's re-posts (last count: 21 on the post above, 66 on Ramanan's second take down, and 124 on its predecessor), perhaps it's time for a point-by-point reply (I originally called it a "point-by-pint" reply, which is perhaps a measure of what was on my mind while I was writing it).

Let me start by saying that I'll refer mostly to this paper, since I had something to do with the notation and ideas presented in it, rather than to Steve's presentation at the UMKC conference, thought I might occasionally refer to it too. Let me also say that said paper (which is being refereed and therefore can sill improve quite a lot), could use a great deal of clarifications. Many of the ideas that were in the back of our minds as we were writing it clearly didn't make it to the printed page, so I welcome the opportunity to elaborate.

With these in mind, here are the essential points:

(1) Our "closed" economy does consist of firms, households, and banks, but we find it useful to

**separate the banking sector from the rest of the private sector**. We do this explicitly on pages 18 to 23, but leave it implicit on page 15, which contains the passages that Ramanan has a beef with. So our "change in debt" is really change in debt of the non-bank private sector to the banking sector, which obviously does not need to cancel out in the aggregate (i.e excluding banks). This is in contrast with the view that "one person's asset is another person's liability", which underlines the view that firm's debt is mirrored by household's savings.

(2)

**Debt only matter after it has been spent**. This is the point of equation (1.5): we assume that investment is financed by retained earnings plus change in debt. If new debt is not spent, it doesn't finance anything, so we don't count it in the model.

(3)

**Accounting rules!**The whole point of the Appendix in the paper is to show that recorded income equals recorded expenditure at the end of a given period (say one year). We don't use continuous mathematics to upset accounting identities, but rather as "a simple way to represent the conceptual difference between spending plans and current received income".

Observe that all 3 points are intimately connected with the idea of endogenous money, which is what I meant by my "two words" zinger. The effects of endogenous money only become apparent when banks are disaggregated from the rest of the private sector (1), capitalists finance new investment above and beyond savings by creating deposits through endogenous money (2), and spending plans exceed current received income for the same reason, even if this is not apparent when one measures recorded income and recorded expenditure (3).

In the end, what we are tying to capture is the idea expressed on page 10, namely that "the essence of endogenous money hypothesis is that banks create spending power for borrowers without reducing the spending power of savers."

Judging by the criticism, we haven't quite succeeded yet, but we'll keep on trying.

## Thursday, October 11, 2012

### Income, expenditure, and endogenous money

My supervisor Ray Streater used to quote an example of a physicist's version of proof by

I was reminded of that as I read this take down of my paper with Steve Keen, where we claim that expenditure is income plus change in debt. In an attempt to show that we are wrong, Ramanan shows that our model violates the "savings equals investment" identity. What an absurd!

Hmm, actually, not only this is not an "inconsistency" of the model (we never claim that this was true, or rely on it to show anything else in the paper), but it is rather essential: in our model, investment is equal to savings plus change in debt. This is the essence of equation (1.5) in the paper, and we were always acutely aware of it.

So much for the rather bombastic conclusion that we must be wrong because we violate the sacrosantity of "savings = investment" -- this is a feature of the model, not a bug!

Having said that, it is nevertheless a feature that ought to be defended, together with the other criticisms raised in both the take down mentioned above and its predecessor.

As it turns out, without engaging in a point-by-pint reply (which would be a very boring read for anyone not called Grasselli, Keen, or Ramanan), it suffices to say that all the criticisms can be defended in two words: endogenous money!

*reductio ad absurdum*that goes more or less like this: "Assume that asymptotic completeness doesn't hold in quantum field theory. What an absurd!"I was reminded of that as I read this take down of my paper with Steve Keen, where we claim that expenditure is income plus change in debt. In an attempt to show that we are wrong, Ramanan shows that our model violates the "savings equals investment" identity. What an absurd!

Hmm, actually, not only this is not an "inconsistency" of the model (we never claim that this was true, or rely on it to show anything else in the paper), but it is rather essential: in our model, investment is equal to savings plus change in debt. This is the essence of equation (1.5) in the paper, and we were always acutely aware of it.

So much for the rather bombastic conclusion that we must be wrong because we violate the sacrosantity of "savings = investment" -- this is a feature of the model, not a bug!

Having said that, it is nevertheless a feature that ought to be defended, together with the other criticisms raised in both the take down mentioned above and its predecessor.

As it turns out, without engaging in a point-by-pint reply (which would be a very boring read for anyone not called Grasselli, Keen, or Ramanan), it suffices to say that all the criticisms can be defended in two words: endogenous money!

## Tuesday, October 2, 2012

### Further thoughts on mathematics and economics

After almost 3 years of blogging to a reduced audience, I observed a sudden spike in my stats after my post about the UMKC conference got re-blogged (or re--tweeted, who knows) in the legendarily active econo blogosphere. So I thought I should ride the wave and elaborate on my remarks.

First of all, as pointed out by a commenter, I only attended the last day of the conference, which according to the unwritten laws of scheduling was

bound to be the least exciting (especially on a Saturday!). I guess I should expect that - after all that was the spot on the schedule reserved for me... But still,

after being blown away by the vibrance and intellectual rigour of the likes of Stephanie Kelton and Scott Fulwiller when I met them at Fields a few months ago, I had great

expectations about the event, and was disappointed to find out that most of the people I wanted to meet had already left (his lordship notwithstanding).

But on a more substantive note, I see from the comments (here and elsewhere), that the role of mathematics in economics is a bit of a raw nerve, hence the urge to elaborate.

Despite being a mathematician, I do not think that mathematical modelling is the most important part of economics, but I do think that it is somewhat essential. Here I'm reminded of the famous

saying that "logic is to mathematics what hygiene is to life: it's clearly essential, but not what it's all about". The same goes for mathematics and economics: historical awareness, acute observations, and empirical plausibility come first in economic reasoning, but I don't see how much progress can be made without mathematics. Notice that this is not about pedagogy, but about being able to even formulate crucial statements.

To borrow from other fields, it would be nearly impossible to even conceptualize something like the basic reproductive number in epidemiology without a mathematical model, and this means the difference between being able to handle a pandemic or not. For another example, no amount of analogy or logical thinking can pinpoint the phenomenon of bifurcations. The fact that smooth changes in some underlying parameter can cause a system to completely change its qualitative behaviour is not something that is predicated by logic and hard thinking alone - it needs mathematics. Needless to say, the list could go on and on.

Finally, just in case you associate mathematics too heavily with neoclassical economics, remember that the final blow to general equilibrium was dealt by a mathematical result: the SMD theorems essentially tell us that the whole framework (and much of neoclassical economics with it) is only guaranteed to work for the trivial case of one agent and one good.

## Sunday, September 30, 2012

### Having fun with economics and Lord Skidelsky

I participated in the 11th International Post Keynesian Conference in Kansas City this weekend and presented a paper on the effect of government intervention in a macro model based on a system of ODE first developed by Steve Keen, who presented just before me.

The conference itself was a bit of a sleepy affair, with most of the other talks I attended being more literary criticism (e.g several quotes from Keynes, Minsky, and others, strung together and compared with recent events) than actual modelling.

My overall impression is that if this is all that heterodoxy has to offer as an alternative to mainstream economics, then the profession is in deeper trouble than I thought.

But the highlight was the keynote after dinner speech by none other than Lord Robert Skidelsky (his business card actually says "Lord Skidelsky", no joking!)

He started his speech with a joke saying that when John Kenneth Galbraith wrote an economic report to Lyndon Johnson, the president allegedly said "You know Ken, talking about economics is like peeing in you pants, it feels hot to you, but leaves everyone else in the cold", then proceeded to say that he was reminded of this story by listening to mathematicians earlier in the afternoon, who appeared to be having to much fun with his models (Steve and me!). Then at the Q&A session, his advice to an economics grad student included "don't use math". Finally when I asked him directly what his advice to mathematician trying to contribute to economics would be, he offered this gem: "well sometimes you just need to tell what hullaballoo is all about".

So apart from all the fun to be had with economics (which I'll continue to do), I think that between the hyper mathematical (but incorrect!) DSGE guys on the mainstream and the math phobics in the fringes, this is going to be an uphill battle.

The conference itself was a bit of a sleepy affair, with most of the other talks I attended being more literary criticism (e.g several quotes from Keynes, Minsky, and others, strung together and compared with recent events) than actual modelling.

My overall impression is that if this is all that heterodoxy has to offer as an alternative to mainstream economics, then the profession is in deeper trouble than I thought.

But the highlight was the keynote after dinner speech by none other than Lord Robert Skidelsky (his business card actually says "Lord Skidelsky", no joking!)

He started his speech with a joke saying that when John Kenneth Galbraith wrote an economic report to Lyndon Johnson, the president allegedly said "You know Ken, talking about economics is like peeing in you pants, it feels hot to you, but leaves everyone else in the cold", then proceeded to say that he was reminded of this story by listening to mathematicians earlier in the afternoon, who appeared to be having to much fun with his models (Steve and me!). Then at the Q&A session, his advice to an economics grad student included "don't use math". Finally when I asked him directly what his advice to mathematician trying to contribute to economics would be, he offered this gem: "well sometimes you just need to tell what hullaballoo is all about".

So apart from all the fun to be had with economics (which I'll continue to do), I think that between the hyper mathematical (but incorrect!) DSGE guys on the mainstream and the math phobics in the fringes, this is going to be an uphill battle.

## Friday, July 6, 2012

### Steve Keen at Fields: a memorable month

Steve Keen left Toronto today after spending the last five weeks visiting the Fields Institute. As always happens when one is having so much fun, time went by very fast, but we collectively managed to get an awful lot done. Here are the highlights:

- we had weekly seminars where people with backgrounds in economics, financial mathematics, mathematical biology, history, and statistics gathered to present and discuss results related to a novel (nonlinear) dynamical systems approach to macroeconomics;

- Steve gave a public lecture hosted by the Canadian Centre for Policy Alternatives, which was interesting in its own right, but also served to put Fields and CCPA on a path for future collaborations;

- we held a very successful closed discussion session on a scorching hot Canada Day public holiday, with the purpose of reconciling Modern Monetary Theory (MMT) and Monetary Circuit Theory (MCT);

- Steve gave another public lecture at Fields, which was not only streamed over the web using our FieldsLive system, but also recorded for an episode of TVO's show Big Ideas.

- we had weekly seminars where people with backgrounds in economics, financial mathematics, mathematical biology, history, and statistics gathered to present and discuss results related to a novel (nonlinear) dynamical systems approach to macroeconomics;

- Steve gave a public lecture hosted by the Canadian Centre for Policy Alternatives, which was interesting in its own right, but also served to put Fields and CCPA on a path for future collaborations;

- we held a very successful closed discussion session on a scorching hot Canada Day public holiday, with the purpose of reconciling Modern Monetary Theory (MMT) and Monetary Circuit Theory (MCT);

- Steve gave another public lecture at Fields, which was not only streamed over the web using our FieldsLive system, but also recorded for an episode of TVO's show Big Ideas.

## Saturday, June 9, 2012

### Steve Keen on endogenous money

We started our Sojourns in Nonlinear Economics this week with Steve Keen giving a thorough introduction to his circuit theory for endogenous money creation in the banking system.

Steve is visiting Fields this month and we organized these informal seminar to get the most out of his stay. Personally I'm anticipating that his highly original ideas and unorthodox models will provide research material for myself and other mathematicians for many years to come.

Stay tuned!

Steve is visiting Fields this month and we organized these informal seminar to get the most out of his stay. Personally I'm anticipating that his highly original ideas and unorthodox models will provide research material for myself and other mathematicians for many years to come.

Stay tuned!

## Saturday, June 2, 2012

### Myron Scholes at McMaster

The McMaster Alumni Association celebrated the 50 years of graduation for the class of 1962 by inviting their most distinguished alumnus back to campus.

I attended the luncheon in his honour, where he treated us to personal recollections of his time as a student at McMaster and some insights on the future of economics. At question time I asked if he thought new techniques were necessary to deal with what I call mesoeconomic problems, such as banking regulation, which do not easily fit into either micro or macroeconomic toolboxes. He said "Yes, sure" and proceeded to explain an Ornestein-Uhlenbeck model for production, with a mean-reverting rate that had to be adjusted after each macroeconomic shock. Okay...

At the public talk that followed, Myron accurately diagnosed the current economic woes of the world as a "debt-deflation" crisis, with all the symptoms associated with it. His prescription, however, was to get the government out of financial markets altogether. He used the analogy of putting out small forest fires for many decades only to discover that the practice created the perfect conditions for uncontrollable big fires.

Now, apart from the obvious fact that this means that Greenspan should NOT have orchestrated the bailout of LTCM (something everyone in the room was too polite to point out), the prescribed medicine happens to be the opposite of what the doctor should call for given the correct diagnoses.

But oh well, it was a day to celebrate, not to get bogged down in technical discussions.

I attended the luncheon in his honour, where he treated us to personal recollections of his time as a student at McMaster and some insights on the future of economics. At question time I asked if he thought new techniques were necessary to deal with what I call mesoeconomic problems, such as banking regulation, which do not easily fit into either micro or macroeconomic toolboxes. He said "Yes, sure" and proceeded to explain an Ornestein-Uhlenbeck model for production, with a mean-reverting rate that had to be adjusted after each macroeconomic shock. Okay...

At the public talk that followed, Myron accurately diagnosed the current economic woes of the world as a "debt-deflation" crisis, with all the symptoms associated with it. His prescription, however, was to get the government out of financial markets altogether. He used the analogy of putting out small forest fires for many decades only to discover that the practice created the perfect conditions for uncontrollable big fires.

Now, apart from the obvious fact that this means that Greenspan should NOT have orchestrated the bailout of LTCM (something everyone in the room was too polite to point out), the prescribed medicine happens to be the opposite of what the doctor should call for given the correct diagnoses.

But oh well, it was a day to celebrate, not to get bogged down in technical discussions.

## Sunday, May 20, 2012

### Office of Financial Research

Just after the crisis, a group of scholars associated with quantitative finance (mathematicians, finance professors, financial economists, etc) got together with practitioners and regulators and formed a committee to lobby for the creation of a National Institute of Finance. The idea was to gather data and expertise to monitor, analyze and potentially prevent future crises.

Against all odds, the idea was enshrined in the Dodd-Frank act and became the Office of Financial Research. It has the potential to create an unprecedentedly useful database and spur paradigm shitting research with far reaching implications to the way finance is understood and practiced, and has already attracted some of the best minds in the field.

Naturally, the Republicans want to kill it.

Against all odds, the idea was enshrined in the Dodd-Frank act and became the Office of Financial Research. It has the potential to create an unprecedentedly useful database and spur paradigm shitting research with far reaching implications to the way finance is understood and practiced, and has already attracted some of the best minds in the field.

Naturally, the Republicans want to kill it.

### Mathematics of New Financial Systems

That was the theme of the recent Hot Topics Workshop at the IMA in Minneapolis.

I was delighted to see many old friends doing some really interesting mathematics on relatively new topics, like systemic risk and stochastic portfolio theory.

My small contribution to the workshop was on the Keen-Minsky model for the dynamics of credit for the economy as a whole (see paper here). Because the workshop was open-minded by construction, many of the economists in attendance refrain from attacking the model too heavily, despite some of my provocative remarks (e.g "macroeconomics is too important to be left at the hands of macroeconomists").

All in all I came away encouraged and optimistic about the prospect of future contributions of mathematics to economics.

I was delighted to see many old friends doing some really interesting mathematics on relatively new topics, like systemic risk and stochastic portfolio theory.

My small contribution to the workshop was on the Keen-Minsky model for the dynamics of credit for the economy as a whole (see paper here). Because the workshop was open-minded by construction, many of the economists in attendance refrain from attacking the model too heavily, despite some of my provocative remarks (e.g "macroeconomics is too important to be left at the hands of macroeconomists").

All in all I came away encouraged and optimistic about the prospect of future contributions of mathematics to economics.

## Saturday, March 24, 2012

### Last IJTAF special issue

Over the past year the International Journal for Theoretical and Applied Finance published 3 special issues with papers presented during the Thematic Program on Quantitative Finance: Foundations and Applications, held at Fields in 2010.

The first issue was devoted to Foundations of Mathematical Finance, the second to Computational Finance, and the third and last, which has just appeared, to Financial Derivatives and Risk Management. Each corresponds to one of the major workshops in the thematic program.

I had the privilege of being a guest editor for each of the issues and writing the prefaces together with my distinguished co-editors.

The first issue was devoted to Foundations of Mathematical Finance, the second to Computational Finance, and the third and last, which has just appeared, to Financial Derivatives and Risk Management. Each corresponds to one of the major workshops in the thematic program.

I had the privilege of being a guest editor for each of the issues and writing the prefaces together with my distinguished co-editors.

## Sunday, February 19, 2012

### Summer school in Cape Town

I'm writing these words while looking out the window of my temporary office at AIMS and gazing at a point not far from where two oceans meet.

I was here to give a series of lectures in the 5th Summer School in Financial Mathematics.

It is hard to describe the striking beauty of Cape Town, the friendly atmosphere at the Institute, and the excitement surrounding the lectures and discussions that follow. In fact it's so hard that I'll resort to pictures instead:

I was here to give a series of lectures in the 5th Summer School in Financial Mathematics.

It is hard to describe the striking beauty of Cape Town, the friendly atmosphere at the Institute, and the excitement surrounding the lectures and discussions that follow. In fact it's so hard that I'll resort to pictures instead:

## Saturday, January 28, 2012

### Dupire at Fields

The first talk of 2012 in the Quantitative Finance Seminar series was given this week by Bruno Dupire.

As I mentioned when I was introducing his talk, the Dupire equation is not the best known result in mathematical finance - that honour has to go to the Black-Scholes equation. But the Black-Scholes equation is an oversimplification of reality that most people try to stay away from as much as possible, whereas the Dupire equation, in the words of Alex Lipton who spoke at Fields in November, is the "most effective result in mathematical finance."

Bruno spoke about yet another remarkable innovation that he has produced in recent years, what he calls Functional Ito Calculus, and its uses in the context of path dependent options. I think it's a fascinating result and have asked him to expand his paper into a short book format to be published as one of the first volumes of the soon to be launched Springer Briefs in Quantitative Finance.

As I mentioned when I was introducing his talk, the Dupire equation is not the best known result in mathematical finance - that honour has to go to the Black-Scholes equation. But the Black-Scholes equation is an oversimplification of reality that most people try to stay away from as much as possible, whereas the Dupire equation, in the words of Alex Lipton who spoke at Fields in November, is the "most effective result in mathematical finance."

Bruno spoke about yet another remarkable innovation that he has produced in recent years, what he calls Functional Ito Calculus, and its uses in the context of path dependent options. I think it's a fascinating result and have asked him to expand his paper into a short book format to be published as one of the first volumes of the soon to be launched Springer Briefs in Quantitative Finance.

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