Archive for the ‘Financial Mathematics’ Category

While we’re on the topic of economics, here’s an interesting graphic I found at The Big Picture blog –

What we see here is that, although China and Japan do have substantial holdings of U.S. government debt, they are far from being majority stakeholders.


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For those who are economically minded, Paul Krugman has a great article in the upcoming Times Sunday Magazine (Jan. 16) on the creation of the euro and the benefits and difficulties involved in this process.

At its heart, his comparison of Ireland and Nevada highlights the issues very clearly,

Climate, scenery and history aside, the nation of Ireland and the state of Nevada have much in common. Both are small economies of a few million people highly dependent on selling goods and services to their neighbors. (Nevada’s neighbors are other U.S. states, Ireland’s other European nations, but the economic implications are much the same.) Both were boom economies for most of the past decade. Both had huge housing bubbles, which burst painfully. Both are now suffering roughly 14 percent unemployment. And both are members of larger currency unions: Ireland is part of the euro zone, Nevada part of the dollar zone, otherwise known as the United States of America.

But Nevada’s situation is much less desperate than Ireland’s.

First of all, the fiscal side of the crisis is less serious in Nevada. It’s true that budgets in both Ireland and Nevada have been hit extremely hard by the slump. But much of the spending Nevada residents depend on comes from federal, not state, programs. In particular, retirees who moved to Nevada for the sunshine don’t have to worry that the state’s reduced tax take will endanger their Social Security checks or their Medicare coverage. In Ireland, by contrast, both pensions and health spending are on the cutting block.

Also, Nevada, unlike Ireland, doesn’t have to worry about the cost of bank bailouts, not because the state has avoided large loan losses but because those losses, for the most part, aren’t Nevada’s problem. Thus Nevada accounts for a disproportionate share of the losses incurred by Fannie Mae and Freddie Mac, the government-sponsored mortgage companies — losses that, like Social Security and Medicare payments, will be covered by Washington, not Carson City.

And there’s one more advantage to being a U.S. state: it’s likely that Nevada’s unemployment problem will be greatly alleviated over the next few years by out-migration, so that even if the lost jobs don’t come back, there will be fewer workers chasing the jobs that remain. Ireland will, to some extent, avail itself of the same safety valve, as Irish citizens leave in search of work elsewhere and workers who came to Ireland during the boom years depart. But Americans are extremely mobile; if historical patterns are any guide, emigration will bring Nevada’s unemployment rate back in line with the U.S. average within a few years, even if job growth in Nevada continues to lag behind growth in the nation as a whole.

Over all, then, even as both Ireland and Nevada have been especially hard-luck cases within their respective currency zones, Nevada’s medium-term prospects look much better.

What does this have to do with the case for or against the euro? Well, when the single European currency was first proposed, an obvious question was whether it would work as well as the dollar does here in America. And the answer, clearly, was no — for exactly the reasons the Ireland-Nevada comparison illustrates. Europe isn’t fiscally integrated: German taxpayers don’t automatically pick up part of the tab for Greek pensions or Irish bank bailouts. And while Europeans have the legal right to move freely in search of jobs, in practice imperfect cultural integration — above all, the lack of a common language — makes workers less geographically mobile than their American counterparts.

The issue here is that the difficulty of creating a monetary union without a corresponding political fiscal union creates huge problems for the member states and their citizens.

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I’ve finally gotten around to reading Nassim Nicholas Taleb‘s The Black Swan.  I’m about halfway through and I’m enjoying it very much.

It’s part math, part economics, part philosophy, and part quirky anecdotes.

I’m not sure what the rest of the book will be like and in the spirit of the book itself I am OK with that!

What it has made me think about so far is mainly that the real issue is not predicting the future because human beings are notoriously and demonstrably bad at predicting the future.

Rather, I think that our energies are better spent trying to develop plans and methods for continuity and adaptability in the face of the inherent and disruptive unpredictable nature of future events.

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Last February, I wrote about the current financial crisis and the role that mathematical trading strategies played in the meltdown.

I recently finished reading The Quants by Wall Street Journal reporter Scott Patterson.  In The Quants Patterson tells the story of computer trading strategies at the big NY investment banks and various hedge funds.  He covers some of the history of quantitative analysis of financial markets as well as the meltdown that started in August of 2007.

One paragraph in particular caught my eye as I was reading:

The market moves PDT and other quant funds started to see early that week defied logic.  The fine-tuned models, the bell curves and random walks, the calibrated correlations – all the math and science that had propelled the quants to the pinnacle of Wall Street – couldn’t capture what was happening.  It was utter chaos driven by pure human fear, the kind that can’t be captured in a computer model or complex algorithm.  The wild fat-tailed moves discovered by Benoit Mandelbrot in the 1950s seemed to be happening on an hourly basis.  Nothing like it had ever been seen before.  This wasn’t supposed to happen. (Italics in original)

Things that the mathematical trading models had predicted to be so unlikely as to happen only once in 10,000 years happened three days in a row in August 2007.  This is not just like flipping a coin and getting 10 heads in a row, it’s like flipping a coin and having it land on edge 10 times in a row.

[By the way, if your model predicts that something will happen once in 10,000 years and then it happens three days in row – NEWSFLASH – there is something deeply flawed in your model!]

Part of the problem was laid out by Nassim Nicholas Taleb, author of the book The Black Swan.  The mathematics used in the world of physics and other hard sciences uses standard bell curves.  If you measure the height of 1,000 people off the street, even if you include a few NBA players, the average won’t change all that much.

The problem in finance is that the scale of the financial world is radically different from the scale of the physical world.  If, instead of measuring the height of 1,000 people, you are measuring the financial net worth of 1,000 people, having someone like Bill Gates in your sample of 1,000 can have extreme effects on the average.

This plays out in the financial world in the effect that very large pools of money inevitably have on markets as they move in and out of various trading positions.  Standard statistical models that were developed to deal with the natural world don’t account for the effect these outsized moves have on prices and markets.

Another idea that I found interesting in The Quants was that, as quantitative trading strategies caught on through the 90s and 00s, more and more people began using similar strategies, which made these strategies less profitable.

There are only so many trades to go around, so if, instead of $1 billion chasing these slight pricing inefficiencies, you suddenly have $100 billion and more chasing these same slight inefficiencies, there is less and less profit in each trade.

The solution – LEVERAGE, LEVERAGE, LEVERAGE.  In other words, once the profits on these strategies started becoming smaller and smaller (because they were effectively being split between more and more traders using similar strategies), they needed to use bigger and bigger trades to get the same amount of profit out.

Many of the hedge funds and investment banks were leveraged 10, 20 or even 30 to 1.  For example, in The Quants, Patterson describes the hedge fund Citadel Group as having $140 billion in assets on only $15 billion in actual capital.  This is about a 9 to 1 leverage ratio.

What makes this so important is that if these trades go bad on the hedge funds, they stand to lose A LOT more money than they can conceivably pay back, especially if what they’re invested in turns out to be worthless.

So, when things turned bad in August of 2007, all of a sudden, all the hedge funds that were making these trades with vast amounts of borrowed money were scrambling for the exits all at once.  They were all highly leveraged and they were all in almost the same trading strategies which meant they were all trying to get out of the same doorway at the same time.

In a 60 minutes interview with Steve Kroft last year, Frank Partnoy pointed out that “You can’t model human behavior with math.”

Another way of saying this is that finance and economics are social sciences – not hard sciences.

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This isn’t really mathematical, but there has been a flurry of articles recently about the financial crisis – both its lingering effects and the day to day issues it raised last September in the Federal Government and the large, money center banks and invesment banks.

Andrew Ross Sorkin has an article in the new Vanity Fair that is an excerpt from his book Too Big to Fail, which comes out later this month.  In it, he details the events of the third week of September 2008.

On September 7th, Fannie Mae and Freddie Mac were taken over by the government.  Bear Stearns had been taken over by JPMorgan Chase in May 2008 (with a $30 billion government guarantee serving as a spoonful of sugar to make the deal go down easier) and Lehman Brothers went into bankruptcy on Monday September 15th.

Sorkin’s story opens on Wednesday September 17, 2008.

The panic was already palpable in John Mack’s office at Morgan Stanley’s Times Square headquarters. Sitting on his sofa with his lieutenants, Walid Chammah, 54, and James Gorman, 50, drinking coffee from paper cups, Mack was railing: the major news on Wednesday morning, he thought, should have been the strength of Morgan Stanley’s earnings report, which he had released the afternoon before, a day early, to stem any fears of the firm’s following in Lehman’s footsteps….

Apart from the general nervousness about investment banks, he was facing a more serious problem than anyone on the outside realized: at the beginning of the week, Morgan Stanley had had $178 billion in the tank—money available to fund operations and lend to its hedge-fund clients. But in the past 24 hours, more than $20 billion of it had been withdrawn by anxious hedge-fund clients, in some cases closing their prime-brokerage accounts entirely.

“The money’s walking out of the door,” Chammah told Mack….

‘What’s wrong?” Mack asked in alarm as Colm Kelleher walked into his office later in the day, his face ashen. “John, we’re going to be out of money on Friday,” Kelleher said with his staccato British inflection. He had been nervously watching the firm’s tank—its liquid assets—shrink, the way an airline pilot might stare at the fuel gauge while circling an airport, waiting for landing clearance.

“That can’t be,” Mack said anxiously. “Do me a favor: go back to the financing desk—go through it again.”

Kelleher returned to Mack’s office 30 minutes later, less shaken, but only slightly. After finding some additional money trapped in the system between trades that hadn’t yet settled, he revised his prognosis: “Maybe we’ll make it through early next week.”

The article details the machinations of the major economic players in last September’s meltdown – Former Treasury Secretary Henry Paulson and then-Head of the New York Federal Reserve Bank and current Treasury Secretary Timothy Geithner, as well Morgan Stanley’s John Mack, Goldman Sach’s Lloyd Blankfein, JPMorgan Chase’s Jamie Dimon, Citibank’s Vikram Pandit, Bank of America’s Ken Lewis and many others.

Another article, this one a blog post from Chris Whalen at The Big Picture economics blog goes over some of the problems that came out of the wave of takeovers late last year as Bank of America took over Countrywide Mortgage and Merrill Lynch, Wells Fargo took over Wachovia Bank, and Bear Stearns was taken over by JPMorgan Chase.  Whalen is a professional bank analyst (who also worked as a republican congressional staffer in the 1990s) and his description is pretty heavy on banking jargon and details.

Oil Trading Denominated in Dollars

Also today, there is an article in the British newspaper The Independent written by Robert Fisk about a plan to sell oil using a “basket” of currencies instead of the U.S. dollar.

Ever since Richard Nixon removed the gold standard in the early 1970s, the U.S. dollar has remained a reserve currency for the world mainly due its status as the currency in which oil trading is denominated.  This is set to change.

In the most profound financial change in recent Middle East history, Gulf Arabs are planning – along with China, Russia, Japan and France – to end dollar dealings for oil, moving instead to a basket of currencies including the Japanese yen and Chinese yuan, the euro, gold and a new, unified currency planned for nations in the Gulf Co-operation Council, including Saudi Arabia, Abu Dhabi, Kuwait and Qatar.

Secret meetings have already been held by finance ministers and central bank governors in Russia, China, Japan and Brazil to work on the scheme, which will mean that oil will no longer be priced in dollars.

The plans, confirmed to The Independent by both Gulf Arab and Chinese banking sources in Hong Kong, may help to explain the sudden rise in gold prices, but it also augurs an extraordinary transition from dollar markets within nine years….

Ever since the Bretton Woods agreements – the accords after the Second World War which bequeathed the architecture for the modern international financial system – America’s trading partners have been left to cope with the impact of Washington’s control and, in more recent years, the hegemony of the dollar as the dominant global reserve currency.

This could devalue the dollar and make imports (like oil itself) more expensive.

In addition to the article, The Independent also included an editorial about the proposed change:

Last autumn’s global financial crisis set off an economic earthquake. And we are still feeling the tremors. The latest sign of the ground shifting beneath our feet is our report today of plans by Gulf states, China, Russia, France and Japan to end their practice of conducting oil deals in US dollars, switching instead to a diverse basket of currencies.

It is not hard to see the motivation for oil exporters to move away from the dollar. The value of the US currency has fallen sharply since last year’s meltdown. And fears are growing, in the light of a spiralling US government deficit, that a further depreciation is likely. They do not want to sell their wares in return for a currency with an uncertain future.


Last on the list is a story about the New York State Pension Fund and a guilty plea in a case of corruption reportedly involving the New York State comptroller Alan Hevesi

ALBANY — Raymond B. Harding, one of the last of New York’s political bosses, admitted on Tuesday that he had accepted more than $800,000 in exchange for doing favors for Alan G. Hevesi, the former state comptroller; among the favors was a scheme to secure an Assembly seat for Mr. Hevesi’s son….

The case has focused on the state’s $116.5 billion pension fund, and how people close to Mr. Hevesi exploited their relationship with the former comptroller to enrich themselves. The comptroller serves as the fund’s sole trustee, a relatively unusual arrangement that gives him ultimate authority over what firms are allowed lucrative contracts to manage the fund’s money…

Mr. Cuomo’s office also said on Tuesday that Saul Meyer of Aldus Equity, a Dallas firm that consulted with the pension fund, had pleaded guilty to a similar charge that had been sealed since Friday. Mr. Meyer admitted to violating his fiduciary duty to pensioners in both New York and New Mexico and taking part in schemes allegedly orchestrated by Mr. Morris. He also faces four years in jail, but is cooperating with the investigation.

“These guilty pleas vividly depict the depth and breadth of corruption involving the New York State pension fund,” Mr. Cuomo said. “In one case, we see New York’s state pension fund looted to reward a political boss with hundreds of thousands of dollars in improper payments.”

“In the other, we see a pension fund adviser — the outside ‘gatekeeper’ who is supposed to safeguard the integrity of the pension fund process — recommending deals based on pressure from pension officials and politically connected people.”

This type of corruption is at the root of the problems in our financial system.  The recently released inspector general’s report on the investigation (or lack of investigation) into Bernard Madoff’s massive ponzi scheme details the failure of the Securities and Exchange Commission to look into repeated reports of fraud.

Cleaning up corrupt practices in both business and government must precede any meaningful resurgence of the U.S. economy.

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I went to Lincoln City this past weekend for the ORMATYC Conference.  ORMATYC is the Oregon Mathematical Association of Two Year Colleges and is a part of the larger group AMATYC, the American Mathematical Association of Two Year Colleges.

Just about every year, we have a conference in Lincoln City – this is the fifth one I’ve attended.  It’s a great experience to get together with other community college math instructors from Oregon to talk about math and math education.  It’s a valuable forum to find out what other schools are doing in their math classes and how we compare with those other schools.

In attending this conference every year, I have learned more about both math and math education.  In some cases it was about how to present a particular topic in class, while in other cases it was something that provided additional background information about the topics I teach so that I can share these ideas with my students.

In addition to being exposed to new ideas about math and math education, another positive aspect of this conference is being part of a community.  After 5 years, I have gotten to know some of the other instructors and have a better idea about which talks to attend while I’m there.

This year, I saw Jim Ballard of OIT Klamath Falls give a talk on the mathematics of finance.  There was a lot of discussion about the current economic situation and he didn’t really get a chance to talk about the Black-Scholes equations which are somewhat controversial, but have been used in mathematical finance for over 20 years.  I wrote about math and finance in an earlier post.

I also attended a session with Ron Wallace of Blue Mountain CC about deciding which topics to teach and which to leave out in the math curriculum.  He asked if anyone there had used the quadratic formula in their lives outside of teaching in the past five years.  I was the only one to raise my hand.

I did use the quadratic formula a few years ago when my Mom asked me about the cost of Medicare Part D programs.  The formula for pricing in Medicare Part D is quadratic in that it initially becomes more expensive the longer you wait to enroll, but the money you save by not enrolling right away can offset the higher premiums you end up paying.

On Friday afternoon I went to a presentation by Art Peck of Lane CC.  I had seen his talk last year about the connection between the Fibonacci Sequence and the Mandelbrot Set, which was excellent.  This year, he talked about applications of mathematics to environmental problems, including alternative energy.  There is a lot of mathematics involved in scientific research that is focused on the environment.  For particular examples, he mentioned a textbook and companion website that have been developed and have some great application questions.

This is an important time for alternative energy generation and research directed at the environment in general.  I wrote an earlier post about the Solar Tres project.  The development of electric car motors and batteries has reached a point that production of “all-electric” car models is happening now.  One of my calculus textbooks has a cover page addressed to the instructor saying “The first person to invent a car that runs on water may be sitting right in your classroom.”

On Saturday morning, I saw a presentation by Geza Laszlo called “Rational Approximations of Roots of Polynomials.”  This is a very interesting topic.  It has connections to some of the material we cover in MTH 111 about roots of polynomials, but it is more closely related to the ideas we discuss in MTH 116 about using Newton’s method to approximate a square root.

Newton’s method uses calculus, but the method itself was known to the Greeks, even though they did not have formal knowledge of the methods of calculus.  The idea of approximating irrational numbers with rational numbers is also of great importance in constructing a musical scale.  Attempting to approximate (log3)/(log2) with a rational number determines how an octave will be separated into notes and how accurate the scale will be.

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Because the math is really complicated people assume it must be right.” — Nigel Goldenfeld, whose company sells derivatives software.(from a NYTimes article March 9, 2009)

The cover article in this month’s Wired magazine has to do with the applications of mathematics to finance and investing.  Specifically, it talks about what is called the Gaussian Copula function, which supposedly allowed investment banks, hedge funds, and other high-flying investment professionals a quick, straightforward way to determine the risk in a pool of investments.

The article starts out explaining that the mathematician who came up with the Gaussian Copula function was widely celebrated and that many people even thought that he might win a Nobel Prize.

“A year ago, it was hardly unthinkable that a math wizard like David X. Li might someday earn a Nobel Prize. After all, financial economists—even Wall Street quants—have received the Nobel in economics before…”

What the article fails to mention, and I think that this is a particularly egregious omission, is that the last mathematical/economics investment wizards to win the Nobel Prize were the team behind the Black-Scholes equation –

“Robert C. Merton and Myron S. Scholes have, in collaboration with the late Fischer Black, developed a pioneering formula for the valuation of stock options. Their methodology has paved the way for economic valuations in many areas. It has also generated new types of financial instruments and facilitated more efficient risk management in society.”

Great! So the mathematicians have finally figured out how to feed a computer equations and get it to spit out money in return!


From Wikipedia

“Together with Myron Scholes, Merton was among the board of directors of Long-Term Capital Management (LTCM), a hedge fund that failed spectacularly in 1998 after losing US$4.6 billion in less than four months.  The Federal Reserve was so concerned about the potential impact of LTCM’s failure on the financial system that it arranged for a group of 19 banks and other firms to provide sufficient liquidity for the banking system to survive.”

Long-Term Capital Management was like a dress rehearsal for today’s financial meltdown.  I think that it would have served the author well to at least mention that the last widely celebrated financial risk assessment mathematicians were also spectacularly wrong.

Here’s the problem – people want a short-cut.  Math has some great short-cuts.  Think of the quadratic formula in Algebra, or the power rule for differentiation in Calculus.  But, for some things, there is no short-cut.  There is only the grunt work of going in and examining and analyzing enough pieces of what you’re working with so that you have a solid overall understanding of the big picture.

When David X. Li came out with his new short-cut for assessing investment risk…

“The effect on the securitization market was electric. Armed with Li’s formula, Wall Street’s quants saw a new world of possibilities. And the first thing they did was start creating a huge number of brand-new triple-A securities. Using Li’s copula approach meant that ratings agencies like Moody’s—or anybody wanting to model the risk of a tranche—no longer needed to puzzle over the underlying securities. All they needed was that correlation number, and out would come a rating telling them how safe or risky the tranche was.”

Normally, ratings agencies and/or investors have to do a lot of analysis to determine the quality of investments.  This is what is known as DUE DILIGENCE.  This is a very important legal term that essentially means doing your homework and NOT taking short-cuts.

Two mathematicians who have their own ideas about applications of math to investing are Benoit Mandelbrot and Nassim Nicholas Taleb.  I wrote a blog entry about these two back in October.

Mandelbrot is the mathematical father of fractals and fractal analysis.  He published a book a few years ago about the application of the ideas of fractal analysis to the financial markets.

Nassim Nicholas Taleb’s training and education were focused on the application of mathematics and statistics to business and finance.  He wrote the book The Black Swan, about how improbable occurrences  that we are unaware of can have a dramatic impact on what we expect the future to look like.

Both of these mathematicians point out the severe limitations of both the Black-Scholes model and the Gaussian Copula model.  The Wired article discussed Taleb’s take on the role the Gaussian Copula function played in the recent financial meltdown.

Nassim Nicholas Taleb, hedge fund manager and author of The Black Swan, is particularly harsh when it comes to the copula. ‘People got very excited about the Gaussian copula because of its mathematical elegance, but the thing never worked,’ he says. ‘Co-association between securities is not measurable using correlation,’ because past history can never prepare you for that one day when everything goes south. ‘Anything that relies on correlation is charlatanism.’ “

On the other hand, some people have used the Black-Scholes model successfully by understanding its limitations.

Paul Wilmott, who is quoted in the Wired article, has also been critical of attempts to quantify risk.  But, he has used the Black-Scholes model in the past to develop trading strategies.

“A couple of years after leaving academia I became a partner in a volatility arbitrage hedge fund, and this was the start of phase three. In this fund we had to price and risk manage many hundreds of options series in real time. As much as I would have liked to, we just weren’t able to use the ‘better’ models that I’d been working on in phase two. There just wasn’t the time. So we ended up streamlining the complex models, reducing them to their simplest and most practical form. And this meant using good ol’ constant volatility Black-Scholes, but with a few innovations since we were actively looking for arbitrage opportunities. From a pragmatic point of view I developed an approach that used Gaussian models for pricing but worst-case scenarios for risk management of tail risk. And guess what? It worked. Sometimes you really need to work with something that while not perfect is just good enough and is understandable enough that you don’t do more harm than good. And that’s Black-Scholes.”

The problem is not necessarily the mathematics but that people often take mathematics (and science) as some kind of all-seeing oracle that will make decisions for them.  This blinds them to what is actually happening right before their eyes and they assume that they don’t have to do the (intellectual) heavy lifting, because the equations and computers are doing that for them.

Remember the fundamental mantra of computer programming –Garbage In, Garbage Out

The Wired article says that

“Bankers should have noted that very small changes in their underlying assumptions could result in very large changes in the correlation number. They also should have noticed that the results they were seeing were much less volatile than they should have been—which implied that the risk was being moved elsewhere. Where had the risk gone?

They didn’t know, or didn’t ask. One reason was that the outputs came from “black box” computer models and were hard to subject to a commonsense smell test. Another was that the quants, who should have been more aware of the copula’s weaknesses, weren’t the ones making the big asset-allocation decisions. Their managers, who made the actual calls, lacked the math skills to understand what the models were doing or how they worked. They could, however, understand something as simple as a single correlation number. That was the problem.”

And with the money rolling in, nobody wants to ask questions.

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