Kill all DJs

Below some traffic lights in Rio de Janeiro, you can find a button; it’s implied that pressing it will somehow signal to the control system that you want to cross the street and change the traffic light dynamics somehow. (Click the picture to see better) Now, I don’t know if those buttons even do a thing; I don’t know anyone in the traffic control authorities or anything. So suppose I gather the following data

1. The full time of “red” traffic light (”green” for pedestrians) when no one presses the button n times. Call this dataset . 
2. Every day arriving at a random time to work and finding a green light (red for pedestrians), press the button and measure the time until lights turn red. Call this dataset . Optionally record the days when I found red lights and never pressed the button.

Here’s the puzzle, somewhat open-ended: How many days do I have to press the button before I know it’s having any effect?  A sub-puzzle: what if I don’t have the dataset T?

There are a few different approaches here. Off the top of my head there’s a solution from the classical test of hypothesis framework and an incremental bayesian solution. I’ll post my ideas on this next monday.

Posted in Statistics | No Comments »

Sadly, metablogging. I want to minimize this kind of thing, but I always end up feeling I have a half cent to add. I should make a GIF with a warning: “Achtung! No useful knowledge was generated in the posting below the fold.”

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Ladies and gentlemen browsing the blogs in the first days of 2009. If there’s one thing I can honestly advise, it’s this: never trust your money to someone who can’t recite the Lyapunov condition for a central limit theorem. But I’ve been gathering some bonus thoughts. Needless to say, I am or have been an infringer in most of the “don’t”s — which is why I know them to be so. Likewise, I’m often lacking in the very things I advocate; advice comes from the heart, not the elbows. But one thing I have not (at least up  to my control) done is trusting my money a man who can’t recite the Lyapunov condition for a central limit theorem. At least not yet.

1.  Never trust your money to someone who can’t recite the Lyapunov condition for a central limit theorem by heart. Memorize it yourself, for it may save ye tushy

where are the third central moments,which (individually) divided by the variances   are the asymmetry coefficients. The three first central moments must also be finite for the condition to imply a CLT. But despair not: this is an amazingly powerful thought: if things turn out more or less symmetrical they will, at one point, become normal. If the person at hand doesn’t realize that the “classical”, pre-Lyapunov/Lindeberg central limit theorem requires identically distributed variables all around the sample (do all stocks in your portfolio have the same kurtosis?), fire him/her.

2. Beware of kurtosis, but don’t worship it.  Everyone -is taught about the Chebyshev upper bound for standardized deviations from the mean — one of the most important rules of thumb you can know, possibly second only to the chain rule in the calculus. Much less well-known is Markov’s inequality, often found unassumingly as a lemma in derivations of the Chebyshev bounds. It is, however, a quantum of solace in the mathematics of fear and chance. Quite simply, being the usual notation for absolute value, it holds for any random variable X that

– regardless of the variance or whether a finite variance even exists. 

Run the numbers: if 1% is the average daily change in some price series, the likelihood of a 8% drop or rise is at tops 12.5% even in the funkiest scenarios conceivable to frustrated measure theorists. Keep that in mind — and  that the Lyapunov condition is about convergence to symmetry, with no discussion of fourth moments. Excess kurtosis is an occupational hazard that needs to be taken seriously, but not pampered to.

3. Mind the mathematical assumptions. People will try to shift blame to the theoretical or conceptual assumptions, but that’s seldom the case. The problems in quant finance are as predicated on the efficient market hypothesis as they are dependent on the axiom of choice. The problems with applied partial equilibrium analysis are mostly about naïveté or carelessness in aggregation (next time someone quotes an elasticity, ask them if they’re willing to posit at least reductibility to  Gorman’s polar form on the choice functions, just to begin with), not about the basic notions of convex rationality that have been under fire since at least satisficing theory and have now been toughly challenged by Nobel-awarded prospect theory. The problems in  finite-sample statistical inference are more about misunderstanding basic real analysis (limits of series) than about overusing asymptotics in small-sample scenarios (there is an even less demanding CLT based on the epsilon-delta-style Lindeberg condition — which goes all back to knowing what a limit really means.)

It’s less ego-bruising to fence naïve faith in platonic axioms as an ostensive explanation for the bad performance of models than to admit faulty reasoning. Because of such psychological mechanisms, entire theories take the flak that actually belong to influential individuals overlooking or misunderstanding technical (wonkish) details. Because of the da Vinci-Heinlein postulate (”you can never be too knowledgeable on any area that crosses your path”; I just made that up combining the life off the first and the quotations of the second), if you eat, you should understand the entire biochemistry of digestion to the point of understanding the technical journals. In real life there’s never enough time, but an economist of any persuasion or occupation has no business calling himself an economist or quant analyst if he doesn’t understand the content of the first four chapters of MWG — in fact, you should have known what MWG means before clicking the link.

4. Don’t trust any dynamic implications resulting from static models. Remember that, unless for very harsh conditions of error sphericity, the “basic” linear regression models are all about cross-sections; the available datasets are fitted a quasi-fictional data generating process that generates timeless, possible-worlds static scenarios. Using regressions to compare different points in time, even on stationarity assumptions (does anybody even bother to check second-order stationarity so error terms are in fact spherical?) is a huge leap of faith. (That’s why they made Kalman filters)

The major dudes — the ones influential in policy-making — are right now using linear back-of-the-envelope calculations based on estimated coefficients to figure out the size of trillion dollar-scale governmental programs in the largest economy in the world. That’s just screwed-up. Even the most basic back-of-the-envelope calculations should use some kind of small-dataset vector autorergressions (maybe positing some cointegration relations from their understanding of long-run macroeconomic theory) and impulse-response curves, and even that is just way too crude for policy formulation.  

The main issue with using regression-like models on observations taken at different points of time is that you’re implicitly assuming that coefficients are constant. All kinds of cobbled together solutions have been found for that, from ad-hocky instrumental variables to way-too-deep-in-the-underlying-theory dynamic-panel GMM estimators, but you have to pick your beliefs carefully and always do proper sensitivity analysis on them.

5. Development theory is whack — all kinds of development theory. Old-school dependency theory is predicated on long-term trends on relative prices who have deviated from their necessary (not predicted, necessary) course, which pretty much makes it safe to discard all remains of it. “Solow theory” (including all kinds of dynamic maximization problems) is based on representative agent models; refer to MWG chap.4  to see if that’s comfortable enough for you. (It’s not just about Gorman polar forms; these are just a minimal condition for a commodity demand function to even be conceived as reasonable.) What’s more, long-term growth has long been recognized to be an imperfect correlate of the development of free (”open” in the sense of Karl Popper) societies, and happiness measures like the Human Development Index are a farce. (Get a bag ready  to puke on and check the methodology out). 

Self-sustaining development is about people being able to reach their higher potential while not sacrificing their basic human inclinations: it’s about self-actualization and the desire to contribute to human progress. It has to do with education — maybe not in a sweep-out form, but at least in a form that picks up the best and the brightest with fair criteria. I think I would go on a limb and say that human-centered urbanism is about as important as education and should receive about the same amount of attention and cash. No amount of premature calculus lessons will cure our sick cities; it’s just something the market can’t take care of well (check out Christaller-Lösch theory). 

6. QQPlots are your friends. Cherish them. Enshrine them, even if you’re at ga-ga-copulas. Post them together with mean performance time-series, even if you have to explain qqplots to confused investors. Statistical moments are deceiving — even third and fourth momets. If your distribution is heading towards a bimodal form, you mght not be able to meet the Lyapunov or Lindeberg conditions.

7.  : If you are modelling a virtual trade on a synth market, remember that there are always three components in a trade: what person A gets, what person B gets and what price gets agreed upon.

Synth markets are pointless if there’s no such notional win-win scenario, even if day-to-day operations are in secondary markets that eschew the actual motivations. Have a model ready for and given the motivation arrows. Know that, notionally, people should have been dealing in such instruments in an organic market were not financial markets incomplete. If your derivatives aren’t the functional equivalent of a synthetic market, then you might as well be card-counting in a Vegas blackjack table.

8. Get a healthy adrenalin source. Finance is boring. Find an adrenalin-intensive sport: surf, downhill skateboarding, hang gliding, whatever. Attempting to find satisfaction where your responsibilities lie is futile and dangerous. If I had a dime for every time I shot myself in the foot trying to shoehorn intensive programming into an otherwise simple job, er, I’d have change for pop-corn, I guess.

9. Dress down in marginal increments. Only you are able to know what’s expected in your workplace, but what are the odds that someone will mind you’re wearing sneakers instead of stiff uncomfortable shoes? Try an extra epsilon now and then.

10. Cut the jargon. It only makes death by semantics more likely. Cf. Alone’s blog for how psychiatry has been in a deadlock for thirty years because of their fancy ontologies that began as simplified explanations for the general public and ended up in the decision trees of practicing clinicians.

11. Realize that things never “are”, they’re just in the process of becoming. The deductive-nomological approach got physicists in the ridiculous position of spending millions trying to photograph a particle they don’t actually know to exist — just  because it would make their model finally fit. Quant finance is where economics breaks out of the deductive-nomological gridlock, and the rest of the profession should follow its example. Don’t listen to anyone arguing from first principles — whether savings precedes investment, whether supply precedes demand, yadda yadda yadda. Focus on well-formed problems — like the soap bubble whose shape comes from minimizing surface tension. The goldilocks point of scientific methodologies probably lies somewhere between lagrangian and hamiltonian mechanics: somewhere between shedding all the metaphysics about first principles and trying to shoehorn all science into one model. General theories never are. 

12. Because things never “are”, advice never applies to you.  All of these points are just things to watch out when modelling your second moments. Your expected returns will always have to be modelled by your gracious self; no matter how much technology we throw at it, each and everyone of us is in constant flux and so are the markets and societies we face. But trust me on the Lyapunov condition: it’s as basic as 2+2=5, and if you don’t get the basics right, you’ll always be solving the wrong problem.

Posted in Economics, Finance, Statistics | No Comments »

(Rant alert: no useful applicable knowledge was generated in the making of this posting. I try to keep this blog constructive, but everybody’s losing their minds and it scares me. Some wide-eyed idealism can be found at the end.)

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A recurring theme of this blog has been the role of derivative contracts as the actualizing event of a synthetic market, ex ante virtual. (Add delandian differential-ontology-as-modal semantic and there’s an entire philosophy PhD thesis to be written about that proposition.) But derivatives are not a necessary condition for synthetic markets to come into existence. An acquaintance who lives in Bolivia has reminded me today of the anticrectic contract — something that had been burning me in the back of my mind as “irrational” (NB: I’m a recovering neowalrasian from the Arrow-Hahn-Debreu strain). 

Anticresis appears to exist as a housing contract form (HCF, just for this post) in spanish and portuguese-speaking countries, but only in Bolivia it has really flourished. This is not only interesting because the housing finance problem is the political and cultural background that begat the specific kind of synth market that everyone else is talking about — CDSes and the resulting synth CDOs — but because anticresis is a “folk”-originated HCF solution that falls no short of Wall Street concoctions in sophistication and creativity, fully functional and widely available even in the absence of a “proper” pricing model in terms of quant finance.

The general concept  is a bit strange outside bolivian culture, lying somewhere between a supply-side mortgage and a lump-sum fixed-period rental. Though the legal definition is a little wider, what usually happens is as follows: a homeowner “gives his house in anticresis” (dar en anticretico) in exchange of a lump-sum loan; the lender/tenant can move until the contract expires, time upon which the loan must be repaid, adding the (implicit) interest derived from the time value of money and subtracting the (implicit) rental costs of the house that generated the contract. Renewals often agreed upon, generally extending the terms of the loan rather than involving the full principal.

Now, this is a real and frequent  folk arrangement that exists as an option to ownership and rental as HCF options; the terms of the loan and the options for renewal are usually negotiated between individuals, with no secondary market other than direct subletting. Nevertheless, the “fundamental price” of such contracts is implicitly linked with the primary mortgage market and the secondary rental market for equivalent housing. This relationship is convoluted enough to warrant a quick examination of the underlying microeconomics in a fully deterministic context. I’ll try to hint at the stochastics and how it relates to current quant technology as knowledge and imagination permits, given that I’ve just started to think about it today, but remember that this folk-synth market is fully functional out there — so deterministic equilibrium can either be interpreted in terms of an Arrow-Debreu-type timepath X commodity space or a Herbert Simon-type “satisficing” solution based on heuristics.

 The issuer of an anticretic contract wants to get immediate money and have it generate some surplus in time for anticresis expiration time, while the demand side is buying housing services and loaning money at the same time (in marxian terms, one could say that through anticresis real estate becomes a proxy of the means of production twice — one by being an input to labour, and one by being wealth turned into productive capital –, which would mean folk bolivian capitalists have really come up with interesting leverage here). Given some kind of wicksellian/pigovian sense of a “money market”, the arrows in the expression below can be exchanged for equality operators. In practice, people enter anticresis precisely because they have different degrees of exposition to the costs and risks of each side.

The middle term (the agreed-upon terms of the contract) relates to the project financing of the issuer (left term) and the habitation services flow enjoyed by the buyer (right term). In reality, what is usually agreed upon are the lump-sums and to be transferred at entering and completing the contract, but these are more interestingly expressed, with no change in semantics, as and . I haven’t got much concrete information on current values, but I’m thinking and could easily be larger than 1 in a credit crunch.

On the project finance side, there’s an arbitrary cashflow, time-discounted by , expressed in terms of the return rate for a given fraction of the original lump-sum invested at a given period. On the habitation side, the tenant is promised the time value-discounted lump , and enjoys the habitation services equivalent to what could be obtained by means of rental of equivalent housing. Rent — unless regulated — is usually a idiosyncratic fraction of the house value; it can fluctuate as real estate is seen as a more or less attractive form of investment — generally, in countries that haven’t experienced a housing bubble (that is, most of the non-US world), an increase in risk aversion will lead to an increase in demand for housing as an investment and a decrease in the demand for house rentals — thus a decrease in the rental coefficients Of course, anticresis as a third option perturbs this reasoning; on one hand, living in anticresis is a form of habitation without ownership (competing with rentals), and therefore an increase in risk aversion would decrease its demand; on the other hand, anticresis competes with homeowning as a fixed return investment, and  is similarly devalued when risk aversion goes down. 

So how is  the market for anticrectic housing a synth market? It’s a swap between project finance risk and rental-value risk. 

By choosing anticresis as a HCF, the tenant/loaner gets to know beforehand his housing costs for the entire contract period and longs a fixed return investment, at the opportunity cost of the cashflow that could be generated by a project equivalent to the one undertaken by the homeowner/issuer. In a small enough economy, if you choose not to loan you can undertake the project yourself while preventing the other party from doing it; in a large enough economy, there is enough financing and enough market for both the would-be issuer having alternative sources of financing and the would-be tenant-now-entrepreneur having almost all the return he’d have if the would-be issuer hadn’t entered the market. Conversely, the homeowner gets a loan at a fixed rate approximately equivalent to  at the opportunity cost of renting out a house that could have higher-than-planned rental coefficients over the contract time, plus the cost of living in an equivalent house (maybe mortgages, maybe rent, maybe there’s even people doing anticresis arbitrage) — both of which probably have significant positive correlation — it’s all cost of living — possibly making the habitation risk quadratic.

So where are the stochastic knobs here?

• -> On one hand, there’s the entire issue of project finance; we’ve just modified the WACC equation. 
• -> On the other hand, there is the problem of housing costs — real-estate fluctuations, how they relate to rental value (the volatility of the rental coefficient ) and to alternative habitation costs.  Part of these can be smashed in an institutional setting where mass packaging is done in a fashion similar to the one that was done with mortgages in the american CDO scene. 
• -> On the third hand, anticresis is like a reverse mortgage in that the tenant/loaner gets to keep the house until the debt is settled, in a default/bankruptcy scenario. 
• -> On the fourth hand, it’s important to consider of how the uncertain project returns are affected if anticresis is never agreed upon, the would-be issuer goes on to pursue alternative sources of financing and the would-be tenant enters the market — that is, the unilateral risk borne by the would-be issuer if the contract is never signed.  (Market concentration analysis is the beginning of an answer, but QWERTY competition can never be assumed away  in the contemporary economy.) In contrast to the housing costs side, this is seldom an issue in the “folk” scale anticresis is currently done — more often than not, tenants entering anticresis are not interested in entrepreneurialism, they’re just buying habitation — but could escalate in a scenario of mass packaging. Then again, such general equilibrium factors are seldom, if ever, considered in quant finance.

How do we make this mathematically tractable? Well, it may depend on whether someone (wink, wink) meets my reserve price for consulting rates (wink, wink), but a simple CAPM-level approach can just put variances on everything,  simplify the time-discount calculations by using the cheapest fixed return instrument available, pray hard on the project return rate/real-estate value covariance and estimate the three equations that result from the “chemical” formula above. At Black-Scholes-like complexity, fix the time-discount rate, add brownian motion to house prices and do something random-y with project returns, preferably grounded on something about the real economy. If going ga-ga-copulas, ditch the gaussian assumption and just use general elliptic copulas (there’s little to gain from gaussian assumptions at this level, and the entire project return/real-estate value covariance problem is tractable as radial asymmetry); just don’t forget the default scenario. 

The good/bad news here is that unlike most synth scenarios, this is being done on an everyday basis, and hopefully Schiller-corrected time-series could be constructed so there’s something to test and/or the models against, but unfortunately, all the experience in this market comes from Bolivia, where the chaotic macro shocks might smash whatever sample variance one had after  taking care of the “equivalent housing” thing. Anyway, this can be done on a folk scale by virtue of the same kind of processes that make local microbanking such a big deal in developing countries: local trust and local knowledge.

Now here’s the kicker: if one can price this stuff semi-credibly, one can synthetize it trivially, and the remaining problem is in streamlining and structuring for the secondary markets. Now, anticresis is effectively a way of  injecting liquidity in the secondary housing markets in a ZIRP environment, by virtue of making homes a source of project finance — and moreover, by leveraging homeowner (or debt package market) knowledge on project viability, injecting liquidity into projects with economic potentials without the incentive distortion (you know, the WACCy stuff?) that comes with the systemic wash-out that the Fed can’t do anyway without stretching the dollar way too far.

Hello, mike test, is anyone out there listening? Housing market relief?

Oh, finishing note: the Royal Spanish Academy dictionary equates anticresis to anti-cresis, and the best etymological hint I could find  about cresis is a footnote on Kittel, Friedlich and Bromiley’s “Theological Dictionary of the New Testament”, which translates it to “loan” on an interestingly general context: God is the source of all power, and we men “have it only [..] as a loan”. I’m not a religious person (I’m borderline anti-religious, really), but aren’t credit default swaps a “loan” on the “power” that everyone ultimately has to default and run away to Barbados?

Posted in Finance | 1 Comment »

I’ve written before about how derivatives are actually about creating synthetic markets where no organic counterpart had arisen.  Some people have  bemoaned that modern derivative operations  are too complex for non-insiders to understand, and moreover that their pricing is too dependent on theoretical models that mix unrealistic assumptions with heavyset abstract mathematics (which is rather untrue — it’s just a little measure theory and sprinkled arbitrary jargon). Others have noted that traders, quants and accountants need to be better supervised, and competition between capital managers is extremely weak and therefore not able to induce best practices in controlling for potential principal-agent problems. Apparently, there’s something very new and quite toxic about these advanced practices in contemporary finance 

But consider short-selling — something that’s been around for many centuries. Going short  (borrowing the tradable asset and selling it immediately, hoping to later repurchase it at lower prices) and long (buying the paper) by the same quantity of equities, you’re basically exposing yourself to the market for a null gain. On the other hand, going slightly more heavily on your long/short position is a softer bet on a price increase/decrease than being naked long/short. You could just buy (or short) the “correct” quantity of equities to begin with, but a mixed long/short strategy allows for fine-tuning over time — for instance, going increasingly long if you start to feel that a price fall is starting to revert but won’t all-out give up your initial strategy.

Call and put options are just a step ahead from that. Like short-selling, vanilla options have a maturity date on which the actual trade has to happen. Unlike short-selling, there are markets for options at different prices, and if the Black-Scholes model is to be believed, different deltas depending on how close to the current price (how “in the money”) — calls have positive deltas and puts have negative deltas. Like with a long/soft strategy, beliefs about the price at maturity date can be used to calibrate the implicit delta in the  combined instrument. Options can go further , though, allowing traders to place bets on price intervals (because of the different strike prices), on volatility and on short-term price dynamics (calibrating on the implicit time decay factor). It’s quite possible, for example, to synthesize an instrument that is entirely independent of current price — that is, a combined instrument with zero delta — trading on the volatility while controlling for charm and color.

We’re just talking technology here — 1970s navigation techniques updating on those used in the 1620s when short-selling began.  Then what is so different about derivatives (like synth CDOs or CDSes) on debt packages? There are three  complementary  answers here, one from stochastic processes, one from economic incentives and one from market structure.

Vanilla options (calls and puts) are priced around the present-date uncertainty about the underlying asset’s price at the maturity date. Most commonly brownian motion is used — which just means that daily variations  are normally distributed and independent from those of the previous day. (Quants love to borrow jargon). Similarly, debt packages are as valuable as the debts themselves are good and carries their uncertainty. A simple way to value a debt portfolio is to assume defaults happen following a Poisson process — no two debts are defaulted “at the same time” and there’s no time-dependence on the probability of a default event (this breaks down if there are underlying reasons for defaulting that collectively affect a portion of the debtors). It’s a fragile little model that can  work well in times of peace, and I’ve been investigating its properties via large simulations in “discrete but granular” time for a slightly different context — an extended Cobb-Douglas production function and machine breakdowns instrad of credit defaults. (I’m exploring project finance beyond  naïve DCF but short of going copula for some very awkward regulatory breakage in the brazilian electric utility sector.)

Poisson defaults are an assumption that’s somewhat strong, but the characteristics that make it a good model — stationary independence on the increments and  allowance for left-jumps — are found in Lévy processes (PDF link), a general class that includes Poisson increments (as well as brownian motion) and  has a single representation as a characteristic function provided you use the measure space corresponding to your statistical model — and moreover, one that satisfies the condition for elliptical copulas, in which quantcore geeks have found their new religion, casting aside brownian motion for the ordinary folk to trade with. Think of copulas as a very general way to specify different types of “correlation” metrics (the linear correlation not being one; copulas are invariant under monotonic transformations); with gaussian copulas (a member of the elliptical family) much of the cross-dependence structure of the multivariate normal structure is kept while allowing for a much wider range of marginals, including all Lévy processes. The problem, of course, is that elliptical copulas are restricted to radial asymmetry, which means there cannot be a stronger dependence between big losses than between big gains — which while controlling for excess kurtosis, is still subject to some modified form of Taleb’s critique. 

So this is the wonky part of how complexity escalates from a vanilla call to a debt package instrument. The economic part of the question arises from the core problem of getting a loan: information asymmetry. The “capital market” (a little wicksellian imagination here with me, please) is “incomplete”, meaning you can’t always get what you want at any price, because the borrowers always know a little more about their own ability to honor their debt than the loaners, no matter how honest everyone is.  It’s the market for lemons problem; in practice, banks don’t price down riskier loans after a threshold, and what we end up having in the loan market is the soviet solution: quotas. Debt repackaging allows for the risk to be transferred by those who believe they have better knowledge (in the very same sense political futures are bought and sold)  than that expressed in the sell price. This is the reason why debt repackaging increases overall liquidity in the system: it makes it possible to issue loans that until recently were too risky to be priced. 

Then there’s the market structure issue. This is where things became rotten, and probably where the government regulation everyone wants to see should happen — bring in transparency and “meta-Basel” requirements  to satisfy the populace’s bloodthirst and let the differential equations do the job they’ve been doing since there’s been compound interest in the world.

This is where my head starts to hurt, and one of the reasons I’m not actually able to reverse engineer gaussian copulas to look at the implicit parameters and assumptions in the current information about collateralized debt instruments. (The main reason being, of course, that there’s no fun in dealing with such real-world contractual complications and I already have to do all kinds of scrubbing in my day job — in short, I’m too lazy). You see, investment banks group these collateralized debts into “tranches”, which have both different risk characteristics — probably not leveraging the gaussian copula technology — and different contractual structures (who gets paid first in case of institutional meltdown? I’m confused about this.)

This is something that just drives me mad: all this technology of generalized correlation metrics and the bankers can’t stop lettersouping everything. (Have you ever seen the Moody’s  sovereign debt rating manual?). My first thought is always that this kind of preference for fuzzy set because of some misplaced mathematical anxiety, but apparently there are numerous accounting benefits to such muddying up of the waters. Thus synth CDOs can be useful accounting tricks, specially with inter-tranche arbitrage, and cash-backed CDOs are really not that clear anyway.

Summarizing, the quant geeks have actually come up with a major improvement on the rather unrealistic brownian motion story behind the Black-Scholes-like models, but the technology has outpaced not only the government regulatory knowlege but even the corporate governance of investment banks and other institutions. This is really a scenario out of Accelerando!, and if it wasn’t for a sudden credit crunch (most likely sparked by the hangover after the great liquidity party) the synth CDOs would soon be pedestrian instruments to be held as a minority share of variable-return investment in a retirement fund. For one, copulas aren’t being used to their full potential.

And while everyone’s looking at sudden “book losses” from panicked mark-to-market results and a cascade of downgrades on synth instruments, I think that ultimately the failure to prevent the excessive expansion of liquidity after decoupling the dollar from hard metal has led to structural problems in the real economy. I predict  two or three years of creative destruction that will leave us with a better, saner global economy — with yet more synth package instruments to patch the information asymmetry  problems of the purely organic credit market. And I’m not above calling radical naysayers neo-luddites.

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For most of the history of applied mathematics, automatic computing machines weren’t widely available, let alone general-purpose computing machines. Large-scale engineering was done with logarithm tables and slide rules. Convergent series play a huge role in such a scenario, specially if they collapse fast. For example, the Taylor polynomials about a number converge to the value of a smooth function, and approximations like  allowed table-makers to list the values of a number of useful functions. (The sine function, if your trigonometrics is gone, is the “height” associated with an angle in the euclidian plane. It’s also one of two component pieces of a general model for periodic signals, but let’s not get carried away here.)

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An interesting convergent series is that of the binomial probability distribution. Everyone has heard the coin-toss story. If there’s a probability p on heads, the probability of getting four heads and then three tails is exactly is , and so is the probability of getting two heads, then two tails, then two more heads and one more tail. How many ways one can get four heads and three tails, regardless of order? Well, throwing back to your high school combinatorics, this is  and the probability of any particular “k heads out of 7 coin tosses” is . Now, the part with the exponents can be obtained pretty easily by the experienced user of logarithm tables and slide rules, but the combinatorial part needs time-expensive methods like factorials or Pascal triangles. What’s the probability of getting 61 heads out of 100 “fair coin” tosses? As you increase the size of the stakes, a familiar bell-shaped curve comes up.

The bell curve had already been showing up in empirical observations for quite a while — and remember this is from a time where the laws of nature were thought to be, well, laws — and finding out that it was the limit to which the binomial distribution — the result of this very basic probabilistic event, do or not do — threw people in a loop. Further results were found about what seemed to be the very equation of God — not only it is the limit of coin-tossing processes as the number went to infinity, it is the limit of the mean of the weight of randomly-sized objects taken out of an infinite pool, as long as the probability distribution on each object is the same. And, well, if you get n to be large enough, it’s kinda not a big deal if you don’t really know the distribution of each random variable. Or is it?

Well, if you know a random process to be binomial, get n to be very large but have a very small p, you might find the convergence to the normal distribution is much, much slower. And if p is not kept constant, you might be heading onto a Poisson distribution, and never get to be even close to a normal distribution at all. And hey, we were talking binomial here. What about stuff where probability distribution you don’t really know to be identical or very similar? That’s all terra incognita. But it seems that the normal distribution is just so common we can look at zero-centered values (just subtract your mean) and know pretty much everything is between -3? and +3?.  Anything outside that strip looks unbelievably strange.

Consider this. Since on the seventh day God hath spoken , we have gotten used to consider ? the measure of “actual value spread” or dispersion par excellence. Actually one of the reasons that made the normal distribution omnipresent is that it’s nicely ammenable to affine transforms — sum/subtract/multiply/divide a randomly distributed value by a constant number and you still have a randomly distributed value with the mean and ? trivially transformed. But there is a number of other aspects of dispersion that are usually handwaved away. Kurtosis, for one — Nassim Taleb has built an entire career out of explaining kurtosis to an unsuspecting, gaussian-brainwashed public. Then there’s entropy, a fun measure by many reasons. But ? is not entirely useless: given any distribution — and this time we really mean “any” — we can know that a value will exceed ?+k? in at most 1/k² of the times. Scarily enough, this means that the probability of exceeding ?+3? (around 0.13% if you assume a normal distribution) can reach as much as 11%. You do not want to assume normality on your financial losses. You do not want to assume normality on your financial losses.

Actually, we’ve known that the returns on the stock market exhibit considerable excess kurtosis since someone has started keeping tabs on the statistical structure of stock market returns. This one tidbit is mentioned in pretty much every “real” (maybe not in “… with Microsoft Excel” business-school fare) introductory probability textbook along with the fun fact that the Lorentz distribution, which usually arises in physics with forced resonance (think guitar distortion), has no well-defined ? or ? at all.

So, erm, what’s with all the bell curves being thrown around? For one, many statistical techniques that used to depend on normality assumptions no longer require such gross approximations — the example hitting closer to home here is that much of econometrics can be restated in terms of the generalized method of the moments. On the other hand, there has been a relatively rich stream of “post-modern” central limit theorems that extend the Word and the Glory to “mixing” (which is this very complicated concept that amounts to ’so far apart that they’re almost independent’) processes, some types of trigonometric series martingales and most recently (all of this in 2007 and 2008) convex bodies, gaussian polytopes, orthogonal matrices. So the bass keeps on running running running running.

But let us not be misled by the volume of real mathematics around convergence to normality: quantitative finance has by large “just assumed” gaussian processes, sometimes renamed for convenience — “brownian motion” — or worked up into interesting contraptions. The gaussian normal distribution looms over statistics like a proverbial 800-pound gorilla that was fed over the centuries by the need to have good analytic approximations at hand. There’s no accounting for taste, but we’ve been using replication methods like the bootstrap and the jackknife in limited context with great success for many decades now, and maybe we should stop feeding the monster. Go read on the method of simulated moments (pdf) and double-check whether you really need those ex-ante assumptions on distributions. (Or alternatively grow a pair and go bayesian.) With some careful, unrestrained by tradition, modeling, Nate Small has done a trick and a half on us by eschewing analytic derivations and using full-on the amazing computing capability we all now have at our fingertips. Have you seen a logarithm table recently?

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Complex derivatives have been denounced as “financial weapons of mass destruction”, but they are in fact the opposite: by virtue of theoretical modeling, virtual markets are priced into existence. Now that markets both ‘organic’ and ’synthetic’ are melting, some have chosen to blame the ‘quants’, modelers who came up with the virtual prices on which derivative markets for illiquid assets are predicated.

It can be truthfully contended that contemporary quantitative stochastic finance (QSF) has oversold its functional grounding on measure theory as the mathematical basis for a better, synthetic capitalism where every aspect of risk — from gammas to volgas — can be actualized as concrete value flows. What Wilmott and others overlook, however, is the counterpart of QSF-derived prices for synthetic markets: market-process derived prices on organic markets. 

Are gaussian copula models for collateralized debt products that much less efficient than the massively parallel, error-prone neural network that would power the individual investor buying directly into the organic mortgage market — as some societies have seen in the late 1800s? As with nukes, technology is just that: technology, not weapon. Furthermore, nuclear technology is a cat that’s either irreversibly out of the bag or goldilocked into nonexistence, while the destructive potential in QSF technology comes from trust, and thus can be effectively calibrated into utopia.

Stochastic calculus is probably oversold. The gist of Moore’s Law — that we all have supercomputers at our fingertips, right now — allows for massive simulation techniques on much more realistic stochastic discounted cash flow models allowing for beta-delta pseudo-hyperbolic intertemporal preferences and many kinds of business-model specificities (random asset replacement times instead of depreciation models, for one) to be simulated hundreds of thousands of time until convergence results are achieved.

In fact, when one looks at what else could be done, it becomes clear where QSF went wrong: it allowed itself to be parameterized by organic pricing, and thus became infected by the error-prone redundant neural networks that power organic markets. The massively simulated cash flow (MSCF) approach just shines a light on  business-model knobs, taking a decisively more technocratic turn at valuing cash flows than QSF approaches that allowed its knobs to be marked to market. 

I’ve been thinking obsessively about how MSCF methods can be brought into the table — specially when it comes to replacing the stone-age method of scenarioized internal rates of return, but also as a replacement for the more fragile parts of QSF like copula CDO models. I haven’t seen yet how MSCF valuation methods can outperform QSF as a technique for modeling synthetic markets into existence, though, and I really would prefer a future where trading on the volgas is possible rather than the technocratic flavor of business-model knobs.

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