rd 1 embrechts

Up the Devil’s Staircase, Down the Fi i l AbFinancial Abyss

Paul EmbrechtsPaul EmbrechtsDepartment of MathematicsDepartment of Mathematics

Director of RiskLab, ETH Zurich Senior SFI Chair

www math ethz ch/~embrechtswww.math.ethz.ch/ embrechts

“Ceterum censeo Carthaginem esse delendam” *esse delendam

(N.H. Bingham, 20/5/2011)

M. Porcius Cato – Cato The ElderRome, 234 - 149 B.C.

* “Furthermore, I think that Carthage must be destroyed!"

The title in pictorial formThe title in pictorial form

(M.C. Escher’s Stairs)

The Devil’s staircase: in Mathematics

A monotonic increasing "staircase" for which the simplest rational numbers have the largest steps. The Devil's staircase continuously maps the interval [0,1] onto [0,1] , but is constant almost everywhere (i.e., 

except on a Cantor set). 

The Devil’s staircase: on Wall StreetAbacus 2700‐AC1

ABS Corr. Trade(1) Fractional Reserve Banking

(2) BSM papers in 1973 creating «The Market»

Cash CDO Synthetic CDO

CDO**2/3

(3)Hybryd CDO

(2) BSM papers in 1973, creating «The Market»through dynamic replication and impliedvolatility 

Futures

Options (Credit) DerivativesABS/CDS

(2)

Stocks

Bonds Mortgages

Futures

Money

Commodities(3) 2006 h i f f h l

FRB(1)

Goods(3) «2006 saw the creation of one of the most unnatural

and destructive financial  products that the world hasever seen, the synthetic CDO, it turned the keg of dyna‐mite into the financial equivalent of a nuclear bomb »mite into the financial equivalent of a nuclear bomb. »(«All the devils are there», B. McLean & J. Nocera,Portfolio/Penguin, New York, 2010,  Chapter 17) 

For a musical re‐creation of Wall Street’s folly, you may want to listen toyou may want to listen toGyörgy Ligeti, Étude No. 13,

L’ li d di bl«L’escalier du diable»(The Devil's Staircase) 

A sobering thoughtA sobering thought

• D.J. Hand: «On Wall Street, gain‐maximisation often leads to constrained optimisation problems moving ever closer to (*) the ethical boundary»

• An example: John Paulson’s shorting of AAA‐An example: John Paulson s shorting of AAAtranches of a specially constructed synthetic CDO, (1) «It didn’t pass Bear Stearn’s ethic standards»(1) «It didn t pass Bear Stearn s ethic standards», (2) Goldman‐Sachs allowed JP to rent (15 Mio $ fee) the Abacus brand name and go ahead with the dealthe Abacus brand name and go ahead with the deal through GS’s employee Fabrice Tourre.   

(*) beyond? 

Leading to some of Wall Street’s mostsickening acts on so called «Frankenstein derivatives»sickening acts on so‐called «Frankenstein‐derivatives»

• “Well, what if we created a ‘thing,’ which has no purpose, which is absolutely conceptual p p y pand highly theoretical and which nobody knows how to price?’” (FT 29/1/07)knows how to price?  (FT, 29/1/07)

• “I’ve managed to sell a few Abacus bonds to h h hwidows and orphans that I ran into at the 

airport, apparently these Belgians adore synthetic ABS CDO2” (FT, 13/6/07)

The «Fabulous Fab»The «Fabulous Fab»

b d h f h l’• Remember, we stand at the «top» of the Devil’s Staircase, early 2007:

• FT’s January 23, 2007, ominous e‐mail: «More and more leverage in the system The wholeand more leverage in the system. The whole building is about to collapse anytime now ... O l t ti l i th f b l F b[ iOnly potential survivor, the fabulous Fab[rice Tourre], standing in the middle of all these complex, highly leveraged, exotic trades he created without necessarily understanding all of y gthe implications of those monstruocities!!!» 

S th ti l l hiSome mathematical soul searching,from L.C.G. Rogers (Cambridge, UK),from L.C.G. Rogers (Cambridge, UK), 

SPA Plenary Lecture on «Mathematical Finance, the P&L» 

Osaka JapanOsaka, JapanSeptember 6, 2010

Begin of quote:

• On some fundamental results from mathematical finance (MF): «they take us somewhere we don’t want to go startingsomewhere we don t want to go, starting from a place we didn’t need to start»A littl f h t MF h d h b• «A little of what MF has done has been

VERY useful ... Most of MF has addressed completely irrelevant issues, and misapplied the energies and talents of amisapplied the energies and talents of a young generation in the process!»

On Mathematics, Finance and Banking,Rogers continued:

• In maths, a result is right iff it’s , gproved

• In math finance, a result is right iff it’s publishedit s published

• In banking, a result is right iff it’sIn banking, a result is right iff it s profitable

End of quote.

My own view, in line with statements made by Hans Föllmer and others 

But first 

Mathematics is of key importance forMathematics is of key importance for• understanding and clarifying models and pricesunderstanding and clarifying models and prices

used in finance, insurance and economics• making heuristic methods mathematically precise,making heuristic methods mathematically precise,

and asking for clear, unambiguous definitions! • highlighting model conditions and restrictions onhighlighting model conditions and restrictions on

applicability• working out numerous explicit examplesworking out numerous explicit examples • leading the way for stress testing and robustness

propertiesproperties• and it would be bad if the current crisis would

induce a shying away from mathematics!induce a shying away from mathematics!

I do hope that:I do hope that:

• Other academic fields like micro-, macro-economics, econometrics, banking, , , g,corporate finance, law, accounting ... do a similar soul searchingsimilar soul searching

• And then I have not yet started with the other actors: industry, politicians, regulators, investors, the press ... g pthis is really crucial as

f i i bl !• we are facing a very serious problem!

Interludium:

From trillions of $ to picoseconds:• 1 tri $ = 1 000 000 000 000 $

$ p$ 000 000 000 000 $

• World GDP = 58 tri $, US GDP = 14.5 tri $ (US deficit 1 35 tri $ debt 13 6 tri $)(US deficit = 1.35 tri $, debt = 13.6 tri $)

• Nominal amount CDS (6/10) = 30 tri $, but ( )62 tri $ at its 06-07 peak

• Nominal amount of OTC (6/10) = 583 tri $• Nominal amount of OTC (6/10) = 583 tri $• CDO volume 2006: 2.7 tri $• Moving from 1 trillion $ to 1 trillionth of a

second (a picosecond): high-frequencysecond (a picosecond): high-frequency trading co-location! Consequences?

An early warning of things to come?The Flash-Crash of May 6, 2010!

- 998.50/9.2%- 600 in 5’

Hence new Risk Management challenges!

From «Not so fast», Buttonwood,The Economist, August 6-12, p. 55

• NASDAC 1 trade/250 micro sec• HFT ~ 2/3-3/4 total Wall St volumeHFT 2/3 3/4 total Wall St volume• Smaller spreads, liquidity sales pitch• HFT and best capital allocation?• Average holding time of stocks ↓ 4 months• Average holding time of stocks ↓ 4 months• «Boundary play» (PE), «spoof orders»• Positive correlation (volume, volatility)• Potential dislocation between share prices• Potential dislocation between share prices

and economic fundamentals ...

On the role of Mathematics:

The two dimensions of (Q)RM((Quantitative) Risk Management)

• Dimension 1: ScopeMi th fi t di fl th li tMicro: the firm, trading floor, the clientMacro: the worldwide systemy

Di i 2 Ti• Dimension 2: TimeShort (HFT): << 1 year (or quarter)S o ( ) yea (o qua e )Medium (Solvency 2/Basel III): ~ 1 yearLong (Social/Life Insurance): >> 1 year

Two examples (micro short):Two examples (micro-short):

• Modelling of Extremes

• Beware of (micro ) Correlation• Beware of (micro-) Correlation

Modelling of Extremesg

RM too often frequency oriented ...

‐ every so often (rare event)

t i d 1 i t‐ return period, 1 in x‐year event

‐ Value‐at‐Risk (VaR)( )

... rather than more relevant severity orientation

h if‐ what if

‐ loss size given the occurence of a rare eventg

‐ Expected Shortfall E[X I X > VaR]

This is not just about theory but a RM attitude!

The Peaks Over Threshold (POT) Method( )

Crucial point!

99%‐quantile with 95% aCI (Profile Likelihood):99% quantile with 95% aCI (Profile Likelihood): 27.3 (23.3, 33.1)  

99% C diti l E E( X I X 27 3) ith 95% CI99% Conditional Excess: E( X I X >  27.3) with a 95% CI

58.2

99%‐quantile 99%‐conditional  excess 

27.3

Several extensions of 1 d EVT exist:Several extensions of 1‐d EVT exist:

i i• Non‐stationarity• Co‐variable modelling within POTg• Beware of discrete data, non‐standard theory• Multivariate extremes: definitions• Multivariate extremes: definitionsSeveral, question dependent approaches exist

• Dynamic, stochastic process models• Diagnostic and graphical toolsDiagnostic and graphical tools• Important: Communicating extreme events

f l !!!• Warning: often very slow convergence!!! 

Beware of (micro‐)Correlation

micro-

The normal distributionThe normal distribution

E t ttExtremes matter

Correlation matters

As a consequence:As a consequence:

Th i i ( d h d i ) f i• The pricing (and hedging) of super-senior AAA CDO tranches has substantial model uncertainty (= MU), “What are your marks?”

• Beware of correlation tradingg• Pricing of CDO**2/**3, synthetic CDOs,

besides being more than questionable frombesides being more than questionable from an economic point of view, is quantitatively near impossible ( MU)near impossible ( MU)

• Hence beware of warehousing such risks!• Similar examples with other products …

With respect to the crisis,in particular:

• Personalities: too many Gordon Geckos out there, at all levels

• The near total failure of risk management (of the type «get out of the way»)yp g y )

• The belief in the existence of a lender of last resort government «guarantees»resort, government «guarantees»

• Collateral calls/triggers ( embedded)N l l d f LTCM 1998• No lessons learned from LTCM 1998

• Near criminal accounting (e.g. REPO 105) and regulatory arbitrage (between BB and TB)

Some very basic RM rules:y• If you don’t understand it, don’t buy/sell it

Speak to “the guys in the boiler room”• Speak to “the guys in the boiler room”• Beware of “new” paradigms, like the New

Economy, the New Risk Management: “new” usually means that tried and trusted measures of the past are being ignored

• Always understand your gains and bewareAlways understand your gains and beware of volume (even/especially AAA)

• Concerning Basel II+ or III: do not try to• Concerning Basel II+ or III: do not try to reinvent the wheel, check countries and institutions that came through the crisisinstitutions that came through the crisis less harmed, understand why!!!

Thank you!y