Anti

19 Pages·2014·0.6 MB·English

Mathematical Definition, Mapping, and Detection of (Anti)Fragility
Nassim Nicholas Taleb and Raphael Douady †
⇤
November 2013
Forthcoming,QuantitativeFinance
Weprovideamathematicaldefinitionoffragilityandantifragilityasnegativeorpositivesensitivitytoasemi-measureof
dispersionandvolatility(avariantofnegativeorpositive"vega")andexaminethelinktononlineareffects. Weintegrate
modelerror(andbiases)intothefragileorantifragilecontext. Unlikerisk,whichislinkedtopsychologicalnotionssuch
as subjective preferences (hence cannot apply to a coffee cup) we offer a measure that is universal and concerns any
object that has a probability distribution (whether such distribution is known or, critically, unknown). We propose a
detectionoffragility,robustness,andantifragilityusingasingle"fast-and-frugal",model-free,probabilityfreeheuristic
thatalsopicksupexposuretomodelerror. Theheuristiclendsitselftoimmediateimplementation,anduncovershidden
risksrelatedtocompanysize,forecastingproblems,andbanktailexposures(itexplainstheforecastingbiases). While
simpletoimplement,itimprovesonstresstestingandbypassesthecommonflawsinValue-at-Risk.
1 Introduction
Prob Density
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The notions of fragility and antifragility were introduced in Taleb (2012). In short, fragility is related to how a
systemsuffersfromthevariabilityofitsenvironmentbeyondacertainpresetthreshold(whenthresholdisK,itiscalled
K-fragility),whileantifragilityreferstowhenitbenefitsfromthisvariability—inasimilarwayto“vega”ofanoptionor
anonlinearpayoff,thatis,itssensitivitytovolatilityorsomesimilarmeasureofscaleofadistribution.
Simply,acoffeecuponatablesuffersmorefromlargedeviationsthanfromthecumulativeeffectofsomeshocks—
conditional on being unbroken, it has to suffer more from “tail” events than regular ones around the center of the
distribution,the“atthemoney”category. Thisisthecaseofelementsofnaturethathavesurvived: conditionalonbeing
in existence, then the class of events around the mean should matter considerably less than tail events, particularly
whentheprobabilitiesdeclinefasterthantheinverseoftheharm,whichisthecaseofallusedmonomodalprobability
distributions. Further,whathasexposuretotaileventssuffersfromuncertainty;typically,whensystems–abuilding,a
bridge,anuclearplant,anairplane,orabankbalancesheet–aremaderobusttoacertainlevelofvariabilityandstress
FormerTrader,flaneur,andNYU-Poly
⇤
†Centred’ÉconomiedelaSorbonne
1
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TALEBandDOUADY MathematicalMappingof(Anti)Fragility
butmayfailorcollapseifthislevelisexceeded, thentheyareparticularlyfragiletouncertaintyaboutthedistribution
ofthestressor,hencetomodelerror,asthisuncertaintyincreasestheprobabilityofdippingbelowtherobustnesslevel,
bringing a higher probability of collapse. In the opposite case, the natural selection of an evolutionary process is
particularlyantifragile,indeed,amorevolatileenvironmentincreasesthesurvivalrateofrobustspeciesandeliminates
thosewhosesuperiorityoverotherspeciesishighlydependentonenvironmentalparameters.
Figure 1 show the “tail vega” sensitivity of an object calculated discretely at two different lower absolute mean
deviations. Weuseforthepurposeoffragilityandantifragility,inplaceofmeasuresinL2 suchasstandarddeviations,
whichrestrictthechoiceofprobabilitydistributions,thebroadermeasureofabsolutedeviation,cutintotwoparts: lower
anduppersemi-deviationabovethedistributioncenter⌦.
Thisarticleaimsatprovidingapropermathematicaldefinitionoffragility,robustness,andantifragilityandexamining
howtheseapplytodifferentcaseswherethisnotionisapplicable.
Intrinsic and Inherited Fragility: Our definition of fragility is two-fold. First, of concern is the intrinsic fragility,
theshapeoftheprobabilitydistributionofavariableanditssensitivitytos-,aparametercontrollingtheleftsideofits
owndistribution. Butwedonotoftendirectlyobservethestatisticaldistributionofobjects,and,ifwedid,itwouldbe
difficulttomeasuretheirtail-vegasensitivity. Nordoweneedtospecifysuchdistribution: wecangaugetheresponse
ofagivenobjecttothevolatilityofanexternalstressorthataffectsit. Forinstance,anoptionisusuallyanalyzedwith
respecttothescaleofthedistributionofthe“underlying”security,notitsown;thefragilityofacoffeecupisdetermined
as a response to a given source of randomness or stress; that of a house with respect of, among other sources, the
distribution of earthquakes. This fragility coming from the effect of the underlying is called inherited fragility. The
transfer function, which we present next, allows us to assess the effect, increase or decrease in fragility, coming from
changesintheunderlyingsourceofstress.
Transfer Function: A nonlinear exposure to a certain source of randomness maps into tail-vega sensitivity (hence
fragility). Weprovethat
InheritedFragility Concavityinexposureontheleftsideofthedistribution
andbuildH,atransferfunctiongivin,ganexactmappingoftailvegasensitivitytothesecondderivativeofafunction.
Thetransferfunctionwillallowustoprobepartsofthedistributionandgenerateafragility-detectionheuristiccovering
bothphysicalfragilityandmodelerror.
1.1 Fragility As Separate Risk From Psychological Preferences
Avoidance of the Psychological: We start from the definition of fragility as tail vega sensitivity, and end up with
nonlinearityasanecessaryattributeofthesourceofsuchfragilityintheinheritedcase—acauseofthediseaserather
than the disease itself. However, there is a long literature by economists and decision scientists embedding risk into
psychologicalpreferences—historically,riskhasbeendescribedasderivedfromriskaversionasaresultofthestructureof
choicesunderuncertaintywithaconcavityofthemuddledconceptof“utility”ofpayoff,seePratt(1964),Arrow(1965),
Rothchild and Stiglitz(1970,1971). But this “utility” business never led anywhere except the circularity, expressed by
MachinaandRothschild(2008),“riskiswhatrisk-avertershate.” Indeedlimitingrisktoaversiontoconcavityofchoices
isaquiteunhappyresult—theutilitycurvecannotbepossiblymonotoneconcave,butrather,likeeverythinginnature
necessarilyboundedonbothsides,theleftandtheright,convex-concaveand,asKahnemanandTversky(1979)have
debunked,bothpathdependentandmixedinitsnonlinearity.
Beyond Jensen’s Inequality: Furthermore, the economics and decision-theory literature reposes on the effect of
Jensen’s inequality, an analysis which requires monotone convex or concave transformations —in fact limited to the
expectationoperator. Theworldisunfortunatelymorecomplicatedinitsnonlinearities. Thankstothetransferfunction,
whichfocusesonthetails,wecanaccommodatesituationswherethesourceisnotmerelyconvex,butconvex-concave
and any other form of mixed nonlinearities common in exposures, which includes nonlinear dose-response in biology.
For instance, the application of the transfer function to the Kahneman-Tversky value function, convex in the negative
domain and concave in the positive one, shows that its decreases fragility in the left tail (hence more robustness) and
reduces the effect of the right tail as well (also more robustness), which allows to assert that we are psychologically
“morerobust”tochangesinwealththanimpliedfromthedistributionofsuchwealth, whichhappenstobeextremely
fat-tailed.
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TALEBandDOUADY MathematicalMappingof(Anti)Fragility
Accordingly,ourapproachreliesonnonlinearityofexposureasdetectionofthevega-sensitivity,notasadefinition
of fragility. And nonlinearity in a source of stress is necessarily associated with fragility. Clearly, a coffee cup, a house
orabridgedon’thavepsychologicalpreferences, subjectiveutility, etc. Yettheyareconcaveintheirreactiontoharm:
simply,takingzasastressleveland⇧(z)theharmfunction,itsufficestoseethat,withn>1,
⇧(nz)<n⇧(z)forall0<nz<Z⇤
where Z isthelevel(notnecessarilyspecified)atwhichtheitemisbroken. Suchinequalityleadsto⇧(z)havinga
⇤
negativesecondderivativeattheinitialvaluez.
So if a coffee cup is less harmed by n times a stressor of intensity Z than once a stressor of nZ, then harm (as a
negativefunction)needstobeconcavetostressorsuptothepointofbreaking;suchstrictureisimposedbythestructure
ofsurvivalprobabilitiesandthedistributionofharmfulevents,andhasnothingtodowithsubjectiveutilityorsomeother
figments. Justaswithalargestonehurtingmorethantheequivalentweightinpebbles, if, forahuman, jumpingone
millimetercausedanexactlinearfractionofthedamageof,say,jumpingtothegroundfromthirtyfeet,thentheperson
wouldbealreadydeadfromcumulativeharm. Actuallyasimplecomputationshowsthathewouldhaveexpiredwithin
hoursfromtouchingobjectsorpacinginhislivingroom,giventhemultitudeofsuchstressorsandtheirtotaleffect. The
fragilitythatcomesfromlinearityisimmediatelyvisible,soweruleitoutbecausetheobjectwouldbealreadybroken
andthepersonalreadydead. Therelativefrequencyofordinaryeventscomparedtoextremeeventsisthedeterminant.
Inthefinancialmarkets,thereareatleasttenthousandtimesmoreeventsof0.1%deviationsthaneventsof10%. There
arecloseto8,000micro-earthquakesdailyonplanetearth,thatis,thosebelow2ontheRichterscale—about3million
ayear. Thesearetotallyharmless,and,with3millionperyear,youwouldneedthemtobeso. Butshocksofintensity
6 and higher on the scale make the newspapers. Accordingly, we are necessarily immune to the cumulative effect of
small deviations, or shocks of very small magnitude, which implies that these affect us disproportionally less (that is,
nonlinearlyless)thanlargerones.
Modelerrorisnotnecessarilymeanpreserving. s-,thelowerabsolutesemi-deviationdoesnotjustexpresschanges
inoveralldispersioninthedistribution,suchasforinstancethe“scaling”case,butalsochangesinthemean,i.e. when
theuppersemi-deviationfrom⌦toinfinityisinvariant,orevendeclineinacompensatorymannertomaketheoverall
meanabsolutedeviationunchanged. Thiswouldbethecasewhenweshiftthedistributioninsteadofrescalingit. Thus
thesamevega-sensitivitycanalsoexpresssensitivitytoastressor(doseincrease)inmedicineorotherfieldsinitseffect
oneithertail. Thuss (l)willallowustoexpressthesensitivitytothe“disordercluster”(Taleb,2012): i)uncertainty,ii)
�
variability,iii)imperfect,incompleteknowledge,iv)chance,v)chaos,vi)volatility,vii)disorder,viii)entropy,ix)time,
x)theunknown,xi)randomness,xii)turmoil,xiii)stressor,xiv)error,xv)dispersionofoutcomes.
DETECTION HEURISTIC
Finally,thankstothetransferfunction,thispaperproposesariskheuristicthat"works"indetectingfragilityevenif
weusethewrongmodel/pricingmethod/probabilitydistribution. Themainideaisthatawrongrulerwillnotmeasure
theheightof achild;butitcancertainlytellusif heisgrowing. Sincerisksinthetailsmaptononlinearities(concavity
of exposure), second order effects reveal fragility, particularly in the tails where they map to large tail exposures, as
revealedthroughperturbationanalysis. Moregenerallyeverynonlinearfunctionwillproducesomekindofpositiveor
negativeexposurestovolatilityforsomepartsofthedistribution.
1.2 Fragility and Model Error
Aswesawthisdefinitionoffragilityextendstomodelerror,assomemodelsproducenegativesensitivitytouncertainty,
in addition to effects and biases under variability. So, beyond physical fragility, the same approach measures model
fragility, based on the difference between a point estimate and stochastic value (i.e., full distribution). Increasing the
variability(say,variance)oftheestimatedvalue(butnotthemean),mayleadtoone-sidedeffectonthemodel—justas
anincreaseofvolatilitycausesporcelaincupstobreak. Hencesensitivitytothevolatilityofsuchvalue,the“vega”ofthe
modelwithrespecttosuchvalueisnodifferentfromthevegaofotherpayoffs. Forinstance, themisuseofthin-tailed
distributions (say Gaussian) appears immediately through perturbation of the standard deviation, no longer used as
pointestimate,butasadistributionwithitsownvariance. Forinstance,itcanbeshownhowfat-tailed(e.g. power-law
tailed) probability distributions can be expressed by simple nested perturbation and mixing of Gaussian ones. Such a
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TALEBandDOUADY MathematicalMappingof(Anti)Fragility
Figure1: Disproportionateeffectoftaileventsonnonlinearexposures,illustratingthenecessarycharacterofthenonlinearityoftheharm
functionandshowinghowwecanextrapolateoutsidethemodeltoprobeunseenfragility.
representationpinpointsthefragilityofawrongprobabilitymodelanditsconsequencesintermsofunderestimationof
risks,stresstestsandsimilarmatters.
1.3 Antifragility
It is not quite the mirror image of fragility, as it implies positive vega above some threshold in the positive tail of the
distributionandabsenceoffragilityinthelefttail,whichleadstoadistributionthatisskewedright.
FragilityandTransferTheorems
Table1introducestheExhaustiveTaxonomyofallPossiblePayoffsy=f(x)
ThecentralTable,Table1introducestheexhaustivemapofpossibleoutcomes,with4mutuallyexclusivecategories
of payoffs. Our steps in the rest of the paper are as follows: a. We provide a mathematical definition of fragility,
robustnessandantifragility. b. Wepresenttheproblemofmeasuringtailrisksandshowthepresenceofseverebiases
attending the estimation of small probability and its nonlinearity (convexity) to parametric (and other) perturbations.
c. Weexpresstheconceptofmodelfragilityintermsoflefttailexposure,andshowcorrespondencetotheconcavityof
thepayofffromarandomvariable. d. Finally,wepresentoursimpleheuristictodetectthepossibilityofbothfragility
andmodelerroracrossabroadrangeofprobabilisticestimations.
Conceptually,fragilityresidesinthefactthatasmall–oratleastreasonable–uncertaintyonthemacro-parameter
ofadistributionmayhavedramaticconsequencesontheresultofagivenstresstest,oronsomemeasurethatdepends
onthelefttailofthedistribution, suchasanout-of-the-moneyoption. Thishypersensitivityofwhatweliketocallan
“outofthemoneyputprice”tothemacro-parameter, whichissomemeasureofthevolatilityofthedistributionofthe
underlyingsourceofrandomness.
Formally,fragilityisdefinedasthesensitivityoftheleft-tailshortfall(non-conditionedbyprobability)belowacertain
thresholdKtotheoverallleftsemi-deviationofthedistribution.
Examples
i- Aporcelaincoffeecupsubjectedto randomdailystressorsfromuse.
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TALEBandDOUADY MathematicalMappingof(Anti)Fragility
Table1: PayoffsandMixedNonlinearities
Type Condition Left Tail Right Tail Nonlinear Derivatives Effect of fa-
(Loss (Gain Do- Payoff Func- Equivalent tailedness of
Domain) main) tion y = f(x) (Taleb,1997) f(x) compared
"derivative" toprimitivex.
where x is a
random vari-
able
Type Fragile Fat (reg- Fat Mixed concave Long up-vega, More fragility
1 (type1) ular or left,convexright shortdown-vega if absorbing
absorbing (fence) barrier, neutral
barrier) otherwise
Type Fragile Thin Thin Concave Shortvega Morefragility
2 (type2)
Type Robust Thin Thin Mixed convex Short up - vega, Noeffect
3 left, concave longdown-vega
right (digital,
sigmoid)
Type Antifragile Thin Fat Convex Longvega More an-
4 (thicker tifragility
thanleft)
ii- Taildistributioninthefunctionofthearrivaltimeofanaircraft.
iii- Hidden risks of famine to a population subjected to monoculture —or, more generally, fragilizing errors in the
applicationofRicardo’scomparativeadvantagewithouttakingintoaccountsecondordereffects.
iv- Hiddentailexposurestobudgetdeficits’nonlinearitiestounemployment.
v- Hiddentailexposurefromdependenceonasourceofenergy,etc. (“squeezabilityargument”).
1.4 Tail Vega Sensitivity
Weconstructameasureof“vega”inthetailsofthedistributionthatdependsonthevariationsof s,thesemi-deviation
belowacertainlevelW,choseninthe L1norminordertoensureitsexistenceunder“fattailed”distributionswithfinite
firstsemi-moment. InfactswouldexistasameasureeveninthecaseofinfinitemomentstotherightsideofW.
LetX bearandomvariable,thedistributionofwhichisoneamongaone-parameterfamilyofpdf f�,� I R. We
considerafixedreferencevalue⌦and,fromthisreference,theleft-semi-absolutedeviation: 2 ⇢
⌦
s�(�)= (⌦ x)f�(x)dx
�
Z
�1
Weassumethat� s–(�)iscontinuous,strictlyincreasingandspansthewholerangeR =[0, + ),sothatwemay
+
use the left-semi-a!bsolute deviation s– as a parameter by considering the inverse function �(s) :1R I, defined by
+
s�(�(s))=sfors R+. !
Thiscondition2isforinstancesatisfiedif,foranygivenx<⌦,theprobabilityisacontinuousandincreasingfunction
of�. Indeed,denoting
x
F�(x)=Pf�(X <x)= f�(t)dt,
Z
�1
anintegrationbypartyields:
⌦
s�(�)= F�(x)dx
Z
�1
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TALEBandDOUADY MathematicalMappingof(Anti)Fragility
Thisisthecasewhen�isascalingparameter,i.e.,X ⌦+ �(X1 ⌦)indeedonehasinthiscase
⇠ �
x ⌦
F�(x)=F1 ⌦+ �� ,
Å ã
@@F��(x)= ⌦��2x f�(x)ands�(�)=�s�(1).
It is also the case when � is a shifting parameter, i.e. X X0 � , indeed, in this case F�(x) = F0(x +�) and
@@s��(x)=F�(⌦). ⇠ �
ForK <⌦ands R+,let:
2
K
⇠(K,s�)= (⌦ x)f�(s )(x)dx
� �
Z
�1
Inparticular,⇠(⌦,s–)=s–. Weassume,inafirststep,thatthefunction⇠(K,s–)isdifferentiableon( ,⌦] R .
+
TheK-left-tail-vegasensitivityofX atstresslevelK <⌦anddeviationlevels�>0forthepdf f� is: �1 ⇥
V(X,f�,K,s�)= @@s⇠(K,s�)= ⌦ (⌦ x)@@f��)dx dds�� �1 (1)
� ÇZ � åÅ ã
�1
Astheinmanypracticalinstanceswherethresholdeffectsareinvolved,itmayoccurthat⇠doesnotdependsmoothly
ons–. Wethereforealsodefineafinitedifferenceversionofthevega-sensitivityasfollows:
1
V(X,f�,K,s�)= 2�s ⇠(K,s�+�s) ⇠(K,s� �s)
� �
� �
K
= (⌦ x)f�(s�+�s)(x)� f�(s���s)(x)dx
� 2�s
Z
�1
Henceomittingtheinput�simplicitlyassumesthat�s 0.
Notethat⇠(K,s�)=�E(X|X <K)Pf�(X <K). Itcanbe!decomposedintotwoparts:
⇠ K,s�(�) =(⌦ K)F�(K)+P�(K)
�
� � K
P�(K)= (K x)f�(x)dx
�
Z
�1
Wherethefirstpart(⌦ K)F�(K)isproportionaltotheprobabilityofthevariablebeingbelowthestresslevelKandthe
secondpartP�(K)isthe�expectationoftheamountbywhichX isbelowK(counting0whenitisnot). Makingaparallel
withfinancialoptions,whiles–(�)isa“putat-the-money”,⇠(K,s–)isthesumofaputstruckatK andadigitalputalso
struckatK withamount⌦–K;itcanequivalentlybeseenasaputstruckat⌦withadown-and-inEuropeanbarrierat
K.
Letting�=�(s–)andintegratingbypartyields
K ⌦
⇠ K,s�(�) =(⌦ K)F�(K)+ F�(x)dx = F�K(x)dx (2)
�
Z Z
� � �1 �1
Where F�K(x)=F�(min(x,K))=min(F�(x),F�(K)),sothat
@⇠ ⌦ @@F��K(x)dx
V(X,f�,K,s�)= @s(K,s�)= R�⌦1 @@F��(x)dx (3)
�1
Forfinitedifferences R
1 ⌦
V(X,f�,K,s�,�s)= 2�s �F�K,�s(x)dx
Z
�1
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TALEBandDOUADY MathematicalMappingof(Anti)Fragility
Where�+ and� aresuchthats(�+)=s +�s,s(� )=s �sand�FK (x)=FK (x) FK (x).
s �s s� � �s� �� �,�s �s+ � �s�
2 Mathematical Expression of Fragility
In essence, fragility is the sensitivity of a given risk measure to an error in the estimation of the (possibly one-sided)
deviationparameterofadistribution, especiallyduetothefactthattheriskmeasureinvolvespartsofthedistribution
–tails–thatareawayfromtheportionusedforestimation. Theriskmeasurethenassumescertainextrapolationrules
that have first order consequences. These consequences are even more amplified when the risk measure applies to a
variablethatisderivedfromthatusedforestimation,whentherelationbetweenthetwovariablesisstronglynonlinear,
asisoftenthecase.
2.1 Definition of Fragility: The Intrinsic Case
Thelocalfragilityofarandomvariable X dependingonparameter�,atstresslevelKandsemi-deviationlevels–(�)with
�
pdf f� isitsK-left-tailedsemi-vegasensitivityV(X,f�,K,s�).
The finite-difference fragility of X� at stress level K and semi-deviation level s�(�) �s with pdf f� is its K-left-tailed
finite-differencesemi-vegasensitivityV(X,f�,K,s�,�s). ±
Inthisdefinition,thefragilityreliesintheunsaidassumptionsmadewhenextrapolatingthedistributionofX from
�
areas used to estimate the semi-absolute deviation s–(�), around ⌦, to areas around K on which the risk measure ⇠
depends.
2.2 Definition of Fragility: The Inherited Case
Next we consider the particular case where a random variable Y = '(X) depends on another source of risk X, itself
subject to a parameter �. Let us keep the above notations for X, while we denote by g� the pdf of Y,⌦Y = '(⌦) and
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TALEBandDOUADY MathematicalMappingof(Anti)Fragility
u (�)theleft-semi-deviationofY.Givena“strike”level
�
L='(K),letusdefine,asinthecaseofX :
K
⇣ L,u�(�) = (⌦Y y)g�(y)dy
�
Z
The inherited fragility of Y with respect to X�at stress�level�L1= '(K) and left-semi-deviation level s–(�) of X is the partial
derivative:
@⇣ K @g ds 1
VX Y,g�,L,s�(�) = @s L,u�(�) = (⌦Y Y) @��(y)dy d�� � (4)
ÇZ � åÅ ã
� � � � �1
NotethatthestresslevelandthepdfaredefinedforthevariableY,buttheparameterwhichisusedfordifferentiation
is the left-semi-absolute deviation of X, s–(�). Indeed, in this process, one first measures the distribution of X and its
left-semi-absolutedeviation,thenthefunction' isapplied,usingsomemathematicalmodelofY withrespecttoX and
theriskmeasure⇣isestimated. Ifanerrorismadewhenmeasurings–(�),itsimpactontheriskmeasureofYisamplified
bytheratiogivenbythe“inheritedfragility”.
Onceagain,onemayusefinitedifferencesanddefinethefinite-differenceinheritedfragilityofY withrespecttoX,
byreplacing,intheaboveequation,differentiationbyfinitedifferencesbetweenvalues�+ and�–,wheres–(�+)=s–+
�sands–(�–)=s– –�s.
3 Effect of Nonlinearity on Intrinsic Fragility
Let us study the case of a random variable Y = '(X); the pdf g� of which also depends on parameter �, related to
a variable X by the nonlinear function '. We are now interested in comparing their intrinsic fragilities. We shall say,
for instance, that Y is more fragile at the stress level L and left-semi-deviation level u (�) than the random variable
�
X, at stress level K and left-semi-deviation level s�(�) if the L-left-tailed semi-vega sensitivity of Y� is higher than the
K-left-tailedsemi-vegasensitivityofX :
�
V(Y,g�,L,µ�)>V(X,f�,K,s�)
Onemayusefinitedifferencestocomparethefragilityoftworandomvariables:V(Y,g�,L,�µ)>V(X,f�,K,�s). In
thiscase,finitevariationsmustbecomparableinsize,namely�u/u– =�s/s–.
Let us assume, to start, that ' is differentiable, strictly increasing and scaled so that ⌦Y = '(⌦) = ⌦. We also
assumethat,foranygivenx<⌦, @@F��(x)>0.
Inthiscase,asobservedabove,� s–(�)isalsoincreasing.
LetusdenoteGy(y)=Pg�(Y < y).!Wehave:
G�(�(x))=Pg�(Y <�(y))=Pf�(X <x)=F�(x).
Hence,if⇣(L,u–)denotestheequivalentof⇠(K,s–)withvariable(Y,g�)insteadof(X,f�),wehave:
⌦ d�
⇣ L,u�(�) = F�K(x)dx(x)dx
Z
� � �1
Because' isincreasingandmin('(x),'(K))='(min(x,K)).Inparticular
⌦ d�
µ�(�)=⇣ ⌦,µ�(�) = F�K(x)dx (x)dx
Z
� � �1
TheL-left-tail-vegasensitivityofY istherefore:
⌦ @@F��K(x)dd�x(x)dx
V Y,g�,L,u�(�) = R�⌦1 @@F��(x)dd�x(x)dx
� � �1
Forfinitevariations: R
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TALEBandDOUADY MathematicalMappingof(Anti)Fragility
1 ⌦ d�
V(Y,g�,L,u�(�),�u)= 2�u �F�K,�u(x)dx(x)dx
Z
�1
Where�+ and� aresuchthatu(�+ )=u +�u,u(�+ )=u �uand FK (x)=FK (x) FK (x).
Next,Theu�orem1�up�roveshowaconcauv�etran�sformation'u�(x)of�a�randomvar�i,a�bulexpro�d+uuces�frag��uility.
THEOREM 1(FRAGILITY TRANSFER THEOREM)
Let, with the above notations, ' : R R be a twice differentiable function such that '(⌦) = ⌦ and for any x < ⌦,
dd'x(x)>0. TherandomvariableY ='(!X)ismorefragileatlevelL='(K)andpdf glambda thanXatlevelKandpdf
f�if,andonlyif,onehas:
⌦ d2'
HK(x) (x)dx <0
� dx2
Z
�1
Where
@PK @PK @P @P
HK(x)= � (x) � (⌦) �(x) �(⌦)
� @� @� @� @�
�
� �
andwhere
x
P�(x)= F�(t)dt
Z
�1
isthepriceofthe“putoption”onX�with“strike”xand
x
PK(x)= FK(t)dt
� �
Z
�1
isthatofa“putoption”with“strike”xand“Europeandown-and-inbarrier”atK.
H can be seen as a transfer function, expressed as the difference between two ratios. For a given level x of the
random variable on the left hand side of ⌦, the second one is the ratio of the vega of a put struck at x normalized by
thatofaput“atthemoney”(i.e. struckat⌦),whilethefirstoneisthesameratio,butwhereputsstruckatxand⌦are
“Europeandown-and-inoptions”withtriggeringbarrieratthelevelK.
Proof
LetIX� = ⌦ @@F��(x)dx,IXK� = ⌦ @@F��K(x)dx,andIY� = ⌦ @@F��(x)dd'x(x)dx. OnehasOnehasV(X,f�,K,s�(�))=
IXK� IX�andVR(Y�1,g�,L,u�(�))=IYL�RI�Y1�hence: R�1
¿ ¿ IL IK IK IL I
V(Y,g�,L,u�(�))�V(X,f�,K,s�(�))= IYY�� � IXX�� = IXY�� ÇIXYK�� � IXY��å (5)
Therefore,becausethefourintegralsarepositive,Therefore,becausethefourintegralsarepositive,
V(Y,g�,L,u�(�)) V(X,f�,K,s�(�))
�
IL IK I I .
Y� X�� Y� X�
Ontheotherhand,wehave IX� = @@P��(⌦)IXK� = @@P��K(⌦)¿and �
⌦ @F d' @P d' ⌦ @P d2'
I = �(x) (x)dx = �(⌦) (⌦) �(x) (x)dx (6)
Y� @� dx @� dx � @� dx2
Z Z
�1 �1
⌦ @FK d' @PK d' ⌦ @PK d2'
IL = � (x) (x)dx = � (⌦) (⌦) � (x) (x)dx (7)
Y� @� dx @� dx � @� dx2
Z Z
�1 �1
c 2013N.N.TalebandR.Douady 9 November2013
�
TALEBandDOUADY MathematicalMappingof(Anti)Fragility
Anelementarycalculationyields:
IYL� IY� = @P�K(⌦) �1 ⌦ @P�K(x)d2'dx+ @P�(⌦) �1 ⌦ @P�(x)d2'dx = ⌦ HK(x)d2'dx. (8)
IXK� � IX� �✓ @� ◆ Z @� dx2 ✓ @� ◆ Z @� dx2 �Z � dx2
�1 �1 �1
LetusnowexaminethepropertiesofthefunctionH�K(x). For x K,wehave @@P��K(x)= @@P��(x)>0(thepositivityis
aconsequenceofthatof @@F��),thereforeH�K(x)hasthesamesignas
@P @PK
�(⌦) � (⌦).
@� @�
�
Asthisisastrictinequality,itextendstoanintervalontherighthandsideofK,say(˘ ,K]withK <K <.Butonthe
otherhand: 1
@P @PK ⌦ @F @F
�(⌦) � (⌦)= �(x)dx (⌦ K) �(K)
@� @� @� @�
� ZK � �
ForKnegativeenough, @@F��(K)issmallerthanitsaveragevalueovertheinterval[K,⌦],hence
@P @PK
�(⌦) � (⌦)>0.
@� @�
�
Wehaveproventhefollowingtheorem.
THEOREM 2(FRAGILITY EXACERBATION THEOREM)
Withtheabovenotations,thereexistsathreshold⇥�<⌦suchthat,if K ⇥� then H�K(x)>0forx (– ,�]with
K<lambda<⌦. Asaconsequence,ifthechangeofvariable'isconcave_on ( ,�]andlinearon2[�1,⌦],thenY
ismorefragileatL='(K)thanXatK. �1
Onecanprovethat,foramonomodaldistribution,⇥�<�<⌦(seediscussionbelow),sowhateverthestresslevel
K belowthethreshold⇥�,itsufficesthatthechangeofvariable' beconcaveontheinterval(– ,⇥�]andlinearon
[⇥�,⌦]forY tobecomemorefragileatLthanX atK.Inpractice,aslongasthechangeofvaria1bleisconcavearound
thestresslevelK andhaslimitedconvexity/concavityawayfromK,thefragilityofY isgreaterthanthatofX.
Figure 3 shows the shape of HK(x) in the case of a Gaussian distribution where � is a simple scaling parameter
�
(� is the standard deviation �) and ⌦ = 0. We represented K = –2� while in this Gaussian case, ⇥� = –1.585�.
DISCUSSION
Monomodalcase
Wesaythatthefamilyofdistributions(f�)isleft-monomodalifthereexistsK�<⌦suchthat @@f�� æ0on(– ,�]and
@@f�� ∂0on[µ�,⌦]. Inthiscase @@P�� isaconvexfunctiononthelefthalf-line(– ,µ�],thenconcaveaftert1heinflexion
pointµ�. ForK µ�,thefunction @@P��K coincideswith @@P�� on(– ,K],thenis1alinearextension,followingthetangent
to the graph of @@P�� in K (see graph below). The value of @@P��K(⌦1) corresponds to the intersection point of this tangent
@P
withtheverticalaxis. ItincreaseswithK,from0whenK – toavalueabove @��(⌦)whenK =µ�. Thethreshold
@PK ! 1@P @P @P
⇥�correspondstotheuniquevalueofK suchthat @��(⌦)= @��(⌦). WhenK <⇥�thenG�(x)= @��(x) @��(⌦)and
G�K(x)= @@P��K(x) @@P��K(⌦) are functions such that G�(⌦)= G�K(⌦)=1 and which are proportional for x ¿K, the latter

.
c 2013N.N.TalebandR.Douady 10 November2013
�