Bitcoins Explained_ Fungibility, Double Coincidence of Wants
The Problem With Commodities · 2007-03-27 · The Problem With Commodities Nassim Nicholas Taleb....
Transcript of The Problem With Commodities · 2007-03-27 · The Problem With Commodities Nassim Nicholas Taleb....
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The Problem With Commodities
Nassim Nicholas Taleb
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Commodities
• Jumps
• Fat tails ->Wild randomness
• Fungibility
• Non-dynamically hedgeable
• Modern finance DOES not work -yet wetrade them
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Dealing with “Infinite” Variance
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Volatility Shmolatility
“The term “price volatility” is used to describeprice fluctuations of a commodity. Volatility ismeasured by the day-to-day percentagedifference in the price of the commodity.”–United States Department of Energy
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Verbal Definition
What is it that people call “volatility”?
Traders with 20 years experience!
Verbal description deliver MAD not at allstandard deviation
The Economist, WSJ, NYT
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Not too natural
It is a matter of norms
Ignoring the mean; the di erence between
and
can be monstrous with “fat tails”
2
1
1( )
n
i
i
Xn =
1
1n
i
i
Xn =
1/
1p
p
xn
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Concentration
Now take an extreme case of concentration, avector extremely large, containing, say, amillion observations, all 0 but for one:X={0,0,....,0,106}
MAD around 1, its STD ~ 1000 times MAD
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No such thing as natural variance
An option delivers aconditional meanaverage deviation
MAD is delivered whenthe option is ATM(assume 0 interestrates)
ATMStraddle =
(K So
K) (S) dS
+ (S KK
) (S) dS
= E[Abs( S)]
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Even to replicate
One cannot replicate “variance” withoptions. Products like “variance swaps”have an unfeasible replicating portfolioTypical static replication:
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Ideal Static Replication
Nonfeasible parts:
Lower tails. We need a small number of infinitelyhigh strikes
Upper tails. We need an infinite number of lowstrikes
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Piecewise linearity
Using an option as abuilding block maynot be possible
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Dynamic Hedging
Dynamic hedging requires finite variancebut it is not necessary (Derman & Taleb2005). Static hedging arguments canproduce option prices (Bachelier-styleequation).
Moral: we do not need sigma for options…we only need “volatility”
Next… the possible families of models
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The Supreme Law of Mediocristan
• Imagine 1000 personsstanding on a stadium.Think that the heaviestconceivable man youknow is among them. Howmuch of the total does herepresent?
• .8%?
• The supreme law ofMediocristan: as thenumber ofobservationsbecomes very large,not a single elementcan be consequentialto the total
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The Supreme Law of Leptocristan
• Imagine the samestadium, the same people.Include the wealthiestperson you know in there–say Bill Gates. How muchdoes he weigh comparedto the total?
• 99.999%?
• Economic variables do notwork like weight, height,calories consumed, etc
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Just a Few Days
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Forget Histograms
• Graphs: daily returns of theSP500 in the real worldcompared to Gaussianmodels
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Pareto
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Mandelbrot
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A Scalable
Exceedant Freq Number of Observations
• 1,000,000 4,000,000
• 2,000,000 1,000,000
• 4,000,000 250,000
• 8,000,000 62,500
• 16,000,000 15,625
• 32,000,000 3,906
• 1,000,000,000 4
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The exponent problem
P(> x) = K xPower law
Alpha <2 “infinite” variance
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More realistic
• 1,000,000 5,600,000
• 2,000,000 2,274,307
• 4,000,000 923,656
• 8,000,000 375,121
• 16,000,000 152,346
• 32,000,000 61,872
• 1,000,000,000 705
• 2,000,000,000 286
• 100,000,000,000 2
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Ito’s Lemma with a FiniteVariance --but Scalable
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N=5000, Gaussian
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N=5000, Scalable (Cubic Alpha)
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Levi-Stable Infinite Variance
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The Whole Idea of Central Limit
The theorem that, under aggregation, thesum of random variables with finitevariance converges to a Gaussian
Levi-Stability
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The Problem of Risk-NeutralPricing
Option theory is based not on actuarialexpectation but on risk-neutral pricing
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The consequences
Convincing work by Gabaix GopilkrishnanPlerou & Stanley (latest, 2005)
Some disagreements (minor): at the mostalpha =3 (“cubic)
Is it much better?
Consider the following: infinite kurtosis
Exploding higher moments
E ects on mean variance
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The Essence of Portfolio Theory
If you
know
“E”and “V”
etc…
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Two basins
limx >
F(x)N
F(N x)
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Odds of Exceeding with aGaussian
• 3 sigmas: 1 in 740 times
• 4 sigmas :1 in 32,000 times
• 5 sigmas :1 in 3,500,000 times
• 6 sigmas :1 in 1,000,000,000 times
• 10 sigmas :1 in 130,000,000,000,000,000,000,000 times
• 20 sigmas: 1 in 3600000000000000000000000000
• 0000000000000000000000000000000000000000000000000000000000000 times
Notice the acceleration
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“What Should We Do?”