I S IT SO B AD THAT WE C ANNOT R ECOGNIZE B LACK S WANS ? Fuad Aleskerov and Lyudmila Egorova National Research University Higher School of Economics Zurich, 2011

I NTRODUCTION A Black Swan: is an outlier, and nothing in the past can convincingly point to its possibility; carries an extreme impact; has retrospective (though not prospective) predictability. Nassim Nicolas Taleb «The Black Swan. The Impact of The Highly Improbable» 2

I NTRODUCTION «The central idea of this book concerns our blindness with respect to randomness, particularly the large deviations: Why do we, scientists or nonscientists, hotshots or regular Joes, tend to see the pennies instead of the dollars? Why do we keep focusing on the minutiae, not the possible significant large events, in spite of the obvious evidence of their huge influence?» Nassim Nicolas Taleb «The Black Swan. The Impact of The Highly Improbable» 3

P ROBLEM We will model economic fluctuations representing them as a flows of events of two types: Q -event reflects the normal mode of an economy; R -event is responsible for a crisis. The number of events in each time interval has a Poisson distribution with constant intensity. is the intensity of the flow of regular events Q. is the intensity of the flow of crisis events R. >> holds (that is, Q -type events are far more frequent than the R -type events). 4

P ROBLEM We will model economic fluctuations representing them as a flows of events of two types: Q -event reflects the normal mode of an economy; R -event is responsible for a crisis. The number of events in each time interval has a Poisson distribution with constant intensity. is the intensity of the flow of regular events Q. is the intensity of the flow of crisis events R. >> holds (that is, Q -type events are far more frequent than the R -type events). 5

P ROBLEM 6

X Q R The problem of correct identification (recognition) Unknown State of nature 7

P ROBLEM X Q R Q Q R R Player´s perceived identification of the state of nature 8

P ROBLEM X Q R Q Q R R a Payoff of correct identification of regular event 9

P ROBLEM X Q R Q Q R R -b Incorrect identification of regular event 10

P ROBLEM X Q R Q R Q R d>>b -d c c>>a 11

P ROBLEM right wrong 12

P ROBLEM How large will be the sum of payoffs received up to time t? 13

S OLUTION Random value Z=X of the total sum of the received payoffs during the time t is a compound Poisson type variable. We give the expression for the expectation of a random variable payoff: E(Z) = [λ(p 1 a – q 1 b) + μ(p 2 c – q 2 d)]t 14

A PPLICATION TO REAL DATA We consider a stock exchange and events Q and R which describe a business as usual and a crisis, respectively. The unknown event X can be interpreted as a signal received, e.g. by an economic analyst or by a broker, about the changes of the economy that helps him to decide whether the economy is in a normal mode or in a crisis. 15

P ARAMETERS FOR S&P

P ARAMETERS FOR S&P

P ARAMETERS FOR S&P

P ARAMETERS FOR S&P 500 Estimates for indices with the threshold 6% Index λμ a, %-b, %c, %-d, % S&P ,6-0,62,8-2,9 Dow Jones24640,6-0,61,9-2,4 CAC ,8-0,83,0-2,5 DAX239110,8-0,92,1-2,5 Nikkei ,8-0,92,6-3,2 Hang Seng24190,9-0,92,6-3,0 19

P ARAMETERS FOR S&P 500 In fact, it is enough to identify regular Q -events in almost half of the cases to ensure a positive outcome of the game ( q ). E(Z) 0 Probability of incorrect identification of crisis R-event Probability of incorrect recognition of regular Q-event 20

P ROBLEM 21 k

M ODEL WITH STIMULATION if 22

M ODEL WITH STIMULATION 23

M ODEL WITH LEARNING if 24

C ONCLUSION We showed in a very simple model that with a small reward for the correct (with probability slightly higher than ½) identification of the routine events (and if crisis events are identified with very low probability) the average player's gain will be positive. In other words, players do not need to play more sophisticated games, trying to identify crises events in advance. 25

T HANK YOU FOR YOUR ATTENTION ! 26