B EHAVIORAL M ODEL OF S TOCK E XCHANGE Fuad Aleskerov, and Lyudmila Egorova National Research University Higher School of Economics University of Rome Tor Vergata November 29, 2011

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» 2

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). 3

P OISSON DISTRIBUTION 4

P ROBLEM 5

6

7

8

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

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

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

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

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

P ROBLEM 14 right wrong

P ROBLEM How large will be the sum of payoffs received up to time t? 15

P ROBLEM 16

P ROBLEM 17 Q

P ROBLEM 18 Q

P ROBLEM 19 Q R

S OLUTION Random value Z 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:

A NALYSIS OF THE SOLUTION 21

A NALYSIS OF THE SOLUTION What conditions should the values of q 1 and q 2 satisfy for the expected value E ( Z ) to be nonnegative with all other parameters being fixed? 22 if

A NALYSIS OF THE SOLUTION 23 E(Z) 0

E XAMPLE Let q 2 =1. How often the player can fail to identify regular event to have still positive or at least zero average gain E ( Z )? 24 =>

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.

P ARAMETERS FOR S&P

C OMPARISONS 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 27

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

P ARAMETERS FOR S&P 500 Indeed, if we choose the horizon of 1 year and the error probability for events Q and R being q 1 = 0.46, q 2 =1 (i.e. even when crises are not at all correctly identified), the expected gain is still positive (although almost zero). 29

N EW MODELS 30 k

M ODEL WITH STIMULATION if

M ODEL WITH STIMULATION 32

M ODEL WITH LEARNING if

C ALCULATIONS 34 If error probabilities are equal to 0.46 and 1 respectively in the basic model, then the expectation of the gain is equal to E(Z)=0. Model with stimulation Model with training

C ALCULATIONS 35 Model with training

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.

C ONCLUSION We considered new models adding rewards for successful behavior as increases in gain and as increases in the probability of correct identification, which means that the player can learn from his past actions and accumulate experience. Both of these models allows the player to enlarge the total gain and to make more mistakes, because he/she can get more in the sequence of correctly identified events. 37

T HANK YOU !