b ehavioral m odel of s tock e xchange fuad aleskerov, and lyudmila egorova national research...
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BEHAVIORAL MODEL OF STOCK EXCHANGE
Fuad Aleskerov,and Lyudmila Egorova
National Research UniversityHigher School of Economics
University of Rome “Tor Vergata”November 29, 2011
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INTRODUCTION
«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
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PROBLEM
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).
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POISSON DISTRIBUTION
The Poisson distribution is a discrete probability distribution that expresses the probability of a number k of events occurring in a fixed period of time if these events occur with a known average rate λ and independently of the time since the last event.
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PROBLEM
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PROBLEM
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PROBLEM
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PROBLEM
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PROBLEM
X
Q
RThe problem of correct identification (recognition)
Unknown
State of nature
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PROBLEM
X
Q
R
Q
Q
R
R
Player´s perceived identification of the state of nature
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PROBLEM
X
Q
R
Q
Q
R
R
aPayoff of correct identification of regular event
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PROBLEM
X
Q
R
Q
Q
R
R -bIncorrect identification of regular event
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PROBLEM
X
Q
R
Q
R
Q
R
d>>b-d
cc>>a
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PROBLEM
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right
right
wrong
wrong
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PROBLEM
How large will be the sum of payoffs received up to time t?
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PROBLEM
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PROBLEM
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Q
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PROBLEM
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Q
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PROBLEM
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Q R
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SOLUTION
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:
𝐸 (𝑍 )=𝜆𝑡 ( (1−𝑞1 )𝑎−𝑞1𝑏)+𝜇𝑡 ( (1−𝑞2 )𝑐−𝑞2𝑑 )
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ANALYSIS OF THE SOLUTION
What conditions should the values of q1 and q2 satisfy for the expected value E(Z) to be nonnegative with all other parameters being fixed?
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ANALYSIS OF THE SOLUTION
What conditions should the values of q1 and q2 satisfy for the expected value E(Z) to be nonnegative with all other parameters being fixed?
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if
if
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ANALYSIS OF THE SOLUTION
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E(Z)0
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EXAMPLE
Let q2=1.
How often the player can fail to identify regular event to have still positive or at least zero average gain E(Z)?
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=>
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APPLICATION 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.
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PARAMETERS FOR S&P 500
02/0
8/19
99
22/1
1/19
99
13/0
3/20
00
03/0
7/20
00
23/1
0/20
00
12/0
2/20
01
04/0
6/20
01
24/0
9/20
01
14/0
1/20
02
06/0
5/20
02
26/0
8/20
02
16/1
2/20
02
07/0
4/20
03
28/0
7/20
03
17/1
1/20
03
08/0
3/20
04
28/0
6/20
04
18/1
0/20
04
07/0
2/20
05
30/0
5/20
05
19/0
9/20
05
09/0
1/20
06
01/0
5/20
06
21/0
8/20
06
11/1
2/20
06
02/0
4/20
07
23/0
7/20
07
12/1
1/20
07
03/0
3/20
08
23/0
6/20
08
13/1
0/20
08
02/0
2/20
09
25/0
5/20
09
14/0
9/20
090
200
400
600
800
1000
1200
1400
1600
1800
0
2
4
6
8
10
12
14
21.09.2001
23.07.2002
25.10.2002 21.01.2008
16.10.2008
09.03.2009
30.07.2009
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COMPARISONS
Estimates for indices with the threshold 6%
Index λ μ a, % -b, % c, % -d, %
S&P 500 246 4 0,6 -0,6 2,8 -2,9
Dow Jones 246 4 0,6 -0,6 1,9 -2,4
CAC 40 243 7 0,8 -0,8 3,0 -2,5
DAX 239 11 0,8 -0,9 2,1 -2,5
Nikkei 225 245 5 0,8 -0,9 2,6 -3,2
Hang Seng 241 9 0,9 -0,9 2,6 -3,0
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PARAMETERS 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
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PARAMETERS FOR S&P 500
Indeed, if we choose the horizon of 1 year and the error probability for events Q and R being q1=0.46, q2=1 (i.e. even when crises are not at all correctly identified), the expected gain is still positive (although almost zero).
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NEW MODELS
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k
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MODEL WITH STIMULATION
if
if
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MODEL WITH STIMULATION
32
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MODEL WITH LEARNING
if
if
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CALCULATIONS
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
Critical
0.464
0.477
0.462
0.466
0.461
0.461
0.464
0.497
0.462
0.473
0.461
0.461
0.464
0.522
0.462
0.485
0.461
0.462
0.464
0.554
0.462
0.504
0.461
0.465
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CALCULATIONS
35
Model with
training
Critical
0.490 0.477 0.470 0.466
0.496 0.481 0.472 0.467
0.500 0.483 0.473 0.468
0.523 0.497 0.482 0.473
0.526 0.499 0.483 0.473
0.529 0.502 0.485 0.472
0.540 0.509 0.490 0.478
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CONCLUSION
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.
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CONCLUSION
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.
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THANK YOU!