BEHAVIORAL MODEL OF STOCK EXCHANGE

Fuad Aleskerov,and Lyudmila Egorova

National Research UniversityHigher School of Economics

University of Rome “Tor Vergata”November 29, 2011

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

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

5

PROBLEM

6

PROBLEM

7

PROBLEM

8

PROBLEM

X

Q

RThe problem of correct identification (recognition)

Unknown

State of nature

PROBLEM

X

Q

R

Q

Q

R

R

Player´s perceived identification of the state of nature

PROBLEM

X

Q

R

Q

Q

R

R

aPayoff of correct identification of regular event

PROBLEM

X

Q

R

Q

Q

R

R -bIncorrect identification of regular event

PROBLEM

X

Q

R

Q

R

Q

R

d>>b-d

cc>>a

PROBLEM

14

right

right

wrong

wrong

PROBLEM

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

15

PROBLEM

16

PROBLEM

17

Q

PROBLEM

18

Q

PROBLEM

19

Q R

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 )

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?

22

if

if

ANALYSIS OF THE SOLUTION

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E(Z)0

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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=>

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.

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

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

30

k

MODEL WITH STIMULATION

if

if

MODEL WITH STIMULATION

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MODEL WITH LEARNING

if

if

CALCULATIONS

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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

CALCULATIONS

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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

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.

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!