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What is determinism?
A system = its equations + its inital conditions
Time is only a cursor.
Time is not creative
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It is not possible to describe everything with determinism :
- Quantum physics
- Market quotes
Need of another kind of modelization :
Statistics
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Definition :
Statistic is a set of tools used for organization, analysis and interpretation of data
2 purposes :
- Description : describe characteristics of groups (= variables). This is descriptive
statistics and it is used to reveal the distribution of the data in each variable.
- Prediction : by generalizing observations on a part of a group. This is Inferential
statistics.
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Descriptive Statistics
Variables
A variable is a set of numbers linked to a characteristic of an object (person, thing,
etc.)
- Qualitative variable : classify data into categories. Ex: male or female, red or black,
maried or single, etc.
- Quantitative variable : records numeric values. Ex : weight, size, etc.
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Distribution :
A distribution is a graphic representation of data.
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Analyzing variables
First determine the distribution.
Measures of central tendency :
- Mean = the average score of a set of data.
- Median = the middle number of a set
- Mode = the most frequent score in a data set.
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Measures of variation:
- The range = distance between the lowest and the highest number
- Variance = Measure how data are dispersed around their mean
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- Standard deviation : is the square root of the variance.
Linear transformations
Are mathematical changes that are applied on a distribution to transform it into a
Standard Normal Distribution. Once we have extracted the needed data, we can
apply backward the changes and get the data about our real distribution.
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Dispersion percentages
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Dependance between 2 variables
Correlation : statistical dependency between 2 variables
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Positive correlation
BNPParibas
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Negative correlation
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The portfolio theory
Minimize risk for a given level of expected return by carefully choosing the proportions
of various assets.
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Modeling stock prices
The random walk
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Brownian Motion
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in 2 dimensions
in 1 dimension
30
20
10
0
-10
-20
-30
-40
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>8
7.5 à 8
7 à 7.5
6.5 à 7
6 à 6.5
5.5 à 6
5 à 5.5
4.5 à 5
4 à 4.5
3.5 à 4
3 à 3.5
2.5 à 3
2 à 2.5
1.5 à 2
1 à 1.5
0.5 à 1
0 à 0.5
-0.5 à 0
-1 à -0.5
-1.5 à -1
-2 à -1.5
-2.5 à -2
-3 à -2.5
-3.5 à -3
-4 à -3.5
-4.5 à -4
-5 à -4.5
-5.5 à -5
-6- à -5.5
-6.5 à -6
-7 à -6.5
-7.5 à -7
-8 à -7.5
>-8
Model vs Reality
35000
30000
25000
20000
15000
10000
5000
0
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From the Brownian Motion to the Îto Process
Wiener Process :
dx =
Arithmetical Brownian Motion:
dx=
Geometrical Brownian Motion:
dx=
Îto process:
dx = a(x,t) dt + b(x,t) dz
Where
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dz
µ dt +
s dz
µ x dt + s x dz
dz   dt
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Quotes simulations
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4
4
Density Estimate
-4
-4
-2
Brownian motionBt
0
2
Final value of Brownian motionB1
-2
0
2
Normal
0.0
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0.2
0.4
t
0.6
0.8
1.0
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0.0
0.3
0.2
0.1
Probability density
0.4
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