Les milieux financiers: cours boursiers, entre instabilité et prévisions

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Ahmed Ammar Rebai PhD

Présentation de TIPE de l'étudiant Aymen Ben AbdAllah encadré par Ahmed Rebai

Ingénierie

Les milieux financiers:
cours boursiers, entre instabilité et
prévisions
Milieux : interactions,
interfaces, homogénéité,
ruptures

Objectifs
• Prévoir les variations des prix dans les marchés
• Etudier la fiabilité des modèles de prédiction sur des données réelles

Contributions
• Prise de contact avec une banque d’investissement
• Implémentation d'un programme de résolution
• Vérification des résultats théoriques sur des données réelles

Plan
I- Milieux d’étude
1) Ecosystème chaotique (brown)
2) Analogie avec le milieu financier
II- Modélisation
1) Approche probabiliste
2) Vers la limite du continu et Black and Scholes
III- Résolutions et fiabilités de modèles
1) Résolutions analytique et numérique
2) Fiabilité et Cox-Ross-Rubinstein

I- Milieux d’étude
1) Ecosystème chaotique (brown)

Observations de Robert Brown
• Mouvement irrégulier
• Trajectoire sans tangentes.
• Indépendant de la nature de la
particule.
• Le mouvement est d’autant plus
erratique que la particule est
petite, la température élevée, la
viscosité faible.
• Incessant.

I- Milieux d’étude
2) Analogie avec le milieu financier

L’analogie de Bachelier
Pollen en interaction (milieu physique) Marché financier en interaction
𝑔 𝐸 (force externe)
Goldman
Sachs
BNP
SG
Hedge
Funds 𝐵𝑎𝑛𝑞𝑢𝑒 𝑐𝑒𝑛𝑡𝑟𝑎𝑙𝑒
(𝑓𝑎𝑐𝑡𝑒𝑢𝑟 𝑒𝑥𝑡𝑒𝑟𝑛𝑒)
Prixenfonctiondu
tempsNasdaq2018
(périodede3mois)

Inadéquation avec l’époque
• Siècle de la mécanique Newtonienne et du déterminisme Laplacien
• Mouvement brownien non perçu dans le cadre de la physique
• Approche probabiliste

Approche probabiliste
• Innovations de Bachelier
 Introduction de la diffusion
 Utilisation des probabilités (5 ans avant Einstein)
• Einstein utilise la densité de probabilité
• Il modélise la marche aléatoire

II- Modélisation
1) Approche probabiliste

La marche aléatoire
• Temps de collision moyen
tc≈2.10-10 s
• temps de mesure de l’appareil 𝜏
≈1.10-2 (par rapport a tc)
• Discrétisation du temps

En une dimension…
• Deplacements de
• Intervalles de temps de
• ℙ
• ℙ
• Avec et
• ℙ (k 𝜖 ℤ)

II- Modélisation
2) Vers la limite du continu et Black and Scholes

Vers la limite du continu…
• k 𝜖 ℤ
• Δx
• kΔx
•
ℙ(kΔx, n𝜏)
Δ
• ℝ
• 𝑑𝓍
• 𝓍
• ℙ(𝓍, t) densité de
probabilité
• 𝓍.ℙ(𝓍, t) 𝑑𝓍
Δ⟶0 𝜏⟶0

Vers une équation de diffusion
• ℙ(𝓍, t)=
1
4𝜋𝐷𝑡
𝑒−
𝓍2
4𝐷𝑡 vérifie:
•
߲
߲ 𝑡
ℙ(𝓍, t)=𝐷
߲2
߲ 𝓍2 ℙ(𝓍, t)
߲
߲ 𝑡
ℙ(𝓍, t)=𝐷. ∆ℙ(𝓍, t)
EN 3
DIMENSIONS…

Equivalence avec Black and Scholes
En posant…
On obtient:
Black and Scholes
Equation de
diffusion avec
coefficients
non constants

III- Résolutions et fiabilités de modèles
1) Résolutions analytique et numérique

Résolution analytique
Pic de Dirac
• ℙ(𝓍, t)=
1
4𝜋𝐷𝑡
𝑒−
𝓍2
4𝐷𝑡
Gaussienne
• ℙ(𝓍, t)=
2𝜎
4𝜋𝐷𝑡+2𝜎
𝑒−
𝓍
2
4𝐷𝑡+2𝜎

III- Résolutions et fiabilités de modèles
2) Fiabilité et Cox-Ross-Rubinstein

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