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Options sur Futures et Courbes de Volatilités.

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Chapitres 17 et 19 Semaine 10

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Options sur Futures et Courbes de Volatilités.

  1. 1. Options sur Futures et Courbe de Volatilités. Chapitres 17 et 19 1
  2. 2. Exos  Chapitre 17  17.1 17.2 17.4 17.7  Chapitre 19  19.1 à 19.7, 19.17 2
  3. 3. 3 Mécanique pour un call sur futures Quand un Call Futures est exercé, le détenteur du call réçoit: 1. Une position longue sur un futures. 2. Un cash équivalent @ F-K Où F= prix futures le plus récent K=Strike prices.
  4. 4. 4 Mécanique pour un Put sur futures Le Put option est excercé et le détenteur obtient 1. Une position short futures. 2. Un cashflow K-F
  5. 5. 5 Parité Put-Call pour les options sur Futures Considérer 2 portefeuilles 1. Euro call + Ke-rT en Cash 2. Euro put + une position long futures + un cash égal @ F0e-rT Les deux positions doivent avoir la même valeur à l’expiration T donc C + Ke-rT= p + F0 e-rT
  6. 6. Créer des positions synthétiques avec la relation Put-Call parité. C + Ke-rT= p + F0 e-rT Synthetic call C = - Ke-rT +p + F0 e-rT Sell the strike buy a put buy the futures Possible de créer d’autres positions 6
  7. 7. F0 e-rT – C = Ke-rT- p  Covered Call  (long futures, short Call) 7 Covered Call Synthetic Covered Call Protective Put Synthetic Protective Put p + F0 e-rT = Protective Put : (long futures, long put) C + Ke-rT
  8. 8. 8 Évaluation des options sur futures par les arbres binomiaux (excellent pour la détermination du prix des options américaine) Rouge: futures Vert: valeur de l’option U: up D: down F0u ƒu F0d ƒd F0 ƒ
  9. 9. 9 Probabilité risque-neutre d’une hausse de prix futures (p) p d   1 u d  Probabilité risque-neutre d’une baisse de prix futures= (1-P)
  10. 10. Exo 17.4  Le prix futures= 50$  Dans 6 mois, il sera soit à 56, soit à 46. le taux sans risque est de 6%. Quel est la valeur d’un call euro à 6 mois et avec K=50 U=1.12 D= 0.92  L’évaluation risque neutre du call donne  e -0.06*0.5(0.4*6 + 0.6*0)= 2.33 10
  11. 11. 11 Modèle de Black (  Pour les Options Euro sur Futures     rT  c  e F N d  K N d ( ) ( ) 0 1 2 p  e K N  d  F N  d ln( F / K ) 2 T / 2 F K T ecartyperetoursurfutures  Fo prixfutures d T T d T d rT               1 0 2 0 1 2 0 1 ln( / ) 2 / 2 où ( ) ( )
  12. 12. Notes sur le modèle de Black  Pas a mémoriser  On revient sur BSM et Black à la dernière semaine du cours.  Problème du modèle: la volatilité n’est pas constante en réalité comme le dit le modèle.  Ce modèle n’est bon que pour les euro options, les arbres binomiaux sont meilleurs pour les options de types américaines. 12
  13. 13. Plusieurs formules Put-Call parités 13 Indices:   c  K e  p  S e Devises :   c  K e  p  S e Futures: rT qT 0 rT r T 0 rT rT c K e p F e 0 f     
  14. 14. Théorie vs Pratique  Rappelez-vous, la réalité peut-être différente de la théorie. 14

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