Mod`ele `a volatilit´e stochastique de Barndorff-Nielsen and Shephard
Premi`ere question On consid`ere la dynamique suivante pour la volatilit´e: dσt2=−kσ2
tdt + dZt, (B.4)
o`u Z est un processus de Poisson compos´e d’intensit´e λ, dont les sauts ont une distribution exponentielle de densit´e exponentielle f (z) = ηe−ηz1z>0. On note vt= σ2
t.
1. En appliquant la formule d’Itˆo `a ektvt, r´esoudre l’´equation (B.4).
2. Calculer la fonction caract´eristique φt(u) = E[eiuZt] et la transform´ee de Laplace Lt(u) = E[e−uZt] de Zt. 3. Calculer
Ehe−R0Tf (s)dZsi
pour une fonction continue born´ee positive f . Indication: commencer par considerer une fonction f constante par morceaux puis passer `a la limite.
4. En d´eduire la transform´ee de Laplace de σ2
t et IT :=RT 0 σ2
Deuxi`eme question On consid`ere maintenant le mod`ele du comportement d’un actif financier
dSt= StσtdWt, (B.5)
o`u σ est donn´ee par l’´equation (B.4) et W est un mouvement brownien standard sous la probabilit´e risque-neutre Q, ind´ependant de Z. Le taux d’int´erˆet est nul.
1. En utilisant la formule d’Itˆo, d´eduire l’EDS satisfaite par Xt:= log St.
2. Montrer que XT a la mˆeme loi qu’un mouvement Brownien avec drift, chang´e de temps par un processus `
a pr´eciser. En utilisant l’ind´ependance entre Z et W et le point 4 de la premi`ere question, calculer la fonction caract´eristique de XT.
3. En d´eduire les indices d’explosion de moments (pour T fix´e) ˜
p = sup{p : E[ST1+p] <∞} et ˜q = sup{q : E[ST−q] <∞} 4. Calculer les asymptotiques de la volatilit´e implicite pour T fix´e et K grand/petit.
Troisi`eme question (ind´ependante des deux premi`eres) Dans cette question on s’int´eresse `a la couver-ture d’une option europ´eenne dans le mod`ele (B.5).
1. Le march´e, est-il complet? Peut-on construire un portefeuille de replication parfaite en utilisant une action et un autre actif liquide? Justifier votre r´eponse.
2. On consid`ere la couverture approch´ee utilisant uniquement les actions.
• Ecrire la dynamique du prix Ct d’une option europ´eenne, en consid´erant que Ct est une fonction d´eterministe de St et vt. On exprimera Ct comme integrale stochastique par rapport `a W et `a la mesure al´eatoire de Poisson de Z, not´ee par JZ.
• Ecrire la dynamique d’un portefeuille autofinan¸cant. • Calculer E[ε2
T] par isom´etrie d’Itˆo, o`u εT est l’erreur de couverture.
• Calculer la strat´egie de couverture quadratique optimale; montrer qu’elle coincide avec la strat´egie de couverture en delta et expliquer ce r´esultat.
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