[Prime de risque et prix de l'option sous l'hypothèse de volatilité stochastique]
Nous résolvons le problème de la détermination du prix de l'option sous un modèle général de volatilité stochastique. Pour une fonction d'utilité HARA, « la prime de risque », i.e. le prix du marché associé au risque de volatilité, est déterminé en utilisant la solution d'une EDP non linéaire. De la même façon, le prix de l'option est déterminé comme solution d'un système découplé d'une EDP non linéaire et d'une EDP de type Black–Scholes.
We have solved the problem of finding (HARA) fair option price under a general stochastic volatility model. For a given HARA utility, the ‘risk premium’, i.e., the ‘market price of volatility risk’ is determined via a solution of a certain nonlinear PDE. Equivalently, the fair option price is determined as a solution of an uncoupled system of a non-linear PDE and a Black–Scholes type PDE.
Accepté le :
Publié le :
@article{CRMATH_2005__340_7_551_0,
author = {Stojanovic, Srdjan D.},
title = {Risk premium and fair option prices under stochastic volatility: the {HARA} solution},
journal = {Comptes Rendus. Math\'ematique},
pages = {551--556},
publisher = {Elsevier},
volume = {340},
number = {7},
year = {2005},
doi = {10.1016/j.crma.2004.11.002},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2004.11.002/}
}
TY - JOUR AU - Stojanovic, Srdjan D. TI - Risk premium and fair option prices under stochastic volatility: the HARA solution JO - Comptes Rendus. Mathématique PY - 2005 SP - 551 EP - 556 VL - 340 IS - 7 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2004.11.002/ DO - 10.1016/j.crma.2004.11.002 LA - en ID - CRMATH_2005__340_7_551_0 ER -
%0 Journal Article %A Stojanovic, Srdjan D. %T Risk premium and fair option prices under stochastic volatility: the HARA solution %J Comptes Rendus. Mathématique %D 2005 %P 551-556 %V 340 %N 7 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2004.11.002/ %R 10.1016/j.crma.2004.11.002 %G en %F CRMATH_2005__340_7_551_0
Stojanovic, Srdjan D. Risk premium and fair option prices under stochastic volatility: the HARA solution. Comptes Rendus. Mathématique, Tome 340 (2005) no. 7, pp. 551-556. doi : 10.1016/j.crma.2004.11.002. http://www.numdam.org/articles/10.1016/j.crma.2004.11.002/
[1] Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, 2000
[2] The Monge–Ampère Equations, Birkhäuser, Boston, 2001
[3] A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, Volume 6 (1993) no. 2, pp. 327-343
[4] Stochastic volatility (Hand, D.; Jacka, S., eds.), Applications of Statistics Series, Arnold, London, 1998
[5] Hypoelliptic second order differential equations, Acta Math., Volume 119 (1967), pp. 147-171
[6] The pricing of options on assets with stochastic volatility, J. Finance, Volume 42 (1987) no. 2, pp. 281-300
[7] On the pricing of contingent claims under constraints, Ann. Appl. Probab., Volume 6 (1996) no. 2, pp. 321-369
[8] Utility-based derivative pricing in incomplete markets (Geman, H.; Madan, D.; Pliska, S.R.; Vorst, T., eds.), Mathematical Finance—Bachelier Congress 2000, Springer, Berlin, 2002
[9] Optimum consumption and portfolio rules in a continuous-time model, J. Economic Theory, Volume 3 (1971), pp. 373-413
[10] Continuous-Time Finance, Blackwell, Cambridge, 1992
[11] Equilibrium state prices in a stochastic volatility model, Mathematical Finance, Volume 6 (1996) no. 2, pp. 215-236
[12] Computational Financial Mathematics using Mathematica®: Optimal Trading in Stocks and Options, Birkhäuser, Boston, 2003
[13] Optimal momentum hedging via hypoelliptic reduced Monge–Ampère PDEs, SIAM J. Control Optimization, Volume 43 (2004) no. 4, pp. 1151-1173
Cité par Sources :
