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Quadratic hedging in affine stochastic volatility models. (English) Zbl 1168.91463

Summary: We determine the variance-optimal hedge for a subset of affine processes including a number of popular stochastic volatility models. This framework does not require the asset to be a martingale. We obtain semiexplicit formulas for the optimal hedging strategy and the minimal hedging error by applying general structural results and Laplace transform techniques. The approach is illustrated numerically for a Lévy-driven stochastic volatility model with jumps as by P. Carr, H. Geman, D. B. Madan and M. Yor [Math. Finance 13, No. 3, 345–382 (2003; Zbl 1092.91022)].

MSC:

91B70 Stochastic models in economics

Citations:

Zbl 1092.91022
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References:

[1] Barndorff-Nielsen O., Shephard N. (2001) Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. Journal of the Royal Statistical Society, Series B 63: 167–241 · Zbl 0983.60028 · doi:10.1111/1467-9868.00282
[2] Carr P., Geman H., Madan D., Yor M. (2003) Stochastic volatility for Lévy processes. Mathematical Finance 13: 345–382 · Zbl 1092.91022 · doi:10.1111/1467-9965.00020
[3] Černý A. (2007) Optimal continuous-time hedging with leptokurtic returns. Mathematical Finance 17: 175–203 · Zbl 1186.91202 · doi:10.1111/j.1467-9965.2007.00299.x
[4] Černý A., Kallsen J. (2007) On the structure of general mean-variance hedging strategies. The Annals of Probability 35: 1479–1531 · Zbl 1124.91028 · doi:10.1214/009117906000000872
[5] Černý A., J. Kallsen (2008) Mean-variance hedging and optimal investment in Heston’s model with correlation. Mathematical Finance 18: 473–492 · Zbl 1141.91413 · doi:10.1111/j.1467-9965.2008.00342.x
[6] Cont R., Tankov P. (2004) Financial modelling with jump processes. Chapman & Hall/CRC, Boca Raton · Zbl 1052.91043
[7] Cont R., Tankov P., Voltchkova E. et al (2007) Hedging with options in models with jumps. In: Benth F. (eds) Stochastic analysis and applications. The Abel symposium 2005. Springer, Berlin, pp 197–217 · Zbl 1151.91496
[8] Dieudonné J. (1960) Foundations of modern analysis. Academic Press, New York · Zbl 0100.04201
[9] Duffie D., Filipovic D., Schachermayer W. (2003) Affine processes and applications in finance. The Annals of Applied Probability 13: 984–1053 · Zbl 1048.60059 · doi:10.1214/aoap/1060202833
[10] Filipović D. (2005) Time-inhomogeneous affine processes. Stochastic Processes and their Applications 115: 639–659 · Zbl 1079.60068 · doi:10.1016/j.spa.2004.11.006
[11] Filipovic, D., & Mayerhofer, E. (2009). Affine diffusion processes: Theory and applications. Preprint. · Zbl 1205.91068
[12] Goll T., Kallsen J. (2000) Optimal portfolios for logarithmic utility. Stochastic Processes and their Applications 89: 31–48 · Zbl 1048.91064 · doi:10.1016/S0304-4149(00)00011-9
[13] Gourieroux C., Laurent J., Pham H. (1998) Mean-variance hedging and numéraire. Mathematical Finance 8: 179–200 · Zbl 1020.91024 · doi:10.1111/1467-9965.00052
[14] Heath D., Platen E., Schweizer M. (2001) A comparison of two quadratic approaches to hedging in incomplete markets. Mathematical Finance 11(4): 385–413 · Zbl 1032.91058 · doi:10.1111/1467-9965.00122
[15] Heston S. (1993) A closed-form solution for options with stochastic volatilities with applications to bond and currency options. The Review of Financial Studies 6: 327–343 · Zbl 1384.35131 · doi:10.1093/rfs/6.2.327
[16] Hubalek F., Krawczyk L., Kallsen J. (2006) Variance-optimal hedging for processes with stationary independent increments. The Annals of Applied Probability 16: 853–885 · Zbl 1189.91206 · doi:10.1214/105051606000000178
[17] Jacod J., Shiryaev A. (2003) Limit theorems for stochastic processes (2nd ed). Springer, Berlin · Zbl 1018.60002
[18] Kallsen J. (2004) {\(\sigma\)}-localization and {\(\sigma\)}-martingales. Theory of Probability and Its Applications 48: 152–163 · Zbl 1069.60042 · doi:10.1137/S0040585X980312
[19] Kallsen J. (2006) A didactic note on affine stochastic volatility models. In: Kabanov Y., Liptser R., Stoyanov J. (eds) From stochastic calculus to mathematical finance. Springer, Berlin, pp 343–368 · Zbl 1104.60024
[20] Kallsen, J., & Muhle-Karbe, J. (2008). Exponentially affine martingales, affine measure changes and exponential moments of affine processes. Preprint. · Zbl 1185.60045
[21] Kallsen J., Pauwels A (2009a).Variance-optimal hedging for time-changed Lévy processes. Preprint. · Zbl 1232.91668
[22] Kallsen, J. & Pauwels, A. (2009b). Variance-optimal hedging in general affine stochastic volatility models. Preprint. · Zbl 1189.91231
[23] Kallsen J., Shiryaev A. (2002) Time change representation of stochastic integrals. Theory of Probability and Its Applications 46: 522–528 · Zbl 1034.60055 · doi:10.1137/S0040585X97979184
[24] Muhle-Karbe, J. (2009). On utility-based investment, pricing and hedging in incomplete markets. Ph.D. dissertation (TU München), München.
[25] Pauwels, A. (2007). Varianz-optimales Hedging in affinen Volatilitätsmodellen. Ph.D. dissertation (TU München), München.
[26] Pham H. (2000) On quadratic hedging in continuous time. Mathematical Methods of Operations Research 51: 315–339 · Zbl 0977.91035 · doi:10.1007/s001860050091
[27] Sato K. (1999) Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge · Zbl 0973.60001
[28] Schweizer M. (2001) A guided tour through quadratic hedging approaches. In: Jouini E., Cvitanic J., Musiela M. (eds) Option pricing, interest rates and risk management. Cambridge University Press, Cambridge, pp 538–574 · Zbl 0992.91036
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