Perturbative Expansion of FBSDE in an Incomplete Market with Stochastic Volatility
AbstractIn this work, we apply our newly proposed perturbative expansion technique to a quadratic growth FBSDE appearing in an incomplete market with stochastic volatility that is not perfectly hedgeable. By combining standard asymptotic expansion technique for the underlying volatility process, we derive explicit expression for the solution of the FBSDE up to the third order of volatility-of-volatility, which can be directly translated into the optimal investment strategy. We compare our approximation with the exact solution, which is known to be derived by the Cole-Hopf transformation in this popular setup. The result is very encouraging and shows good accuracy of the approximation up to quite long maturities. Since our new methodology can be extended straightforwardly to multi-dimensional setups, we expect it will open real possibilities to obtain explicit optimal portfolios or hedging strategies under realistic assumptions.
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Bibliographic InfoPaper provided by arXiv.org in its series Papers with number 1202.0608.
Date of creation: Feb 2012
Date of revision: Sep 2012
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Web page: http://arxiv.org/
This paper has been announced in the following NEP Reports:
- NEP-ALL-2012-02-15 (All new papers)
- NEP-CTA-2012-02-15 (Contract Theory & Applications)
- NEP-FDG-2012-02-15 (Financial Development & Growth)
- NEP-ORE-2012-02-15 (Operations Research)
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- Christian Kahl & Peter Jackel, 2006. "Fast strong approximation Monte Carlo schemes for stochastic volatility models," Quantitative Finance, Taylor and Francis Journals, vol. 6(6), pages 513-536.
- Ulrich Horst & Ying Hu & Peter Imkeller & Anthony Reveillac, 2011. "Forward-backward systems for expected utility maximization," SFB 649 Discussion Papers SFB649DP2011-061, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
- Masaaki Fujii & Seisho Sato & Akihiko Takahashi, 2012. "An FBSDE Approach to American Option Pricing with an Interacting Particle Method," Papers 1211.5867, arXiv.org.
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