Perturbative Expansion of FBSDE in an Incomplete Market with Stochastic Volatility
In 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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- Horst, Ulrich & Hu, Ying & Imkeller, Peter & Réveillac, Anthony & Zhang, Jianing, 2014.
"Forward–backward systems for expected utility maximization,"
Stochastic Processes and their Applications,
Elsevier, vol. 124(5), pages 1813-1848.
- 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.
- Christian Kahl & Peter Jackel, 2006. "Fast strong approximation Monte Carlo schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 513-536.
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