Perturbative Expansion of FBSDE in an Incomplete Market with Stochastic Volatility

Abstract

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 for its level, and the fourth order for its diffusion part that 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.