具有随机波动率的不完全市场中FBSDE的扰动展开
Perturbative Expansion of FBSDE in an Incomplete Market with Stochastic Volatility
摘要：
在这项工作中，我们应用我们新提出的扰动扩展技术的二次增长FBBundle出现在一个不完全的市场与随机波动率，是不完全对冲。结合标准的渐近展开技术的潜在波动率过程，我们得到了明确的表达式的FB的解决方案，直到三阶波动率的波动率，它可以直接转化为最优投资策略。我们比较我们的近似与精确解，这是已知的是由科尔-霍普夫变换在这个流行的设置。结果是非常令人鼓舞的，并显示出良好的精度的近似相当长的到期日。由于我们的新方法可以直接扩展到多维设置，我们预计它将打开真正的可能性，以获得明确的最佳投资组合或对冲策略下现实的假设。
英文摘要：
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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