We present an algorithm to approximate moments for forward rates under a displaced lognormal forward-LIBOR model (DLFM). Since the joint distribution of rates is unknown, we use a multi-dimensional full weak order 2.0 Ito–Taylor expansion in combination with a second-order Delta method. This more accurately accounts for state dependence in the drift terms, improving upon previous approaches. To verify this improvement we conduct quasi-Monte Carlo simulations. We use the new mean approximation to provide an improved swaption volatility approximation, and compare this to the approaches of Rebonato, Hull–White and Kawai, adapted to price swaptions under the DLFM. Rebonato and Hull–White are found to be the least accurate. While Kawai is the most accurate, it is computationally inefficient. Numerical results show that our approach strikes a balance between accuracy and efficiency.

中文翻译：

---

适用于掉期的置换远期 LIBOR 利率的矩近似

我们提出了一种算法来近似在位移对数正态正向 LIBOR 模型 (DLFM) 下的正向利率矩。由于速率的联合分布是未知的，我们使用多维全弱阶 2.0 Ito-Taylor 展开与二阶 Delta 方法相结合。这更准确地解释了漂移项中的状态依赖性，改进了以前的方法。为了验证这种改进，我们进行了准蒙特卡罗模拟。我们使用新的均值近似来提供改进的掉期期权波动率近似值，并将其与适用于 DLFM 下的价格掉期期权的 Rebonato、Hull-White 和 Kawai 的方法进行比较。Rebonato 和 Hull-White 被认为是最不准确的。虽然 Kawai 是最准确的，但它的计算效率很低。