LIBOR 市場模型的遞迴三元樹

Abstract

LIBOR市場模型（LMM）是定價利率衍生性金融商品的重要框架。然而，在統一測度下模擬多個遠期LIBOR利率時，因爲對長天期漂移項的依賴，其數值計算面臨挑戰。雖然已有研究提出高效的重組三元樹來近似LMM動態，但該方法在波動較高的情況下，由於近似誤差累積而有失準。本論文提出了一種創新的遞迴元樹，把誤差最高、機率權重最大的中央節點當作新子樹的根節點，而非與現有節點重組，透過選擇性地解除中央節點的重組限制，以提升定價準確性。此「遞迴」結構重置了沿著重組樹中央高機率路徑所累積的局部動差近似誤差。數值實驗驗證了此方法的有效性，顯示其在定價普通選擇權與離散障礙選擇權時達到了顯著提升的精準度，優於先前方法。因此，遞迴三元樹為LIBOR模型下的利率衍生工具定價提供了另一個選擇，在效率與精確度之間取得平衡。

The LIBOR Market Model (LMM) is a fundamental framework for pricing interest rate derivatives. However, modeling multiple forward LIBOR rates under a common measure introduces complex, state-dependent drift terms that complicate numerical valuation. When recombining trinomial trees are applied to the LMM, the model's state-dependent drift terms prevent the asymptotic matching of both local mean and variance at the majority of nodes. This mismatch between the tree's local moments and those dictated by the LMM results in convergence issues. Numerical experiments reveal that significant deviations in local moments, and consequently potential pricing inaccuracies, concentrate along the tree's central region. This tree's central nodes initiate sub-trees periodically after every predetermined number of time steps. This recursion resets the accumulated errors from mismatched local moments. Numerical experiments confirm its effectiveness in raising precision for the prices of vanilla and discrete barrier options compared to the standard recombining trinomial tree.