A Valuation of Quanto Constant Maturity Swap Products under the Three-Factor BGM Model

Abstract

匯率連動固定期利率交換（Quanto Constant Maturity Swaps，以下簡稱Quanto CMS）商品可做爲管理國外利率交換利差風險的輔助工具。以往對Quanto CMS商品的評價通常是利用蒙地卡羅模擬法(Monte Carlo Simulation)來模擬進行，但這樣的評價方式通常較耗時。本文應用國外遠期交換利率近似於國外遠期LIBOR利率之線性組合的特徵來設定BGM模型下國外遠期交換利率的近似動態過程。基於國外遠期交換利率的近似動態，本文推導出三因子BGM模型下評價Quanto CMS利差選擇權(Quanto Constant Maturity Swaps Spread Option)及Quanto CMS輪棘選擇權(Quanto Constant Maturity Swaps Ratchet Option)的近似解析公式。數值分析的結果顯示上述兩種商品在不同履約價下近似解析公式解法對應蒙地卡羅模擬法的相對誤差都很小且近似解析公式解法之計算時間遠少於蒙地卡羅模擬法。

Quanto constant maturity swaps (Quanto CMS) products can be used to manage the spread risk of foreign interest rate swap. Monte Carlo simulation is usually used to evaluate Quanto CMS products, but it`s often time consuming to use Monte Carlo simulation method. In this paper we derive an approximated dynamic process of the foreign forward swap rate under the three-factor BGM model with the characteristic which the foreign forward swap rate is approximated to the linear combination of the foreign forward LIBOR rate. We use no-arbitrage analytical formula to evaluate Quanto CMS products under the three-factor BGM model. Then we apply this approximated formula to evaluate Quanto CMS Spread option and Quanto CMS Ratchet option. The numerical analysis shows that the relative errors between the Monte Carlo simulations and the approximated analytic formulas are very small for the both examined option products. Moreover, the calculation time of the analytic formulas method is much smaller than the Monte Carlo simulation method for both products.