以SABR-LMM 模型進行利率衍生性金融商品評價之探討

Pricing Interest Rate Derivatives with SABR-LMM Models

選擇權市場中隱含波動度微笑曲線是相當常見的現象，但加入隨機波動度後的評價模型往往難以得到封閉解。本研究嘗試使用Mercurio and Morini (2009) 所提出的SABR-LMM模型評價利率衍生性金融商品，除了可以有效地解釋隱含波動度所出現的微笑曲線或偏斜情況，並且可以獲得近似封閉解。本研究並發展出一套獨特的校準工作程序，使該一複雜的模型能成功地用實際市場資料進行參數校準，也推導出此模型下考慮了利率與波動度間相關性後的避險參數，該參數更能提供交易員一個較明確的方向來判斷利率變動與波動度變動之間的交互作用。透過實證研究，我們發現SABR-LMM 模型能夠完整地校準出市場上利率交換選擇權資料所代表的波動度曲面。另外，我們也模擬出SABR-LMM 模型下遠期利率以及隨機波動度的動態過程，並以數值案例作評價與分析，也說明SABR-LMM 模型與傳統的LIBOR 市場模型在避險比率的計算上有何差異及相對優點。

The appearance of implied volatility smile or skew has been well documented in the options market. However, it is very difficult to obtain closed-form solution if stochastic volatility is taken into account. This study employs the SABR-LMM (LIBOR Market Model) model proposed by Mercurio and Morini (2009) to price interest rate derivatives. The model can derive an approximate closed-form solution and explain volatility smile or skew phenomenon very well. This study also develops a unique procedure to calibrate the SABRLMM model to market transaction data successfully. In addition, we derive the relevant hedge ratios considering the correlations between interest rates changes and volatility changes, which can provide concrete guidance for traders to judge the interactions between interest rate changes and volatility changes. This study also conducts an empirical examination to find that the SABR-LMM model can sufficiently calibrate the whole interest rates swaption volatility surface and accurately capture the volatility skew, which cannot be well dealt with traditional LIBOR market models. Finally, we simulate the forward rates and stochastic volatility dynamics under the SABR-LMM model and use the result to price interest rates products numerically. Also, this study demonstrates the differences and the advantages between the SABR-LMM model and LIBOR market models while calculating hedge ratios.