Abstract

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 SABR-LMM 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.  

Keywords

SABR modelstochastic volatilityLIBOR market model