The Extension Of LMM Model Under The Shifted Log-Normal Assumption

In 1997,Brace,Gatarek,Musiela has developed the LIBOR market model,which is a no-arbitrage interest rate model.The model focusing on the London Interbank Offered Rate has leaded the interest rate modeling to the new direction and is widely used in the pricing of interest rate derivatives.The LMM model is non arbitrage built on the market interest rate,it assumes that the forward interest rate follows log-normal distribution,the log-normal assumption caters the opinion that the interest rate can not go negative.Comparing with short rate model,LMM model has some good features,LMM model is directed based on observable market interest rate,eliminating the process of transforming short-term interest rate or instantaneous forward rate to the market rate,this feature helps us avoided the complex numerical calibration process.Since the financial crisis in 2008,in order to stimulate economic growth,many countries have adopted relatively loose monetary policies in succession.European Central Bank,Bank of Japan,Swiss National Bank,Swedish Central Bank and other global multinational central banks have adopted negative interest rates.In the new negative interest rate environment,the failure of pricing of interest rate derivatives(interest rate caps and interest rate swaps)under the assumption of log-normal distribution forces us to re-adjust the model to better price interest rate derivatives.In this paper,we mainly focus on the LMM model under the new shifted log-normal assumption,and invert the black volatility to new frame implied volatility.After that,we analyze the explanatory power of the implied volatility under the two different frames.And,we studied the Delta hedge efficiency under the two different frames.Finally,the relationship between the two groups of implied volatility term structure and macroeconomic variables is studied by PCA method.