15.2 Implied Volatility Specification


15.2 Implied Volatility Specification

Market prices of traded interest rate derivatives are used to estimate the volatility functions. The choice of number and form of the volatility functions is driven by the users' discretion, desire for the model to exhibit certain characteristics and quality of data available. It is common practice to assume a functional form for the volatilities and then determine the parameters such that the market prices are matched as closely as possible. Consider the volatility functions tested by Amin and Morton [ 2 ]. These are special cases of the general functional form:

which has four parameters ƒ , ƒ 1 , » and ³ . Amin and Morton found that attempting to calibrate a general functional form, i.e. one with more than two parameters, resulted in unstable parameter values with large estimation errors. They found that given the data available, calibration with two parameters proved optimal. Once a suitable parameterisation is chosen , the parameter values are estimated such that model prices best match market prices. This may be done by a simple procedure such as minimising the sum of squared errors, that is:

where

˜

-

vector of parameters specifying the volatility function,

V i ( ˜ )

-

model price of interest rate derivative i , based on parameter values ˜ ,

V i

-

market price of interest rate derivative i ,

k

-

number of interest rate derivatives used in calibration on a given day.




Interest Rate Modelling
Interest Rate Modelling (Finance and Capital Markets Series)
ISBN: 1403934703
EAN: 2147483647
Year: 2004
Pages: 132

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