Calibration of Interest Rate Models:Transition Market

Case

Martin Vojtek

martin.vojtek@cerge-ei.cz

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MOTIVATION

• Need for pricing of interest rates (IR) derivatives in transition countries

• Precise pricing is based on correct calibration of chosen IR models

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MOTIVATION

• No calibration work for transition countries

• Small number of empirical studies dealing with IR markets

• Reasons: chaotic development, not enough data etc.

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PLAN OF WORK

• Setup of a model– Brace, Gatarek, Musiela (1997) model

• Model of LIBOR interest rates – observable quantities at market

• Very powerful model

• Calibration of model– Usually through the implied volatilities

• Not possible to use as there is no liquid market for IR derivatives in transition countries

• Therefore other methodology is needed

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Parameters of BGM model

• Instantaneous volatilities of LIBOR rates and instantaneous correlations among LIBOR rates with various maturities

• For estimation other than using implied volatilities one needs to choose a robust volatility model which can be easily estimated and is numerically efficient

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GARCH models

• Multivariate GARCH models seems to be suitable models for volatilities

• Problems with estimation – large number of parameters (~n2 , if n processes modeled)– Solution: Impose some structure on the

covariance matrix which enables to estimate less number of parameters

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(G)O-GARCH model

• Imposes a structure without danger of mispricing of certain element of market

• Based on the principal components processes of realized returns of (LIBOR) rates

• The returns of some rate are modeled as a linear combination of principal components, which are the same for all rates

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(G)O-GARCH model

• These principal components (they are orthogonal) can be considered as the increases in orthogonal Wiener processes (then actually a BGM specification follows for examined rates)

• Then, the covariance matrix of returns is Var(Y)=WDW’, where W is a vector of weights in mentioned linear combination (known) and D is diag. (because of orthogonolity) matrix of variances processes for PCs

• So, it is enough to model this matrix D

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(G)O-GARCH model

• It can be done by running simple GARCH models for each PC

• In highly correlated systems (as interest rates) can choose just r<n PCs (often 3 are enough – can control for changes in level, slope and shape)

• Then have ~r parameters• Can be generalized, as PCs are uncorrelated only

unconditionally (then get GO-GARCH model)

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(G)O-GARCH model

• Inputs of model: Log-returns of LIBOR rates (as specified by BGM model)

• Output: time evolution of the covariance matrix of LIBOR rates – actually parameters of BGM model

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Empirical part: Data used

• 4 Visegrad countries: Slovakia, Czech Republic, Poland and Hungary

• LIBOR-like interest rates, maturities up to one year (longer rates are not quoted)

• Need to test the approach

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Empirical part

• Model is working good for Czech Republic and Poland – they have probably developed enough markets

• Numerical problems for Slovakia and Hungary due to the markets not developed enough, to often external shocks

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CZK results

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CZK results

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SKK results

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SKK results