1. Calibration of Interest Rate Models:Transition Market Case Martin Vojtek martin.vojtek@cerge-ei.cz

2. 2 MOTIVATION Need for pricing of interest rates (IR) derivatives in transition countries Precise pricing is based on correct calibration of chosen IR models

3. 3 MOTIVATION No calibration work for transition countries Small number of empirical studies dealing with IR markets Reasons: chaotic development, not enough data etc.

4. 4 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

5. 5 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

6. 6 GARCH models Multivariate GARCH models seems to be suitable models for volatilities Problems with estimation – large number of parameters (~n 2, if n processes modeled) –Solution: Impose some structure on the covariance matrix which enables to estimate less number of parameters

7. 7 (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

8. 8 (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

9. 9 (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)

10. 10 (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

11. 11 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

12. 12 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

13. 13 CZK results

14. 14 CZK results

15. 15 SKK results

16. 16 SKK results