Calibration of Interest Rate Models:Transition Market Case

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

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

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. 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. 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

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. (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. (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. (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. 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. 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. CZK results

14. CZK results

15. SKK results

16. SKK results