Adaptive Interest Rate Modelling

Transcription

1 Modelling Mengmeng Guo Wolfgang Karl Härdle Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin

2 1 Motivation 1-1 Interest Rate Essential for pricing derivatives and hedging corresponding risk. A signal of macroeconomic activity. Influenced by macroeconomic variables. Follow unstable dynamic process.

3 1 Motivation 1-2 Classical One-factor Short Rate Models Vasicek Model dr(t) = a{b r(t)}dt + σdw t CIR Model dr(t) = a{b r(t)}dt + σ r(t)dw t r t Mean reversion

4 1 Motivation Figure 1: Moving window estimator â with window sizes, and 7

5 1 1 Motivation Figure 2: Moving window estimator ˆb with window sizes, and 7

6 1 Motivation Figure 3: Moving window estimator ˆσ with window sizes, and 7

7 1 Motivation 1-6 Extended Short Rate Models Three strands of extended interest rate models: Jump-Diffusion Models Regime-Switching Models Time-varying Coefficients Models.

8 1 Motivation 1-7 Outline 1. Motivation 2. Local Adaptive Approach for CIR model 3. Simulations 4. Empirical Study 5. Conclusion

9 1 2-1 Model The time-varying CIR model with θ t = (a t, b t, σ t ) : dr(t) = a t {b t r(t)}dt + σ t r(t)dwt (1) Discretization: r ti+1 r ti = a t {b t r ti } t + σ t rti Z i (2) {Z i } N(, t)

10 1 2-2 Local Parametric Approach (LPA) Given time point t, go back and split time series into K intervals, I I 1 I K, with I k = [t m k + 1, t]. Accept the smallest interval I without change point (i.e. homogeneous interval). Sequentially check the historical intervals to search a change point. Two methods of LPA: Local model selection(lms) and Local change point (LPC).

11 1 2-3 Algorithm of LPC Goal: Find an unknown change point τ in the interval I k. 1. Determine ˆθ = θ. 2. Increase interval to I k, k 1. Get θ Ik. 3. Compare test statistics with critical value. If test statistic is accepted go to step 4, otherwise go to step Let ˆθ = θ Ik, and set k = k + 1, repeat step Detect the change point τ in I k, Iˆk = I k 1 without change point.

12 1 2-4 Why we use LPA? Find the longest stable interval for each t. Allow for structural breaks and jumps in parameter values. Distinguish blooming and declining regimes.

13 1 2-5 Test Statistic Test statistics T Ik+1,τ : T Ik+1,τ = L J ( θ J ) + L J c ( θ J c ) L Ik+1 ( θ Ik+1 ) (3) where J = [τ + 1, t], and J c = [t m k+1, τ], and τ J k = I k \I k 1. Consider the supremum of the test statistics over interval J k : T k = sup τ Jk T Ik+1,τ (4)

14 1 2-6 Test Algorithm T m + k 1 c J T m k τ T mk 1 J T J k + 1 J k I k 1 I k I k +1 1 Figure 4: Construction of the Test Statistics in the Local Change Point Test

15 1 2-7 The criteria for testing homogeneous intervals: T k z k, for k ˆk (5) and T k+1 > zˆk+1. Iˆk is the longest time homogeneous interval for time point t, ˆθ t = θ Iˆk. z k is the critical value, obtained by Monte-Carlo simulations.

16 1 2-8 Risk Bound Parametric risk bound R r (θ ), given the true value θ, for any interval I k, E θ L Ik ( θ Ik, θ ) r R r (θ ) (6) where L Ik ( θ Ik, θ ) = L Ik ( θ Ik ) L Ik (θ ) is the likelihood ratio between the two parameters.

17 1 2-9 Small Modeling Bias Condition The SMB condition for the interval I k, and given some θ Θ: E Ik (θ) (7) and Ik = t I k K{r(t), r(t; θ)} K{r t, r t (θ)} : Kullback-Leibler divergence between P r(t) and P r(t;θ). Oracle Choice k : the largest I k s.t. (7) holds.

18 1 2-1 Propagation Condition and Stability Propagation: E θ L Ik ( θ Ik, ˆθ Ik ) r ρr r (θ ) (8) Stability: I k is accepted interval, then ˆθ Ik = θ Ik L(ˆθ Ik, ˆθ Ik+1 ) z k (9) For fixed, the loss L Ik ( θ Ik, θ ) r stochastically bounded by a constant proportional to e

19 Critical Value Sequential choice of critical values z k. Change point detected at step l k B l : rejection at step l. B l = {T 1 z 1,, T l 1 z l 1, T l > z l } and ˆθ Ik = θ Il 1 on B l, l = 1, 2,, k.

20 Critical Value To determine z 1, max E θ L( θ Ik, θ I ) r 1(B 1 ) ρr r (θ )/K (1) k=1,,k B l only depends on z 1,, z l, controlled by z l. The minimal value ensures max E θ L( θ Ik, θ Il 1 ) r 1(B l ) = ρr r (θ )/K (11) k l R r (θ ) is parametric risk bound.

21 Choice of the Length of Interval I with length m. Interval I k : m k = [m a k ] with a > 1. Results not sensitive to a. r=.5, power of the loss function. ρ=.2, level of the test. m = 4, a = 1.25, and K= 15, m K = 1136.

22 Oracle Property I k is the oracle interval, E Ik (θ), k k, and ˆθ Iˆk close to the oracle estimate θ Ik E log{1 + L I k ( θ Ik, ˆθ Iˆk ) r R r (θ) } ρ + (12) For k > k, the adaptive estimator ˆθ Iˆk satisfies E log{1 + L I k ( θ Ik, ˆθ Iˆk ) r R r (θ) } ρ + + log{1 + zr k R r (θ) } (13)

23 1 Simulation 3-1 Simulation Setup We simulate CIR process with 1 observations and times. t a b σ t [1, ] t [1, ] t [1, 1] Table 1: The parameter settings for simulations of the CIR process

24 1 Simulation Est. a Est. b Figure 5: LPA estimator â and ˆb with simulated CIR paths.

25 1 Simulation Est. sigma Length of Time Homogeneous Interval Figure 6: LPA estimator ˆσ and selected time homogeneous intervals.

26 1 Empirical Study 4-1 Data Yield of 3M US T-Bill from the Federal Reserve Bank of St. Louis from to Mean SD Skewness Kurtosis r t dr t Table 2: Statistical Summary of 3-month T-Bill

27 1 Empirical Study Interest Rate x Change of Interest Rate Figure 7: 3-month Treasure Bill Rate: Top panel: Daily yields. Bottom panel: Changes of daily yields.

28 1 Empirical Study 4-3 MLE Estimator of CIR model Sample Size â ˆb ˆσ Table 3: Estimated parameters of CIR model using MLE

29 1 Empirical Study 4-4 Critical Value Critical Value Log Length of Interval Figure 8: Critical values with m = 4, K=15

30 1 Empirical Study 4-5 Estimator â 1 Est. a Figure 9: Estimated â by LPA

31 1 Empirical Study 4-6 Estimator ˆb Est. b.5.1 Figure 1: Estimated ˆb by LPA

32 1 Empirical Study 4-7 Estimator ˆσ.6 Est. sigma.4.2 Figure 11: Estimated ˆσ by LPA

33 1 Empirical Study 4-8 Homogeneous Intervals 3 1 Figure 12: Selected time homogeneous intervals with ρ =.2, and r =.5

34 1 Empirical Study 4-9 In Sample Fitting Interest Rate Figure 13: Confidence Interval (Red); Real Data (Black); LPA CIR (Blue); CIR (Purple)

35 1 Empirical Study 4-1 Forecasting H: the prediction period horizon, then the absolute prediction error(ape): APE(t) = r t+h ˆr t+h t / H (14) h H APE Ratio = APE LPA(t) APE MW (t) LPA: Local Parametric Approach. MW: Moving Window Estimation.

36 1 Empirical Study 4-11 Forecasting APE Ratio 2 APE Ratip 1 1 Figure 14: The APE ratio between LPA and MW with window size. Left: 1-day ahead forecasting; Right: 1-day ahead forecasting.

37 1 Empirical Study 4-12 Performance of Forecasting Horizon MAE l = l = l = 7 One Day LPA MW Ten Days LPA MW Table 4: The MAE of 1 day and 1 days ahead forecasting of the short rate based on the LPA and MW.

38 1 Conclusion 5-1 Conclusion Interest rate in recession is more volatile. The selected time homogeneous intervals can not last long due to the complexities of macroeconomy. The LPA can detect jumps and structural break points in the interest rate dynamics. The LPA outperforms the moving window estimation especially in long horizon forecasting.

39 1 Conclusion 5-2 References P. Čížek, W. Härdle and V. Spokoiny Adaptive Pointwise Estimation in Time-inhomogeneous Conditional Heteroscedasticity Models Econometric Journal Vol.12(9), pp C. Cox, E. Ingersoll and A. Ross A Theory of the Term Structure of Interest Rates Econometrica Vol.53(1985), pp J. Franke, W. Härdle and C. Hafner Statistics of Financial Markets - 3rd ed. Springer Verlag, 211.

40 1 Conclusion 5-3 References J. Hull and A. White Pricing Interest-Rate-Derivative Securities Review of Financial Studies Vol.3(199), No.4, pp V. Spokoiny Multiscale Local Change Point Detection with Applications to Value-at-Risk The Annals of Statistics Vol.37(9), No.3, pp O. Vasicek An Equilibrium Characterization of the Term Structure Journal of Financial Economics Vol.5(1977), pp

41 Modelling Mengmeng Guo Wolfgang Karl Härdle Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin