CDO Surfaces Dynamics

Transcription

1 D Surfaces Dynamics Barbara Choroś-Tomczyk Wolfgang Karl Härdle stap khrin Ladislaus von Bortkiewicz Chair of Statistics C..S.E. - Center for pplied Statistics and Economics Humboldt-Universität zu Berlin

2 K H R I N H E R D L E Motivation 1-1 itraxx over Time 8 8 Spread 6 4 Spread Time Time Figure 1: Spreads of itraxx tranches, Series 5, maturity 5 (left) and (right) years, data from Tranches: 1, 2, 3, 4, 5. CD Surfaces Dynamics

3 K H R I N H E R D L E Motivation 1-2 itraxx Spread Surface Spread 2 Spread Tranche 2 5 Time to Maturity 5 15 Tranche 2 5 Time to Maturity Figure 2: Spreads of tranches of all series observed on 2899 (left) and (right). CD Surfaces Dynamics

4 K H R I N H E R D L E Motivation 1-3 Research Goals Modelling the dynamics of CD surfaces spread surfaces base correlation surfaces pplications in trading CD Surfaces Dynamics

5 K H R I N H E R D L E Motivation 1-4 Dynamic Semiparametric Factor Model pplications: 1. Implied volatility surfaces in M. R. Fengler, W. Härdle and E. Mammen, JFE (27) and B. Park, E. Mammen, W. Härdle, and S. Borak, JS (29) 2. Risk neutral densities in E. Giacomini, W. Härdle, and V. Krätschmer, St (29) 3. Limit order book in W. Härdle, N., Hautsch, and. Mihoci, JEF (212) 4. Variance swaps in K. Detlefsen and W. Härdle, QF (213) 5. fmri images in. Myšicková, S. Song, P. Majer, P. Mohr, H. Heekeren, W. Härdle, Psychometrika (213) CD Surfaces Dynamics

6 K H R I N H E R D L E utline 1. Motivation 2. CDs 3. DSFM 4. Empirical Study 5. pplications 6. Conclusions CD Surfaces Dynamics

7 K H R I N H E R D L E CDs 2-1 Risk Transfer CD Surfaces Dynamics

8 K H R I N H E R D L E CDs 2-2 itraxx Europe static portfolio of 125 equally weighted CDS on European entities; Sectors: Consumer (3), Financial (25), TMT (2), Industrials (2), Energy (2), uto (); New series of itraxx Europe issued every 6 months (March and September) and the underlying reference entities are reconstituted; Maturities: 3Y, 5Y, 7Y, Y. CD Surfaces Dynamics

9 K H R I N H E R D L E CDs 2-3 Gaussian Copula Model Default times are modelled from the Gaussian vector (X 1,..., X d ) : X i = ρy + 1 ρz i, where Y (systematic risk factor), {Z i } d i=1 are i.i.d. N(, 1). Hence: (idiosyncratic risk factors) with (X 1,..., X d ) N(, Σ), 1 ρ ρ ρ 1 ρ Σ = ρ ρ 1 CD Surfaces Dynamics

10 K H R I N H E R D L E CDs 2-4 Large Portfolio Framework ssume that obligors have the same default probability and LGD, one dependence parameter ρ, d very large. Computations are simplified significantly when the portfolio loss distribution is approximated: { 1 ρφ 1 (x) Φ 1 } (p) P( L x) = Φ. ρ CD Surfaces Dynamics

11 K H R I N H E R D L E CDs 2-5 Correlation s Types Compound correlation ρ(l j, u j ), j = 1,..., J. Compound Correlation.4 Equity Mezzanine Junior Mezzanine Super Senior Senior Tranches Figure 3: Implied correlation smile in the Gaussian one factor model, CD Surfaces Dynamics

12 K H R I N H E R D L E CDs 2-6 Correlation s Types Base correlation (BC) ρ(, u j ), j = 1,..., J. Represent the expected loss E{L (lj,u j )} as a difference: E{L (lj,u j )} = E ρ(,uj ){L (,uj )} E ρ(,lj ){L (,lj )}, j = 2,..., J. of the expected losses of two fictive tranches (, u j ) and (, l j ). Bootstrapping process: E{L (,3%) } is traded on the market, E{L (3%,6%) } = E ρ(,6%) {L (,6%) } E ρ(,3%) {L (,3%) }, E{L (6%,9%) } = E ρ(,9%) {L (,9%) } E ρ(,6%) {L (,6%) },... CD Surfaces Dynamics

13 K H R I N H E R D L E CDs 2-7 Base Correlations over Time 1 Series 5 Maturity 5 1 Series 5 Maturity ρ t ρ t Time Time Figure 4: BC of itraxx tranches, Series 5, maturity 5 (left) and (right) years, data from Tranches: 1, 2, 3, 4, 5. CD Surfaces Dynamics

14 K H R I N H E R D L E DSFM 3-1 Base Correlation Surfaces 1 1 BC.5 BC Time to Maturity Tranche Time to Maturity Tranche Figure 5: Implied base correlations on day 2899 (left) and (right). CD Surfaces Dynamics

15 K H R I N H E R D L E DSFM 3-2 Dynamic Semiparametric Factor Model L Y t,k = m (X t,k ) + Z t,l m l (X t,k ) + ε t,k = Zt ψ(x t,k ) + ε t,k l=1 Y t,k k X t,k m l Z t,l ψ(x t,k ) log-spreads and Z-transformed BC on day t, t = 1,..., T intra-day numbering of BCs on day t, k = 1,..., K t two-dimensional vector of the tranche seniority and the time-to-maturity factor functions, time invariant, nonparametric estimation time series, l =,..., L, dynamic behavior tensor B-spline basis coefficient matrix CD Surfaces Dynamics

16 K H R I N H E R D L E DSFM 3-3 Estimation Using an iterative algorithm: (Ẑt, Â) = arg min T K t Z t, t=1 k=1 { } 2 Y t,k Zt ψ(x t,k ) Selection of L, the numbers of spline knots R 1, R 2 and the orders of splines k 1, k 2 by maximising the explained variance criterion: EV(L, R 1, r 1, R 2, r 2 ) = 1 where m is an empirical mean surface. T { Kt t=1 k=1 Y t,k } 2 L l=1 Z t,lm l (X t,k ) T Kt t=1 k=1 {Y t,j m (X t,k )} 2, CD Surfaces Dynamics

17 K H R I N H E R D L E DSFM 3-4 DSFM without the Mean Factor Reduce the number of factors estimated in the iterative algorithm by first subtracting the empirical mean m and then fitting the DSFM: Y t,k = m (X t,k )+ L Z t,l m l (X t,k )+ε t,k = m (X t,k )+Zt ψ(x t,k )+ε t,k, l=1 where m l are new factor functions, l = 1,..., L. CD Surfaces Dynamics

18 K H R I N H E R D L E Empirical Study 4-1 Data Series 2- Maturities 5, 7, Y 4 days between data points Year 3Y 5Y 7Y Y ll Table 1: Number of observed values of itraxx tranches in the period CD Surfaces Dynamics

19 K H R I N H E R D L E Empirical Study 4-2 Data Preparation Convert the upfront payment quotes of the equity tranche to standard spreads using the Gaussian copula model. Since the data are monotone in the tranche seniority direction and positive, use log-spreads and Z-transformed-BC Time Figure 6: Daily number of curves for every surface during the period CD Surfaces Dynamics

20 K H R I N H E R D L E Empirical Study 4-3 DSFM for Z-transformed-BC EV.93 EV R 2 r 2.99 EV L CD Surfaces Dynamics Figure 7: Proportion of the explained variance as a function of R 2 (up left) with r 2 = 2, as a function of r 2 (up right) with R 2 =, as a function of L (down) for L = 1, L = 2, L = 3, r 1 = 2 and R 1 = 5.

21 K H R I N H E R D L E Empirical Study 4-4 DSFM w/o Mean F. for Z-transformed-BC m.5 m Tranche τ Tranche τ m 2 Z t Tranche τ Time Figure 8: Estimated factor functions and loadings (Ẑt,1, Ẑt,2). CD Surfaces Dynamics pp

22 K H R I N H E R D L E Empirical Study 4-5 DSFM Estimation Results For DSFM for both data types Ẑt,1 is a slope-curvature factor Ẑt,2 is a shift factor Model Log-Spr Z-BC DSFM.16.4 DSFM w/o mean f Table 2: Mean squared error of the in-sample fit. CD Surfaces Dynamics

23 K H R I N H E R D L E Empirical Study 4-6 DSFM without the mean factor Fit Figure 9: In-sample fit of the models to data on 2899 and CD Surfaces Dynamics

24 K H R I N H E R D L E pplications 5-1 Curve Trades So, how can I make money with this? Combine tranches of different time to maturity, see Felsenheimer et al. (24) and Kakodkar et al. (26): Flattener sell a long-term tranche, buy a short-term tranche Example: sell Y 3-6% and buy 5Y 6-9% utlook: bullish long-term, bearish short-term Steepener opposite trade CD Surfaces Dynamics

25 K H R I N H E R D L E pplications 5-2 JP Morgan Trading Loss, May 212 J.P. Morgan s flattener bought 5Y CDX IG 9 index, sold Y CDX IG 9 index in a 3:1 ratio. The final loss reached $6.2 billion. CD Surfaces Dynamics

26 K H R I N H E R D L E pplications 5-3 Flattener Sell protection at s 1 (t ) for the period [t, T 1 ] and buy protection at s 2 (t ) for [t, T 2 ], T 1 > T 2. t t for l = 1, 2: T l MTM l (t )= β(t, t) [s l (t ) te{f l (t)} E{L l (t) L l (t t)}]=. t=t 1 t t > t, the market quotes s l ( t) and MTM l ( t) = {s l (t ) s l ( t)} β( t, t) te{f l (t)}. t= t 1 T l CD Surfaces Dynamics

27 K H R I N H E R D L E pplications 5-4 Curve Trade positive MTM means a positive value to the protection seller. If the protection seller closes the position at time t, then receives from the protection buyer MTM l ( t). Flattener-trader aims to maximize the total MTM value PL( t) = MTM 1 ( t) MTM 2 ( t). CD Surfaces Dynamics

28 K H R I N H E R D L E pplications 5-5 Risk in Curve Trades If one buys 5Y 6-9% and sells Y 6-9%, then the trade is hedged for default until the maturity of the 5Y tranche. Defaults that emerge from Y 6-9% are covered by 5Y 6-9% till it expires. Series differ in the composition of the collateral. If one buys 5Y 6-9% and sells Y 3-6%, then these tranches provide protection of different portion of portfolio risk. If there is any default in Y 3-6%, then we must deliver a payment obligation and incur a loss. CD Surfaces Dynamics

29 K H R I N H E R D L E pplications 5-6 Empirical Study Idea Use DSFM to forecast spread and BC surfaces Calculate forecasted MTM surfaces Recover those tranches that maximise P&L Remarks Because of many missing data and short data histories, the standard econometric methods cannot be used for the forecasting. Consider trades that generate no or a positive carry the spread of the long tranche doesn t exceed the spread of the short tranche. Do not account for default payments (no data of historical defaults in itraxx), do not account for the positive carry. CD Surfaces Dynamics

30 K H R I N H E R D L E pplications 5-7 Forecasting with DSFM in Rolling Windows Let Y t be log-spreads or Z-transformed-BC. Consider a rolling window of w = 25. Estimate the DSFMs using {Y ν } t ν=t w+1 for t = w,..., T h. s a result, we get T w + 1 times m = ( m,..., m L ) and Ẑ t = (Ẑt,,..., Ẑt,L) of length w. Compute h-day forecast of the factor loadings using VR. Due to the fixed issuing scheme, X t+h,k is not forecasted. Calculate the forecast Ŷt+h from the forecast Ẑt+h. Transform Ŷt+h suitably to get ŝ(t + h) or ˆρ(t + h). CD Surfaces Dynamics

31 K H R I N H E R D L E pplications 5-8 Forecasting MTM Surfaces For predicted {ŝ k (t), ˆρ k (t)}, t =w +h,..., T, k =1,..., K t, compute MTM k (t), where the initial spread s k (t ) is observed on t =t h. Figure : MTM surfaces on 2899 (left) and (right) calculated using one-day spread and BC predictions obtained with the DSFM. CD Surfaces Dynamics

32 K H R I N H E R D L E pplications 5-9 Transaction Costs Calculate the ask (bid) spread by increasing (reducing) the observed spread by the following percentage: Maturity Y Y Y Table 3: verage bid-ask spread excess over the mid spread as a percentage of the mid spread for Series 8 during the period CD Surfaces Dynamics

33 K H R I N H E R D L E pplications 5- Trading Strategies Construct a curve trade 1. Fit and forecast the DSFM models to spreads and BC. 2. Calculate h-day forecasts of the MTM surfaces. 3. Recover which two tranches optimize a given strategy. Strategies restrict the choice to a flattener (or a steepener) with 1. a fixed tranche and fixed maturities, 2. a fixed tranche and all maturities, 3. all tranches and fixed maturities, 4. all tranches and all maturities (no restrictions), or allow to combine flatteners and steepeners. CD Surfaces Dynamics

34 K H R I N H E R D L E pplications 5-11 Backtesting Consider the time horizons h = 1, 5, 2 days. For the tranches that optimize a given strategy, check the corresponding historical market spreads, calculate the resulting MTM values, and the realised P&L. CD Surfaces Dynamics

35 K H R I N H E R D L E pplications 5-12 Mean of Daily Gains in Percent DSFM DSFM without the mean factor Strategy 1 day 1 week 1 month 1 day 1 week 1 month LZ Z LZ Z LZ Z LZ Z LZ Z LZ Z FS-llT-llM FS-T2-llM FS-T3-llM FS-T4-llM FS-T5-llM F-T2-llM F-T3-llM F-T4-llM F-T5-llM S-T2-llM S-T3-llM S-T4-llM S-T5-llM F-llT F-llT F-llT S-llT S-llT S-llT Table 4: Calculations based on predictions of log-spreads and Z-transformed BCs marked as LZ; based only on Z-transformed BCs marked as Z. CD Surfaces Dynamics

36 K H R I N H E R D L E pplications 5-13 Investor s Strategy Follow a certain strategy over a year and constantly rebalance the portfolio. t t enter an optimal (according to the DSFM) curve trade for h-day horizon. t t + h chose: 1. keep the current position for the next h-days, 2. close the current position and enter a new one. ssume a margin of % of your notional. Every time the position is closed, add to the margin the realized P&L. If margin, quit the trade. CD Surfaces Dynamics

37 K H R I N H E R D L E pplications 5-14 Investor s Strategy Final PL in % 2 Final PL in % 2 Final PL in % Time Time CD Surfaces Dynamics Time Figure 11: Combined flatteners and steepeners from all tranches and all maturities. Closing profits after one year. Rebalancing after: 1 day (upper left), 1 week (upper right), 1 month (lower). Calculations based on the DSFM predictions of logspreads and Z-transformed BCs.

38 K H R I N H E R D L E pplications 5-15 Investor s Strategy Cumulated PL in % Time Cumulated PL in % Time CD Surfaces Dynamics Cumulated PL in % Time Figure 12: Daily cumulated P&L over one year Rebalancing after: 1 day (upper left), 1 week (upper right), 1 month (lower). Calculations based on the DSFM predictions of logspreads and Z-transformed BCs.

39 K H R I N H E R D L E Summary 6-1 Conclusions Investigated evolution over time of tranche spread surfaces and base correlation surfaces using the DSFM. Empirical study is conducted using an extensive data set of 49,52 observations of itraxx Europe tranches in Proposed a modification to the classic DSFM. Both DSFMs successfully reproduce the dynamics in data. Used DSFM in constructing the curve trades. nalysed the performance of 43 strategies that combine different positions, tranches, and maturities. Backtesting showed high daily gains of the resulting curve trades. CD Surfaces Dynamics

40 References C. Bluhm and L. verbeck Structured Credit Portfolio nalysis, Baskets and CDs Chapman & Hall/Crc Financial Mathematics Series, 26 J. Felsenheimer, P. Gisdakis, and M. Zaiser DJ itraxx: Credit at its best! Credit derivatives special HVB Corporates & Markets, 24 M. R. Fengler, W. K. Härdle, and E. Mammen semiparametric factor model for implied volatility surface dynamics Journal of Financial Econometrics, 27 C. Gourieroux and J. Jasiak Dynamic factor models Econometric Reviews, 21. Kakodkar, S. Galiani, J. G. Jónsson, and. Gallo Credit derivatives handbook, guide to the exotics credit derivatives market Technical report Merrill Lynch, 26 B. Park, E. Mammen, W. K. Härdle, and S. Borak Dynamic Semiparametric Factor Models Journal of the merican Statistical ssociation, 29

41 D Surfaces Dynamics Barbara Choroś-Tomczyk Wolfgang Karl Härdle stap khrin C K H R I N H E R D L E Ladislaus von Bortkiewicz Chair of Statistics C..S.E. - Center for pplied Statistics and Economics Humboldt-Universität zu Berlin

42 K H R I N H E R D L E ppendix 7-1 DSFM for Log-Spreads.2 m m Tranche τ Tranche τ m 2 Z t Tranche τ Time Figure 13: Estimated factor functions and loadings (Ẑt,1, Ẑt,2). CD Surfaces Dynamics Talk

43 K H R I N H E R D L E ppendix 7-2 DSFM without the Mean Factor for Log-Spreads.2 m 5 m Tranche τ Tranche τ m 2 Z t Tranche τ Time Figure 14: Estimated factor functions and loadings (Ẑt,1, Ẑt,2). CD Surfaces Dynamics Talk

44 K H R I N H E R D L E ppendix 7-3 DSFM for Z-transformed-BC.2 m.2 m Tranche τ Tranche τ 8 3 m Tranche τ 8 Z t Time Figure 15: Estimated factor functions and loadings (Ẑt,1, Ẑt,2). CD Surfaces Dynamics Talk