High-Dimensional Time Series Modeling for Factors Driving Volatility Strings

Transcription

1 for Factors Driving Volatility Strings Julius Mungo Institute for Statistics and Econometrics CASE - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin

2 ACF- ACF-z2 ACF- Motivation implied volatility moneyness time to maturity Figure 1: An implied volatility data design for DAX on Solid lines indicate the observed maturities, which move towards expiry.

3 ACF- ACF-z2 ACF- Motivation 1-2 Dimension reduction For practical application a low-dimensional representation of the IVS is required, we apply a dimension reduction technique based on dynamic semiparametric factor models (DSFM), to obtain time varying factor loadings series that describe the IVS dynamics.

4 ACF- ACF-z2 ACF- Motivation 1-3 Objectives 1. to understand the dynamics of the factor loadings, what may deliver insights in the behavior of the IVS, to provide better assessment of market risk. 2. to provide insight into the nature of the underlying economic forces that drive its movements, relate and interpret the factor loadings dynamics w.r.t. economic conditions. 3. to study the general stochastic structure (e.g. long memory).

5 ACF- ACF-z2 ACF- Motivation 1-4 Inference from our research may provide valuable information for: asset pricing, hedging derivatives. forecasting IV distributions (e.g. risk management). relating with macroeconomic indicators (e.g. feedback between risk factors and the economy). measuring global exposure to risk (e.g. volatility trading).

6 ACF- ACF-z2 ACF- Motivation 1-5 Outline 1. Motivation 2. Dynamic Semiparametric Factor Models the multivariate time series of factor loadings 3. PART I (Modeling the factor loadings dynamics) the model framework empirical application 4. PART II (Long-memory in the factor loadings) 5. Conclusion second-order properties of stochastic processes model framework for long-memory estimation empirical analysis

7 ACF- ACF-z2 ACF- Dynamic Semiparametric Factor Model 2-1 DSFM model (Fengler et al. 2007; Borak et al. 2008); Park et al. 2009) Y t,j = L Z t,l m l (X t,j ) + ε t,j = Z t m(x t,j) + ε t,j (1) l=0 (X t,j, Y t,j ) are observed data for j = 1,..., J t, (index of traded options on day t), t = 1,..., T. m( ) is a tuple of basis functions (m 0, m 1,..., m L ) in covariables X t,j The errors ε t,j have zero means and nite second moments Z t = (1, Z t,1,..., Z t,l ) is a multivariate time series.

8 ACF- ACF-z2 ACF- Dynamic Semiparametric Factor Model 2-2 Model estimation Based on Kernel estimator (Fengler et al., 2007), choice of model size (L) based on the residual sum of squares per total variance, choice of bandwidths based on AIC type criteria. we obtain 3 factor loadings that capture around 96% variation of the DAX IVS.

9 ACF- ACF-z2 ACF- Dynamic Data Semiparametric Plot 02/17/09 11:22:03 Factor Model Factor loadings series time z2 Figure 2: Time series plots of the estimated factor loadings from the DSFM 1.0 t for the DAX -Option from to

10 ACF- ACF-z2 ACF- Dynamic Semiparametric Factor Model 2-4 Interpretation: factor loadings series Z t,1 represents overall level shift (mean of log-ivs) of the IVS. Z t,2 represents terms structure changes (maturity slope). Z t,3 represent changes in the IVS curvature (moneyness slope).

11 ACF- ACF-z2 ACF- PART I (Modeling the factor loadings dynamics) 3-1 PART I (Modeling the factor loadings dynamics)

12 ACF- ACF-z2 ACF- PART I (Modeling the factor loadings dynamics) 3-2 The VAR model framework Z t = ν + A 1 Z t A p Z t p + ε t, (2) Z t = (Z t1, Z t2, Z t3 ), vector of loadings series. ν is L 1 vector of intercept parameters. A i, i = 1,..., p are xed L L parameter matrices. ε t = (ε t1,..., ε tl ) are innovations, with mean zero and time-invariant and non-singular covariance matrix, Σ ε = E[ε t ε t ]

13 ACF- ACF-z2 ACF- PART I (Modeling the factor loadings dynamics) 3-3 Inference, Park et al. (2009) the covariance structure of Ẑt converges in probability to the covariance structure of Z t (true unobservable) simulation studies show that this asymptotic equivalence carries over to the VAR framework justies our VAR model application with Ẑt.

14 ACF- ACF-z2 ACF- PART I (Modeling the factor loadings dynamics) 3-4 Stationarity of loadings series Unit root tests: Augmented Dickey-Fuller test: (Dickey & Fuller, 1979) refers to the regression, p Z t,k = φz t 1,k + a i Z t i,k + ε t,k (3) i=1 Point-optimal unit root test: (Elliot et al., 1996) based on quasi-dierences of Z t,k, { 1 if t = 1 d(z t,k a) = Z t,k az t 1,k if t > 1, a, is the point alternative against which unit root is tested. (4)

15 ACF- ACF-z2 ACF- PART I (Modeling the factor loadings dynamics) 3-5 Capturing dynamic relations Impulse Response analysis Generalized Impulse Response, (Pesaran & Shin, 1998) GIR(h, δ, F t 1 ) = E (Z t+h ε j,t = δ, F t 1 ) E ( Z t+h F t 1 ) it integrates out all contemporaneous and future shocks, GIR are unique and invariant to orderings of variables. Causality Analysis: X does not Granger cause Y if removing the past of X from the information set does not change the optimal forecast of Y. testing for zero restrictions of some VAR coecients.

16 ACF- ACF-z2 ACF- PART I (Modeling the factor loadings dynamics) 3-6 Empirical analysis Data set: DAX Option from January 4, 1999 to February 25, Economic indicators: returns in foreign exchange (US dollar per Euro), commodity price (Goldman Sachs), interest rate (12-months German money market) and global stock markets (U.S. S&P 500). volatility in foreign exchange (CME Eurodollar future), commodity price (Anglogold Ltd.), bond (Euro-BUND future) and global stock markets (CME S&P 500).

17 ACF- ACF-z2 ACF- PART I (Modeling the factor loadings dynamics) 3-7 The VAR model setup VAR(7) : Z t = (Z t1, Z t2, Z t3 ), factors loadings-only (sample, to ). VAR(5) : Z t = (Z t1, Z t2, Z t3, EX t, COM t, R12 t, SP500 t ), factor loadings with returns on economic indicators. VAR(8): Z t = (Z t1, Z t2, Z t3, EXV t, GOLDV t, BUNDV t, SP500V t ), factor loadings with implied volatility on economic indicators (sample, to ).

18 ACF- ACF-z2 ACF- PART I (Modeling the factor loadings dynamics) 3-8 VAR modeling results and interpretation VAR(7): Factors loadings-only model Z t1 response negatively to a shock in Z t2 and vice versa. Z t3 is not importantly related to Z t1 and Z t2. Z t1 Granger causes Z t2 and Z t3, Z t2 Granger causes Z t1 and Z t3. Z t1 and Z t2 capture systematic risk faced by a DAX option investor.

19 ACF- ACF-z2 ACF- PART I (Modeling the factor loadings dynamics) 3-9 Response to Generalized One S.D. Innovations ± 2 S.E..04 Response of Z1 to EX.04 Response of Z1 to COM.04 Response of Z1 to R12M.04 Response of Z1 to SP Response of Z2 to EX Response of Z2 to COM Response of Z2 to R12M Response of Z2 to SP Response of Z3 to EX Response of Z3 to COM Response of Z3 to R12M Response of Z3 to SP Figure 3: GIR to a unit std. dev. shock with 95% asymptotic C.I. for VAR(5): Z t = (Z t1, Z t2, Z t3, EX t, COM t, R12M t, SP500 t )

20 ACF- ACF-z2 ACF- PART I (Modeling the factor loadings dynamics) 3-10 VAR modeling results and interpretation VAR(5): loadings with returns on economic indicators. an appreciation of the Euro, a higher interest rate, a shock in SP500 t returns will induce a higher overall risk (increase uncertainty) in the German market. shocks to the factor loadings series have less eect on the SP500 t returns.

21 ACF- ACF-z2 ACF One S.D. Innovations ± 2 S.E. 24 Response to Generalized PART I (Modeling the factor 2 4 6loadings dynamics) to Generalized One S.D. Innovations ± 2 S.E. Response of SP500V to Z Response of Z1 to SP500V V to Z Response.02 of SP500V to Z2 Re.008 Response of SP500V Response to Z2 of Z1 to SP500V Response of Response SP500V to of Z3 Z2 to SP500V Response of Z3 to SP500V Response to Generalized One S.D. Innovations ± 2 S.E. Response of SP500V to Z1 Response of SP500V Response to of Z2Z1 to SP500V Response of Response SP500V of to Z2 Z3to SP500V Re to Z2 Response.008 of SP500V Response to of Z3Z2 to SP500V Response.02 of Z3 to SP500V to Z3 Response of SP500V to Z3 Response of Z3 to SP500V Figure 4: GIR Response Response to of of SP500V Z3 ato unit SP500V to Z2 std..008 dev. shock Response with of Z2 to 95% SP500V asymptotic.0015 C.I for VAR(8):.0010 Z t = (Z t1, Z t2,.004 Z t3, EXV t, GOLDV t, BUNDV t, SP500V t ) High-Dimensional.000 Time Series Modeling

22 ACF- ACF-z2 ACF- PART I (Modeling the factor loadings dynamics) 3-12 VAR modeling results and interpretation VAR(8): factor loadings with IV of economic indicators S&P500 implied volatility is negatively link with changes in the overall risk level (Z t1 ) of the DAX, this may reect investment strategy of international investors, higher investments may invoke high liquidity, lowering risk. factor loadings interaction with foreign exchange, money and bond markets is relatively weak or even insignicant with commodity markets.

23 ACF- ACF-z2 ACF- PART I (Modeling the factor loadings dynamics) 3-13 Measuring global exposures to risk Dependency between nancial markets: DAX and KOSPI a positive shock in the DAX induces a small but positive response (1-2 days) in the KOSPI. response of the DAX to a shock in the KOSPI is statistically insignicant. a higher overall risk in the German market does not translate statistically to greater uncertainty in Korean market.

24 ACF- ACF-z2 ACF- PART II (Long-memory in the factor loadings) 4-1 PART II (Long Memory in the Factor Loadings)

25 ACF- ACF-z2 ACF- 0.5 PART II (Long-memory in the factor loadings) Motivation: an important stylized fact 1.00 acf lag ACF- ACF-z2 ACF- Figure 5: Autocorrelation function (ACF ) plots for the loadings series up to 120 lags.

26 ACF- ACF-z2 ACF- PART II (Long-memory in the factor loadings) 4-3 Motivation: Economic consequences option pricing and forecast performance. better option pricing and forecast properties (Bollerslev & Mikkelsen, 1999; Tsay et al., 2008). trading and investment strategies, long-term investors can hold more equity on the average than short-term investors (Campbell & Viceira, 2005). risk management application (VaR risk measure), ignoring long memory in volatility may underestimate the variation of VaR estimate, (J. Taylor, 2002).

27 ACF- ACF-z2 ACF- PART II (Long-memory in the factor loadings) 4-4 The basic long-memory model framework (1 L) d z t = ε t, (5) ε t (0, σ 2 ε), L is the lag operator, d is the degree of memory in the process, (1 L) d = 1 dl 1 2 d(1 d)l2 1 6 d(1 d)(2 d)l3... For 0 < d < 1, z t is a fractional integrated process, it allows for more exibility than unit root processes, (Robinson, 2003).

28 ACF- ACF-z2 ACF- PART II (Long-memory in the factor loadings) 4-5 Characterizing long-memory processes Time domain: for long-memory the autocorrelations ρ(k) decay at a hyperbolic rate such that, T lim ρ(k) =. (6) T k= T Frequency domain: for long-memory the spectral density, f (λ) = (2π) 1 k= {γ k = cov(z t, z t+k )} has a pole at the origin. e ikλ γ(k) (7)

29 ACF- ACF-z2 ACF- PART II (Long-memory in the factor loadings) 4-6 Characterizing long-memory processes Self-similar processes by scaling law (L ), z t, (t 0) is self-similar if there exist H(Hurst exponent) > 0 such that for c > 0, L (z ct ) = L (c H z t ) (8) in the sense of nite-dimensional distributions. we estimate H by regression based on the approximation, ρ(k) c ρ k (2 2H) for large k and H (0.5, 1) (9) f (λ) c f λ (2H 1) for small frequencies λ. (10) H = d (11)

30 ACF- ACF-z2 ACF- PART II (Long-memory in the factor loadings) 4-7 Test for long-memory Rescale Variance test: (Giraitis et. al, 2003) test statistic, T k V /S(q) = T 2ˆσ 2 (q) (X j X T ) 1 T k (X j X T ), T T k=1 j=1 k=1 j=1 k j=1 (X j X T ) are partial sums of the observations, ˆσ 2 (q) = ˆγ T ( ) q j=1 1 j ˆγ 1+q j is the (Newey-West, 1994) HAC estimator of the variance at truncation lag q. ˆγ 0 is the variance of the process. (12)

31 ACF- ACF-z2 ACF- PART II (Long-memory in the factor loadings) 4-8 Identifying true from spurious long-memory Stress tests: method of Randomized Buckets, randomizing parts of a time series to independently control the amount of correlation at dierent scales, z t is partitioned into sets of buckets of size b and the bucket items are classied as in-bucket and out-bucket pair, bucket randomization algorithms are applied to decouple short-range from Long-range dependence. (Karagiannis, 2002)

32 ACF- ACF-z2 ACF- PART II (Long-memory in the factor loadings) 4-9 Bucket randomization algorithms: Internal randomization: the order of the buckets remains unchanged while the contents of each bucket are randomized. this removes short-term, while preserving long term correlation if the ACF of the internal randomized series shows slow hyperbolic rate of decay, then the original series exhibits long memory.

33 ACF- ACF-z2 ACF- PART II (Long-memory in the factor loadings) 4-10 Bucket randomization algorithms: External randomization: the order of buckets is randomized, whereas the content of each bucket remains intact. this removes long-term correlation, preserving short-term correlation if the ACF of the external randomized series shows no signicant correlation beyond the chosen bucket size b, then the original series exhibits long memory.

34 ACF- ACF-z2 ACF- Long-memory models 5-1 Long-memory modeling Model setup The ARFIMA(p, d m, q) model, (Granger and Joyeux, 1980; Hosking, 1981). α(l)(1 L) dm (z t µ) = β(l)ε t, (13) α(l) = 1 α 1 L α p L p, β(l) = 1 + β 1 L + + β q L q, d m ( 0.5, 0.5), ε t i.i.d(0, σ 2 ε). exact maximum likelihood, (Doornik & Ooms, 2004), with skewed Student-t errors, (Lambert & Laurent, 2001).

35 ACF- ACF-z2 ACF- Long-memory models 5-2 Long-memory modeling The FIGARCH(1, d v, 1) model, (Baillie et al., 1996). the conditional volatility, σ 2 t = { } ω 1 β(l) + α(l)(1 L)dv 1 ε 2 t, (14) 1 β(l) d (0, 1), ω > 0 and α, β < 1. quasi- maximum likelihood (Bollerslev & Wooldridge, 1992), with skewed Student-t errors.

36 ACF- ACF-z2 ACF- Long-memory models 5-3 Long-memory modeling The ARFIMA(1, d m, 1)-FIGARCH(1, d v, 1) model the conditional volatility, σ 2 t = { } ω 1 θ(l) + φ(l)(1 L)dv 1 ε 2 t (15) 1 θ(l) d m and d v capture the degree of persistence in the mean and variance respectively. quasi- maximum likelihood, with skewed Student-t errors.

37 ACF- ACF-z2 ACF- Empirical analysis 6-1 Empirical analysis Long-memory test result q z Table 1: Rescale variance V /S test for I (0) against I (d). q is the truncation lag. at 5% sig. level (critical value,0.1869), results indicate long-memory in the factor loadings series.

38 ACF- ACF-z2 ACF- Empirical analysis 6-2 Bucket randomization results C:\Dokumente und Einstellungen\Mungo\Desktop\BUCKETING\-bucket.gwg 02/01/09 16:15:09 ACF C:\Dokumente und Einstellungen\Mungo\Desktop\BUCKETING\z2-bucket.gwg 02/01/09 16:21: ACF- z C:\Dokumente und Einstellungen\Mungo\Desktop\BUCKETING\-bucket.gwg 02/01/09 16:25: ACF Figure 6: ACF (up to 180 lags) of factor loadings series under unrandom- Page: 1 of 1 Page: 1 of 1 ized (black), internal randomized (blue) and external randomized (red) bucketing with size b = 80.

39 ACF- ACF-z2 ACF- Empirical analysis 6-3 Figure 7: Self-similarity results: periodogram estimates in the log-log plane. The slope of the tted regression line yields an estimated Hurst coecients H, 0.74 (), 0.72 (z2), 0.73 ().

40 ACF- ACF-z2 ACF- Empirical analysis 6-4 Model forecast performance Short versus long-memory models RMSE MAE Level factor () h = 1 h = 5 h = 1 h = 5 ARMA(1, 1) GARCH(1, 1) ARFIMA(1, 0.75, 1) FIGARCH(1, 0.54, 1) ARFIMA FIGARCH Table 2: In-sample model forecast: root mean square (RMSE) and mean absolute (MAE) error for h = 1 step-ahead and h = 5 steps-ahead forecast horizon.

41 ACF- ACF-z2 ACF- Conclusions 7-1 Conclusions the 1-st and 2-nd factor loadings capture systematic risk faced by a DAX investor. the factors loadings exhibit long range dependence shocks to the volatility will persist and are likely to signicantly aect stock prices. fractional integrated models can better accommodate the long-run behavior of the loadings series in a exible way, such models should be taken into account for better pricing or risk valuation involving the DAX options.

42 ACF- ACF-z2 ACF- Conclusions 7-2 Outlook Vega-hedging application the DSFM decomposes the IV as, ( K ) σ t = exp Ẑ tk m k. (16) k=0 the sensitivities can be computed w.r.t. the factor loadings by Vega-hedging of Z t1 and Z t2 : Ẑt1 Ẑt2, an up-and-down shift vega of the IVS., a slope shift vega of the IVS.

43 ACF- ACF-z2 ACF- Conclusions 7-3 Outlook Vega-risk application Interaction between Z t1 and Z t2 may be used to assess market risk in a simplied way, develop portfolio of stocks (S t ) and risk factors Z t1, Z t2, Π (S t, Z t1, Z t2) (17) forecast its distribution in time t + t, Π(S t+ t, Z t1+ t, Z t2+ t). (18) e.g., for a Vega VaR measure.

44 ACF- ACF-z2 ACF- 8-1 THANK YOU FOR YOUR ATTENTION