. Large-dimensional and multi-scale effects in stocks volatility m

Large-dimensional and multi-scale effects in stocks volatility modeling Swissquote bank, Quant Asset Management work done at: Chaire de finance quantitative, École Centrale Paris Capital Fund Management, Paris Journées MAS Toulouse, 27 29 août 2014 Large-dimensional and multi-scale effects in stocks volatility m

Outline Introduction and definitions Large-dimensional and multi-scale effects in stocks volatility m

Definitions Volatility dynamics: clustering and memory Regressive models of conditional volatility dynamics Stationnarity in ARCH(q) Stock prices log-returns: x t = ln P t ln P t+1 Stock s volatility: a measure of typical amplitude or fluctuations Can be understood globally (distributional sense, like empirical standard deviation) or dynamically (time-varying) In this talk: time-varying volatility (several possible estimators: x t, x 2 t, rolling std-dev, Rogers-Satchell, etc) Large-dimensional and multi-scale effects in stocks volatility m

Definitions Volatility dynamics: clustering and memory Regressive models of conditional volatility dynamics Stationnarity in ARCH(q) Clustering 010 005 000 005 010 2000 2002 2004 2006 2008 2010 Large-dimensional and multi-scale effects in stocks volatility m

Definitions Volatility dynamics: clustering and memory Regressive models of conditional volatility dynamics Stationnarity in ARCH(q) Long memory: auto-correlation 000 004 008 012 0 100 200 300 400 500 τ Large-dimensional and multi-scale effects in stocks volatility m

Definitions Volatility dynamics: clustering and memory Regressive models of conditional volatility dynamics Stationnarity in ARCH(q) Long memory: auto-correlation 000 004 008 012 (x 2 t σ 2 )x 2 t τ 0 100 200 300 400 500 τ Large-dimensional and multi-scale effects in stocks volatility m

Definitions Volatility dynamics: clustering and memory Regressive models of conditional volatility dynamics Stationnarity in ARCH(q) Long memory: auto-correlation 000 004 008 012 (x 2 t σ 2 )x 2 t τ power-law fit τ β 0 100 200 300 400 500 τ Large-dimensional and multi-scale effects in stocks volatility m

Definitions Volatility dynamics: clustering and memory Regressive models of conditional volatility dynamics Stationnarity in ARCH(q) x t = σ t ξ t ξ t F ξ stochastic signed residuals (eg Student) σt 2 = F({x t τ }) positive fluctuating volatility Not stochastic vol models, rather conditionally deterministic vol See discussion on Time-reversal asymmetry later q ARCH(q): F({x t τ }) = s 2 + K(τ)x 2 t τ, τ=1 q Large-dimensional and multi-scale effects in stocks volatility m

Definitions Volatility dynamics: clustering and memory Regressive models of conditional volatility dynamics Stationnarity in ARCH(q) x t = σ t ξ t ξ t F ξ stochastic signed residuals (eg Student) σt 2 = F({x t τ }) positive fluctuating volatility Not stochastic vol models, rather conditionally deterministic vol See discussion on Time-reversal asymmetry later q ARCH(q): F({x t τ }) = s 2 + K(τ)x 2 t τ, q τ=1 Leverage: F({x t τ }) = s 2 + L(τ)x t τ + K(τ)x 2 t τ, L < 0 τ>0 τ>0 Large-dimensional and multi-scale effects in stocks volatility m

Definitions Volatility dynamics: clustering and memory Regressive models of conditional volatility dynamics Stationnarity in ARCH(q) x t = σ t ξ t ξ t F ξ stochastic signed residuals (eg Student) σt 2 = F({x t τ }) positive fluctuating volatility Not stochastic vol models, rather conditionally deterministic vol See discussion on Time-reversal asymmetry later ARCH(q): F({x t τ }) = s 2 + q K(τ)x 2 t τ, τ=1 Leverage: F({x t τ }) = s 2 + τ>0 QARCH: F({x t τ }) = s 2 + τ>0 q L(τ)x t τ + τ>0 K(τ)x 2 t τ, L < 0 L(τ)x t τ + τ,τ >0 K(τ, τ )x t τ x t τ Large-dimensional and multi-scale effects in stocks volatility m

Definitions Volatility dynamics: clustering and memory Regressive models of conditional volatility dynamics Stationnarity in ARCH(q) x t = σ t ξ t ξ t F ξ stochastic signed residuals (eg Student) σt 2 = F({x t τ }) positive fluctuating volatility σt 2 = s 2 + = s 2 + σt 4 = q K(τ) x 2 t τ τ=1 q K(τ) σt τ 2 ξt τ 2 τ=1 need to be finite In particular, Tr K < 1/ ξ 2 = 1 Large-dimensional and multi-scale effects in stocks volatility m

Parameter space and criticality Outline Introduction and definitions Large-dimensional and multi-scale effects in stocks volatility m

Parameter space and criticality q = 1 q = 32 q = g 00 02 04 06 08 10 00 02 04 06 08 10 00 02 04 06 08 10 00 05 10 15 20 25 30 α 00 05 10 15 20 25 30 α 00 05 10 15 20 25 30 α Figure: Allowed region in the α, g space for K(τ, τ) = g τ α 1 {τ q} and L(τ) = 0, according to the finiteness of σ 2 and σ 4 Divergence of σ 2 is depicted by 45 (red) hatching, while divergence of σ 4 is depicted by 45 (blue) hatching In the wedge between the dashed blue and solid red lines, σ 2 < while σ 4 diverges Large-dimensional and multi-scale effects in stocks volatility m

Parameter space and criticality Critical α c 1376 where σ 4 diverges as soon as σ 2 diverges Large-dimensional and multi-scale effects in stocks volatility m

Parameter space and criticality Critical α c 1376 where σ 4 diverges as soon as σ 2 diverges a long-ranged power-law decaying correlation function (0 < β < 1) can be obtained theoretically with a power-law volatility-feedback kernel with exponent α = (3 β)/2 (1, 15) Large-dimensional and multi-scale effects in stocks volatility m

Parameter space and criticality Critical α c 1376 where σ 4 diverges as soon as σ 2 diverges a long-ranged power-law decaying correlation function (0 < β < 1) can be obtained theoretically with a power-law volatility-feedback kernel with exponent α = (3 β)/2 (1, 15) Empirically, the estimated (exponentially truncated) power-law kernel is found to have g 0081 and α 111 Large-dimensional and multi-scale effects in stocks volatility m

Parameter space and criticality q = 00 02 04 06 08 10 00 05 10 15 20 25 30 α Large-dimensional and multi-scale effects in stocks volatility m

Spectral interpretation of QARCH Examples of non-diagonal quadratic kernels Model calibration Outline Introduction and definitions Large-dimensional and multi-scale effects in stocks volatility m

Spectral interpretation of QARCH Examples of non-diagonal quadratic kernels Model calibration F({x t τ }) = s 2 + τ,τ >0 K(τ, τ )x t τ x t τ ( ) q λ n v n (τ )v n (τ ) r t τ r t τ n τ,τ =1 n λ n r v n 2 t The square volatility σ 2 t picks up contributions from various past returns eigenmodes The modes associated to the largest eigenvalues λ are those which have the largest contribution to volatility spikes Large-dimensional and multi-scale effects in stocks volatility m

Spectral interpretation of QARCH Examples of non-diagonal quadratic kernels Model calibration Examples of non-diagonal quadratic kernels (0) ARCH(q): purely diagonal [engle1982autoregressive, bollerslev1986generalized, bollerslev1994arch] K(τ, τ ) q σt 2 = s 2 + K(τ, τ)x 2 t τ τ=1 Large-dimensional and multi-scale effects in stocks volatility m

Spectral interpretation of QARCH Examples of non-diagonal quadratic kernels Model calibration Examples of non-diagonal quadratic kernels (1) Correlation between past 1-day returns and q-days weighted trends K(τ, τ ) q σt 2 = ARCH + x t 1 k LT (τ)x t τ τ=1 Large-dimensional and multi-scale effects in stocks volatility m

Spectral interpretation of QARCH Examples of non-diagonal quadratic kernels Model calibration Examples of non-diagonal quadratic kernels (2) Past squared 2-days returns over q lags K(τ, τ ) q 2 σt 2 = ARCH + k 2 (τ)[r (2) t τ ]2 where R (l) t τ=0 l τ=1 x t τ Large-dimensional and multi-scale effects in stocks volatility m

Spectral interpretation of QARCH Examples of non-diagonal quadratic kernels Model calibration Examples of non-diagonal quadratic kernels (3) Squared last l-days trends [borland2005multi] K(τ, τ ) σ 2 t = ARCH + q l=1 k BB (l)[r (l) t ] 2 Large-dimensional and multi-scale effects in stocks volatility m

Spectral interpretation of QARCH Examples of non-diagonal quadratic kernels Model calibration Examples of non-diagonal quadratic kernels (4) Correlations between past l-days trends [zumbach2010volatility] K(τ, τ ) q/2 σt 2 = ARCH + k Z (l)r (l) t R (l) t l l=1 Large-dimensional and multi-scale effects in stocks volatility m

Spectral interpretation of QARCH Examples of non-diagonal quadratic kernels Model calibration Estimation methods Method of Moments pros: no distributional hypothesis, computationally easy (inverting a linear system) cons: very noisy, in particular with high moments Maximum Likelihood pros: does not rely on noisy moment estimates cons: need to specify a residual distribution, emphasis on the core of the distribution, computationally (very) intensive Large-dimensional and multi-scale effects in stocks volatility m

Spectral interpretation of QARCH Examples of non-diagonal quadratic kernels Model calibration Estimation methods Method of Moments pros: no distributional hypothesis, computationally easy (inverting a linear system) cons: very noisy, in particular with high moments Maximum Likelihood pros: does not rely on noisy moment estimates cons: need to specify a residual distribution, emphasis on the core of the distribution, computationally (very) intensive Compromise: one-step ML with GMM prior Large-dimensional and multi-scale effects in stocks volatility m

Spectral interpretation of QARCH Examples of non-diagonal quadratic kernels Model calibration Dataset Daily stock prices for N = 280 names: universality hypothesis Present in the SP500 index during 2000 2009 (T = 2515 days) Removing market low-frequency fluctuations (separate calibration for volatility of the index) Large-dimensional and multi-scale effects in stocks volatility m

Spectral interpretation of QARCH Examples of non-diagonal quadratic kernels Model calibration Results 5 10 15 20 5 10 15 001 0005 0 0005 001 000 002 004 006 008 0 20 40 60 80 100 τ 20 Large-dimensional and multi-scale effects in stocks volatility m

Spectral interpretation of QARCH Examples of non-diagonal quadratic kernels Model calibration Results 5 10 15 20 5 10 15 20 001 0005 0 0005 001 no evident structure diagonal dominates still significant off-diag content Large-dimensional and multi-scale effects in stocks volatility m

Time-reversal asymmetry Conclusion References Outline Introduction and definitions Large-dimensional and multi-scale effects in stocks volatility m

Time-reversal asymmetry Conclusion References Explicitly backward looking construction: σ 2 t = F({x t τ }) Large-dimensional and multi-scale effects in stocks volatility m

Time-reversal asymmetry Conclusion References Explicitly backward looking construction: σ 2 t = F({x t τ }) Generates a too large Time-reversal asymmetry: C (2) (l) = (σ 2 t σ 2 )x 2 t l 00 05 10 15 (τ) = τ t=1 C~(2) (t) C ~(2) ( t) 005 003 001 000 5 10 15 20 010 015 020 025 030 (τ) = τ t=1 C~(2) (t) C ~(2) ( t) 005 003 001 000 5 10 15 20 0 5 10 15 20 τ 5 10 15 20 τ Large-dimensional and multi-scale effects in stocks volatility m

Time-reversal asymmetry Conclusion References Explicitly backward looking construction: σ 2 t = F({x t τ }) Generates a too large Time-reversal asymmetry: C (2) (l) = (σ 2 t σ 2 )x 2 t l 00 05 10 15 (τ) = τ t=1 C~(2) (t) C ~(2) ( t) 005 003 001 000 5 10 15 20 010 015 020 025 030 (τ) = τ t=1 C~(2) (t) C ~(2) ( t) 005 003 001 000 5 10 15 20 0 5 10 15 20 τ 5 10 15 20 τ Put in more randomness: ARCH mechanism + TRI stochastic volatility! Large-dimensional and multi-scale effects in stocks volatility m

Time-reversal asymmetry Conclusion References Conclusions: Extensions: Large-dimensional and multi-scale effects in stocks volatility m

Time-reversal asymmetry Conclusion References Conclusions: Large-dimensional requirements and criticality Extensions: Large-dimensional and multi-scale effects in stocks volatility m

Time-reversal asymmetry Conclusion References Conclusions: Large-dimensional requirements and criticality sub-dominant but statistically significant Extensions: Large-dimensional and multi-scale effects in stocks volatility m

Time-reversal asymmetry Conclusion References Conclusions: Large-dimensional requirements and criticality sub-dominant but statistically significant Feedback structure not obvious Extensions: Large-dimensional and multi-scale effects in stocks volatility m

Time-reversal asymmetry Conclusion References Conclusions: Large-dimensional requirements and criticality sub-dominant but statistically significant Feedback structure not obvious Extensions: specific day/night self- and cross-excitement effects Large-dimensional and multi-scale effects in stocks volatility m

Time-reversal asymmetry Conclusion References Conclusions: Large-dimensional requirements and criticality sub-dominant but statistically significant Feedback structure not obvious Extensions: specific day/night self- and cross-excitement effects similarities with Hawkes modelling Large-dimensional and multi-scale effects in stocks volatility m

Introduction and definitions Pierre Blanc,, and Jean-Philippe Bouchaud The fine structure of volatility feedback II: Overnight and intra-day effects Physica A: Statistical Mechanics and its Applications, 402:58 75, 2014 Tim Bollerslev Generalized autoregressive conditional heteroskedasticity Journal of Econometrics, 31(3):307 327, 1986 Tim Bollerslev, Robert F Engle, and Daniel B Nelson ARCH models, pages 2959 3038 Volume 4 of Engle and McFadden [engle1986handbook], 1994 Lisa Borland and Jean-Philippe Bouchaud On a multi-timescale statistical feedback model for volatility fluctuations The Journal of Investment Strategies, 1(1):65 104, December 2011 and Jean-Philippe Bouchaud The fine-structure of volatility feedback I: Multi-scale self-reflexivity Physica A: Statistical Mechanics and its Applications, 410:174 195, 2014 Robert F Engle Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation Econometrica: Journal of the Econometric Society, pages 987 1007, 1982 Robert F Engle and Daniel L McFadden, editors Handbook of Econometrics, volume 4 Elsevier/North-Holland, Amsterdam, 1994 Gilles O Zumbach Volatility conditional on price trends Quantitative Finance, 10(4):431 442, 2010 Time-reversal asymmetry Conclusion References Large-dimensional and multi-scale effects in stocks volatility m