Risk-parameter estimation in volatility models Christian Francq - PowerPoint PPT Presentation

Conditional risk in volatility models Risk parameter in volatility models Estimating the risk parameter Risk-parameter estimation in volatility models Christian Francq Jean-Michel Zakoïan CREST and University Lille 3, France SFdS-JdS 2013, 29 May 2013 Toulouse This work was supported by the ANR via the Project ECONOM&RISK (ANR 2010 blanc 1804 03) Francq, Zakoian Risk-parameter estimation in volatility models

Conditional risk in volatility models Risk parameter in volatility models Estimating the risk parameter Model risk/Estimation risk Risk assessment framework defined by "Pillar II" directives: panel of risks including market risk. In July 2009, the Basel Committee issued a directive requiring that financial institutions quantify "model risk": "Banks must explicitly assess the need for valuation adjustments to reflect two forms of model risk: the model risk associated with using a possibly incorrect valuation methodology; and the risk associated with using unobservable (and possibly incorrect) calibration parameters in the valuation model." This talk is about quantifying the estimation risk in some dy- namic models. Francq, Zakoian Risk-parameter estimation in volatility models

Conditional risk in volatility models Risk parameter in volatility models Estimating the risk parameter Outline 1 Conditional risk in volatility models Properties of financial time series Models for the volatility Risk measures 2 Risk parameter in volatility models Model and basic assumptions Standard estimators of the volatility parameter Risk parameter 3 Estimating the risk parameter QML estimators of general risk parameters One-step VaR estimation Comparison with two-step VaR estimators Francq, Zakoian Risk-parameter estimation in volatility models

Conditional risk in volatility models Properties of financial time series Risk parameter in volatility models Models for the volatility Estimating the risk parameter Risk measures Main properties of daily stock indices Non stationarity of the prices. Illustration Possible unpredictability of the returns (martingale difference assumption), but non-independence. Illustrations Volatility clustering. Strong positive autocorrelations of the squares or of the absolute values (even for large lags). Illustrations Leptokurticity of the marginal distribution. Illustrations Asymmetries (leverage effects). Illustrations Francq, Zakoian Risk-parameter estimation in volatility models

Conditional risk in volatility models Properties of financial time series Risk parameter in volatility models Models for the volatility Estimating the risk parameter Risk measures Volatility Models Almost all the volatility models are of the form ǫ t = σ t η t where ( η t ) is iid, σ t > 0 , σ t and η t are independent. For GARCH-type (Generalized Autoregressive Conditional Heteroskedasticity) models, σ t ∈ σ ( ǫ t − 1 , ǫ t − 2 ,...) . See Bollerslev (Glossary to ARCH (GARCH), 2009) for an impressive list of more than one hundred GARCH-type models. Francq, Zakoian Risk-parameter estimation in volatility models

Conditional risk in volatility models Properties of financial time series Risk parameter in volatility models Models for the volatility Estimating the risk parameter Risk measures Examples Standard GARCH ( p , q ) (Engle (82), Bollerslev (86)): q p � � σ 2 α 0 i ǫ 2 β 0 j σ 2 t = ω 0 + t − i + t − j i = 1 j = 1 Asymmetric Power GARCH model: for δ > 0 , q p � � t − i ) δ + α 0 i − ( − ǫ − t − i ) δ + σ δ β 0 j σ δ α 0 i + ( ǫ + t = ω 0 + t − j i = 1 j = 1 ARCH( ∞ ) (Robinson (91)), introduced to capture long memory: � ∞ σ 2 ψ 0 i ǫ 2 t = ψ 00 + t − i i = 1 Francq, Zakoian Risk-parameter estimation in volatility models

Conditional risk in volatility models Properties of financial time series Risk parameter in volatility models Models for the volatility Estimating the risk parameter Risk measures Examples Standard GARCH ( p , q ) (Engle (82), Bollerslev (86)): q p � � σ 2 α 0 i ǫ 2 β 0 j σ 2 t = ω 0 + t − i + t − j i = 1 j = 1 Asymmetric Power GARCH model: for δ > 0 , q p � � t − i ) δ + α 0 i − ( − ǫ − t − i ) δ + σ δ β 0 j σ δ α 0 i + ( ǫ + t = ω 0 + t − j i = 1 j = 1 ARCH( ∞ ) (Robinson (91)), introduced to capture long memory: � ∞ σ 2 ψ 0 i ǫ 2 t = ψ 00 + t − i i = 1 Francq, Zakoian Risk-parameter estimation in volatility models

Conditional risk in volatility models Properties of financial time series Risk parameter in volatility models Models for the volatility Estimating the risk parameter Risk measures Examples Standard GARCH ( p , q ) (Engle (82), Bollerslev (86)): q p � � σ 2 α 0 i ǫ 2 β 0 j σ 2 t = ω 0 + t − i + t − j i = 1 j = 1 Asymmetric Power GARCH model: for δ > 0 , q p � � t − i ) δ + α 0 i − ( − ǫ − t − i ) δ + σ δ β 0 j σ δ α 0 i + ( ǫ + t = ω 0 + t − j i = 1 j = 1 ARCH( ∞ ) (Robinson (91)), introduced to capture long memory: � ∞ σ 2 ψ 0 i ǫ 2 t = ψ 00 + t − i i = 1 Francq, Zakoian Risk-parameter estimation in volatility models

Conditional risk in volatility models Properties of financial time series Risk parameter in volatility models Models for the volatility Estimating the risk parameter Risk measures Examples (continued) Models on the log-volatility: EGARCH (Nelson, 1991) or Log-GARCH (see Sucarrat and Escribano, 2010) : � q � p log σ 2 α i log ǫ 2 β j log σ 2 t = ω + t − i + t − j i = 1 j = 1 MIDAS model of Ghysels, Santa-Clara and Valkanov (2006) GAS model of Creal, D., Koopmans, S.J. and A. Lucas (2012) Beta-t-EGARCH model of Harvey and Sucarrat (2012) Francq, Zakoian Risk-parameter estimation in volatility models

Conditional risk in volatility models Properties of financial time series Risk parameter in volatility models Models for the volatility Estimating the risk parameter Risk measures Examples (continued) Models on the log-volatility: EGARCH (Nelson, 1991) or Log-GARCH (see Sucarrat and Escribano, 2010) : � q � p log σ 2 α i log ǫ 2 β j log σ 2 t = ω + t − i + t − j i = 1 j = 1 MIDAS model of Ghysels, Santa-Clara and Valkanov (2006) GAS model of Creal, D., Koopmans, S.J. and A. Lucas (2012) Beta-t-EGARCH model of Harvey and Sucarrat (2012) Francq, Zakoian Risk-parameter estimation in volatility models

Conditional risk in volatility models Properties of financial time series Risk parameter in volatility models Models for the volatility Estimating the risk parameter Risk measures Examples (continued) Models on the log-volatility: EGARCH (Nelson, 1991) or Log-GARCH (see Sucarrat and Escribano, 2010) : � q � p log σ 2 α i log ǫ 2 β j log σ 2 t = ω + t − i + t − j i = 1 j = 1 MIDAS model of Ghysels, Santa-Clara and Valkanov (2006) GAS model of Creal, D., Koopmans, S.J. and A. Lucas (2012) Beta-t-EGARCH model of Harvey and Sucarrat (2012) Francq, Zakoian Risk-parameter estimation in volatility models

Conditional risk in volatility models Properties of financial time series Risk parameter in volatility models Models for the volatility Estimating the risk parameter Risk measures Examples (continued) Models on the log-volatility: EGARCH (Nelson, 1991) or Log-GARCH (see Sucarrat and Escribano, 2010) : � q � p log σ 2 α i log ǫ 2 β j log σ 2 t = ω + t − i + t − j i = 1 j = 1 MIDAS model of Ghysels, Santa-Clara and Valkanov (2006) GAS model of Creal, D., Koopmans, S.J. and A. Lucas (2012) Beta-t-EGARCH model of Harvey and Sucarrat (2012) Francq, Zakoian Risk-parameter estimation in volatility models

Conditional risk in volatility models Properties of financial time series Risk parameter in volatility models Models for the volatility Estimating the risk parameter Risk measures Value at Risk and other risk measures 0.3 Distribution of the returns 0.2 0.1 α 0.0 − VaR t ( α ) −4 0 4 Other risk measures, for instance � α ES t ( α ) = α − 1 0 VaR t ( u ) du . Conditional versus marginal distribution. Francq, Zakoian Risk-parameter estimation in volatility models

Conditional risk in volatility models Properties of financial time series Risk parameter in volatility models Models for the volatility Estimating the risk parameter Risk measures Conditional risk Modern financial risk management focuses on risk measures based on distributional information. Traditional approaches: marginal distributions of (log) returns risk = a parameter More sophisticated approaches: conditional distributions of (log) returns risk = a stochastic process Francq, Zakoian Risk-parameter estimation in volatility models

Conditional risk in volatility models Properties of financial time series Risk parameter in volatility models Models for the volatility Estimating the risk parameter Risk measures Conditional VaR for a simulated process Returns 6 4 2 0 −2 −4 0 200 400 600 800 1000 Conditional and marginal distributions and VaR’s 0.0 0.1 0.2 0.3 0.4 0.5 − VaR t ( 0.01 ) 0 −9.5 9.5 Francq, Zakoian Risk-parameter estimation in volatility models