ARCH and GARCH models Fulvio Corsi SNS Pisa 5 Dic 2011 Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 1 / 21
Asset prices S&P 500 index from 1982 to 2009 1600 1400 1200 1000 800 600 400 200 0 1980 1985 1990 1995 2000 2005 2010 asset prices are typically integrated of order one processes I(1) Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 2 / 21
Asset returns the standard solution is to take the first difference of prices. Two different type of returns: 1 simple net return R t = Pt P t P t 2 log return or continuously compounded returns r t = log P t log P t = p t p t over short horizon, r t is typically small ( r t << 10%) so that R t r t being 1+R t = exp(r t) = 1+r t + 1 2 r2 t +... The main advantage of the log return is that a k-period return r t(k) is simply: r t(k) = p t p t k = r t (k 1) +...+ r t + r t hence multi-period log returns are simply the sum of single-period log returns. Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 3 / 21
Asset returns dynamics 6 USD/CHF 1989 2001 4 2 std deviations 0 2 4 6 8 0 500 1000 1500 2000 2500 3000 6 Gaussian noise 4 2 std deviation 0 2 4 6 8 0 500 1000 1500 2000 2500 3000 6 S&P 500 1990 2003 4 2 std deviations 0 2 4 6 8 0 500 1000 1500 2000 2500 3000 Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 4 / 21
Time changing volatility Dynamics in the volatility of asset returns has paramount consequences in important finance applications: asset allocation risk management derivative pricing What makes volatility change over time? Still unclear. event-driven volatility : different information arrival rate, consistent with EMH error-driven volatility : due to over- and underreaction of the market to incoming information price-driven volatility : endogenously generated by trading activities of heterogeneous agents strong positive correlation between volatility and market presence Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 5 / 21
Different volatility notions Different types of volatility approaches: Parametric: volatility measure is model-dependent - Discrete-time: ARCH/GARCH models - Continuous-time: Stochastic Volatility models Non-Parametric: volatility measure is model-independent (or model-free) - Realized Volatility (exploiting the information in High Frequency data) Different notions of volatility: ex-ante conditional volatility spot/instantaneous volatility ex-post integrated volatility Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 6 / 21
Basic Structure and Properties of ARMA model standard time series models have: Y t = E[Y t Ω t 1 ]+ɛ t E[Y t Ω t 1 ] = f (Ω t 1 ;θ) [ ] Var[Y t Ω t 1 ] = E ɛ 2 t Ω t 1 = σ 2 hence, Conditional mean: varies with Ω t 1 Conditional variance: constant (unfortunately) k-step-ahead mean forecasts: generally depends on Ω t 1 k-step-ahead variance of the forecasts: depends only on k, not on Ω t 1 (again unfortunately) Unconditional mean: constant Unconditional variance: constant Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 7 / 21
AutoRegressive Conditional Heteroskedasticity (ARCH) model Engle (1982, Econometrica) intruduced the ARCH models: Y t = E[Y t Ω t 1 ]+ɛ t E[Y t Ω t 1 ] = f (Ω t 1 ;θ) ] Var[Y t Ω t 1 ] = E [ɛ 2 t Ω t 1 = σ(ω t 1 ;θ) σt 2 hence, Conditional mean: varies with Ω t 1 Conditional variance: varies with Ω t 1 k-step-ahead mean forecasts: generally depends on Ω t 1 k-step-ahead variance of the forecasts: generally depends on Ω t 1 Unconditional mean: constant Unconditional variance: constant Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 8 / 21
ARCH(q) How to parameterize E [ ɛ 2 t Ω t 1] = σ(ωt 1 ;θ) σ 2 t? ARCH(q) postulated that the conditional variance is a linear function of the past q squared innovations q σt 2 = ω + α i ɛ 2 t i = ω +α(l)ɛ2 t 1 i=1 Defining v t = ɛ 2 t σ 2 t, the ARCH(q) model can be written as ɛ 2 t = ω +α(l)ɛ 2 t 1 + vt Since E t 1 (v t) = 0, the model corresponds directly to an AR(q) model for the squared innovations, ɛ 2 t. The process is covariance stationary if and only if the sum of the positive AR parameters is less than 1 i.e. q i=1 α i < 1. Then, the unconditional variance of ɛ t is Var(ɛ t) = σ 2 = ω/(1 α 1 α 2... α q). Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 9 / 21
ARCH and fat tails Note that the unconditional distribution of ɛ t has Fat Tail. In fact, the unconditional kurtosis of ɛ t is where the numerator is E(ɛ 4 t ) E(ɛ 2 t )2 [ ] E ɛ 4 t [ ] = E E(ɛ 4 t Ω t 1) [ ] = 3E σt 4 = 3[Var(σ 2 t )+E(σ 2 t ) 2 ] = 3[Var(σ t 2 }{{} ) +E(ɛ 2 t )2 ] >0 > 3E(ɛ 2 t) 2. Hence, Kurtosis(ɛ t) = E(ɛ4 t) E(ɛ 2 t) 2 > 3 Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 10 / 21
ARCH and fat tails: intuition r t N(µ,σ t) is a mixture of Normals with different σ t fatter tails Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 11 / 21
AR(1)-ARCH(1) Example: the AR(1)-ARCH(1) model Y t = φy t 1 +ɛ t σ 2 t = ω +αɛ 2 t 1 ɛ t N(0,σ 2 t ) - Conditional mean: E(Y t Ω t 1 ) = φy t 1 - Conditional variance: E([Y t E(Y t Ω t 1 )] 2 Ω t 1 ) = ω +αɛ 2 t 1 - Unconditional mean: E(Y t) = 0 - Unconditional variance: E(Y t E(Y t)) 2 = 1 ω (1 φ 2 ) (1 α) Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 12 / 21
Other characteristics of ARCH(1) for the ARCH(1) model Y t = µ+ɛ t σ 2 t = ω +αɛ 2 t 1 ɛ t N(0,σ 2 t ) we also have: - Excess kurtosis: kurtosis is equal to 3 iff α = 0 Kurtosis(ɛ t) = E(ɛ4 t ) 1 α2 E(ɛ 2 = 3 t )2 1 3α 2 - Stationarity condition for finite variance of ɛ 2 (or for finite kurtosis of ɛ) α < 1 3 0.577 - Autocorrelation of σ t: Corr(σ t,σ t h ) = α h difficult to replicate empirical persistence of σ t. ARCH(q) models but... Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 13 / 21
GARCH(p,q) ARCH(q) problem: empirical volatility very persistent Large q i.e. too many α s Bollerslev (1986, J. of Econometrics) proposed the Generalized ARCH model. The GARCH(p,q) is defined as q p σt 2 = ω + α i ɛ 2 t i + β j σt j 2 = ω + α(l)ɛ2 t 1 +β(l)σ2 t 1 i=1 j=1 As before, defining v t = ɛ 2 t σt 2, the GARCH(p,q) can be also rewritten as ɛ 2 t = ω +[α(l)+β(l)]ɛ 2 t 1 β(l)v t 1 + v t which defines an ARMA[max(p, q),p] model for ɛ 2 t. Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 14 / 21
GARCH(1,1) By far the most commonly used is the GARCH(1,1): ɛ t = σ tz t with z t i.i.d.n(0, 1), σt 2 = ω +α ɛ 2 t 1 +β j σt 1 2 with ω > 0,α > 0,β > 0 Stationarity conditions: being and hence, σ 2 t = ω +(αz 2 t +β)σ 2 t 1 then the process is covariance stationary iff E[σ 2 t ] = ω +(α +β)e[σ2 t 1 ] α+β < 1 Unconditional variance: Var(ɛ) = ω 1 (α +β) Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 15 / 21
GARCH(1,1) By recursive substitution, the GARCH(1,1) may be written as the following ARCH( ): σt 2 = ω(1 β)+α β i 1 ɛ 2 t i i=1 which reduces to an exponentially weighted moving average filter for ω = 0 and α+β = 1 (sometimes referred to as Integrated GARCH or IGARCH(1,1)). Moreover, GARCH(1,1) implies an ARMA(1,1) representation in the ɛ 2 t ɛ 2 t = ω +(α +β)ɛ 2 t 1 βv t 1 + v t From the ARMA(1,1) representation we can guess that The precise calculations give: ρ h Corr(σ t,σ t h ) (α+β) h. ρ 1 = α(1 β2 αβ) 1 β 2 2αβ, ρ h = (α+β)ρ h 1 for h > 1 Forecasting. Denoting the unconditional variance σ 2 ω(1 α β) 1 we have: ˆσ 2 t+h t = σ2 +(α+β) h 1 (σ 2 t+1 σ2 ) showing that the forecasts of the conditional variance revert to the long-run unconditional variance at an exponential rate dictated by α+β Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 16 / 21
Asymmetric GARCH In standard GARCH model: σ 2 t = ω +αr 2 t 1 +βσ2 t 1 σ 2 t responds symmetrically to past returns. The so called news impact curve is a parabola conditional variance 8 6 4 2 0 News Impact Curve symmetric GARCH asymmetric GARCH 5 0 5 standardized lagged shocks Empirically negative r t 1 impact more than positive ones asymmetric news impact curve GJR or T-GARCH σ 2 t = ω +αr 2 t 1 +γr2 t 1 D t 1 +βσ 2 t 1 with D t = - Positive returns (good news): α - Negative returns (bad news): α + γ - Empirically γ > 0 Leverage effect Exponential GARCH (EGARCH) ln(σt 2 ) = ω +α r t 1 σ +γ r t 1 +βln(σt 1 2 t 1 σ ) t 1 { 1 if rt < 0 0 otherwise Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 17 / 21
GARCH in mean In the GARCH-M (Garch-in-Mean) model Engle, Lilien and Robins (1987) introduce the (positive) dependence of returns on conditional variance, the so called risk-return tradeoff. The specification of the model is: r t σ 2 t = µ+γσ 2 t +σ tz t = ω +αr 2 t 1 +βσ2 t 1 Given the inherent noise of financial returns r t, the estimates of γ are often very difficult, typically long time series are required to find significant results. Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 18 / 21
Engle test for heteroscedasticity or ARCH test The ARCH test of Engle assesses the null hypothesis that a series of residuals ɛ t exhibits no conditional heteroscedasticity (ARCH effects), The test is performed by running the following regression ɛ 2 t = c+a 1 ɛ 2 t 1 + a 2ɛ 2 t 2 +...+ a Lɛ 2 t L then computes the Lagrange multiplier statistic T R 2, where T is the sample size and R 2 is the coefficient of determination of the regression. Under the null, we have that T R 2 χ 2 L i.e. the asymptotic distribution of the test statistic is chi-square with L degrees of freedom. Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 19 / 21
Estimation A GARCH process with gaussian innovation: has conditional densities: f(r t Ω t 1 ;θ) = 1 σt 1 2π r t Ω t 1 N(µ t(θ),σ 2 t (θ)) ( (θ) exp 1 2 ) (r t µ t(θ)) σt 2 (θ) using the prediction error decomposition of the likelihood L(r T, r T 1,..., r 1 ;θ) = f(r T Ω T 1 ;θ) f(r T 1 Ω T 2 ;θ)... f(r 1 Ω 0 ;θ) the log-likelihood becomes: log L(r T, r T 1,..., r 1 ;θ) = T T 2 log(2π) logσ 1 T t(θ) 2 t=1 t=1 Non linear function in θ Numerical optimization techniques. When innovations not Normal PMLE standard errors ( sandwich form ) (r t µ t(θ)) σ 2 t (θ) Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 20 / 21
Stochastic Volatility models (idea) In ARCH-GARCH models, the variance at time t, σ 2 t is completely determined by the information at time t 1, i.e. σ 2 t it is conditionally deterministic or F t 1 measurable Another possibility is to have σ 2 t being also a (positive) stochastic process i.e. variance is also affected by an idiosyncratic noise term Stochastic Volatility models Example: the Heston model dp(t) = µp(t)dt + h(t)p(t)dw P (t) h(t) = k(θ h(t)) +ν h(t)dw h (t) CIR process where dw P (t) and dw h (t) are two (possibly correlated) Brownian processes. Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 21 / 21