Symmetric btw positive & negative prior returns. where c is referred to as risk premium, which is expected to be positive.

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1 Advantages of GARCH model Simplicity Generates volatility clustering Heavy tails (high kurtosis) Weaknesses of GARCH model Symmetric btw positive & negative prior returns Restrictive Provides no explanation Not sufficiently adaptive in prediction GARCH-M model: r t = µ + cσt + a t, a t = σ t ϵ t, σt = α 0 + α 1 a t 1 + β 1 σt 1, where c is referred to as risk premium, which is expected to be positive. EGARCH model: Asymmetry in responses to + and - returns: g(ϵ t ) = θϵ t + [ ϵ t E( ϵ t )], with E[g(ϵ t )] = 0. To see asymmetry of g(ϵ t ), rewrite it as g(ϵ t ) = An EGARCH(m, s) model: { (θ + 1)ϵt E( ϵ t ) if ϵ t 0, (θ 1)ϵ t E( ϵ t ) if ϵ t < 0. a t = σ t ϵ t, ln(σ t ) = α 0 + m α i g(ϵ t i ) + s β i ln(σt i). Some features of EGARCH models: uses log trans. to relax the positiveness constraint 1

2 asymmetric responses. Consider an EGARCH(1,1) model a t = σ t ϵ t, (1 βb) ln(σ t ) = α 0 + αg(ϵ t 1 ), Under normality, E( ϵ t ) = /π and the model becomes (1 βb) ln(σ t ) = where α = α 0 α /π. { α + α(θ + 1)ϵ t 1 if ϵ t 1 0, α + α(θ 1)ϵ t 1 if ϵ t 1 < 0, This is a nonlinear fun. similar to that of the threshold AR model of Tong (1978, 1990). Specifically, we have where θ 1 = θα. { σt = σ β exp[(θ1 + α)ϵ t 1 exp(α ) t 1 ] if a t 1 0, exp[(θ 1 α)ϵ t 1 ] if a t 1 < 0, The coefficients (θ 1 +α) and (θ 1 α) show the asymmetry in response to positive and negative a t 1. The model is, therefore, nonlinear if θ 1 0. Thus, θ 1 (or θ) is referred to as the leverage parameter. It shows the effect of the sign of a t 1 whereas α denotes the magnitude effect. See Nelson (1991) for an example of EGARCH model. ARMA-GARCH model: r t = p ϕ i r t i + a t a t = η t σ t, σ t = α 0 + m α i a t i + q θ j a t j, j=1 s β j σt j. j=1 r t is called ARMA(p, q) GARCH(r, s) model.

3 ARIMA-GARCH model: ϕ p (B)(1 B) log P t = θ q (B)a t, a t = η t σ t, σ t = α 0 + m s α i a t i + β j σt j. j=1 log P t is called ARIMA(p, 1, q) GARCH(m, s) model. Estimation: Maximum Likelihood Estimation We consider the case with p = 1 and q = 0 and s = m = 1. Denote Z t = (r t, r t 1, ). Given Z t, we can calculate: a t (ϕ) = r t ϕr t 1, σ t (λ) = α 0 + α 1 (y t 1 ϕy t ) + β 1 σ t 1(λ), where ϕ = ϕ 1 and λ = (ϕ, α 0, α 1, β 1 ). The conditional probability of r t is ( ) f(r t Z 1 t 1 ) = πσ t (λ) exp a t (ϕ). σt (λ) The conditional joint probability of (r n, r n 1,, r 1 ): { f(r t,, r 1 Z n ( ) } 1 0 ) = πσ t (λ) exp a t (ϕ). σt (λ) t=1 Log-conditional joint probability of (r n, r n 1,, r 1 ): L(λ) ln f(r t,, r 1 Z 0 ) = n ln(π) 1 { } ln σt (λ) + a t (ϕ). σt (λ) The MLE of λ is the maximizer of L(λ), denote by λ. If Eε t <, then t=1 λ λ as n. Diagnostic Checking and Model selection 3

4 The formal method is not provided in SAS. AIC is a main tool for model selection. We can use Ljung-Box test for squared standardized residuals. Forecasting The forecast value r t (l) of r t+l is calculated by r t (l) = E(r t+l r t, r t 1, ). The formulas is the same as the one in the ARMA model with a constant variance. The one-step forecast interval for the ARIMA-GARCH model: where N α [ ] r t (1) N α σ t(ˆλ), r t (1) + N α σ t(ˆλ) is the α/ quantile of N (0, 1). SAS only provides the forecasting value and forecast interval for AR- GARCH model. You have to write your own programs in practice. 4

5 Other GARCH-type Models The Threshold GARCH (TGARCH) or GJR Model A TGARCH(s,m) or GJR(s,m) model is defined as r t = µ t + a t, a t = σ t ϵ t, s m σt = α 0 + (α i + γ i N t i )a t i + β j σt j, j=1 where N t i is an indicator variable such that N t i = 1 if a t i < 0, and = 0 otherwise One expects γ i to be positive so that prior negative returns have higher impact on the volatility. The CHARMA Model Make use of interaction btw past shocks. A CHARMA model is defined as r t = µ t + a t, a t = δ 1t a t 1 + δ t a t + + δ mt a t m + η t, where {η t } is iid N(0, ση), {δ t } = {(δ 1t,, δ mt ) } is a sequence of iid random vectors D(0, Ω), {δ t } and {η t } are independent. The model can be written as a t = (a t 1,, a t m )δ t + η t, with conditional variance σ t = σ η + (a t 1,, a t m )Ω(a t 1,, a t m ), where Ω = Cov(δ t δ t). RCA Model A time series r t is a RCA(p) model if r t = ϕ 0 + p (ϕ i + δ it )r t i + a t. 5

6 For the model, we have µ t = E(a t F t 1 ) = p ϕ i a t i, σ t = σ a + (r t 1,, r t p )Ω δ (r t 1,, r t p ). Stochastic volatility model A (simple) SV model is a t = σ t ϵ t, (1 α 1 B α m B m ) ln(σ t ) = α 0 + v t, where ϵ t s are iid N(0, 1), v t s are iid N(0, σv), {ϵ t } and {v t } are independent. Alternative Approaches to Volatility Two alternative methods: Use of high-frequency financial data Use of daily open, high, low and closing prices Use of High-Frequency Data Purpose: monthly volatility Data: Daily returns Let rt m be the t th month log return. Let {r t,i } n be the daily log returns within the t th month. Using properties of log returns, we have r m t = r t,i. Assuming that the conditional variance and covariance exist, we have V ar(r m t F t 1 ) = V ar(r t,i F t 1 ) + i<j 6 Cov[(r t,i, r t,j ) F t 1 ],

7 where F t 1 = the information available at month t 1 (inclusive). Further simplification possible under additional assumptions. If {r t,i } is a white noise series, then V ar(r m t F t 1 ) = nv ar(r t,1 ), where V ar(r t,1 ) can be estimated from the daily returns {r t,i } n by ˆσ = n (r t,i r t ), n 1 where r t is the sample mean of the daily log returns in month t (i.e., r t = n r t,i/n). The estimated monthly volatility is then ˆσ m = n n 1 (r t,i r t ). If {r t,i } follows an MA(1) model, then V ar(r m t F t 1 ) = nv ar(r t,1 ) + (n 1)Cov(r t,1, r t, ), which can be estimated by ˆσ m = n n 1 Advantage: Simple Weaknesses: n 1 (r t,i r t ) + (r t,i r t )(r t,i+1 r t ). Model for daily returns {r t,i } is unknown. Typically, 1 trading days in a month, resulting in a small sample size. Realized integrated volatility If the sample mean r t is zero, then σ m r t,i. Use cumulative sum of squares of daily log returns within a month as an estimate of monthly volatility. 7

8 Apply the idea to intrdaily log returns and obtain realized integrated volatility. Assume daily log return r t = n r t,i. The quantity RV t = r t,i, is called the realized volatility of r t. Advantages: simplicity and using intraday information Weaknesses: Effects of market microstructure (noises) Overlook overnight return Use of Daily Open, High, Low and Close Prices: See text book. 8