Financial Econometrics and Volatility Models Stochastic Volatility

Transcription

1 Financial Econometrics and Volatility Models Stochastic Volatility Eric Zivot April 26, 2010

2 Outline Stochastic Volatility and Stylized Facts for Returns Log-Normal Stochastic Volatility (SV) Model SV Model with Student-t Errors Asymmetric SV Model Multivariate SV Model

3 Reading APDVP, chapters 8 and 11 FMUND, chapter 4 (section 7)

4 Stochastic Volatility and Stylized Facts for Returns Assume daily cc returns can be described as where r t = μ + σ t u t 1. σ t is a positive random variable s.t. var(σ t r t 1,r t 2,...) > 0 2. {σ t } is stationary, E[σ 4 t ] < and ρ τ,σ 2 =corr(σ 2 t,σ2 t+τ ) > 0 for all τ 3. u t iid (0, 1) 4. {u t } and {σ t } are independent

5 SV vs. ARCH The ARCH model is expressed as r t = μ + σ t u t σ 2 t = a 0 + a 1 rt 1 2 However, var(σ 2 t r t 1,r t 2,...,)=E[σ 4 t r t 1,r t 2...] E[σ 2 t r t 1,r t 2...] 2 = E[(a 0 + a 1 rt 1 2 )2 r t 1,r t 2...] E[a 0 + a 1 rt 1 2 r t 1,r t 2...] 2 =(a 0 + a 1 rt 1 2 )2 ³ a 0 + a 1 rt =0 so that there is no unpredictable volatility component.

6 SV vs ARCH SV specification can be motivated by economic theory Discrete-time SV specification has continuous-time diffusion representation SV fits nicely into continuous-time finance theory

7 Properties of Returns in SV Model Key result: Because {u t } and {σ t } are independent, for any functions f 1 and f 2 we have E[f 1 (σ t,σ t 1,...,)f 2 (u t,u t 1,...,)] = E[f 1 (σ t,σ t 1,...,)]E[f 2 (u t,u t 1,...,)] Moments E[r t μ] = E[σ t u t ]=E[σ t ]E[u t ]=0 var(r t ) = E[(r t μ) 2 ]=E[σ 2 t u 2 t ]=E[σ 2 t ]E[u 2 t ]=E[σ 2 t ]

8 Moments continued E[(r t μ) 4 ] = E[σ 4 t u 4 t ]=E[σ 4 t ]E[u 4 t ]=kurt(u t )E[σ 4 t ] kurt(r t ) = E[(r t μ) 4 ] E[σ 2 = kurt(u t)e[σ 4 t ] t ]2 E[σ 2 t ]2 = kurt(u t ) Ã 1+ var(σ2 t ) E[σ 2 t ]2! > kurt(u t ) Autocovariances and Autocorrelations γ τ,r = cov(r t,r t+τ )=cov(σ t u t,σ t+τ u t+τ ) = E[σ t u t σ t+τ u t+τ ] E[σ t u t ]E[σ t+τ u t+τ ] = E[σ t σ t+τ ]E[u t u t+τ ] E[σ t ]E[u t ]E[σ t+τ ]E[u t+τ ]=0

9 Define s t =(r t μ) 2 = σ 2 t u2 t. Then γ τ,s = cov(s t,s t+τ )=cov(σ 2 t u 2 t,σ 2 t+τu 2 t+τ) = E[σ 2 t u 2 t σ 2 t+τu 2 t+τ] E[σ 2 t u 2 t ]E[σ 2 t+τu 2 t+τ] = E[σ 2 t σ 2 t+τ]e[u 2 t ]E[u 2 t+τ] E[σ 2 t ]E[u 2 t ]E[σ 2 t+τ]e[u 2 t+τ] = E[σ 2 t σ 2 t+τ] E[σ 2 t ]E[σ 2 t+τ] = cov(σ 2 t,σ 2 t+τ) =γ τ,σ 2 > 0 Positive dependence in squared returns result from positive dependence in σ 2 t Note: ρ τ,s = cov(s t,s t+τ ) var(s t ) = cov(σ t,σ t+τ ) var(σ 2 t ) var(σ 2 t ) var(s t ) = ρ τ,σ 2 " var(σ 2 t ) var(s t ) #

10 Define a t = r t μ = σ t u t Then for τ>0 E[a p t ap t+τ ] = E[σp t σp t+τ u t p u t+τ p ] = E[σ p t σp t+τ ]E[ u t p ] 2 E[a p t ]E[ap t+τ ] = E[σp t ]E[σp t+τ ]E[ u t p ] 2 and cov(a p t,ap t+τ ) = E[ap t ap t+τ ] E[ap t ]E[ap t+τ ] = ³ E[σ p t σp t+τ ] E[σp t ]E[σp t+τ ] E[ u t p ] 2 = γ τ,σ pe[ u t p ] 2 for τ>0

11 Note: var(a p t ) = E[a2p t ] E[ap t ]2 = E[σ 2p t u t 2p ] E[σ p t u t p ] 2 = E[σ 2p t ]E[ u t 2p ] E[σ p t ]2 E[ u t p ] 2

12 Autocorrelations of a t p Define A(p) = E[σ2p t ] E[σ p and B(p) =E[ u t 2p ] t ]2 E[ u t p ] 2 Taylor (1994) derived the following result ρ τ,a p = corr(a p t,ap t+τ )=cov(ap t,ap t+τ ) var(a p t ) = γ τ,σ pe[ u t p ] 2 E[σ 2p t ]E[ u t 2p ] E[σ p t ]2 E[ u t p ] 2 = C(p)ρ τ,σ p C(p) = A(p) 1 A(p)B(p) 1 1 B(p)

13 Result: If u t iid N(0, 1) then E[ u t p ] = 2 p/2 π 1/2 Γ((p +1)/2) Γ(u) = Z 0 x u 1 e x dx, u > 0 Γ(1/2) = π, Γ(1) = 1, Γ(u +1)=uΓ(u) Γ(n) = (n 1)! if n is an integer If u t iid Student s t with v df then E[ u t ]= 2 v 2Γ((v +1)/2) π(v 1)Γ(v/2)

14 The Log-Normal AR(1) Stochastic Volatility Model Note r t = μ + σ t u t ln(σ t ) α = φ(ln(σ t 1 α)+η t, φ < 1 Ã! ÃÃ! Ã!! ut iid N, 0 0 σ 2 η η t ln(σ t ) N(α, β 2 ) β 2 = σ 2 η 1 φ 2 σ2 η = β 2 (1 φ 2 )

15 Log Normal Distribution Definition: If ln(y ) N(μ, σ 2 ) then Y LN(μ, σ 2 ) such that f(y μ, σ 2 1 ) = yσ 2π exp 1 Ã ln(y) μ 2 σ E[Y n ] = exp µnμ n2 σ 2 E[Y ] = exp µμ σ2 var(y ) = exp ³ 2μ + σ 2 ³ exp(σ 2 ) 1 In the Log-Normal AR(1) SV model σ t LN(α, β 2 )! 2, y > 0

16 Alternative Parameterization Some authors specify the log-normal SV model as Here It follows that r t = μ +exp(w t /2)u t w t α w = φ(w t 1 α w )+η w,t, η w,t iid N(0,σ 2 η w ) w t =ln(σ 2 t )=2ln(σ t ) and σ t =exp(w t /2) α w = E[w t ]=2E[ln(σ t )] = 2α β 2 w = var(w t )=var(2ln(σ t )) = 4var(ln(σ t )) = 4β 2 σ 2 η w = β 2 w(1 φ 2 )=4β 2 (1 φ 2 )

17 Basic Properties {r t } is strictly stationary All moments of r t are finite kurt(r t )=3exp(4β 2 ) cov(r t,r t+τ )=0( {r t μ, I t } is a MDS ) cov(s t,s t+τ ) > 0 when φ>0, s t =(r t μ) 2 ACF of a p t = r t μ p behaves like ACF of s t

18 Extensions of Standard SV Model Fat tailed distribution for u t (e.g. Student s t) Dependence between u t and η t to capture leverage effect Long memory behavior for ln(σ t ) Multivariate formulation

19 Density and Moments r t μ = σ t u t = log-normal Normal no closed form expression for density Derivation of Moments Exploit independence between {σ t } and {u t } Utilize moments of log-normal distribution Absolute Moments E[ r t μ p ]=E[a p t ]=E[σp t up t ]=E[σp t ]E[ u t p ]

20 Now ln(σ p t ) = p ln(σ t) N(pμ, p 2 β 2 ) σ p t LN(pμ, p2 β 2 ) Hence E[σ p t ]=exp µ pα p2 β 2 Furthermore, recall for u t iid N(0, 1) Therefore, E[ r t μ p ]=E[σ p t ]E[ u t p ]=exp E[ u t p ]=2 p/2 π 1/2 Γ((p +1)/2) µ pα p2 β 2 2 p/2 π 1/2 Γ µ p +1 2

21 For p =1and p =2 E[ r t μ ] = exp µα β2 q 2/π E[ r t μ 2 ] = var(r t )=exp ³ 2α +2β 2 2 π Γ = exp ³ 2α +2β 2 µ 3 2 because Γ µ 3 2 µ 1 = Γ 2 +1 = 1 2 Γ µ 1 2 = 1 2 π Straightforward algebra gives kurt(r t )= E[ r t μ 4 ] var(r t ) 2 =3exp ³ 4β 2

22 Autocorrelations {r t μ} = {σ t u t } is a MDS {r t } is an uncorrelated processes {σ t } is autocorrelated because ln(σ t ) follows an AR(1) process a t = r t μ = σ t u t,l t =lna t =ln(σ t )+ln( u t ) and s t =(r t μ) 2 are autocorrelated and behave similarly to {σ t }

23 Autocorrelations of l t,σ t,a t,ands t Autocorrelations of l t =ln(a t )=ln(σ t )+ln( u t ) cov(l t,l t+τ ) = cov(ln(σ t )+ln( u t ), ln(σ t+τ )+ln( u t+τ )) = cov(ln(σ t ), ln(σ t+τ )) (b/c u t is iid) = φ τ β 2 (b/c ln(σ t ) follows an AR(1)) Then ρ τ,l =corr(l t,l t+τ )= cov(l t,l t+τ ) var(l t ) = φ τ β 2 var(ln(σ t )+ln( u t ))

24 Now var(ln(σ t )) = β 2, var(ln( u t )) = π 2 /8 Hence ρ τ,l = φ τ β 2 β 2 + π 2 /8 = 8φτ β 2 8β 2 + π 2 = C(0,β) sign(ρ τ,l ) = sign(φ)

25 Autocorrelations of σ p t As ln(σ p t ) is a Gaussian AR(1) process, with mean αp, variance p2 β 2 and autoregressive coefficient φ it can be shown that ln(σ p t )+ln(σp t+τ )=ln(σp t σp t+τ ) N(2pα, 2(1 + φ τ )p 2 β 2 ) This follows since E[ln(σ p t )+ln(σp t+τ )] = pα + pα =2pα var(ln(σ p t )+ln(σp t+τ )) = var(ln(σp t )) + var(ln(σp t+τ )) +2cov(ln(σ p t ), ln(σp t+τ )) = p 2 β 2 + p 2 β 2 +2p 2 β 2 φ τ = 2(1+φ τ )p 2 β 2

26 Hence σ p t σp t+τ LN(2pα, 2(1 + φ τ )p 2 β 2 )) It follows that E[σ p t σp t+τ ] = exp³ 2pα +(1+φ τ )p 2 β 2 ρ τ,σ p = exp(p2 β 2 φ τ ) 1 exp(p 2 β 2 ) 1

27 Autocorrelations of a p t = r t μ p Previous we stated that ρ τ,a p = corr(a p t,ap t+τ )=C(p)ρ τ,σ p C(p) = A(p) 1 A(p)B(p) 1,A(p) =E[σ2p t ] E[σ p and B(p) =E[ u t 2p ] t ]2 E[ u t p ] 2 Hence, it can be shown that When p =2,wehave ρ τ,a p = exp(p2 β 2 φ τ ) 1 B(p)exp(p 2 β 2 ) 1 ρ τ,s = exp(4β2 φ τ ) 1 3exp(4β 2 ) 1

28 Log-Normal AR(1) SV Model with Student-t Errors r t = μ + σ t u t, u t iid St(v), v > 2 ln(σ t ) α = φ(ln(σ t 1 α)+η t, η t iid N(0,σ 2 η) u t is independent of η t for all t Here f(u v) = c(v) c(v) = " 1+ u2 Γ ³ v+1 2 v 2 Γ ³ v 2 qπ(v 2) # (v+1)/2, v > 2 E[u] = 0, var(u) =1, kurt(u) = 3(v 2) v 4

29 Note: by definition Then we can write u t = v t wt v t iid N(0, 1) (v 2)wt 1 x 2 v r t μ = σ t = σ t u t = σ t wt v t = σ t v t σ t wt ln σ t = lnσ t ln w t = AR(1) + WN(0,σ 2 ln w ) = ARMA(1, 1)

30 Moments a t = r t μ = σ t u t u t iid St(v) Then E[a p t ] = E[σp t ]E[ u t p ] = exp µpα p2 β 2 E[ u t p ] < for p<v Example E[a t ] = exp µα + 1 2(v 2)Γ ³ v+1 2 β2 2 ³ π(v 1)Γ v2, v > 1 E[a 2 t ] = exp(2α +2β 2 ), v > 2 E[a 4 t ] = exp ³ 4α +8β 2 3(v 2) v 4, v > 4

31 Note: Moment existence depending on v causes problems for GMM estimation of v.