Inference Methods for Stochastic Volatility Models

Transcription

1 Intrnational Mathmatical Forum, Vol 8, 03, no 8, Infrnc Mthods for Stochastic Volatility Modls Maddalna Cavicchioli Cá Foscari Univrsity of Vnic Advancd School of Economics Cannargio 3, Vnic, Italy Abstract In th prsnt papr w considr stimation procdurs for stationary Stochastic Volatility modls, making infrncs about th latnt volatility of th procss W show that a squnc of gnralizd last squars rgrssions nabls us to dtrmin th stimats Finally, w mak infrncs itrativly by using th Kalman Filtr algorithm Mathmatics Subjct Classification: 6M0, 6M0, 9B84, 93E03 Kywords: Stochastic Volatility, Gnralizd Last Squars, Kalman Filtr Introduction Stochastic Volatility (SV) modls hav bn rcivd a growing intrst in tim sris analysis sinc thy find many financial applications as, for xampl, option pricing, asst allocation and risk managmnt For a comprhnsiv discussion on SV modls s, for xampl, Taylor [3] and Tsyplakov [4] Lt us considr th basic stochastic volatility modl givn by { } y t = xp () h t(θ) u t h t (θ) =μ + ρh t (θ)+v t whr th rror trms u t IIN (0, ) and v t IIN (0,σv ) ar assumd to b indpndnt of on othr, and th paramtr vctor θ =(μρσv )isina compact paramtr st Θ (0, + ) 3 To nsur stationarity, w always st ρ < In this cas w hav () h t (θ) =μ( ρ) + ρ i v t i

2 370 M Cavicchioli hnc E(h t )=μ( ρ) and Var(h t )=σ v ( ρ ) Squaring () and taking logs, w gt th stat-spac rprsntation (3) x t = α + h t (θ)+ t h t (θ) =μ + ρh t (θ)+v t whr x t = log y t, α = E[log u t ] is a ral constant ( 7) and t = log u t α is a martingal diffrnc but not normal, i, t IID(0,σ ), whr σ = π / (hr log dnots th natural logarithm) S Bridt and Carriquiry [] Following Kim-Nlson [], Chp3, w prsnt two altrnativ ways of making infrncs about h t (volatility) conditional on information availabl up to tim t (givn θ) In Sction w show that a squnc of gnralizd last squars (GLS) rgrssions nabls us to dtrmin h t t = E(h t Ψ t ), whr Ψ t dnots th information st up to tim t In Sction 3 w mak infrncs about h t by mploying th Kalman Filtr algorithm Gnralizd Last Squars Estimation As usual, w approximats th SV modl in (3) by a Gaussian stat-spac modl From (3) w gt t () h t = μ( ρ) ( ρ t )+ρ t h + ρ i v t i for t Thus h ρ t+ ρ v + ρ v ρ t+ v t + ρ t+ v t h ρ t+ ρ v ρ t+3 v t + ρ t+ v t = h t μ h t ρ ρ v t 0 h t Dfin: and ρ ρ ρ t+ ρ t+ 0 ρ ρ t+3 ρ t+ B t = ρ a t =(ρ t+ ρ t+ ρ ) ρ t+ ρ ρ t+ ρ ρ 0 α μ( ρ) (ρ t+ ) α μ( ρ) (ρ t+ ) C t = α μρ α

3 Infrnc mthods for stochastic volatility modls 37 Using th masurmnt quation in (3) and th abov matrix rlation, w hav () x t = C t + a t h t + ɛ t whr x t =(x x x t ), ɛ t = B t (v v t ) + t and t =( t ) Thn w hav (3) E(ɛ t ɛ t)=σ I t + σ vb t B t =Ω t On could apply GLS to modl () for t =,,T Thn w gt (4) h t t =(a t Ω t a t ) a t Ω t (x t C t ) hnc (5) h t t = h t +(a t Ω t a t ) a t Ω t ɛ t Thn w hav and Dfin E(h t t )=E(h t )=μ( ρ) P t t = E[h t h t t ] =(a t Ω t a t ) b t = B ta t a t = ρ t ρ t ( ρ ) ρ t+ ( ρ 4 ) ρ ( ρ t+ ) Thorm With th abov notation, w hav whr a t = ρ t ρ b t = P t t = σ a t + σ v b t ρ 4t+ ρ ( ρ )( ρ t ) t ρ t+ ( ρ t ) Proof W apply th Shrman-Morrison-Woodbury (SMW) formula, i, if A and C ar invrtibl matrics, thn (A + BCD) = A A B(C + DA B) DA Stting A = σ I t, B = B t, C = σv I t and D = B t, from (3) w gt Ω t = σ I t σ B t (σ I t + σ v B tb t ) σ B t

4 37 M Cavicchioli hnc a t Ω t a t = σ a t a t σ a t B t(σ v I t + σ B t B t) σ B t a t Apply again th SMW formula with A = σ a t a t, B = σ a t B t, C = (σv I t + σ B tb t ) and D = σ B ta t Thn w hav P t t =(a t Ω t a t ) = σ (a t a t) (a t a t) a t B t (σ v I t + σ B tb t σ B ta t (a ta t ) a tb t ) B ta t (a ta t ) = σ a t +(a t a t) a t B t(σv I t ) B t a t(a t a t) = σ a t + σv b t which givs th rsult of th statmnt Thorm P = T P t t = σv ( ρ ) = var(h t ) t= Proof To comput th partial sums of th sris w took advantag of th softwar Mathmatica W hav (rcall that ρ < ) a t = t= ρ ρ = (ρ (0) )ψρ (T +)+( ρ )ψ (0) ρ () t ρ log(ρ ) t= whr ψ q (0) (z) =ψ q (z) = log Γ q (z)/ z dnots th q-digamma function Sinc (6) T ψ(0) ρ (T +)=0 w gt (7) T a t =0 t= Furthrmor, w hav ρ 4t+ ρ ( ρ )( ρ t ) = ψ () ρ (T +) ρ log (ρ ) + (ρ (0) +)ψρ (T +) ρ (ρ ) log(ρ ) + T ρ + c t= whr c is th numrical constant c = ψ() ρ () ρ log (ρ ) (ρ (0) +)ψρ () ρ (ρ ) log(ρ )

5 Infrnc mthods for stochastic volatility modls 373 and ψ q () (z) dnots th first drivativ of th q-digamma function Now (6) and (8) T ψ() ρ (T +)=0 imply (9) T t= ρ 4t+ ρ ( ρ )( ρ t ) = ρ It rmains to considr th sris t= t ρ t+ ( ρ t ) = t= t ρ (ρ t ρ t ) W tak th first Taylor xpansion around ρ 0 (0, ) of th function (ρ t ρ t ) and us th following squnc of inqualitis 0 < t (ρ t ρ t ) t (ρ t 0 ρ t 0 ) +t(ρ t 0 ρ t 0 )(ρt 0 + ρ0 t )(ρ ρ 0 ) < (ρ t 0 ρ t 0 )(ρt 0 + ρ0 t )(ρ ρ 0 ) for 0 <ρ<ρ 0 But w hav (ρ t ρ t )(ρ t + ρ t ) = ρψ (0) ρ (T +) ρψ ρ (0) (T iπ 4 log(ρ) t= whr d is th numrical constant log(ρ) +) + ρψ(0) ρ (T + iπ log(ρ) +)+ρψ(0) ρ (T iπ +) log(ρ) + d 4 log(ρ) d = ρψ(0) ρ () + ρψ ρ (0) ( iπ log(ρ) )+ρψ(0) ρ ( + iπ ) log(ρ) ρψ(0) ρ ( iπ 4 log(ρ) Thn w hav as T ψ(0) ρ T t= (T + ) = (ρ t ρ t )(ρ t + ρ t ) =0 T ψ(0) ρ = T ψ(0) ρ (T ± iπ log(ρ) +) (T iπ log(ρ) +)=0 log(ρ) )

6 374 M Cavicchioli This implis hnc T (0) T t= t= t ρ (ρ t ρ t ) =0 b t = ρ Finally, from (7) and (0), w gt th rsult of th statmnt 3 Estimation by th Kalman Filtr Sinc Modl (3) is in linar stat-spac form, prdictd filtrd and smoothd valus of h t can b computd rcursivly via th Kalman Filtr algorithm S Kim-Nlson [], Sc3 Dfin h t τ = E[h t Ψ τ ], P t τ = E[h t h t τ ], x t τ = E[x t Ψ τ ], η t τ = x t x t τ and f t τ = E[η t τ ] for τ T For th on-stp-ahad prdiction, w hav h t t = μ + ρh t t P t t = ρ P t t + σ v η t t = x t x t t = x t α h t t =(h t h t t )+ t f t t = P t t + σ Th initial stats of th rcursion ar Th updating is givn by whr h 0 0 = μ( ρ) and P 0 0 = σ v ( ρ ) h t t = h t t + K t η t t = h t t + K t (x t α h t t ) P t t = P t t K t P t t = P t t Pt t f t t K t = P t t f t t = P t t (P t t + σ ) is th Kalman gain From ths rcursions, on can construct th (quasi) Gaussian log-liklihood l(θ Ψ T )= T log(π) log f t t Th smoothd stimats and thir variancs ar givn by t= t= h t T = h t t + P t (h t+ T μ ρh t t ) P t T = P t t +[P t ] (P t+ T P t+ t ) η t t f t t

7 Infrnc mthods for stochastic volatility modls 375 whr P t = ρp t t P t+ t Solving th diffrnc quations in h t t and P t t givs th stimats Mor prcisly, w hav t t P t t = σv ρt ( ρ ) ( K t i )+σv h t t = μ( ρ) + ρ i K t i η t i t i η t t = α t (L)v t + β t (L) t k=0 ρ k k ( K t j ) whr α t (L) =+ t r= ρr r s= ( K t s)l r and β t (L) =α t (L)( ρl) (hr L dnots th lag oprator) j=0 Rfrncs [] FJ Bridt and AL Carriquiry, Quasi-Maximum Liklihood Estimation for Stochastic Volatility Modls, in Modlling and Prdiction, Honoring Symour Gisl (A Zllnr, JS L, ds), Springr Vrlag, (996) [] CJ Kim and CR Nlson, Stat-Spac Modls with Rgim Switching Classical and Gibbs-Sampling Approachs with Applications, Th MIT Prss, Cambridg MA, (999) [3] SJ Taylor, Modlling Stochastic Volatility, Mathmatical Financ, 4 (994), [4] A Tsyplakov, Rvaling th arcan: an introduction to th art of Stochastic Volatility modls, MPRA, no55, 00 Rcivd: Novmbr, 0