Intrnational Mathmatical Forum, Vol 8, 03, no 8, 369-375 Infrnc Mthods for Stochastic Volatility Modls Maddalna Cavicchioli Cá Foscari Univrsity of Vnic Advancd School of Economics Cannargio 3, Vnic, Italy maddalnacavicchioli@univit 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
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 3 + + ρ t+ v t + ρ t+ v t h ρ t+ ρ v 3 + + 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 = 0 0 0 ρ 0 0 0 0 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
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
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 )
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) )
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
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), 83-04 [4] A Tsyplakov, Rvaling th arcan: an introduction to th art of Stochastic Volatility modls, MPRA, no55, 00 Rcivd: Novmbr, 0