1 September 22nd, 2008 Multivariate L´ evy driven Stochastic Volatility Models Robert Stelzer Chair of Mathematical Statistics Zentrum Mathematik Technische Universit¨ at M¨ unchen email: rstelzer@ma.tum.de http://www.ma.tum.de/stat/ Parts based on joint work with O. E. Barndorff-Nielsen and Ch. Pigorsch c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
2 September 22nd, 2008 Outline of this talk • Motivation from finance and the univariate model • Matrix subordinators • Positive semi-definite Ornstein-Uhlenbeck type processes (based on Barndorff-Nielsen & St., 2007; Pigorsch & St., 2008a) • Multivariate Ornstein-Uhlenbeck type stochastic volatility model (based on Pigorsch & St., 2008b) • Multivariate COGARCH(1,1) (based on St., 2008) c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
3 September 22nd, 2008 Stylized Facts of Financial Return Data • non-constant, stochastic volatility • volatility exhibits jumps • asymmetric and heavily tailed marginal distributions • clusters of extremes • log returns exhibit marked dependence, but have vanishing autocorrelations (squared returns, for instance, have non-zero autocorrelation) Stochastic Volatility Models are used to cover these stylized facts. c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
4 September 22nd, 2008 Univariate BNS Model I • Logarithmic stock price process ( Y t ) t ∈ R + : dY t = ( µ + βσ t − ) dt + σ 1 / 2 t − dW t with parameters µ, β ∈ R and ( W t ) t ∈ R + being standard Brownian motion. • Ornstein-Uhlenbeck-type volatility process ( σ t ) t ∈ R + : dσ t = − λσ t − dt + dL t , σ 0 > 0 with parameter λ > 0 and ( L t ) t ∈ R + being a L´ evy subordinator. c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
5 September 22nd, 2008 Univariate BNS Model II • Usually E (max(log | L 1 | , 0)) < ∞ and σ is chosen as the unique stationary solution to dσ t = − λσ t − dt + dL t given by � t e − λ ( t − s ) dL s . σ t = −∞ • Closed form expression for the integrated volatility � t σ s ds = 1 λ ( L t − σ t + σ 0 ) . 0 Derivative Pricing via Laplace transforms possible. c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
6 September 22nd, 2008 The Need for Multivariate Models Multivariate models are needed • to study comovements and spill over effects between several assets. • for optimal portfolio selection and risk management at a portfolio level. • to price derivatives on multiple assets. Desire: Multivariate models that are flexible, realistic and analytically tractable. c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
7 September 22nd, 2008 Some Matrix Notation • M d ( R ) : the real d × d matrices. • S d : the real symmetric d × d matrices. • S + d : the positive-semidefinite d × d matrices (covariance matrices) (a closed cone). • S ++ : the positive-definite d × d matrices (an open cone). d • A 1 / 2 : for A ∈ S + d the unique positive-semidefinite square root (functional calculus). c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
8 September 22nd, 2008 Matrix Subordinators • Definition: An S d -valued L´ evy process L is said to be a matrix subordinator , if L t − L s ∈ d for all s, t ∈ R + with t > s . S + (Barndorff-Nielsen and P´ erez-Abreu (2008)). • The paths are S + d -increasing and of finite variation. • The characteristic function µ L t of L t for t ∈ R + is given by � � �� � � e i tr( XZ ) − 1 � µ L t ( Z ) = exp t i tr( γ L Z ) + ν L ( dX ) , Z ∈ S d , S + d \{ 0 } where γ L is the drift and ν L the L´ evy measure. c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
9 September 22nd, 2008 Examples of Matrix Subordinators • Analogues of univariate subordinators can be defined via the characteristic functions: e.g. (tempered) stable, Gamma or IG matrix subordinators • Diagonal matrix subordinators, i.e. off-diagonal elements zero, diagonal elements univariate subordinators • Discontinuous part of the Quadratic (Co-)Variation process of any d - evy process ˜ dimensional L´ L : [˜ L, ˜ � ∆˜ L s (∆˜ L s ) T L ] d t = s ≤ t c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
10 September 22nd, 2008 Linear Operators Preserving Positive-Semidefiniteness Proposition Let A : S d → S d be a linear operator. Then e A t ( S + d ) = S + d for all t ∈ R , if and only if A is representable as X �→ AX + XA T for some A ∈ M d ( R ) . � One has e A t X = e At Xe A T t for all X ∈ S d . In the above setting σ ( A ) = σ ( A ) + σ ( A ) . Hence, A has only eigenvalues of strictly negative real part, if and only if this is the case for A . c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
11 September 22nd, 2008 Positive-semidefinite OU-type Processes Theorem Let ( L t ) t ∈ R be a matrix subordinator with E (max(log � L 1 � , 0)) < ∞ and A ∈ M d ( R ) such that σ ( A ) ⊂ ( −∞ , 0) + i R . Then the stochastic differential equation of Ornstein-Uhlenbeck-type d Σ t = ( A Σ t − + Σ t − A T ) dt + dL t has a unique stationary solution � t e A ( t − s ) dL s e A T ( t − s ) Σ t = −∞ � t −∞ e ( I d ⊗ A + A ⊗ I d )( t − s ) d vec( L s ) . or, in vector representation, vec(Σ t ) = Moreover, Σ t ∈ S + d for all t ∈ R . � c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
12 September 22nd, 2008 Stationary Distribution Theorem Let γ L be the drift of the driving matrix subordinator L and ν L its L´ evy measure. The stationary distribution of the Ornstein-Uhlenbeck process Σ is infinitely divisible (even operator self-decomposable) with characteristic function � � � ( e i tr( Y Z ) − 1) ν Σ ( dY ) µ Σ ( Z ) = exp ˆ i tr( γ Σ Z ) + , Z ∈ S d , S + d \{ 0 } where � ∞ � I E ( e As xe A T s ) ν L ( dx ) ds γ Σ = − A − 1 γ L and ν Σ ( E ) = S + 0 d \{ 0 } for all Borel sets E in S + d \{ 0 } . A − 1 is the inverse of the linear operator A : S d ( R ) → S d ( R ) , X �→ AX + XA T which can be represented as vec − 1 ◦ (( I d ⊗ A ) + ( A ⊗ I d )) − 1 ◦ vec . � c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
13 September 22nd, 2008 Strict Positive-definiteness Proposition If γ L ∈ S ++ or ν L ( S ++ ) > 0 , then the stationary distribution P Σ d d of Σ is concentrated on S ++ , i.e. P Σ ( S ++ ) = 1 . � d d evy process in R d with L´ Theorem Let ˜ L be a L´ evy measure ν ˜ L � = 0 and assume that ν ˜ L is absolutely continuous (with respect to the Lebesgue measure on R d ). Then the stationary distribution of the Ornstein-Uhlenbeck type process Σ t driven by the discontinuous part of the quadratic variation [˜ L, ˜ L ] d t is absolutely continuous with respect to the Lebesgue measure. Moreover, the stationary distribution P Σ of Σ t is concentrated on S ++ , i.e. P Σ ( S ++ ) = 1 . � d d c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
14 September 22nd, 2008 Marginal Dynamics Assume that A is real diagonalisable and let U ∈ GL d ( R ) be such that UAU − 1 =: D is diagonal. • M t := UL t U T is again a matrix subordinator. �� t � t −∞ e D ( t − s ) d ( UL s U T ) e D ( t − s ) � • ( U Σ t U T ) ij = −∞ e ( λ i + λ j )( t − s ) dM ij,s . ij = the individual components of U Σ t U T • Hence, are stationary one- dimensional Ornstein-Uhlenbeck type processes with associated SDE d ( U Σ t U T ) ij = ( λ i + λ j )( U Σ t U T ) ij dt + dM ij,t . M ii for 1 ≤ i ≤ d are necessarily subordinators and ( U Σ t U T ) ii have to be positive OU type processes. • The individual components Σ ij,t of Σ t are superpositions of (at most d 2 ) univariate OU type processes. The individual OU processes superimposed are in general not independent. c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
15 September 22nd, 2008 Second Order Structure Theorem Assume that the driving L´ evy process is square-integrable. Then the second order moment structure is given by � γ Σ − A − 1 yν ( dy ) = − A − 1 E ( L 1 ) E (Σ t ) = S + d \{ 0 } −A − 1 var(vec( L 1 )) var(vec(Σ t )) = e ( A ⊗ I d + I d ⊗ A ) h var(vec(Σ t )) , cov(vec(Σ t + h ) , vec(Σ t )) = where t ∈ R and h ∈ R + , A : M d ( R ) → M d ( R ) , X �→ AX + XA T and A : M d 2 ( R ) → M d 2 ( R ) , X �→ ( A ⊗ I d + I d ⊗ A ) X + X ( A T ⊗ I d + I d ⊗ A T ) . � The individual components of the autocovariance matrix do not have to decay exponentially, but may exhibit exponentially damped sinusoidal behaviour. c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
16 September 22nd, 2008 The Integrated Volatility Theorem The integrated Ornstein-Uhlenbeck process Σ + t is given by � t Σ t dt = A − 1 (Σ t − Σ 0 − L t ) Σ + t := 0 for t ∈ R + . � c Advanced Modeling in Finance and Insurance, RICAM, Linz � Robert Stelzer
Recommend
More recommend
