The L evy-driven Continuous-Time Garch Model Claudia Kl uppelberg - PowerPoint PPT Presentation

The L´ evy-driven Continuous-Time Garch Model Claudia Kl¨ uppelberg Technische Universit¨ at M¨ unchen email: cklu@ma.tum.de http://www.ma.tum.de/stat/ Joint work with Alexander Lindner, Ross Maller, Vicky Fasen, Stefan Haug, Gernot M¨ uller

Question: How to model the volatility ( σ t ) t ≥ 0 . 15 10 5 0 −5 −10 0 1000 2000 3000 4000 5000 6000 12 10 8 6 4 2 0 0 1000 2000 3000 4000 5000 6000 Figure 1: Deseasonalised 5 minutes log-returns of Intel (February 1 - May 31, 2002) and estimated volatility.

Stylized facts of volatility: (1) volatility is random; (2) volatility has heavy-tailed marginals (higher moments do not exist); (3) volatility has skewed marginals (leverage effect); (4) volatility is a stochastic process with long-range dependence effect; (5) volatility is a stochastic process with clusters in the extremes.

Recall discrete time GARCH(1,1) model Y n = σ n Z n i.i.d. innovations ( Z n ) n ∈ N 0 , Define for σ 2 the random recurrence equation Volatility process: σ 2 n = β + λY 2 n − 1 + δσ 2 n − 1 , n ∈ N . Reorganise and iterate the recurrence: σ 2 β + λY 2 n − 1 + δσ 2 n − 1 = β + ( δ + λZ 2 n − 1 ) σ 2 = n n − 1 n − 1 n − 1 n − 1 � � � ( δ + λZ 2 j ) + σ 2 ( δ + λZ 2 = β j ) ( 1 ) 0 i =0 j = i +1 j =0 = β � ∞ � i d d σ 2 σ 2 j =1 ( δ + λZ 2 Under appropriate conditions: → j ) . n ∞ i =0

Continuous time GARCH(1,1) Idea: start with (1) and replace the sum by an integral       � n [ s ] n − 1 � �  + σ 2  exp σ 2  β  − log( δ + λZ 2  log( δ + λZ 2  ⇔ n = exp j ) ds j ) 0 0 j =0 j =0 Replace Z j by jumps of a L´ evy process L and take β, η = − log δ, ϕ = λ/δ . Then for a finite r.v. σ 2 0 define the volatility process � � � t σ 2 e X s ds + σ 2 e − X t t = β t ≥ 0 . 0 0 with auxiliary process � log(1 + ϕ (∆ L s ) 2 ) X t = tη − t ≥ 0 . 0 <s ≤ t

Ee isL t = e tψ L ( s ) , s ∈ R , with Recall: ( L t ) t ≥ 0 is L´ evy process if � s 2 ( e isx − 1 − isxI {| x | < 1 } )Π L ( dx ) , ψ L ( s ) = iγ L s − τ 2 2 + s ∈ R . L R � | x | <ε x 2 Π L ( dx ) < ∞ . ( γ L , τ L , Π L ) characteristic tripel , Π L L´ evy measure : Define the COGARCH(1,1)process by � G t = σ t − dL t t ≥ 0 . (0 ,t ] (Note: this defines the martingale part of the price process.)

100 50 G t 0 −50 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 10 G (1) i 0 −10 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 10 8 σ 2 6 i 4 2 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 50 L t 0 −50 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 First: Simulated VG driven COGARCH(1,1) process with β = 0 . 04 , η = 0 . 053 and ϕ = 0 . 038 ; second: differenced COGARCH process ( G (1) t ) ; third: volatility process ( σ t ) ; last: VG process ( L t ) with characteristic function Ee iuL 1 = (1 + u 2 / (2 C )) − C and C = 1 ;

Properties • G jumps at the same times as L with jump size ∆ G t = σ t ∆ L t . • ( X t ) t ≥ 0 is spectrally negative, has drift η , no Gaussian part, L´ evy measure � ( e x − 1) ϕ } ) for x > 0 . Π X ([0 , ∞ )) = 0 Π X (( −∞ , − x ]) = Π L ( {| y | ≥ t − d [ L, L ] ( d ) • dσ 2 t = ( β − ησ 2 t − ) dt + ϕ σ 2 t = � 0 <s ≤ t (∆ L s ) 2 and where [ L, L ] ( d ) t � t � σ 2 t = σ 2 σ 2 σ 2 s − (∆ L s ) 2 0 + βt − η s ds + ϕ t ≥ 0 . ( 2 ) 0 0 <s ≤ t � � 1 + ϕx 2 � log Π L ( dx ) < η ⇐ ⇒ EX 1 > 0 • � ∞ R d d e − X t dt. σ 2 → σ 2 ⇐ ⇒ = β t ∞ 0

Sample path behaviour • From ( 2 ) we know that σ 2 t has only upwards jumps. • If ( L t ) t ≥ 0 is compound Poisson with jump times 0 = T 0 < T 1 < . . . , � � t = β T j − β σ 2 σ 2 e − ( t − T j ) η , η + t ∈ ( T j , T j +1 ) . η ∞ ≥ β • For the stationary process, we have σ 2 η a.s.

9 8 7 6 5 4 3 2 1 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Sample paths of σ 2 σ 2 t (solid line) and b t (dotted line) of one simulation of a VG process.

Theorem Suppose that EL 1 = 0 , var( L 1 ) = 1 . Define Ee − sX t = e t Ψ X ( s ) . � i Assume that the volatility process is stationary, and define G (1) := i − r σ s − dL s . i If Ψ X (1) < 0 , then rβ i ) 2 = EG (1) = 0 , E ( G (1) 1 and corr( G (1) i , G (1) − Ψ X (1) EL 2 i + h ) = 0 . i � R x 3 ν L ( dx ) = 0 , then for k, p > 0 If EL 4 1 < ∞ , Ψ X (2) < 0 and corr(( G (1) i ) 2 , ( G (1) i + h ) 2 ) = ke − hp , h ∈ N . ✷ Theorem Assume that L 1 is symmetric and that there exists κ > 0 such that | L 1 | κ log + | L 1 | < ∞ and Ψ X ( κ/ 2) = 1 . Then a stationary version of the volatility process exists with P ( σ t > x ) ∼ cx − κ/ 2 , x → ∞ . ✷

Stylized facts of volatility: (1) volatility is random; (2) volatility has heavy-tailed marginals (higher moments do not exist: K., Lindner and Maller (2004), Fasen, K., Lindner (2004)); (3) volatility has skewed marginals (leverage effect introduced in Haug et al.) (4) volatility is a stochastic process with long-range dependence effect (acf decreases geometrically: K., Lindner and Maller (2004)); (5) volatility is a stochastic process with clusters in the extremes: Fasen: Extremes of genOU processes (2006, 2007).

Question: Can we find a discrete time skeleton, which approximates the COGARCH(1,1) process, and is a GARCH(1,1) process. The following approximation, called first jump approximation shows that (under some technical conditions) the solution of a L´ evy-driven SDE can be approximated arbitrarily close, by replacing the L´ evy process with its first jump approximation. Theorem [Szimayer and Maller (2007), Haug and Stelzer (2007)] evy process in R d , which has no Brownian part, drift γ L and L´ Let L be a L´ evy measure Π L and satisfies EL 2 (1) = 1 . For n ∈ N let 1 > ε ( n ) ↓ 0 and 0 = t ( n ) < t ( n ) < t ( n ) · · · ↑ ∞ . 0 1 2 Set δ ( n ) := sup i ∈ N ( t ( n ) i − 1 ) and assume that lim n →∞ δ ( n ) = 0 . Assume that − t ( n ) i n →∞ δ ( n ) (Π( { x ∈ R d : | x | > ε ( n ) } ) 2 = 0 . lim ( 3 )

Define for all n ∈ N � γ ( n ) := γ L − x Π L ( dx ) ε ( n ) < | x |≤ 1 τ ( n ) inf { t : t ( n ) i − 1 < t ≤ t ( n ) , | ∆ L t | > ε ( n ) } := ∀ i ∈ N i i � L ( n ) � γ ( n ) t + := ∆ L τ ( n ) ∀ t ≥ 0 t i { i ∈ N : τ ( n ) ≤ t } i ( n ) L ( n ) � L := . t t ( n ) i − 1 Then L ( n ) → L ( n ) , L ) P � in ucp as n → ∞ and d S ( L → 0 n → ∞ . ✷

Whenever one of the sequences ( δ ( n ) ) or ( ε ( n ) ) are given, one can Remark (i) always choose the other such that ( 3 ) holds. (ii) Note that the time grid is not necessarily equidistant. The construction allows for discrete sampling of a continuous-time L´ evy-driven model. This is useful for high-frequency data. (iii) The construction allows also the embedding of a discrete-time model into a continuous-time jump model. ✷ Example [COGARCH(1,1) and its GARCH(1,1) approximation] Maller, M¨ uller and Szimayer (2007) specify this approach and apply it to: (1) Parameter estimation by pseudo MLE. (2) Option pricing using the approach of Ritchken and Trevor (1999). For an alternative approach, see Kallsen and Vesenmayer (2007).

Example [COGARCH(1,1) and its GARCH(1,1) approximation, Maller, M¨ uller and Szimayer (2007)] We use the notation as in the theorem and assume that all assumptions hold. For n ∈ N set ∆ t i ( n ) := t ( n ) − t ( n ) i − 1 and define ∆ L τ ( n ) as the first jump of size i i larger than ε ( n ) in ( t ( n ) i − 1 , t ( n ) ] . Define i − ν ( n ) } < ∞ ∆ L τ ( n ) 1 { τ ( n ) i i i Z i,n = , i ∈ N . ξ ( n ) i By the strong Markov property ( 1 { τ ( n ) < ∞} ∆ L τ ( n ) ) i ∈ N is an iid sequence with i i distribution � 1 − e − η ∆ t i ( n )Π( { x ∈ R d : | x | >ε ( n ) } ) � Π( dx ) 1 {| x | >ε ( n ) } , x ∈ R \ { 0 } . Π( { x ∈ R d : | x | > ε ( n ) }

Then ( Z i,n ) i ∈ N is an iid sequence with mean 0 and variance 1. Now recall � t − d [ L, L ] ( d ) dσ 2 t = ( β − ησ 2 t − ) dt + ϕ σ 2 and G t = σ t − dL t t > 0 . t (0 ,t ] We discretise as follows: for G 0 ,n = G 0 = 0 set � G i,n − G i − 1 ,n = σ i − 1 ,n ∆ t i ( n ) Z i,n , i ∈ N , and � � σ 2 1 + ϕ ∆ t i ( n ) Z 2 e − η ∆ t i ( n ) σ 2 i,n = β ∆ t i ( n ) + i − 1 ,n , i ∈ N . i,n This defines a discrete time GARCH(1,1) random recurrence equation; cf. p. 4.

Follow the construction as before and introduce continuous-time versions (piecewise constant) of the auxiliary process X i,n , σ 2 i,n and G i,n . Then with the usual technical efforts, it is shown that P d S (( G n , σ 2 n ) , ( G, σ 2 )) → 0 n → ∞ . ✷