Affine processes on positive semidefinite matrices

Affine processes on positive semiefinite matrices Josef Teichmann (joint work with C. Cuchiero, D. Filipović an E. Mayerhofer) ETH Zürich Zürich, June 2010 C. Cuchiero, D. Filipović, E. Mayerhofer, J. Teichmann Affine processes on positive semiefinite matrices

1 Introuction Overview Motivation: Multivariate stochastic covariance moels Literature 2 Definition of affine processes on S + Affine processes on S + are Feller an regular 3 Main theorem Amissible parameters Ieas of the proof 4 1 / 69

Overview Motivation: Multivariate stochastic covariance moels Literature Introuction Affine processes on S +, the cone of positive semiefinite matrices, are stochastically continuous time-homogeneous Markov processes with state space S +, whose Laplace transform have exponential-affine epenence on the initial state, E x [e Tr(uXt)] = e φ(t,u) Tr(ψ(t,u)x). Aim: Full characterization of this class of processes Necessary conitions on the parameters of the infinitesimal generator implie by the efinition of an affine process. 4 / 69

Overview Motivation: Multivariate stochastic covariance moels Literature Motivation: Affine stochastic covariance moels One-imensional affine stochastic (co)variance moels (Heston [15], Barnorff-Nielsen Shepar moel [1], etc.). Risk neutral ynamics for the log-price process Y t an the R + -value variance process X t : X t = (b + βx t t + σ X t W t + J t, X 0 = x, ( Y t = r X ) t t + X t B t, Y 0 = y. 2 B, W : correlate Brownian motions, J: pure jump process, r: constant interest rate. 8 / 69

Overview Motivation: Multivariate stochastic covariance moels Literature Multivariate affine stochastic covariance moels Extension to multivariate stochastic covariance moels with the aim to......capture the epenence structure between ifferent assets,...obtain a consistent pricing framework for multi-asset options such as basket options, 12 / 69

Overview Motivation: Multivariate stochastic covariance moels Literature Multivariate affine stochastic covariance moels Multivariate stochastic covariance moels consist of a -imensional logarithmic price process with risk-neutral ynamics ( Y t = r1 1 ) 2 X t iag t + X t B t, Y 0 = y, an stochastic covariation process. 0 X = Y, Y. B: -imensional Brownian motion. r: constant interest rate 1: the vector whose entries are all equal to one. X iag : the vector containing the iagonal entries of X. 14 / 69

Overview Motivation: Multivariate stochastic covariance moels Literature Multivariate affine stochastic covariance moels The following affine ynamics for X have been propose in the literature: X t = (b + HX t + X t H )t + X t W t Σ + Σ Wt Xt + J t, X 0 = x S +, b: suitably chosen matrix in S +, H, Σ: invertible matrices, W a stanar -matrix of Brownian motions possibly correlate with B, J a pure jump process whose compensator is an affine function of X. 16 / 69

Overview Motivation: Multivariate stochastic covariance moels Literature Multivariate affine stochastic covariance moels Analytic tractability: Uner some (mil) technical conitions, the following affine transform formula hols: [ ] E x,y e Tr(zXt)+v Y t = e Φ(t,z,v)+Tr(Ψ(t,z,v)x)+v y for appropriate arguments z S is an v C. The functions Φ an Ψ solve a system of generalize Riccati ODEs. Option pricing via Fourier methos v y in the moment generating function correspons to a homogeneity assumption an implies that the covariation process X is a Markov process in its own filtration, which motivates the analysis of affine processes on S +. 18 / 69

Overview Motivation: Multivariate stochastic covariance moels Literature Relate Literature Affine processes on R m + R n : D. Duffie, D. Filipović an W. Schachermayer [11]: Characterization of affine processes on R m + R n. D. Filipović an E. Mayerhofer [12]: Characterization of affine Diffusion processes. M. Keller-Ressel, W. Schachermayer an J. Teichmann [16]: Regularity of affine processes.... many other authors. Affine processes on S + - Theory of Wishart processes: M. Bru [3]: Existence an uniqueness of Wishart processes of type X t = (δi )t + X t W t + Wt Xt, X 0 S +, for δ 1. C. Donati-Martin, Y. Doumerc, H. Matsumoto an M. Yor [10]: Properties of Wishart processes. 20 / 69

Overview Motivation: Multivariate stochastic covariance moels Literature Relate Literature Affine processes on S + Applications from mathematical Finance: O. Barnorff-Nielsen an R. Stelzer [2]: Matrix-value Lévy riven Ornstein-Uhlenbeck processes of finite variation. B. Buraschi et al. [5, 4]: Correlation risk an portfolio optimization. J. Da Fonseca et al. [6, 7, 8, 9]: Multivariate stochastic covariance an option pricing. C. Gourieroux an R. Sufana [13, 14]: Wishart quaratic term structure moels. M. Leippol an F. Trojani [17]: S + -value affine jump iffusions an financial applications (multivariate option pricing, interest rate moels, etc.) 21 / 69

Definition of affine processes on S + Affine processes on S + are Feller an regular Setting an notation S : symmetric -matrices equippe with scalar prouct x, y = Tr(xy). 22 / 69

Definition of affine processes on S + Affine processes on S + are Feller an regular Setting an notation S : symmetric -matrices equippe with scalar prouct x, y = Tr(xy). S + : cone of symmetric -positive semiefinite matrices. S ++ : interior of S + in S, the cone of strictly positive efinite matrices. Partial orer inuce by the cones: x y y x S + an x y y x S ++. X time-homogeneous Markov process with state space S +. (P t ) t 0 : associate semigroup acting on boune measurable functions f : S + R, P t f (x) := E x [f (X t )] = f (ξ)p t (x, ξ), x S +. S + 27 / 69

Definition of affine processes on S + Affine processes on S + are Feller an regular Definition Definition The Markov process X is calle affine if 1 it is stochastically continuous, that is, lim s t p s (x, ) = p t (x, ) weakly on S + x S +, an for every t an 2 its Laplace transform has exponential-affine epenence on the initial state: P t e u,x = e u,ξ p t (x, ξ) = e φ(t,u) ψ(t,u),x, S + for all t an u, x S +, for some functions φ : R + S + R + an ψ : R + S + S +. 29 / 69

Definition of affine processes on S + Affine processes on S + are Feller an regular Regularity an Feller property Definition The affine process X is calle regular if the erivatives φ(t, u) ψ(t, u) F (u) =, R(u) = (1) t t t=0+ t=0+ exist an are continuous at u = 0. 30 / 69

Definition of affine processes on S + Affine processes on S + are Feller an regular Remarks on the proof The proof relies on several properties of the functions φ an ψ: Semi-flow property: For all t, s R + φ(t + s, u) = φ(t, u) + φ(s, ψ(t, u)), (2) ψ(t + s, u) = ψ(s, ψ(t, u)). (3) Orer relations: For all u, v S + with v u an for all t 0, φ(t, v) φ(t, u) an ψ(t, v) ψ(t, u). 33 / 69

Definition of affine processes on S + Affine processes on S + are Feller an regular Remarks on the proof Remark on the proof of the above theorem: Regularity can be prove by methos on the regularity of (semi-)flows from Montgommery an Zippin. The { Feller property follows from regularity an the fact that e u,x u S ++ } is ense in C0 (S + ) (Stone-Weierstrass). 37 / 69

Definition of affine processes on S + Affine processes on S + are Feller an regular Consequences of regularity: From regularity we can conclue local characteristics: Since the following limit exists, Ae u,x = e u,x 1 lim t 0 t E( e u,xt x 1 ), we know that the limit is of Lévy-Khintchine form on the cone S + Rx. Due to the affine property the previous equation is also an equation for the erivatives of φ an ψ. By the Markov property (semigroup property) these equations turn into ODEs for φ an ψ. 38 / 69

Definition of affine processes on S + Affine processes on S + are Feller an regular The function φ an ψ are solutions of ODEs: Hence we obtain φ(t, u) = F (ψ(t, u)), t φ(0, u) = 0, (4) ψ(t, u) = R(ψ(t, u)), t ψ(0, u) = u S +, (5) where F an R are efine in (1). Due to the particular form of F an R we shall call them generalize Riccati equations. 39 / 69

Definition of affine processes on S + Affine processes on S + are Feller an regular The functions F an R the question of existence of affine processes Theorem Moreover, φ(t, u) an ψ(t, u) solve the ifferential equations (4) an (5), where F an R have the following form F (u) = b, u + c (e u,ξ 1)m(ξ), S + \{0} R(u) = 2uαu + B (u) + γ ( ) µ(ξ) e u,ξ 1 + χ (ξ), u ξ 2 1. S + \{0} Conversely, let (α, b, β ij, c, γ, m, µ) be an amissible parameter set. Then there exists a unique affine process on S + with infinitesimal generator (6). 40 / 69

Main theorem Amissible parameters Ieas of the proof Infinitesimal generator Theorem If X is an affine process on S +, then its infinitesimal generator is of affine form: Af (x) = 1 A ijkl (x) 2 f (x) + b + B(x), f (x) (c + γ, x )f (x) 2 x ij x kl i,j,k,l + (f (x + ξ) f (x)) m(ξ) (6) S + \{0} + (f (x + ξ) f (x) χ(ξ), f (x) ) M(x, ξ), S + \{0} for some truncation function χ an amissible parameters ( α, b, B(x) = i,j β ij x ij, c, γ, m(ξ), M(x, ξ) = where A ijkl (x) = x ik α jl + x il α jk + x jk α il + x jl α ik. ) x, µ, ξ 2 1 41 / 69

Main theorem Amissible parameters Ieas of the proof Relation to semimartingales Corollary Let X be a conservative affine process on S +. Then X is a semimartingale. Furthermore, there exists, possibly on an enlargement of the probability space, a -matrix of stanar Brownian motions W such that X amits the following representation ( t ) X t = x + b + χ(ξ)m(ξ) + B(X s) s, 0 S + \{0} t ( ) + XsWsΣ + Σ W s Xs 0 t ( ) + χ(ξ) µ X (t, ξ) (m(ξ) + M(X t, ξ)) t 0 S + \{0} t + (ξ χ(ξ))µ X (t, ξ), 0 S + \{0} where Σ is a matrix satisfying Σ Σ = α an µ X enotes the ranom measure associate with the jumps of X. 42 / 69

Main theorem Amissible parameters Ieas of the proof Amissible parameters linear iffusion coefficient: α S +, 43 / 69

Main theorem Amissible parameters Ieas of the proof Amissible parameters linear iffusion coefficient: α S +, linear jump coefficient: -matrix µ = (µ ij ) of finite signe measures on S + \ {0} with µ(e) S + for all E B(S + \ {0}) M(x, ξ) := x,µ(ξ) satisfies ξ 2 1 S + \{0} χ(ξ), u M(x, ξ) < for all x, u S + with x, u = 0, linear rift coefficient: the linear map B(x) = i,j βij x ij with β ij = β ji S satisfies B(x), u χ(ξ), u M(x, ξ) 0 S + \{0} for all x, u S + with x, u = 0, 45 / 69

Main theorem Amissible parameters Ieas of the proof Remark on the amissible parameters No constant iffusion part, linear part is of very specific form u, A(x)u = 4 x, uαu. 50 / 69

Main theorem Amissible parameters Ieas of the proof Remark on the amissible parameters No constant iffusion part, linear part is of very specific form u, A(x)u = 4 x, uαu. Very remarkable constant rift conition: b ( 1)α. For 2 the bounary of S + is curve an implies this relation between linear iffusion coefficient α an rift part b. Jumps escribe by m are of finite variation, for the linear jump part we have finite variation for the irections orthogonal to the bounary while parallel to the bounary general jump behavior is allowe. 52 / 69

Main theorem Amissible parameters Ieas of the proof Remark on the amissible parameters Parameters of prototype equation are amissible: X t = (b + HX t + X t H )t + X t W t Σ + Σ W t Xt + J t, 53 / 69

Main theorem Amissible parameters Ieas of the proof Remark on the amissible parameters Parameters of prototype equation are amissible: X t = (b + HX t + X t H )t + X t W t Σ + Σ W t Xt + J t, Σ R : α = Σ Σ S +, b ( 1)Σ Σ, B(x) = Hx + H x: Hx + H x, u = 0 if u, x = 0, c = 0, γ = 0, J a compoun Poisson process with intensity l + λ, x an jump istribution ν supporte on S + : m(ξ) = lν(ξ), M(x, ξ) = λ, x ν(ξ). 58 / 69

Main theorem Amissible parameters Ieas of the proof What are the new results Full characterization an exact assumptions uner which affine processes on S + actually exist. 59 / 69

Main theorem Amissible parameters Ieas of the proof What are the new results Full characterization an exact assumptions uner which affine processes on S + actually exist. Necessity an sufficiency of the remarkable rift conition b ( 1)α. Extension of the moel class. General linear rift part B(x) = ij βij x ij. This allows epenency of the volatility of one asset on the other ones which is not possible for B(x) = Hx + xh. Example: = 2 an ( ) x22 x B(x) = 12. x 12 x 11 62 / 69

Main theorem Amissible parameters Ieas of the proof Ieas an methos applie in the proof Necessary conitions: Since the limit exists Ae u,x = e u,x 1 lim t 0 t E( e u,xt x 1 ), we know that the limit is of Lévy-Khintchine form on the cone S + Rx. The structure of infinitely ivisible ranom variables on cones implies the form of F an R an it etermines the form of the infinitesimal generator since Ae u,x = ( F (u) R(u), x )e u,x. In orer to erive the conition on b we consier et(x t ). Invariance conitions for R + -value process imply then b ( 1)α. 64 / 69

Main theorem Amissible parameters Ieas of the proof Ieas an methos applie in the proof Sufficient conitions Problems: Unboune coefficients, x is not Lipschitz continuous at the bounary, infinitely active jumps. Regularization of the coefficients (replace x by x + ɛi ɛi ). Through stochastic invariance, we establish existence of an S + -value solution of the martingale problem for the generator of the regularize process Existence of an S + -value solution of the martingale problem for A. Uniqueness an existence of global solutions φ an ψ of the generalize Riccati equations (4) an (5) yiel uniqueness of the solution of the martingale problem Existence of an affine (Markov) process on S +. 65 / 69

On S +, the laws of X t = δi t + X t W t + Wt Xt are those of non central Wishart istributions an are not infinitely ivisible. As a consequence of the conition on the constant rift, we obtain Corollary X is affine an its one-imensional marginal istributions are infinitely ivisible if an only if α = 0 or = 1. 66 / 69

Conclusion an Outlook Conclusion Characterization of S + -value affine processes. Exhaustive moel specification. Characterization of infinitely ivisible affine processes. Outlook Calibration an parameter estimation of multivariate stochastic covariance moels. Affine processes on symmetric cones. Calculation of stochastic quantities beyon marginals in the affine setting. Infinite imensional affine processes. 67 / 69

[1] Ole E. Barnorff-Nielsen an Neil Shephar. Moeling by Lévy processes for financial econometrics. In Lévy processes, pages 283 318. Birkhäuser Boston, Boston, MA, 2001. [2] Ole Eiler Barnorff-Nielsen an Robert Stelzer. Positive-efinite matrix processes of finite variation. Probab. Math. Statist., 27(1):3 43, 2007. [3] Marie-France Bru. Wishart processes. Journal of Theoretical Probability, 4(4):725 751, 1991. [4] B. Buraschi, A. Cieslak, an F. Trojani. Correlation risk an the term structure of interest rates. Working paper, University St.Gallen, 2007. [5] B. Buraschi, P. Porchia, an F. Trojani. Correlation risk an optimal portfolio choice. Working paper, University St.Gallen, 2006. [6] Jose Da Fonseca, Martino Grasseli, an Florian Ielpo. Heging (co)variance risk with variance swaps. Working paper, University St.Gallen, 2006. [7] Jose Da Fonseca, Martino Grasseli, an Florian Ielpo. Estimating the Wishart affine stochastic correlation moel using the empirical characteristic function. Working paper, 2008. [8] Jose Da Fonseca, Martino Grasseli, an Clauio Tebali. Option pricing when correlations are stochastic: an analytical framework. Review of Derivatives Research, 10(2):151 180, 2007. [9] Jose Da Fonseca, Martino Grasseli, an Clauio Tebali. A multifactor volatility Heston moel. 68 / 69

J. Quant. Finance, 8(6):591 604, 2008. [10] Catherine Donati-Martin, Yan Doumerc, Hiroyuki Matsumoto, an Marc Yor. Some properties of the Wishart processes an a matrix extension of the Hartman-Watson laws. Publ. Res. Inst. Math. Sci., 40(4):1385 1412, 2004. [11] Darrel Duffie, Damir Filipović, an Walter Schachermayer. Affine processes an applications in finance. Ann. Appl. Prob., 13:984 1053, 2003. [12] D. Filipović an E. Mayerhofer. Affine iffusion processes: Theory an applications. In Avance Financial Moelling, volume 8 of Raon Ser. Comput. Appl. Math. Walter e Gruyter, Berlin, 2009. [13] Christian Gourieroux an Razvan Sufana. Wishart quaratic term structure moels. Working paper, CREST, CEPREMAP an University of Toronto, 2003. [14] Christian Gourieroux an Razvan Sufana. Derivative pricing with Wishart multivariate stochastic volatility: application to creit risk. Working paper, CREST, CEPREMAP an University of Toronto, 2004. [15] S. Heston. A close-form solution for options with stochastic volatility with appliactions to bon an currency options. Rev. of Financial Stuies. [16] Martin Keller-Ressel, Walter Schachermayer, an Josef Teichmann. Affine processes are regular. Preprint, 2009. [17] Markus Leippol an Fabio Trojani. Asset pricing with matrix affine jump iffusions. Working paper, 2008. 69 / 69