Exact and high order discretization schemes. Wishart processes and their affine extensions


 Shon Long
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1 for Wishart processes and their affine extensions CERMICS, Ecole des Ponts Paris Tech  PRES Université Paris Est Modeling and Managing Financial Risks 2011
2 Plan 1 Motivation and notations 2 Splitting operator property Exact simulation 3 Wishart processes Affine processes 4
3 Plan 1 Motivation and notations 2 Splitting operator property Exact simulation 3 Wishart processes Affine processes 4
4 Motivation Wishart process and affine process defined on the symmetric positive cone S + d (R) are a keytool for:  Defining the natural correlation between processes.  A generalization of stochastic volatility in multidimension.  Pricing complex derivatives taking into account the relationship between spot (Outperformer, Best/Worst of options...). Pricing with Fourier transform methods are less efficient in the multi dimension context. To the best of our knowledge, this is the first exact simulation and high order discretization that work without any restriction on parameters.
5 Definitions We say that the process (Xt x ) t 0 is a continuous positive affine process, if it is a solution of the following SDE: X x t = x + Z t 0 (α + B(X x s )) ds + Z t 0 p p X x s dw sa + a T dws T X x s, (1) where (W t, t 0) denotes a dbyd square matrix made of independent standard Brownian motions, x, ᾱ S + d (R), a M d(r), B L(S d (R)) ( where L(S d (R)) is a linear mapping on S d (R)), and x S + d (R) x = odiag(λ 1,..., λ d )o T = x = odiag( λ 1,..., p λ d )o T. Wishart processes correspond to the following case : α 0, ᾱ = αa T a and b M d (R), x S d (R), B(x) = bx + xb T. (2)
6 Application in finance : Gourieroux and Sufana model We consider d risky assets S 1 t,..., S d t. Let (B t, t 0) denote a standard Brownian motion on R d that is independent from (X t) t 0 WIS d (x, α, b, a). Then, we have t 0, 1 l d, ds l t S l t = rdt + ( X tdb t) l. Gourieroux and Sufana model assumes that the Wishart process (X t) t 0 is the Covariance matrix of the spot vector (S t) t 0. Da Fonseca and al. have chosen the adequate correlation between spot vector (S t) t 0 and its Covariance matrix (X t) t 0 to observe the smile effect, and to keep the model affine.
7 Plan Splitting operator property Exact simulation 1 Motivation and notations 2 Splitting operator property Exact simulation 3 Wishart processes Affine processes 4
8 Splitting operator property Exact simulation Composition technique for the exact simulation Proposition If (Yt x ) t 0 is an affine process starting from x and associated to the infinitesimal operator L Y, such that L Y = L Z + L X, and L Z L X = L X L Z. Then Y x t X Z x t t, where (X x t ) t 0 and (Z x t ) t 0 are two affine independent processes associated respectively to two infinitesimal generators L X and L Z. Proof. For some class of functions f E[f (Xt x )] = P k=0 tk L k X f (x)/k! := e tl X h i h h ii (f )(x) E f (X Z t x t ) = E E f (X Z t x t ) Zt x = P h i + t k 1 k 1 =0 E k 1 L k 1! X f (Z t x ) = P + t k 1 +k 2 k 1,k 2 =0 k 1!k 2! Lk 1 X Lk 2 Z f (x) = P t k k=0 (L k! X + L Z ) k f (x) = E[f (Yt x )]
9 Splitting operator property Exact simulation Canonical Wishart process transformation Proposition Let t > 0, a, b M d (R) and α d 1. Let m t = exp(tb), q t = R t 0 exp(sb)at a exp(sb T )ds and n = Rk(q t). Then, there is θ t G d (R) such that q t = tθ tid n θt T, and we have: WIS d (x, α, b, a; t) = Law θ twis d (θ 1 t m txmt T (θt 1 ) T, α, 0, Id n ; t)θt T Remark General /Non central Wishart distribution Canonical/Central Wishart distribution In the case of d = 1, we obtain the usual identity of Bessel and CIR processes WIS 1(x, α, b, a; t) = a 2 e 2bt 1 2btx WIS 1(, α, 0, 1; t). Law 2bt a 2 (1 e 2bt )
10 A remarkable splitting operator Splitting operator property Exact simulation Theorem Let L be the generator associated to the Wishart process WIS d (x, α, 0, Id n) and L i be the generator associated to WIS d (x, α, 0, ed i ) for i {1,..., d}. Then, we have n L = L i and i, j {1,..., d}, L i L j = L j L i, (3) i=1 where 1 i d, 1 k, l d, (e i d ) k,l = 1 {k=l=i}, (I n d ) k,l = 1 {k=l,k n} Remark The operators L i and L j are the same up to the exchange of coordinates i and j. The processes WIS d (x, α, 0, ed i ) and WIS d(x, α, 0, Id n ) are well defined on S + d (R) under the same hypothesis, namely α d 1 and x S+ d (R).
11 Splitting operator property Exact simulation Exact simulation for the canonical Wishart distribution Let us consider t > 0 and x S + d (R). We define iteratively: where, X 1,x i,...xt t X 1,x t WIS d (x, α, 0, e 1 d; t), 2,X 1,x t Xt WIS d (X 1,x t, α, 0, ed; 2 t),... 1,x n,...xt Wishart process starting from X Xt WIS d (X n 1,...X t 1,x t, α, 0, e n d ; t), is sampled according to the distribution at time t of an independent 1,x i 1,...Xt t and with parameters (α, 0, e i d). We have the following result: Proposition Let X 1,x n,...xt t 1,x be defined as above. Then X n,...x t WIS d (x, α, 0, Id n ; t). t
12 Splitting operator property Exact simulation Exact simulation of WIS d (x, α, 0, ed 1 ), with d N Theorem The solution of WIS d (x, α, 0, ed) 1 is given explicitly by: 0 1 (U u P Xt x t ) {1,1} + r ((Ut u ) {1,k+1} ) 2 ((Ut u ) {1,l+1} ) T 1 l r 0 = q B k=1 ((Ut u ) {1,l+1} ) 1 l r I r 0 A qt, where d(ut u ) {1,1} = (α r)dt + 2 p (Ut u ) {1,1} dzt 1 0, d((ut u ) {1,l+1} ) 1 l r = (dzt l+1 ) 1 l r. d((ut u ) {k,l} ) 2 k,l r = d((xt x ) {k,l} ) 2 k,l r = 0, and q 0 c r 0 A. 0 k r I d r 1 TT
13 Splitting operator property Exact simulation Methodology to sample Exactly Wishart distribution WIS d (x, α, b, a; t) θ t WIS d (θt 1 m t xmt T (θt 1 ) T, α, 0, Id n; t)θt t Law 2 n d, WIS d (x, α, 0, Id n ), By composition Technique 2 i d, WIS d (x, α, 0, ed i ), By permutation WIS d (x, α, 0, ed 1). Sampling one square Bessel process and d 1 Brownian motions.
14 Plan Wishart processes Affine processes 1 Motivation and notations 2 Splitting operator property Exact simulation 3 Wishart processes Affine processes 4
15 Wishart processes Affine processes A potential ν order scheme for the operator L 1, d N Theorem By replacing in Transformation ((U u t ) {1,l} ) 2 l d (resp. (U u t ) {1,1} ) with t(ĝ i ) 1 i r (resp. with (Ûu t ) {1,1} ), then ˆX t is a potential νorder scheme for the operator L 1, where : (Ĝ i ) 1 i r is a sequence of independent real variables with finite moments of any order such that: i {1,..., r}, k 2ν + 1, E[(Ĝ i ) k ] = E[G k ], where G N (0, 1). (Ûu t ) {1,1} is sampled independently according to a potential weak νthorder scheme for the CIR process d(ut u ) {1,1} = (α r)dt + 2 p (Ut u ) {1,1} dzt 1 starting from u {1,1}.
16 Wishart processes Affine processes Methodology to build the scheme of order ν ŴIS d (x, α, b, a; t) = θ t ŴIS d (θt 1 m t xmt T (θt 1 ) T, α, 0, Id n; t)θt t 2 n d, ŴIS d (x, α, 0, Id n ), By composition Technique 2 i d, ŴIS d (x, α, 0, ed i ), By permutation ŴIS d (x, α, 0, ed 1). Schemes of order ν for: one square Bessel process and d 1 Brownian motions.
17 Wishart processes Affine processes The third order discretization for Wishart process Theorem Let (Xt x ) t 0 WIS d (x, α, b, a) such that either a G d (R) or a T ab = ba T a, and f Cpol(S d (R)). Let ( ˆX N, 0 i N) be sampled with the scheme defined t i N previously with the third order scheme for the CIR given in Alfonsi2009 and starting from x 0 S + d (R). Then, C, N 0 > 0, N N 0, E[f ( ˆX N t N N )] E[f (X x 0 t )] C/N 3. Remark New extension of the regularity of the function u(t, x) = E [f (X x t )], from the CIR process to Wishart process. (Phd thesis A.Alfonsi 2006)
18 Wishart processes Affine processes Canonical positive affine process transformation Proposition Let (Xt x ) t 0 AFF d (x, ᾱ, B, a) and n = Rk(a) be the rank of a T a. Then, there exist a diagonal matrix δ, and a non singular matrix u G d (R) such that ᾱ = u T δu, and a T a = u T Id n u, and we have: (Xt x ) t 0 = u T AFF d (u 1 ) T xu 1, δ, B u, Id n u, Law where y S d (R), B u(y) = (u 1 ) T B(u T yu)u 1.
19 Wishart processes Affine processes The potential second order discretization for a general affine process defined on S + d (R) It is sufficient to study the affine process AFF d (x, δ, B, I n d ). By splitting operator, if L denotes the infinitesimal generator of AFF d (x, δ, B, I n d ), we conclude then that L = L ODE + L Wishart, L ODE = Tr((δ δ min Id n + B(x))D S ) Xt 1, L Wishart = Tr((δ min Id n )D S ) + 2Tr(xD S Id n D S ) = P n i=1 L i Xt 2. Proposition 1,x t/2 2,X Both schemes X 1,X 2,x 1,x t 1,Xt 2,Xt t/2 and UXt + (1 U)Xt are potential second order scheme for AFF d (x, δ, B, Id n ), where U is an independent Bernoulli variable with parameter 1. 2
20 Wishart processes Affine processes Fast potential second order discretization for a general affine process defined on S + d (R), δ di n d The previous algorithm requires O(d 4 ), on each step time due to Cholesky decomposition of each transformation (L i ) 1 i d. In the case of δ di n d we propose an other scheme that costs only O(d 3 ) : L = L ODE + L Wishart, L ODE = Tr((δ di n d + B(x))D S ), L Wishart = Tr((dI n d )D S ) + 2Tr(xD S I n d D S ) (c + t GI n d )(c + t GI n d ) T, where G is a matrix, in M d (R), made of independent variables that fit the first five moments of normal random variable.
21 Plan 1 Motivation and notations 2 Splitting operator property Exact simulation 3 Wishart processes Affine processes 4
22 Time computation for E [exp(itr(vx x T ))]:Nmc = 106, a = I d, b = 0, x = 10I d, v = 0.09I d and T = 1 N = 10 N = 30 Schemes R. value Im. value Time R. value Im. value Time Exact (1 step) nd order bis nd order rd order Exact (N steps) Corrected Euler α = 3.5, d = 3, R = 10 3, Im = 10 3, exact value R. = and Im.= Exact (1 step) nd order rd order Exact (N steps) Corrected Euler α =2.2, d = 3, R = , Im = , exact value R. = and Im.= Exact (1 step) nd order bis nd order rd order Exact (N steps) Corrected Euler α = 10.5, d = 10, R = , Im = , exact value R. = and Im.= Exact (1 step) nd order rd order Exact (N steps) Corrected Euler α = 9.2, d = 10, R = , Im = , exact value R. = and Im.=
23 Laplace transform E [exp(itr(vx x T ))], d = El O3 Ex Ex O O O Figure: d = 3, 10 7 MC, T = 10. The RV of E[exp( Tr(iv ˆX N t N N ))] in function of T /N. Left: v = 0.05I d and x = 0.4I d, α = 4.5, a = I d and b = 0. Ex.Val.: Right: v = 0.2I d q and x = 0.4I d + 0.2q, α = 2.22, a = I d and b = 0.5I d. Ex.Val: Here, q is the matrix defined by: q i,j = 1 i j. The width of each point represents the 95% confidence interval.
24 Laplace transform E [exp(itr(vxt x ))], d = Ex O O O3 Ex O El Figure: d = 10, 10 7 MC, T = 10. Left: IM of E[exp( Tr(iv ˆX N t N N ))] with v = 0.009I d in function of T /N, x = 0.4I d, α = 12.5, b = 0 and a = I d. Ex.Val: Right: RV of E[exp( Tr(iv ˆX N t N N ))] with v = 0.009I d in function of T /N, x = 0.4I d, α = 9.2, b = 0.5I d and a = I d. Ex.Val The width of each point represents the 95% confidence interval.
25 Trajectory error E [max 0 s T Tr(X x s ))] El El O2 2.5 Ex O2 O O Figure: d = 3, 10 7 MC, T = 1, x = 0.4I d + 0.2q with q i,j = 1 i j, α = 2.2, b = 0 and a = I d. Left, E[max 0 k N Tr( ˆX N t k N )], right: E[max 0 k N Tr( ˆX N t k N )] E[max 0 k N Tr(X x t k N )] in function of T /N. The width of each point gives the precision up to two standard deviations.
26 Gourieroux Sufana Model  Put Best of Option O El Val 15.2 O Val 14.8 El Figure: E[e rt (K max (Ŝ1,N t N N, Ŝ2,N t N N )) + ] in function of T /N. d = 2, T = 1, K = 120, S 0 1 = S2 0 = 100, and r = 0.02, x = 0.04I d q with q i,j = 1 i j, a = 0.2I d, b = 0.5I d and α = 4.5 (left), α = 1.05 (right). The width of each point gives the precision up to two standard deviations (10 6 MC).
27 Summary In this work, we have presented : Exact scheme for Wishart process. Second and third order scheme for Wishart process. Potential second order scheme for a general affine process defined on S + d (R).
28 Thank you!!