Pricing and Hedging Asian Basket Options with Quasi-Monte ... - Infn

More documents

Recommendations

Info

Methodol Comput Appl Probab3 Path-Generation TechniquesFrom the previous discussion it comes out that the pricing of Asian basket optionsby simulation requires an averaging on the sample trajectories of an M-dimensionalBM. In general, if Y ∼ N (0, Y ) and X ∼ N (0, I) are two N-dimensional Gaussianrandom vectors, we will always be able to write Y = CX, whereC is a matrixsuch that: Y = CC T . (13)and the core problem consists in finding the matrix C. Inourcase Y coincides with MN of Eq. 5. The accuracy of the standard MC method does not depend on thechoice of the matrix C because the order of the random variables is not important.However, a choice of C that reduces the nominal dimension would improve theefficiency of the (R)QMC method, and in the following we discuss a few possiblecases.3.1 Cholesky ConstructionThe CH decomposition simply finds the matrix C among all the lower triangularmatrices. In the case of constant volatilities the matrix MN is the Kronecker productof R and , and the Kronecker product is compatibile with a CH decomposition (seePitsianis and Van Loan 1993). In fact, denoting by C MN , C R and C the CH matricesassociated to MN , R and respectively, we haveC MN = C R ⊗ C . (14)This now entails a remarkable reduction of the computational cost: it turns outindeed that a O ( (M × N) 3) computation is reduced to a O ( M 3) + O ( N 3) one.When time-dependent volatilities are considered, however, we can no longer usethese properties of the Kronecker product: in their stead, since MN is a blockboomerang matrix, we can take advantage of the following result:Proposition 1 Let B ∈ R nB×nB be a block boomerang matrix and let (B 1 ,...,B P ) T ,where B h ∈ R D×D , h = 1,...,P with n B = P × D, be its associated elementary blockvector. Then the CH matrix C B associated to B is⎛⎞C 1 0 ... 0.C B =. C .. 2 0⎜⎝... ... ⎟(15).. ⎠C 1 C 2 ... C Pwhere the D × DblocksC h ,h= 1,...,PareC h = Chol (B h − B h−1 ) (16)with Chol denoting the CH factorization. We also assume B 0 = 0.
Methodol Comput Appl ProbabProof Consider the h th row of C B and the m th row of its transposed matrix; wethen have(C 1 ,...,C h , 0,...,0) T · (C h∧m1 T ,...,CT m , 0,...,0) T∑= C l ClTl=1h∧m∑= (B l − B l−1 ) = B h∧ml=1and this concludes the proof.⊓⊔3.2 Principal Component AnalysisAcworth et al. (1998) have proposed a path generation technique based on the PCA.Following this approach we consider the spectral decomposition of MN MN = EE T = ( E 1/2)( E 1/2) T, (17)where is the diagonal matrix of all the positive eigenvalues of MN sorted indecreasing order and E is the orthogonal matrix (EE T = I) of all the associatedeigenvectors. The matrix C solving Eq. 13 then is E 1/2 . The amount of variance∑ ki=1explained by the first k principal components is the ratio: ∑ λid where d is the ranki=1 λiof MN . The PCA construction permits the statistical ranking of the normal factors,while this is not possible by the CH decomposition. For a market with constantvolatilities, the Kronecker product reduces this calculation to the computation ofeigenvalues and eigenvectors of the two smaller matrices R and . All these simplifications,on the other hand, are no longer valid for the time-dependent volatilities.Nevertheless we can still reduce the computational cost of a PCA decomposition ina different way.Let M 1 , M 2 , M 3 and M 4 be respectively p × p, p × q, q × p and q × q matrices,and suppose that M 1 and M 4 are invertible. Assume then( )M1 MM =2M 3 M 4and define S 1 = M 4 − M 3 M −11 M 2 and S 4 = M 1 − M 2 M −14 M 3, namely the Schurcomplements of M 1 and M 4 respectively. Then by Schur’s lemma the inverse M −1 is:(M −1 S= 4 −M −11 M 2S −1 )1−M −14 S−1 4S −1 . (18)1Taking into account the previous result it is possible to prove the followingpropositionProposition 2 Let B ∈ R nB×nB be a block boomerang matrix, and (B 1 ,...,B P ) T —where B h ∈ R D×D , h = 1,...,P with n B = P × D—its associated elementary blockvector: then the inverse of B is symmetric block tri-diagonal. The blocks on the lower(and upper) diagonal are T l =−(B l+1 − B l ) −1 ,l= 1,...,P − 1 while those on the
Page 1 and 2: Methodol Comput Appl ProbabDOI 10.1
Page 3 and 4: Methodol Comput Appl Probabwill be
Page 5: Methodol Comput Appl Probabrepresen
Page 10 and 11: Methodol Comput Appl ProbabWe then
Page 12 and 13: Methodol Comput Appl Probabwhile st
Page 14 and 15: Methodol Comput Appl ProbabThe Mall
Page 16 and 17: Methodol Comput Appl Probabthe PCA,

Pricing and Hedging Asian Basket Options with Quasi-Monte ... - Infn

Create successful ePaper yourself

Delete template?

Save as template?