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

Methodol Comput Appl Probaband remark that by duality k = E [ a(T))δ Sk (G k u) ] k = 1,...,M. (40)This concludes the proof.⊓⊔Proposition 5 allows a certain flexibility in choosing either the process u, or betterz. Wetakethenz h = α k δhk Kr;h, k = 1,...,M, α k = 1, ∀k, and this implies that tocompute the k-th delta we can consider only the k-th term of the Skorohod integral,thus reducing the computational cost. In particular, this choice is motivated by thefact that in this way the localization technique needs to control only δk Sk (·), and henceonly the k-th component of W(t). We now define L k and calculate for k = 1,...,ML k =∫ T0D k s m(T)ds = M ∑i=1N∑w ij S i (t j )j=1∫ t j0σ ik (s)ds, (41)∫ T0D k s G kds =N∑j=1w jk S k (t j )x k∫ t j0σ kk (s)ds =N∑j=1w jk S k (t j )x k∫ t j0σ k (s)ds, (42)∫ T0D k s L kds =N∑j=1(∫ t j 2w ij S i (t j ) σ ik ds), (43)0so that[ ( )] k = E a(T)δkSk Gk, k = 1,...,M. (44)L KDue to the properties of the Skorohod integral we then have for k = 1,...,M( )Gkδ k = G kW k (T) − 1 ∫ T∫ T)(LL K L K L 2 k D k s G kds − G k D k s L kds , (45)k 00Remark that with a different choice of z (for instance z h = α h ) k would linearly dependon the whole M-dimensional BM, eventually making the localization techniqueless efficient.We finally investigate the applicability of the RQMC approach to estimate theexpected value in Eq. 44 for k = 1,...,M. Take the same input parameters as inSection 4, and generate the trajectories (the values S i (t j ), i = 1,...,M, j = 1,...,N)in Eqs. 41–43 as done in that section. Consider then the α im as the elements ofthe CH matrix associated to ρ im , i, m = 1,...,M: the Table 5 compares the deltasobtained only with RQMC, with the same number of scenarios as in Section 4. Weadopted the same LT procedure used to estimate the option price, and not that forthe integrand function in Eq. 44: at first sight this does not seem to be an optimalchoice; but, would we have applied the LT for the integrand function in Eq. 44,M = 10 decomposition matrices (one for each delta) should have been considered.This would have increased the CPU time by at least 1/3 of the total time (or evenmore, due to the larger number of terms to compute) thus making the calculation lessconvenient. Table 5 shows that the PCA, LT and KPA approaches perform almostequally well in terms of total accuracy, with the LT giving better results only forcorrelated assets. In terms of total accuracy (Err), the KPA performs better than