Complexity With n underlyings and time t an option problem comprises n+1 independent variables. Assume that we discretize each of the n+1 axes with M grid points, then Mn+1 nodes are involved. Hence the complexity C of the n-factor model is C := O(Mn+1), which amounts to an exponential growth with the dimension, nicknamed curse of dimension. Depending on the chosen method, the error E is of the order M−, E := O  1 M  . Argue log C = −n + 1  log E + γ for a method-dependent constant γ.

s ϕi(xi)=1 ϕi(x) ≡ 0 for xxi+2 ϕ ∈ C2(−∞, ∞). To construct these ϕi proceed as follows:

a) Construct a spline S(x) that satisfies the above requirements for the special nodes x˜k := −2 + k for k = 0, 1, ..., 4 .

b) Find a transformation Ti(x), such that ϕi = S(Ti(x)) satisfies the requirements for the original nodes.

c) For which i, j does ϕiϕj = 0 hold?