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I have a function noted $u$ which I know the value on N points $s_{1} ,s_{2},.....,s_{N}$ we denote $u_{1},u_{2},...,u_{n}$ the values of u in these points and a grid of strikes $ (K_{i})_{1 \le i \le N_{k}} $

Im looking for the best approximation of $u$ as a sum of functions of the form $g(x)=\sum_{i=1}^{N_{k}}\alpha_{i}(x-K_{i})$ Then I should find as i want to "sur-replicate" that I want to solve the ( wrongly formulated ) following problem
$\min_{\alpha} u(x)-g(x) $ subject to $u(x)<g(x)$ for all $x \in R$ ( We can restrict this condition to some compact of the form $]- S_{min};S_{max}[$ )

How do I perform this ? It seems to be easy but im not getting the thing here. At most i can consider something similar to linear regression , but the constraint changes the nature of the problem.

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  • $\begingroup$ Did you really mean $\sum \alpha_i(x_i-K_i)$ or did you mean $\sum \alpha_i(x_i-K_i)^+$ $\endgroup$
    – nbbo2
    Aug 30, 2020 at 17:29
  • $\begingroup$ Possibly related question quant.stackexchange.com/questions/37419/… $\endgroup$
    – nbbo2
    Aug 30, 2020 at 22:46
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    $\begingroup$ It is completely unclear what you are asking. 1) Your function $u$ is defined on an interval? 2) Best approximation in what norm? 3) How is the second problem with the "min" related to the first one? 4) Do you really want to minimise the difference? It is always negative so there is no lower bound??? One approach for your first problem is just piecewise linear interpolation! Assuming indeed you mean $(x - K_i)^+$ $\endgroup$
    – g g
    Aug 31, 2020 at 20:19

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Multivariate Adaptive Regression Splines (MARS) models might be helpful (I don't see any hockey stick/call payoff in your function $g$, but I'll assume that you want them), as they are built on functionals of type:

$$ g(x)= \sum_{i=1}^n \alpha_i \max (x - K_i, 0). $$

The basis can include products of hockey stick (also called hinge, ramp, or rectifier) functions. (See also implementation packages referenced in the wiki link above.)

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