Approximating Functions with Call Payoff Basis Functions

Summary

The document asks how to approximate a function from values observed at a finite set of points using a weighted sum of call-like payoff functions at selected strikes. It frames the goal as a super-replication problem: find coefficients so that the approximation stays above the target function over a chosen domain. It notes that ordinary linear regression does not directly capture this inequality constraint and recognizes that the stated optimization objective needs reformulation.

The response points to Multivariate Adaptive Regression Splines, whose basis functions include positive-part hinge terms resembling call payoffs. It also notes that MARS can use products of these terms to build richer approximations. However, the answer offers a modeling connection rather than a full solution to the constrained optimization problem: it does not explain how to enforce the global super-replication condition, select strikes or coefficients, or assess approximation error outside the observed points. Those details would need separate treatment.

Key ideas

• The target is to approximate observed function values with a weighted sum of call-like basis functions at chosen strikes.
• The desired super-replication property requires the approximation to dominate the target over the selected domain.
• The document notes that the proposed minimization statement does not correctly express that constrained objective.
• MARS uses hinge functions related to call payoffs and can combine them through products.
• The response suggests a basis family but does not provide a method for enforcing the global inequality constraint.

Tags

# Best approximation of a function as sums of calls


# Best approximation of a function as sums of calls












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.

## Answer by ir7 (score 1)

https://quant.stackexchange.com/a/57689

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.)

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.