I'm trying to solve the Dupire equation using natural cubic splines and Tikhonov regularization. See equation (2.1) page 3 http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.69.9928&rep=rep1&type=pdf
The idea is to compute (for each maturity) a smoothing natural cubic spline u_i s.t. the sum is minimized. The authors refer to two sources but it didn't help. How do I determine u_i such that (2.1) is minimzed? I need an analytical expression of u_i (for each i) because I want to determine the first and second derivative of u_i w.r.t. X to ultimately determine the local volatility.
The idea is to compute (for each maturity) a smoothing natural cubic spline u_i s.t. the sum is minimized. The authors refer to two sources but it didn't help. How do I determine u_i such that (2.1) is minimzed? I need an analytical expression of u_i (for each i) because I want to determine the first and second derivative of u_i w.r.t. X to ultimately determine the local volatility.
