Panel data are modern statistical tools which are commonly used in all kinds of econometric problems under various regularity assumptions. The panel data models with changepoints are introduced together with atomic pursuit methods and they are applied to estimate the underlying option price function. Robust estimates and complex insight into the data are both achieved by adopting the quantile LASSO approach. The final model is produced in a fully data-driven manner in just one single modeling step. In addition, the arbitrage-free scenarios are obtained by introducing a set of well defined linear constraints. The final estimate is, under some reasonable assumptions, consistent with respect to the model estimation and the changepoint detection performance. The finite sample properties are investigated in a simulation study and proposed methodology is applied for the Apple call option pricing problem.