Spectral Calibration of Time-inhomogeneous Exponential Lévy Models: with Asymptotic Normality, Confidence Intervals, Simulations, and Empirical Results

The problem of calibrating time-inhomogeneous exponential Lévy models with finite jump activity based on market prices of plain vanilla options is studied. Belomestny and Reiß introduced an estimation procedure for calibration in the homogeneous case with one maturity. The open-ended question that will be addressed is if we can extend this model to use all listed plain vanilla options with multiple maturities. This opens a way to use all available data to create a time-inhomogeneous model with time-dependent Lévy triplets between subsequent intervals based on the maturities. We establish via an adapted Lévy-Khintchine representation an estimation procedure on the explicit inversion of the option price formulas in the spectral domain with a cut-off scheme for the regularisation of high frequencies. The estimation procedure will be shown to be well-defined and the parameters to be normally distributed asymptotically. Practical implications imply that using the asymptotic variances of the normal distributions leads to insufficient confidence intervals for non-asymptotical cases. We, therefore, construct confidence intervals using an approximated finite sample variance. Monte Carlo simulations are implemented in the computational software R to evaluate the stability and accuracy of the estimation procedure. Furthermore, the finite sample confidence intervals are assessed with coverage probabilities. In the end, the estimation procedure is evaluated by calibrating to market data of plain vanilla S\&P500 options, which contain numerous maturities. The estimated parameters with confidence intervals between maturities support time-dependency and the constructed time-inhomogeneous exponential L{\'e}vy models appear favorable in contrast to the time-homogeneous model.Applied Mathematic