Abstract

Model calibration is essential to pricing and hedging of financial products. A key question
when calibrating a model based on prices of European call and put options is how all
information contained in the option prices can be merged into one model. On the one hand,
the model needs to be large enough to be sufficiently flexible and to be able to integrate
all the available information. On the other hand, the model needs to be identifiable from
the options traded on the market. The frequently observed volatility smile or skew is an
indication that the Black–Scholes model is not flexible enough to account for the prices of
options with different strike prices. Exponential Lévy models are flexible enough to model
the volatility smile or skew and can therefore incorporate the information of options with
different strike prices. However, empirical evidence shows that calibrating exponential Lévy
models by options with different maturities leads to conflicting information. In other words,
the stationarity implicitly assumed in the exponential Lévy model is not satisfied. We
propose an identifiable time-inhomogeneous Lévy model that does not assume stationarity
and that can integrate option prices from different maturities and different strike prices
without leading to conflicting information.
In the time-inhomogeneous Lévy model, we derive the convergence rates and show
confidence intervals for the estimators of the volatility, the drift, the intensity, and the Lévy
density. Previously confidence intervals have been constructed for time-homogeneous Lévy
models in an idealized Gaussian white noise model by Söhl [2]. In the idealized Gaussian
white noise model, it is assumed that the observations are Gaussian and given continuously
across the strike prices. This simplifies the analysis significantly. Here we construct the
confidence intervals in a discrete observation setting for time-inhomogeneous Lévy models
and the only assumption on the errors is that they are sub-Gaussian, in particular, all
bounded errors with arbitrary distributions are covered. Our additional results on the
convergence rates extend the paper by Belomestny and Reiß [1] from time-homogeneous
to time-inhomogeneous Lévy models.

References

[1] Belomestny D., Reiß M., Spectral calibration of exponential Lévy models, Finance and
Stochastics, 10(4) (2006), 449-474.
[2] Söhl J., Confidence sets in nonparametric calibration of exponential Lévy models, Fi-
nance and Stochastics, 18(3) (2014), 617-649.