A Note on Estimating the Parameters of the Diffusion-Jump Model of Stock Returns

A practical approach to semideviation and its time scaling in a jump-diffusion process

“…The task of fitting a jump-diffusion model to real-world data is not as easy as it appears, and this fact has been early recognized, see for instance Beckers (1981) or Honoré (1998). Essentially, the reason lies in the fact that the likelihood function of an infinite mixture of distribution can be unbounded, hence resulting in inconsistencies.…”

Calibration of Jump-Diffusion Option Pricing Models: A Robust Non-Parametric Approach

“…In fact, the gradient of our calibration criterion (computed in section 4.4) vanishes at zero which means that the algorithm does not modify the Lévy density in this region: the intensity of small jumps can not be retrieved accurately. The redundancy of small jumps and diffusion component is well known in the context of statistical estimation on time series [8,26]. Here we retrieve another version of this redundancy in a context of calibration to a cross sectional data set of options.…”

Disentangling Volatility from Jumps

“…Early examples of the use of this or similar formulae for maximum likelihood in finance are contained in Press (1967), Beckers (1981) and Ball and Torous (1983). A non-zero value of the mean jump size β would add skewness.…”

“…Indeed, while the early use on jumps in finance has focused exclusively on Poisson jumps (see Press (1967), Merton (1976), Beckers (1981) and Ball and Torous (1983)), the literature is rapidly moving towards incorporating other types of Lévy processes, such as the Cauchy jumps which are considered here. This is the case either for theoretical option pricing (see e.g., Madan et al (1998), Chan (1999) and Carr and Wu (2003a)), risk management (see e.g., Eberlein et al (1998)), or as a means of providing more accurate description of asset returns data (see e.g., Carr et al (2002)).…”