As is well-documented, time series data of stock quotes can be regarded as the realization of a trajectory associated with a one-dimensional random walk. This chapter utilizes empirical data from the S&P 500 index. The probability density function (PDF) derived from this data exemplifies heavy-tailed distributions, characteristic of real-world phenomena, particularly in the context of the stock market. The analysis introduces the concept of moral hazard, where the potential for rare but substantial losses accompanies significant profits. This phenomenon necessitates a modernized understanding of risk, especially in investment paradigms. Lévy processes are well-suited for describing such behavior, in notable contrast to Gaussian processes, characterized by narrower distributions. Crucially, significant information regarding the statistical properties of the system resides in the tails of the distribution function. Consequently, this leads to infinite variance and kurtosis, rendering traditional risk assessments that rely on variability ineffective. Therefore, it is evident that a robust market risk analysis demands a reconceptualization of risk. More generally, non-Brownian and non-Gaussian stochastic processes observed in financial markets have contributed to a new/contemporary perspective on market risk in the broad sense.