Expected Shortfall and Lévy Processes

Christoffer T.M.B. Christiansen

Student thesis: Master thesis


The purpose of this master’s thesis is to assess market risk via the risk measure Expected Shortfall using varying distributional assumptions. First a discussion revolving around assumptions in modern finance theory illustrates the need for a more flexible distribution than the widely used Gaussian distribution. An empirical analysis gives that daily financial returns are fat tailed with more extreme returns than what a Gaussian distribution can account for. Then an applied introduction of Lévy processes and their basic properties is given. By time-changing a Brownian motion with a inverse Gaussian process one constructs the time deformed normal inverse Gaussian process. As this process has variable higher moments it is found to account well for financial returns. To substantiate the risk measure Expected Shortfall one has to have knowledge of its more well known sibling Value-at-Risk. The coherency of these two measures are subsequently examined to exemplify the Basel Committee on Banking Supervisions’ proposal to remove Value-at-Risk as a regulatory risk measure. Furthermore various non-parametric and parametric methods are applied to examine the necessity for a different distributional assumption than the Gaussian. Finally the risk models are estimated and backtested for a variety of basic assets; stocks, exchange rates, and commodities. The main finding is, that assuming that returns are normal inverse Gaussian distributed provides superior forecasting to a Gaussian risk model. This is due to the normal inverse Gaussian distribution being more flexible as it can exhibit excess kurtosis and skewness. Furthermore, while a Gaussian model will lead to extreme events being incredibly rare, a normal inverse Gaussian will give the events realistic probabilities.

EducationsMSc in Mathematics , (Graduate Programme) Final Thesis
Publication date2013
Number of pages91