An Application of Extreme Value Theory to Finance – An Empirical Study of Whether Stock Shape Parameters and the Differences Between Corresponding Actual and BSM Put Option Values Are Positively Correlated

Abstract

Large stock price movements and financial crises are a common occurrence relative to the belief of normality in markets, and they are becoming more common as the world becomes more interconnected and more technology driven. After large dips in prices and after crises the topic of risk management, specifically tail risk quantification, comes into the focus of not just researchers, but is also put into question by the general public. Recently, Extreme Value Theory, originally targeted at the study of weather and climate, entered the field of risk management and provided a new approach to measuring tail risk, letting the data speak for itself rather than underpinning distributional assumptions as most approaches had done thus far. This thesis employs Extreme Value Theory, specifically the Peaks over Threshold approach or Generalized Pareto Distribution approach, in order to quantify the tail thickness of the left tail of stock return distributions of the S&P 500 index and its constituents. The thesis then assesses the relationship between the tail fatness of the underlying stock and the absolute as well as relative difference between the actual put option value and the put option value according to Black Scholes Merton. The hypothesis is that thicker left tails imply that large negative movements are more probable than under the Gaussian assumption. This in turn could display itself in an undervaluation of put options, in particular deep out of the money put options, as calculated by traditional BSM. In the analysis, neither the absolute nor the relative differences seem to show a relationship to respective shape parameters. However, the averages of the absolute differences of deep out of the money put options do appear to be very small in size, indicating a valuation close to that of the BSM model, while the averages of the relative differences of deep out of the money put options are for the larger part close to 100%, indicating a complete undervaluation of these put options by BSM, regardless of the shape parameter. The lack of relationship between shape parameters and differences begs the question of whether buyers are valuing options correctly and the employed methodology is wrong or whether the thesis methodology is appropriate and buyers are mistaken. A short analysis provides evidence pointing to the latter.