George Skiadopoulos, Nikolaos Panigirtzoglou
This paper first investigates the dynamics of implied probability density functions (PDFs), and the dynamics of implied cumulative distribution functions (CDFs). Subsequently, it presents new algorithms for simulating the probabilities evolution through time. Understanding the dynamics of implied distributions is essential for effective risk management, for ''smile-consistent'' arbitrage pricing with stochastic volatility, and for economic policy decisions. We investigate the number and shape of shocks that drive implied PDFs and CDFs by applying Principal Components Analysis (PCA). Under a variety of criteria, two components are identified. A Procrustes type of rotation is performed on the CDF components and it reveals an intuitive interpretation. The first component affects the location of the implied distribution while the second affects its variance and kurtosis.
The proposed algorithms are arbitrage-free and they can be implemented easily. The only inputs that they require are the known initial implied distribution and the PCA output. An out-of-sample application for Value-at-Risk (VaR) purposes is provided. The application reveals that the algorithms forecast accurately the range within which the future VaR level will lie.
JEL classification: G11, G12, G13.
Keywords: Implied Probability Density Function, Implied Cumulative Distribution Function, Principal Components Analysis, Monte Carlo Simulation, Value-at-Risk.