Risk and Ambiguity in Turbulent Times

This paper focuses on the 2008–2020 period during which two major crises, affecting the economy and the financial markets, occurred. Between 2008 and 2020, there were less extreme tail events, including the lingering Eurozone and Greece crises. In particular, after extremely high stock market volatility and volatility of volatility (VoV) during 2008, the long-run average volatility declined to about 20% and the VoV to around 100%. This paper analyzes this period through the lens of risk and ambiguity (uncertainty). It aims to address the question: what are the financial markets that trade risk — the volatility derivatives markets — telling us? To this end, this paper uses several measures of uncertainty. It reviews the history of volatility and uncertainty measures and discusses their informativeness. It then discusses the information derived from volatility derivatives.

Fast maximum likelihood estimation of parameters for square root and Bessel processes

Explicit formulae for maximum likelihood estimates of the parameters of square root processes and Bessel processes and first and second order approximate sufficient statistics are supplied. Applications of the estimation formulae to simulated interest rate and index time series are supplied, demonstrating the accuracy of the approximations and the extreme speed-up in estimation time. This significantly improved run time for parameter estimation has many applications where ex-ante forecasts are required frequently and immediately, such as in hedging interest rate, index and volatility derivatives based on such models, as well as modelling credit risk, mortality rates, population size and voting behaviour.