Narrow Results

We assume the market price to diffuse in a hierarchical comb of barriers, the heights of which represent the importance of new information entering the market. We find fat tails with the desired exponent for the price change distribution, and effective multifractality for intermediate times.

Starting from the random-walk model, practices of financial markets are included into the random-walk so that fat tail distributions like those in the high frequency data of the SP500 index are reproduced, though the individual mechanisms are modeled by normally distributed data. The incorporation of local correlation narrows the distribution for "frequent" events, whereas global correlations due to technical analysis leads to fat tails. Delay of market transactions in the trading process shifts the fat tail probabilities downwards. Such an inclusion of reactions to market fluctuations leads to mini-trends which are distributed with unit variance.

For square and simple cubic Ising models at T=T_c, we look at the tails of the magnetization distribution for untypically large magnetizations. No indications for a stretched exponential or power law behavior are found.

We analyze a stochastic volatility market model in which volatility is correlated with return and is represented by an Ornstein-Uhlenbeck process. In the framework of this model we exactly calculate the leverage effect and other stylized facts, such as mean reversion, leptokurtosis and negative skewness. We also obtain a close analytical expression for the characteristic function and study the heavy tails of the probability distribution.

A log-normal random walk with parameters that are functions of market stress naturally accounts for volatility clustering and fat-tailed return distributions. Fitting this model to a stock and a bond index we find no evidence of significant misspecification despite the fact that the model has no adjustable parameters. This model can be interpreted as a stochastic volatility model without latent variables. We obtain a closed-form expression for the Value at Risk (VaR) that accommodates returns of any magnitude and discuss several other applications.

Post-mortems of the financial crisis typically mention "black swans" as the rare events that were the Achilles heel of financial models, manifesting themselves as "25 standard deviation events occurring several days in a row". Here, we briefly discuss the implications of "black swan" events in asset pricing and risk management. We then show that the "black swans" problem virtually disappears for S&P Index returns when surprises are measured relative to the standard deviation of the conditional S&P distribution. In our illustration, we use the one-day-lagged VIX as an easy-to-understand measure of that conditional S&P standard deviation.

Low-probability, high-impact risks are critical features of climate change economics; however, there are many unanswered policy and modeling questions about the implications of fat-tailed uncertainty. This paper examines the impact of fat-tailed uncertainty about the climate sensitivity on abatement decisions using a sequential decision-making framework. The results demonstrate how policy prescriptions from integrated assessment models are sensitive to the specifications of uncertainty, learning, and damages. Fat tails alone do not merit immediate and stringent mitigation but require strongly convex damages and slow learning. The analysis illustrates the potential value of midcourse corrections on reducing consumption risks imposed by uncertain damages from climate change and focuses attention on the dynamics of learning.