In order to characterize a statistical probability distribution p(x) of a variable x, the moments of the distribution are used; the first two of which are the mean and standard deviation. The z-score is often used to characterize data points of x (e.g. outliers with large z-scores). Polynomials...

We present an exact analytic solution to the two-dimensional correlated default structural Merton model in the form of a local volatility problem using a conformal square-root transformation of the exact solution to a 2D hybrid barrier problem. We also give an approximation and evaluate it...

This is the third paper in a series devoted to obtaining noise-reduced, stable correlations by smoothing time series using Singular Spectrum Analysis, or SSA. Here we show that the SSA-based correlations are superior in terms of noise reduction, employing a number of simple tests using Random...

We introduce a methodology from geophysics, Singular Spectrum Analysis (SSA), to obtain stable, noise-cleaned correlations for long term risk (e.g. counterparty risk). SSA is applied to time series to smooth them in a robust manner. The SSA-smoothed time series are then used to obtain the...

We summarize new results for estimating correlations for use in risk management. These estimates have better behavior than traditional estimation approaches from both a business standpoint and a technical standpoint. We smooth time series using Singular Spectrum Analysis (SSA) and compute...

We present the framework for a distressed bond model. The utility is as a proxy for calculating the risk of a distressed bond portfolio. We elaborate several possible implementations and give an example

We introduce advanced idiosyncratic risk (“AI-Risk”), a parsimonious correlated residual correction to a predictive stress CAPM-like factor model, aimed to get more accurate stock-stock correlations. We find that AI-Risk can be significant for stock portfolios. Inclusion of AI-Risk gives a...

We construct “Hybrid Value at Risk” (HYVAR) that is an arbitrary mixture of Historical VAR and Monte Carlo VAR. The procedure is capable of retaining both the correlation matrix of the original time series and also jumps/‘fat tails'. For this reason HYVAR provides more realistic scenarios,...

A common scenario risk analysis employs a multiple factor model with assumed changes in the factors to obtain changes in non-factor variables. This analysis is sometimes designated as a “predictive stress scenario”. We choose to designate the factor model as a multifactor “CAPM” model,...

The Macro Micro (MM) model contains different time scales and deals with risk as it occurs in the real world, especially trend risk. A new methodology - Singular Spectrum Analysis (SSA) – is introduced to identify historical trends, trend volatilities, and noise-reduced trend-trend...