Xipei Yang

Non-Leading Eigenvalue Distributions, RMT, and Correlations

We showed that Singular Spectrum Analysis (SSA) applied to time series yields better correlations for risk simulations. This involved comparing SSA-based correlations with standard correlations and to noise, a zero correlation Wishart random matrix (WRM). We complete this testing here. We also present tractable analytic approximate WRM results that we used in the analysis: (1) leading and non-leading eigenvalue distributions of a WRM, (2) eigenvalue spacing of WRMs, and (3) eigenvector components of WRMs.

Predicting Equity Crises, Critical Exponents, and Earthquakes - II

We present further encouraging evidence for the Critical Exponent Earthquake Crisis (CEEC) Model that gives the probabilities of equity crises one year in advance. The CEEC model uses suitable precursor signals and is agnostic regarding dynamical origins of crises. The precursors accumulate in time between crises, like precursors to some earthquakes. The model uses a sophisticated noise filter to separate out crisis signals. The main metric is the anomalous exponent of an equity series describing the difference of the data variance scaling exponent from the Brownian variance scaling exponent of 1. No extra non-equity variables are used. The CEEC Model results for predicting crises are not perfect, but are much better than chance, including out-of-sample tests. The details here supplement our previous CEEC crisis paper (2013). In another paper (2016) we give details showing that various markets - not just equities - that are already in crisis are on the average described by a critical exponent of the nonlinear-diffusion Reggeon Field Theory (RFT), calculated in 1974, with no free parameters. Nonlinear diffusion naturally extends Brownian motion. Rich/cheap crisis behavior is suggested as a paradigm. This supplements our previous CEEC crisis paper (2013). The CEEC Model crisis predictions use the same scaling form described by the anomalous exponent as does the RFT. This consistency for crisis predictions and crisis behavior is significant.

Describing Crises with a Critical Exponent of the Reggeon Field Theory

We present evidence that markets in crisis can be described by a critical exponent of the nonlinear-diffusion Reggeon Field Theory, calculated 40 years ago, with no free parameters, translated to finance. We propose this as a benchmark for average crisis behavior, to which individual crises can be rich or cheap. In another paper we present a quantitative model for the probability of equity crises in advance. An earlier paper contained a summary.

Nearest Neighbor Technique for a Positive Definite Correlation Matrix in Advanced Stressed VAR

We present a new technique for obtaining a positive definite (PD) correlation matrix from a stressed target matrix within the context of Advanced Stressed Value at Risk, (cf. Dash ). The technique uses the spherical decomposition and a “nearest neighbor” technique. The advantage is that control over the scenario specified by the risk manager for the target matrix is achieved in that a maximal number of correlations specified by the target matrix scenario is maintained.

Stable Reduced-Noise 'Macro' SSA - Based Correlations for Long-Term Counterparty Risk Management

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 correlations. We call these “macro�? correlations because they are determined with macroscopic time scales. Stable correlations are desirable to suppress noise from short time scales that make risk measures unstable. If correlations move around, risk measures also move around, making business decisions difficult. SSA-based correlations ameliorate this business problem.

Noise-Reduced Correlations, the Signal to Noise Ratio, and SSA

This is the second paper presenting noise-reduced, stable correlations for long-term risk measurement. We smooth time series using Singular Spectrum Analysis (SSA) and then form the correlations from these smoothed time series. These correlations have superior time stability and are cleaned of noise. Here we show that the Signal-to-Noise Ratio is larger for the SSA-based correlations than usual, and we perform other signal/noise tests. We use new results refining random matrix correlations.

Path Integrals and Smart Monte Carlo - I

“Smart Monte Carlo�? (SMC) improves accuracy and speed. We extend results in an earlier paper, applying SMC to path-dependent deals and multifactor models. Auxiliary results are a path-to-path distance, an analytic approximation for N-dimensional Gaussian integrals, and time interpolation results. We suggest “model perturbation�? using simple approximate models, and introduce a new product “DIAS�? for better mortgage servicing hedging. We prove a consistency condition for the MRG interest-rate model, and discuss real-world vs. risk-neutral simulations.

Macro-Micro, Trends vs. Noise, and SSA - II

We describe some details of extensions of the Macro Micro (MM) model. Applications include a long-term real-world PFE risk simulation, including realistic quasi-random Macro trends. The details elaborated here include the use of the 3rd order skew Green function to obtain micro mean reversion, a random time distribution for the Macro component, sliding down the yield curve, and approximate no-arbitrage in the MM model.

Market Crises, Earthquakes, and the Reggeon Field Theory

This paper contains new results for helping to understand financial crises. First, we present a new model for obtaining the probability of equity crises within one year in advance, and we test it. Second and separately, various markets already in crises appear quantitatively related to a theory of nonlinear diffusion called the Reggeon Field Theory calculated years ago. Details are in a longer companion paper i. The time dynamics of the origins of crises are commonly pictured by bubbles growing and collapsing. Our dynamic model, the “CEEC (Critical Exponent Earthquake Crisis) Model”, has these features , and can provide early warning equity crisis signals to help prevent losses. The CEEC Model uses concepts of “critical exponents” from physics plus a qualitative analogy from earthquakes to describe the build-up of bubbles (increase of “frictional stress”) with subsequent crises from bubble collapses (“earthquakes”) . The only inputs to the CEEC model for an equity index are equity index returns. Tests comparing the CEEC model to data yield encouraging results, much better than chance. Here are results from running the model as the user would have run it in the past: • In 69% of the tests for which the model indicated a crisis in the short term (within one year), a crisis was in fact observed within one year. • 31% of actual crises were missed by the model within one year before the crisis. We give a summary of the CEEC model in this short paper. A longer companion paper describing full details of the model will appear. We also analyze various markets already in crisis (equity, FX, commodities, rates, bonds). We find behavior numerically consistent with a theoretical result with no free parameters for the general theory of nonlinear diffusion in physics, generalizing standard Brownian motion, the Reggeon Field Theory RFT. An anomalous RFT critical exponent translated into finance language is around 0.3, and this number qualitatively describes the average behavior of markets in crisis. This 0.3 RFT anomalous variance exponent is perhaps the first number calculated in advance since the Gaussian Brownian diffusion variance exponent (1.0) used in standard finance, without numerically fitting anything.