This paper collects statistical and mathematical properties of the stable count distribution, with dispersed treatises on economic regime related to volatility. It is hypothesized that the stable count distribution is the marginal distribution of the volatility indices, such as CBOE VIX and...

We show how to construct a composite Hidden Markov Model (HMM) to calculate real-time recession probability, using the jubilee and ldhmm packages in R. The input data is the unemployment rate (UNRATE) which is released monthly by the U.S. government. There are two sub-models: The one-year...

We developed a factor regression model, nicknamed “GUPTY”, to study the business cycles, and their relation to the monetary policy. It covers several major macro-economic quantities, including unemployment rate, GDP, and weekly payrolls in the U.S. after WWII. The model postulates that these...

This tutorial shows how to use the jubilee package in R to forecast the 10-year and 20-year growth trajectory of S&P 500 Index. R programs are provided to run the regressions, make predictions, and plot the charts. Additionally, in order to capture the short-horizon volatility (less than 10...

We propose a long-term forecast model based on linear growth and mean reversion characteristics in the U.S. stock market. It can forecast future returns of the stock market, Treasury yield, and gold price. The “jubilee” name comes from its optimal trend-following window of 45 years. The...

The risk neutral measure is identified as a symmetric location-scale family of distribution in the local regime of the λ model. A partial differential equation is derived as the transformation between the implied volatility surface and such risk neutral probability. Given a well-interpolated...

The R package ldhmm is developed for the study of financial time series using Hidden Markov Model (HMM) with the lambda distribution framework. In particular, S&P 500 index is studied in depth due to its importance in finance and its long history. Major features in the index, such as regime...

We develop a new theory of asset return and volatility based on the stable law and Lihn's former work on the lambda distribution. The accomplishments are twofold: First, the newly discovered stable count distribution models the stationary distribution of VIX with high statistical precision....

A special version of option pricing model based on elliptic distribution is developed to explain the volatility smile for short-maturity options. A skew fractional exponential distribution, called λ distribution, is formulated to facilitate the so-called λ transformation for option pricing....

Closed form solution for pricing SPX options is developed in the quartic λ pricing model aka λ = 4. The solution fits the term structure of SPX volatility smiles very well, from the shortest expiration dates all the way up to one year, on three distinct market conditions. The ATM volatility,...