This working paper studies the quot;skew lognormal cascade distributionquot;, which is proposed by the first time here, as the static solution of the simplified SIBM model (Stephen Lihn 2008, SSRN: 1149142). This distribution exhibits fat-tail, asymmetry tunable by a skew parameter, converges to...

A continuous-time scale-invariant Brownian motion (SIBM) stochastic equation is developed to investigate the dynamics of the stock market. The equation is used to solve the fat tail distribution of the stock universe and the DJIA time series. It is also used to model the volatility clustering in...

This working paper presents the general theory of the higher order "skew lognormal cascade distribution" as a mathematical extension of the previously proposed skew lognormal cascade distribution. In particular, the second order distribution is studied in details, which incorporates the fat...

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...

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....

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...

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,...

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 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...