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...

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....

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...

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,...

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...

A new Poisson subordinated distribution is proposed to capture major leptokurtic features in log-return time series of financial data. This distribution is intuitive, easy to calculate, and converge quickly. Analytic formula exists for all moments for the entire parameter space. It fits well to...

This working paper is an excerpt on the recent finding of the network effect in the stock market. Specifically I summarize the three-half power law which states that the total value of the stock market is proportional to the three-half power of the number of stocks in the market. This power law...

To illustrate that the mathematical form of P. K. Clark's distribution of lognormal-normal increments, with a minor bug fix, is the same as the symmetric lognormal cascade distribution that I studied in 2008-9, although the two were derived from very different stochastic assumptions

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...