The first ever explicit formulation of the concept of the option's probability density functions has been introduced in our publications “Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options - True...

The first ever explicit formulation of the concept of an option's probability density functions has been introduced in our publications “Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options - True Value-at-Risk...

• It is not widely emphasized in the literature that derivatives are complex random quantities which should, by custom, be characterized by their probability density functions. • It is understood that Black-Scholes style of derivatives pricing represents an expected value, i.e. the...

A Hidden Markov Model (HMM) is used to model the VIX (the Cboe Volatility Index). A 4- state Gaussian mixture is fitted to the VIX price history from 1990 to 2022. Using a growing window of training data, the price of the S&P500 is predicted and two trading algorithms are presented, based on the...

Linear GARCH(1,1) and threshold GARCH(1,1) processes are established as regularly varying, meaning their heavy tails are Pareto like, under conditions that allow the innovations from the, respective, processes to be skewed. Skewness is considered a stylized fact for many financial returns...

Several procedures to estimate daily risk measures in cryptocurrency markets have been recently proposed in the literature. Among them, procedures taking into account the presence of extreme observations, as well as procedures that include more than a single regime, have performed substantially...

Volatility forecasts play a central role among equity risk measures. Besides traditional statistical models, modern forecasting techniques, based on machine learning, can readily be employed when treating volatility as a univariate, daily time-series. However, econometric studies have shown that...

We introduce extensions of the Realized Exponential GARCH model (REGARCH) that capture the evident high persistence typically observed in measures of financial market volatility in a tractable fashion. The extensions decompose conditional variance into a short-term and a long-term component. The...

Several procedures to forecast daily risk measures in cryptocurrency markets have been recently implemented in the literature. Among them, long-memory processes, procedures taking into account the presence of extreme observations, procedures that include more than a single regime, as well as...

This paper studies large and moderate deviation properties of a realized volatility statistic of high frequency financial data. We establish a large deviation principle for the realized volatility when the number of high frequency observations in a fixed time interval increases to infinity. Our...