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...

This paper will provide information on what happened in the financial crisis of 2008 and how to graph volatility outside of the option market. We will investigate the causes of the financial crisis, as well as some of the social inequalities that still exist today. We will explore household...

Topographic finance is the study of surfaces to describe financial systems in multiple dimensions. The problem with finance and economics is to describe accurately what is actually governing price dynamics. The price dynamics are behavioral and do not exhibit a rational maximization of a utility...

In this paper we provide an outline of interest rate swaptions and how to price swaptions with different payoff or settlement types. Firstly we review the different settlement styles commonplace in financial markets. Secondly we review the swaption pricing formulae corresponding to each...

In this paper we outline the European interest rate swaption pricing formula from first principles using the Martingale Representation Theorem and the annuity measure. This leads to an expression that allows us to apply the generalized Black-Scholes result. We show that a swaption pricing...

We address the inverse problem of local volatility surface calibration from market given option prices. We integrate the ever-increasing ow of option price information into the well-accepted local volatility model of Dupire. This leads to considering both the local volatility surfaces and their...

Realized volatility computed from high-frequency data is an important measure for many applications in finance. However, its dynamics are not well understood to date. Recent notable advances that perform well include the heterogeneous autoregressive (HAR) model which is economically...

This paper proposes a cluster HAR-type model that adopts the hierarchical clustering technique to form the cascade of heterogeneous volatility components. In contrast to the conventional HAR-type models, the proposed cluster models are based on the relevant lagged volatilities selected by the...

In this paper we consider several time-varying volatility and/or heavy-tailed models to explain the dynamics of return time series and to fit the volatility smile for exchange-traded options where the underlying is the main ‘Borsa Italiana' stock index. Given observed prices for the time...