Frank J. Fabozzi

Financial Models with Lévy Processes and Volatility Clustering: Rachev/Financial

Preface. About the Authors. Chapter 1 Introduction. 1.1 The need for better financial modeling of asset prices. 1.2 The family of stable distribution and its properties. 1.3 Option pricing with volatility clustering. 1.4 Model dependencies. 1.5 Monte Carlo. 1.6 Organization of the book. Chapter 2 Probability distributions. 2.1 Basic concepts. 2.2 Discrete probability distributions. 2.3 Continuous probability distributions. 2.4 Statistic moments and quantiles. 2.5 Characteristic function. 2.6 Joint probability distributions. 2.7 Summary. Chapter 3 Stable and tempered stable distributions. 3.1 alpha-Stable distribution. 3.2 Tempered stable distributions. 3.3 Infinitely divisible distributions. 3.4 Summary. 3.5 Appendix. Chapter 4 Stochastic Processes in Continuous Time. 4.1 Some preliminaries. 4.2 Poisson Process. 4.3 Pure jump process. 4.4 Brownian motion. 4.5 Time-Changed Brownian motion. 4.6 Levy process. 4.7 Summary. Chapter 5 Conditional Expectation and Change of Measure. 5.1 Events, s-fields, and filtration. 5.2 Conditional expectation. 5.3 Change of measures. 5.4 Summary. Chapter 6 Exponential Levy Models. 6.1 Exponential Levy Models. 6.2 Fitting a-stable and tempered stable distributions. 6.3 Illustration: Parameter estimation for tempered stable distributions. 6.4 Summary. 6.5 Appendix : Numerical approximation of probability density and cumulative distribution functions. Chapter 7 Option Pricing in Exponential Levy Models. 7.1 Option contract. 7.2 Boundary conditions for the price of an option. 7.3 No-arbitrage pricing and equivalent martingale measure. 7.4 Option pricing under the Black-Scholes model. 7.5 European option pricing under exponential tempered stable Models. 7.6 The subordinated stock price model. 7.7 Summary. Chapter 8 Simulation. 8.1 Random number generators. 8.2 Simulation techniques for Levy processes. 8.3 Tempered stable processes. 8.4 Tempered infinitely divisible processes. 8.5 Time-changed Brownian motion. 8.6 Monte Carlo methods. Chapter 9 Multi-Tail t-distribution. 9.1 Introduction. 9.2 Principal component analysis. 9.3 Estimating parameters. 9.4 Empirical results. 9.5 Conclusion. Chapter 10 Non-Gaussian portfolio allocation. 10.1 Introduction. 10.2 Multifactor linear model. 10.3 Modeling dependencies. 10.4 Average value-at-risk. 10.5 Optimal portfolios. 10.6 The algorithm. 10.7 An empirical test. 10.8 Summary. Chapter 11 Normal GARCH models. 11.1 Introduction. 11.2 GARCH dynamics with normal innovation. 11.3 Market estimation. 11.4 Risk-neutral estimation. 11.5 Summary. Chapter 12 Smoothly truncated stable GARCH models. 12.1 Introduction. 12.2 A Generalized NGARCH Option Pricing Model. 12.3 Empirical Analysis. 12.4 Conclusion. Chapter 13 Infinitely divisible GARCH models. 13.1 Stock price dynamic. 13.2 Risk-neutral dynamic. 13.3 Non-normal infinitely divisible GARCH. 13.4 Simulate infinitely divisible GARCH. Chapter 14 Option Pricing with Monte Carlo Methods. 14.1 Introduction. 14.2 Data set. 14.3 Performance of Option Pricing Models. 14.4 Summary. Chapter 15 American Option Pricing with Monte Carlo Methods. 15.1 American option pricing in discrete time. 15.2 The Least Squares Monte Carlo method. 15.3 LSM method in GARCH option pricing model. 15.4 Empirical illustration. 15.5 Summary. Index.

A Probability Metrics Approach to Financial Risk Measures: Rachev/A Probability Metrics Approach to Financial Risk Measures

All of these questions are essential to finance and they have one feature in common: measuring distances between random quantities. Problems of this kind have been explored for many years in areas other than finance. In A Probability Metrics Approach to Financial Risk Measures, the field of probability metrics and risk measures are related to one another and applied to finance for the first time, revealing groundbreaking new classes of risk measures, finding new relations between existing classes of risk measures, and providing answers to the question of which risk measure is best for a given problem. Applications include optimal portfolio choice, risk theory, and numerical methods in finance.

A COMPARISON AMONG PERFORMANCE MEASURES IN PORTFOLIO THEORY

Abstract This paper examines some performance measures to be considered as an alternative of the Sharpe Ratio. More specifically, we analyze allocation problems taking into consideration portfolio selection models based on different performance ratios. For each allocation problem, we compare the maximum expected utility observing all the portfolio selection approaches proposed here. We also discuss an ex-post multi-period portfolio selection analysis in order to describe and compare the sample path of the final wealth processes.

Fractional Partial Differential Equation and Option Pricing

There are different approaches for studying the behavior of a stochastic process. A stochastic process can be studied as a stochastic differential equation, a partial integro-differential equation, and a fractional partial differential equation. The efficiency of these different approaches depends on the dynamics of the asset price process and the numerical approach for solving them.

Fractional calculus and fractional processes: an overview

In this monograph we discuss how fractional calculus and fractional processes are used in financial modeling, finance theory, and economics. We begin by giving an overview of fractional calculus and fractional processes, responding upfront to two important questions: 1. What is the fractional paradigm for both calculus and stochastic processes?

Fractional Diffusion and Heavy Tail Distributions: Geo-Stable Distribution

In this chapter, our objective is twofold. First, we establish a connection between the stable distributions with fractional calculus. This is accomplished by defining appropriate fractional diffusion equations, the fundamental solution of which provides the PDF for the univariate and multivariate stable distributions.1 Second, by using some analytic-numerical approaches such as the homotopy perturbation method, the Adomian decomposition method, and the variational iteration method, which are used for solving partial differential equations (PDEs), we obtain some analytic-numerical approximations for the PDF of the univariate and multivariate stable distributions.

Continuous-Time Random Walk and Fractional Calculus

Continuous-time random walk is an extension of the random walk. More specifically, it is constructed by introducing a new source of randomness to the random walk. This new source of randomness is waiting time. In this chapter, we first discuss the continuous-time random walk, and then move on to its applications in financial economics.

Applications of Fractional Processes

Chapter 2 described how fractional calculus can be applied to generate fat-tailed distributions; discussed how to apply fractional processes to the pricing of derivatives. As fractional processes are not semi martingales, violations of the no-arbitrage condition might occur. We have seen how to circumvent this problem.

2 – Fractional Calculus

: Since the early 1960s, there have been a good number of papers related to heavy tail distributions. These papers support the view that the heavy tail property is a stylized fact about financial time series. Stable distributions have infinite variance, a property which is not found in empirical samples where empirical variance does not grow with the size of the sample. As an example of the application of fractional calculus, by extending diffusion equations to fractional order, a connection between stable and tempered stable distribution with fractional calculus can be made.

Practical Applications of Recent Trends in Equity Portfolio Construction Analytics

The growth of big data and advances in data analytics offers asset managers new challenges-and opportunities. Affordable cloud computing services and software tools are now widely available, allowing managers the ability to accessand analyze large data sets, which can be used to more effectively create, implement and manage smart beta and other portfolio optimization strategies. Recent Trends in Equity Portfolio Construction Analytics offers a state-of-the-art review of equity portfolio analytics and lays out the current developments in market analytics and asset allocation models. It was co-written by Dessislava Pachamanova , Associate Professor of Operations Research at Babson College , and Frank Fabozzi , Editor of JPM and Professor of Finance at the EDHEC Business School in Nice, France.