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Über dieses Buch

This book is devoted to the history of Change of Time Methods (CTM), the connections of CTM to stochastic volatilities and finance, fundamental aspects of the theory of CTM, basic concepts, and its properties. An emphasis is given on many applications of CTM in financial and energy markets, and the presented numerical examples are based on real data. The change of time method is applied to derive the well-known Black-Scholes formula for European call options, and to derive an explicit option pricing formula for a European call option for a mean-reverting model for commodity prices. Explicit formulas are also derived for variance and volatility swaps for financial markets with a stochastic volatility following a classical and delayed Heston model. The CTM is applied to price financial and energy derivatives for one-factor and multi-factor alpha-stable Levy-based models.

Readers should have a basic knowledge of probability and statistics, and some familiarity with stochastic processes, such as Brownian motion, Levy process and martingale.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction to the Change of Time Methods: History, Finance and Stochastic Volatility

Abstract
In this Chapter, we provide a historical context for the development of change of time methods (CTM) and their connections with finance in general and stochastic volatility methods in particular.
Anatoliy Swishchuk

Chapter 2. Change of Time Methods: Definitions and Theory

Abstract
In this chapter, we consider the general theory of a change of time method (CTM). One of probabilistic methods which is useful in solving stochastic differential equations (SDEs) arising in finance is the “change of time method”. We give the definition of CTM and describe CTM in martingale, semimartingale, and the SDEs settings. We also point out the association of CTM with subordinators and stochastic volatilities.
Anatoliy Swishchuk

Chapter 3. Applications of the Change of Time Methods

Abstract
In this chapter, we give an overview on applications of change of time methods considered in this book in Chapters 4–8 These applications include yet another (among many) derivation of the Black-Scholes formula; the derivation of option pricing formula for a mean-reverting asset in energy finance; pricing of variance, volatility, covariance, and correlation swaps for the classical Heston model; pricing of variance and volatility swaps in energy markets; pricing of financial and energy derivatives with multifactor Lévy models; and pricing of variance and volatility swaps and hedging of volatility swaps for the delayed Heston model. This chapter not only describes the applications of the change of time method but also constitutes the ultimate difference between Barndorff-Nielsen-Shiryaev’s book (2010) and present book.
Anatoliy Swishchuk

Chapter 4. Change of Time Method (CTM) and Black-Scholes Formula

Abstract
In this Chapter, we consider applications of the CTM to (yet one more time) obtain the well-known Black-Scholes formula for European call options. In the early 1970s, Black-Scholes (1973) made a major breakthrough by deriving a pricing formula for a vanilla option written on the stock. Their model and its extensions assume that the probability distribution of the underlying cash flow at any given future time is lognormal. We mention that there are many proofs of this result, including PDE and martingale approaches, (see Wilmott et al. 1995; Elliott and Kopp 1999). The present approach, using change of time of getting the Black-Scholes formula, was first shown in Swishchuk (2007).
Anatoliy Swishchuk

Chapter 5. CTM and Variance, Volatility, and Covariance and Correlation Swaps for the Classical Heston Model

Abstract
In this chapter, we apply the CTM to price variance and volatility swaps for financial markets with underlying assets and variance that follow the classical Heston (Review of Financial Studies 6, 327–343, 1993) model. We also find covariance and correlation swaps for the model. As an application, we provide a numerical example using S&P60 Canada Index to price swap on the volatility (see Swishchuk (2004)).
Anatoliy Swishchuk

Chapter 6. CTM and the Delayed Heston Model: Pricing and Hedging of Variance and Volatility Swaps

Abstract
In this chapter, we apply the CTM for pricing and hedging of variance and volatility swaps for the delayed Heston model. We present a variance drift-adjusted version of the Heston model which leads to a significant improvement of the market volatility surface fitting (compared to the classical Heston model). The numerical example we performed with recent market data shows a significant reduction of the average absolute calibration error (calibration on 12 dates ranging from 19 September to 17 October 2011 for the FOREX underlying EURUSD). Our model has two additional parameters compared to the Heston model and can be implemented very easily. It was initially introduced for the purpose of volatility derivative pricing. The main idea behind our model is to take into account some past history of the variance process in its (risk-neutral) diffusion. Using a change of time method for continuous local martingales, we derive a closed formula for the Brockhaus and Long approximation of the volatility swap price in this model. We also consider dynamic hedging of volatility swaps using a portfolio of variance swaps.
Anatoliy Swishchuk

Chapter 7. CTM and the Explicit Option Pricing Formula for a Mean-Reverting Asset in Energy Markets

Abstract
In this chapter, we apply the CTM to get the explicit option pricing formula for a mean-reverting asset in energy markets.
Anatoliy Swishchuk

Chapter 8. CTM and Multifactor Lévy Models for Pricing Financial and Energy Derivatives

Abstract
In this chapter, the CTM is applied to price financial and energy derivatives for one-factor and multifactor α-stable Lévy-based models. These models include, in particular, as one-factor models, the Lévy-based geometric motion model and the Ornstein and Uhlenbeck (1930), the Vasicek (1977), the Cox et al. (1985), the continuous-time GARCH, the Ho and Lee (1986), the Hull and White (1990), and the Heath et al. (1992) models and, as multifactor models, various combinations of the previous models. For example, we introduce new multifactor models such as the Lévy-based Heston model, the Lévy-based SABR/LIBOR market models, and Lévy-based Schwartz-Smith and Schwartz models. Using the change of time method for SDEs driven by α-stable Lévy processes, we present the solutions of these equations in simple and compact forms. We then apply this method to price many financial and energy derivatives such as variance swaps, options, forward, and futures contracts.
Anatoliy Swishchuk

Backmatter

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