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立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
Chapter 1 Introduction
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C HAPTER 1
Introduction
Merton (1971) carried out seminal work on intertemporal optimal portfolio choices assuming a Gaussian distribution in asset returns. Following the framework of Merton, diffusion has been the standard continuous-time stochastic process representing uncertainty with regard to invest-ment opportunities for optimizing portfolio choice.
The focus of asset price modeling has shifted to a framework based on non-Gaussian distribu-tion to alleviate problems in underestimating the frequency and magnitude of extreme events, namely, crashes and booms. Particularly, many early empirical evidences have fundamentally shaken assumptions made by the diffusion model, for instance, Mandelbrot (1963), Fama (1965) and Engle (1982). Recently, Jondeau et al. (2007) published an organized treatise on this topic.
Portfolio optimization is a field in continuous flux. When returns are non-Gaussian, the Mer-ton rule may fail to optimize a portfolio due to the first and second moments are no longer fully representing asset return specifications in real financial markets. Furthermore, advanced im-provement in financial modeling provides researchers and participants with insights into portfo-lio optimization, asset pricing, and risk management. Particularly, observations of real asset markets reveal numerous stylized facts. Cont (2001) comprehensively summarized the facts, such as asymmetric volatility, aggregated normality, and absence of serial correlation, etc.
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Among these non-Gaussian specifications, asymmetric volatility and fat tail are relevant con-siderations with regard to asset allocation decisions (see Chunhachinda et al. (1997)). Campbell et al. (1997, Chapter 12) explained asymmetric volatility in terms of the leverage and volatility feedback effects. The leverage effect proposed by Black (1976), Christie (1982) and Nelson (1991), claimed that a negative equity return reduces the leverage firm value and thus increases the risk of holding equity, thus increasing volatility risk. Additionally, the volatility feedback effect proposed by Campbell and Hentschel (1992), Bekaert and Wu (2000) and Wu (2001) ad-vocated that it should be satisfied if return volatility behavior involves persistent clustering, in which case a shock in either direction enhances the anticipated increase in volatility and increase the required rate of return for holding stocks, and furthermore, reduces the asset price to enable higher future returns. Kraus and Litzenberger (1976) demonstrated that investors with the power utility favor positive over negative skewness. Therefore, the leverage and volatility feedback effects significantly influence optimal portfolio choices when asset return has asymmetric vola-tility. While asymmetric volatility is of central interest in the context of portfolio optimization, the area is under-researched and comparatively neglected, particularly for continuous-time asset return models.
The Lévy process has recently been applied to asset price modeling in response to such criti-cisms. Since Lévy processes offer the merit of extending the scope of the distribution via the law of infinite divisibility and/or independent increment property, the price change can be expressed as the result of the aggregation of random economic shocks. Moreover, non-Gaussian or discon-tinuous specifications for stylized facts are characterized by Lévy measure (density) which
de-‧
scribes the arrival rate for jumps of all sizes.
The finance literature has extensively explored the setting of Lévy density, for instance, Va-riance Gamma (VG) was presented by Madan and Seneta (1990) and extended to skewness by Madan et al. (1998), CGMY was adapted by Geman et al. (2001) and Carr et al. (2002), and exponential dampened power law was proposed by Wu (2006). While the literature documents evidence supporting the superior fitting ability of Lévy processes (e.g. Carr et al. (2002) and Geman (2002)), room remains for Lévy processes related to financial modeling, such as the ab-sence of stochastic volatility, stochastic skewness, and predictability of returns or volatility (see Wu (2008)). Consequently, time-changed Lévy processes emerge for these deficiencies.
Time-changed Lévy process is widely utilized due to its probabilistic tractability, and the in-stantaneous rate of stochastic time change is regarded as a state variable (Carr and Wu (2004)).
Furthermore, asset price can also be considered the outcome of interaction among several eco-nomic variables. Under the framework of the time-changed Lévy processes, stochastic time changes are particularly suitable candidates for playing this relevant role1, namely, economic variables. The Lévy process accelerated by an increasing stochastic time is cautiously selected to match the features existing in different financial markets (e.g. Mo and Wu (2007), Carr et al.
(2003) and Carr and Wu (2007)).
Carr et al. (2003) discussed stochastic volatility in relation to the pure-continuous asset price model in three homogeneous Lévy processes, including normal inverse Gaussian (NIG) pre-sented by Barndorff-Nielsen (1998), VG, and CGMY, in the form of a stochastic time change
1 See Bertoin (1996), Sato (1999), Applebaum (2004) and Cont and Tankov (2004).
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independent of the original Lévy processes. Besides stochastic volatility, the instantaneous rate of stochastic time change, a solution to the CIR mean-reverting square root stochastic process, is offered to promote volatility clustering, but without the leverage effect.
Cvitanić et al. (2008) denoted risky asset price dynamics as a pure-jump stochastic process in which underlying uncertainty is described via the state-dependent Lévy density for a VG model.
The of state-dependent Lévy density is fully captured by the state variable followed in the form of a CIR mean-reverting square root stochastic process to investigate the portfolio optimi-zation for investors facing higher moments. However, the research of Cvitanić et al. did not di-rectly invoke stochastic time changes; rather they simply randomized the intensity of the jump structure, namely, transforming the constant Lévy density into a varying one. Hence some inter-esting findings may be sacrificed.
In contrast to early studies, this work further probes the time-changed Lévy processes related to portfolio optimization. In comparison to those of Carr et al. (2003), present study enhances a Brownian motion with drift subordinated by a pure-continuous increasing stochastic process, an integral of a solution to the CIR mean-reverting square root stochastic process, such that the time-changed Lévy process is associated with the state variable. Hence the pure-continuous time-changed asset return model is established. In contrast with Cvitanic ́et al. (2008), this study provides another infinite-jump asset return model obtained by directly applying a one-sided jump process (with finite ) to randomize the clock in which a Brownian motion with drift is run.
The infinite-jump time-changed asset return model represents the risky asset available for in-vestors in my economy.
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Regarding portfolio optimization under the non-Gaussian framework, several studies have recently considered the asset allocation decision, either in studying the influence of jumps or in introducing stochastic volatility. For example, Kallsen (2000) proposed a continuous-time framework for maximizing expected utility based on terminal wealth in a market in which risky asset prices follow an exponential Lévy process. Building on the Merton problem, Benth et al.
(2003) provided a model that includes stochastic volatility in asset returns using a superposition of non-Gaussian Ornstein-Uhlenbeck process (Barndorff-Nielsen and Shephard (2001)). Finally, Gron et al. (2004) examined the effect of stochastic volatility on optimal portfolio choices in both partial and general equilibrium using single period returns.
However, few studies have investigated the implications of asymmetric volatility, particularly for leverage and volatility feedback effects, in relation to optimal portfolio choices. Research on the performance of time-charged Lévy processes in optimal portfolio choice still remains im-mature. This study attempts to fill this gap and enrich the literature. The primary contributions of this work are as follows: First, this study proposes two distinct exponential time-changed Lévy processes with asymmetric volatility for risky assets. Second, this study numerically ex-amines the economic implications of leverage effect and volatility feedback effect for optimiz-ing portfolio. Finally, I adopt the perspective of econometric analysis to apply the proposed general stochastic asymmetric volatility asset return model by calibrating them to S&P500 index returns. To resolve the difficulties in getting an analytical expression for probability density function, this study employs spectral GMM estimation (Chacko and Viceira (2003)) to estimate the parameters of the general asset return model. Based on asymmetric volatility, I examine
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whether the diffusion term needs to be included when modeling asset returns given the ability of infinite-activity jump structure to describe both frequent small moves and occasional large moves.
Following the discussion of the influence of asymmetric volatility on portfolio optimization, this study proposes that the leverage effect directly induces the intertemporal asymmetric vola-tility hedging demand while the volavola-tility feedback effect works indirectly via the leverage ef-fect and thus exerts only a minor influence on asset holding under the pure-continuous time-changed Lévy process. The volatility feedback effect can induce additional hedging de-mand for risky assets when the returns on those assets are negatively correlated with changes in asset volatility, that is, the positive leverage effect exists in the economy. Conversely, hedging demand for risky assets occurs when the volatility feedback effect is not obvious but the nega-tive leverage effect dominates the economy. Otherwise, this study partially explains why inves-tors prefer to hold low-price assets during the periods of high volatility.
Based on the infinite-jump time-changed Lévy process, I claim that the leverage effect induc-es the intertemporal hedging demand. However, the volatility feedback effect just works over the short-term investment horizon. In sum, the leverage effect plays a major role for portfolio optimization.
Empirically, this study concludes that the diffusion term in the general asset return model is an essential determinant when modeling the index dynamics given infinite-activity jump struc-ture.
The rest of the paper is organized as follows: chapter 2 reviews some essential results related
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國立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
Chapter 1 Introduction
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to Lévy processes and time-changed Lévy processes, and further proposes two distinct exponen-tial time-changed Lévy processes with asymmetric volatility for risky assets. Chapter 3 presents a rigorous formulation of the problems associated with dynamic asset allocation and provides some results for optimal portfolio weights together with some relevant numerical examples to investigate the implications of asymmetric volatility, particularly for the leverage and volatility feedback effects. To understand whether the diffusion term provides the critical effect for finan-cial modeling, chapter 4 assesses the asymmetric volatility to explore the proposed general asset return model, which employs stochastic time changes associated with both jump and diffusion components. Chapter 5 presents conclusions.
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國立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
Chapter 2 Time-Changed Lévy Processes with Asymmetric Volatility
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C HAPTER 2
Time-Changed Lévy Processes with Asymmetric Volatility
2.1 Fundamental Properties of Lévy Process
Let Ω, , , denote a filtered complete probability space representing the underly-ing economic uncertainty, where represents a physical probability measure and the filtration
satisfies the usual conditions (cf. Jacob and Shiryaev (2003)). All stochastic processes considered in this study are adapted to this filtration.
A process , 0 with values in such that 0 0 (almost surely) is termed a Lévy process, which includes Poisson process, Brownian motion (simply a Lévy process with a continuous sample path) and compound Poisson process as special cases, if it is right continuous with left limits almost surely and its increments are independent and time-homogeneous. The first can exclude the “calendar effect” due to the impossibility of anticipating a jump prior to time . The second expresses that two non-overlapping increments are independent and the final condition, “time homogeneity”, characterizes the Lévy process that for any time interval ex-ceeding zero, the law of the increment does not depend on .
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Chapter 2 Time-Changed Lévy Processes with Asymmetric Volatility- 9 -
The Lévy-Khintchine formula and Lévy-Itô decomposition can help strengthen understanding of path variation and the distributional property of the Lévy process. From the Lévy-Khintchine formula, the characteristic function of real-valued Lévy process has the form
0,
where √ 1, · denotes the expectation operator under the measure and the characteris-tic exponent , , is given by
1
2 1 1| |
is termed the Lévy measure, which is the expected number, per unit time, of jumps whose size belongs to 0 , 0 . The characteristic triplet , , de-notes the Lévy triplet, where is the constant drift depending on the choice of the truncation function (e. g. 1| | ), denotes the constant diffusion coefficient and describes the arrival rate for jumps of size and satisfies
| | ∞
| | , and | | ∞
For simplicity, I assume a Lévy density exits2. The sample paths of a pure-jump Lévy process excluding diffusion risk display finite (infinite) activity when the integral of the Lévy density is finite (infinite). A finite (infinite) activity jump process generates a finite (infinite) number of jumps within any finite time interval. Finally, the distribution of increments of the Lévy process is infinitely divisible, which describes price changes as resulting from numerous
2 The density of Lévy measure is called the Lévy density, which has the same mathematical requirements as a probability density function, except that is does not need to be integrable and must have zero mass at the origin.
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Chapter 2 Time-Changed Lévy Processes with Asymmetric Volatility- 10 -
economic shocks if the characteristic function is linear at time .
The Lévy-Itô decomposition can be used to divide the Lévy process into three semimartin-gales, a Brownian motion with drift, an infinite superposition of independent Poisson processes and an infinite superposition of independent compensated Poisson processes. Therefore, any Lévy process is also a semimartingale, which represents a good property for the no-arbitrage assumption from a financial standpoint, such as asset pricing, under the physical probability measure . Lévy processes should be an appropriate starting point in representing asset returns (Cartea and Howison (2003); Nunno et al. (2006); Carr and Wu (2008)).
2.2 Stochastic Time Changes for Lévy Processes
To more accurately capture and explore the stylized facts of asset returns, this study applies sto-chastic time changes to alter the clock time used in running the Lévy process. The mapping
can be regarded in the same way as the above procedure. Intuitively, the original time denotes calendar time and the random clock represents business time. Business time runs faster during busy trading periods, implying that business time speed is related to business activ-ity rate, which stands for the intensactiv-ity of trading activactiv-ity. Furthermore, after time-changing a Brownian motion with drift, the Brownian scaling property shifts the focus from scale changes to time changes. Hence, academics immediately replace the role of the diffusion process with the familiar Brownian motion.
Clark (1973) was the first researcher to propose stochastically altering the calendar time in the finance literature. Geman and Ané (1996) and Ané and Geman (2000) subsequently further
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Chapter 2 Time-Changed Lévy Processes with Asymmetric Volatility- 11 -
cidated this concept. Clark proposed the subordinated process derived from stochastically time-changing a Brownian motion through the cumulative volume of traded contracts as the proxy of the speed of business time. On the contrary, Ané and Geman claimed the cumulative number of trades is a better proxy as an increasing stochastic time process than the cumulative volume. Furthermore, the well-known theorem of Monroe (1978) indicated that every semimar-tingale can be presented as a Brownian motion evaluated at a stochastic time change, thus pro-viding the fuels for the subsequent studies to asset price modeling.
Geman (2005) extensively reviewed stochastic time changes and changes of numéraire. A growing number of recent publications and empirical evidences have confirmed the positive contribution of time-changed Lévy processes by extracting and capturing features of asset re-turns in financial markets. Geman (2002) argued that pure-jump Lévy processes, for instance, CGMY and the hyperbolic motion by Eberlein and Keller (1995), Barndorff-Nielsen (1998) and Rydberg (1999), possess better fit than classical diffusion or jump-diffusion models. Carr et al.
(2003) introduced stochastic volatility and distributional skewness into exponential Lévy processes via stochastic time changes. Similarly, Carr and Wu (2004) provided a framework that permits jumps, stochastic volatilities, and the leverage effect in stock prices and which encom-passes all models presented in the literature. Recently, Mendoza et al. (2008) proposed time-changed Markov processes designed for defaultable stocks. The present study can be con-sidered a combination of the extension of mean-reverting stochastic volatility model developed by Carr et al. (2003) and the concept of state-dependent Lévy density proposed by Cvitanic ́et al.
(2008).
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國立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
Chapter 2 Time-Changed Lévy Processes with Asymmetric Volatility
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2.3 Time-Changed Asset Price Processes with Asymmetric Volatility
This subsection comprises two main parts, which respectively consider two models to yield ap-propriate representations of asset return with asymmetric volatility. Mathematical models are developed to make comparison of the two stochastic time changes, including a pure-continuous asset dynamic process and an infinite-jump asset dynamic process. The theoretical setup is based on the essential characteristics of the business time, which is an increasing stochastic process.
As I pointed out at the beginning, asymmetric volatility is the striking empirical regularity in the finance literature. Particularly, Bekaert and Wu (2000) proposed a unified framework to in-vestigate this topic at the firm and the market level and to explore two possible explanations for volatility asymmetry: the leverage effect and the volatility feedback effect. To introduce defini-tion, let , , , , where denotes the information set available at time and , denotes the return of the stock at time 1. Finally, define conditional variances
as , var , .
Definition [Bekaert and Wu (2000)]: A return , exhibits asymmetric volatility if
var , , , 0 , var , , , 0 ,
In simple terms, negative unanticipated innovations in asset return enhance the level of the con-ditional volatility, whereas positive unanticipated innovations compensate the level of the
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Chapter 2 Time-Changed Lévy Processes with Asymmetric Volatility- 13 -
tional volatility.
Campbell et al. (1997) further put forward asymmetric volatility in light of the leverage and volatility feedback effects. The key research by Black (1976), Christie (1982) claimed that a negative equity return reduces the firm value and thus increases the risk of holding equity, thus increasing volatility risk, under the leverage argument. The volatility feedback effect by Camp-bell and Hentschel (1992), Bekaert and Wu (2000) and Wu (2001) advocated that it should be satisfied if return volatility displays persistent clustering and is priced, in which case a shock in either direction enhances the anticipated increase in volatility and increase the required rate of return for holding stocks, and furthermore, reduces the asset price to enable higher future returns.
For this reason, the leverage effect runs counter to the volatility feedback effect: the former claims that return shocks result in changes in return volatility, whereas the latter asserts that re-turn shocks are caused by changes in rere-turn volatility (See Figure1.).
Figure 1: Unanticipated news impact on asset return volatility.
Although much work has been done for continuous-time stochastic volatility model in Lévy processes (e.g. Carr et al. (2003)), more works need to be conducted to ascertain the leverage effects and volatility feedback effect in asymmetric volatility. Moreover, the requirement of
News good news , , Leverage effect ,
persistence ,
Volatility feedback effect
risk premium
bad news , ,
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Chapter 2 Time-Changed Lévy Processes with Asymmetric Volatility- 14 -
non-Gaussian specification can be easily grasped by the jump structure of Lévy processes, but stochastic volatility in asset return need to be considered by altering the clock the original process is run. In this paper, I further propose two models as having the asymmetric volatility by introducing unobservable state variable which describes the economic environment and is re-lated to the intensity of trading.
2.3.1 Pure-Continuous Asset Dynamic Process
Formally, this study denotes the process for the risky asset price by . To model asset price dynamics, the Brownian motion with drift and the stochastic time change are in-cluded as:
0 exp
where the process is as follows3:
denotes a constant, 0 represents the initial asset price, is a standard Brownian
denotes a constant, 0 represents the initial asset price, is a standard Brownian