1.1 Background of the Study
With the emergence of financial derivatives markets in the past two decades, hedging has been of interest to both academicians and practitioners. The goal of hedging is to minimize exposure to unwanted risk. This is carried out by establishing the position of a derivative instrument to offset exposure to price fluctuations opposite to that of underlying assets, such as using futures to hedge a portfolio of risky assets. The primary objective is to estimate the size of the short position that must be held in the futures market (i.e., a proportion of the long position held in the spot market) with minimal risk and specific risk aversion of the hedged portfolio.
Aside from hedge ratio, hedge horizon should also be considered simultaneously because investors, such as regulators and speculative investors, as well as individuals and institutions participating in the stock and futures markets have different hedging horizon and decision making over different time scales. Therefore, ignoring the dependence of the optimal hedge ratio on hedging horizon could lead to investors making inadequate decisions (Geppert, 1995;
Lien and Shrestha, 2007), suggesting problems on the optimal hedge ratio (OHR) decision.
Many methods have been used to decide the OHR. Most studies adopt the mean-variance framework, which measures the risk of the hedged portfolio by standard deviation, and which assumes that OHR simply minimizes the variance of hedged portfolios. Many applications of optimal hedging use the criterion of minimum variance to estimate OHR, such as by regressing the spot market return on the futures market return using ordinary least squares (OLS) (Ederington, 1979; Hill and Schneeweis, 1982; Sener, 1998). However, OHRs estimated via the conventional approach is constant over time. The classical time-invariant OHR appears inappropriate with the time-varying nature of many financial markets. Improvements were
made to capture time-varying features, such as by adopting dynamic hedging strategies based on the bivariate generalized autoregressive conditional heteroskedasticity (GARCH) framework (Kroner and Sultan, 1993; Lien and Luo 1994; Moschini and Myers, 2002;
Choudhry, 2003; Wang and Low, 2003) or the stochastic volatility (SV) model (Anderson and Sorensen 1996; Lien and Wilson 2001). Although these studies are successful in capturing time-varying features, many give little attention to OHR decisions over different time scales.
1.2 Statement of the Problem
The models presented by the authors have several limitations in estimating the multiscale hedge ratio. All these approaches to estimating the abovementioned OHR are based on sample variance and covariance estimators of returns without considering the underlying distribution of data. The conventional OLS approach ignores the conditional distribution of most financial asset returns, which tends to vary over time. To obtain recent information, most research adopt a rolling window scheme to estimate the variance and covariance of spot and futures returns.
However, rolling window estimators use an equally weighted moving average of past squared returns and their cross products. Observations have equal weight in the variance-covariance matrix estimator of the arbitrarily defined estimation period, but they have zero weight beyond the estimation period. GARCH class models are successful in capturing time-varying features for estimating conditional variance-covariance matrices, but they place too much weight on extreme observations (Nelson and Foster, 1996) when the distribution of data is leptokurtic and fat-tailed.
Furthermore, disregarding the dependence of OHR on the hedging horizon is problematic in these conventional approaches for estimation. Only a few studies consider different hedging horizons for hedge ratio estimation, including Howard and D’Antonio (1991), Lien and Luo
(1993, 1994), Geppert (1995), Lien and Wilson (2001), Chen, Lee, and Shrestha (2004), In and Kim (2006), and Lien and Shrestha (2007). However, these models have three problems in incorporating the hedging horizon in OHR estimation. First, the long-horizon OHR estimator based on a handful independent observations generated from long-horizon return series is unreliable (Geppert, 1995). This is because the frequency of data must match the hedging horizon (e.g., weekly or monthly data must be used to estimate the hedge ratio where the hedging horizon is one week or one month, respectively). Low data frequency would result in a substantial reduction in sample size (Lien and Shrestha, 2007). Second, the assumption for the error term of the GARCH/SV model would lead to inaccurate results when estimating the multiperiod hedge ratio (Lien and Wilson, 2001). Third, the assumption for the underlying data-generating process, such as a unit root process, is unsuitable when the assumed condition does not hold true, as evidenced in many commodities markets (Chen, Lee and Shrestha, 2004).
1.3 Purpose of the Study
The main purpose of this paper is to introduce a novel approach for deciding the OHR of different hedging horizons using computational intelligence technique. The new approach uses the growing hierarchical self-organizing map (GHSOM) of Rauber et al. (2002) to cluster time series data, which could decompose financial data into a hierarchical architecture consisting of several familiar clusters. Several applications of cluster analysis to economics and finance time series have been documented in recent literature, including identification of mutual funds styles by analyzing the time series of past returns (Pattarin et al., 2004), discovery of companies that share similar behavior with the Dow Jones industrial average (DJIA) index (Basalto et al., 2007), prediction of value at risk (Karandikar et al., 2007), prediction of oil futures price (Zhu, 2008), and determination of optimal tracking portfolio (Focardi and Fabozzi, 2009).
In this paper, our work employs a different weight for observations in a rolling window OLS estimator of the variance-covariance matrix subsequent to the clustering time series using GHSOM. The weights of observations are determined by the measurement of similar patterns, which are correlated with the sample size of the cluster they belong to in the hierarchy architecture. The observations with different weights in clusters are then used to predict the conditional distribution of spot and futures returns for different hedging horizons in the future.
When the conditional distribution of spot and futures returns is predictable, a more efficient estimate of the OHR can be obtained by conditioning on recent information (Harris and Shen, 2003).
1.4 Significance of the Study
The application of GHSOM to clustering time series to improve the conventional OHR estimator has at least three salient advantages. First, clustering time series does not suffer from the sample reduction problem when matching data frequency to hedging horizon. Second, observations within the cluster with similar patterns provide a way for examining the dependency of observations, which are generated from long-horizon return series and can provide predictable time patterns. The conditional distribution of returns in the next hedging horizon is predictable by aggregating these clustered observations, which are inspired by the well-established features of many asset returns that their conditional distribution is time-varying and tendency display volatility clustering. The final advantage is that the proposed computational intelligence (CI) approach is a non-parametric method, which can avoid too many inappropriate assumptions and restrictions found in conventional parametric models.
By doing this, OHR estimation for different horizons can be achieved. This proposed
approach is also categorized as an analysis tool to investigate the relationship between hedge ratio and hedging horizon, which provides valuable information for reference in OHR decision making.
1.5 Theoretical Framework
Our study focuses on OHR estimation over various hedging horizons to support decision making. Conventional approaches on OHR estimation are based on parametric models that may encounter many issues and be restricted by many inappropriate assumptions. The proposed CI approach is a non-parametric model designed to overcome issues found in conventional models without the underlying assumptions. The theoretical framework is shown in Figure 1-1.
non-parametric models
optimal hedge ratio decision
Figure 1-1. Theoretical framework
1.6 Organization of the Dissertation
This dissertation is presented in five chapters. Chapter 1 includes the background, statement of the problem, purpose of the study, significance of the study, and the theoretical framework. Chapter 2 presents a review of literature. Chapter 3 introduces the proposed model used for this study. Chapter 4 presents the experiments design and results analysis. The concluding remarks and recommendations for further work are provided in Chapter 5.