Asymptotic Dependence and Its Impact on Hedging Effectiveness: An Examination of Stock, Currency, and Commodity Futures

• Published date: 01 November 2024
• Author: Udayan Sharma,Madhusudan Karmakar
• Date: 01 November 2024
• DOI: http://doi.org/10.1002/fut.22546

Journal of Futures Markets

RESEARCH ARTICLE

Asymptotic Dependence and Its Impact on Hedging

Effectiveness: An Examination of Stock, Currency,

and Commodity Futures

Udayan Sharma

1

| Madhusudan Karmakar

2

1

Department of Finance and Accounting, Indian Institute of Management Indore, Rau, Madhya Pradesh, India |

2

Department of Finance and Accounting,

Indian Institute of Management Lucknow, Lucknow, Uttar Pradesh, India

Correspondence: Udayan Sharma (udayans@iimidr.ac.in)

Received: 14 November 2023 | Revised: 8 July 2024 | Accepted: 14 July 2024

Funding: The authors received no specific funding for this work.

Keywords: asymptotic dependence | downside risk| GARCH–EVT–copula | hedging effectiveness

ABSTRACT

This study measures the asymptotic dependence between spot and futures losses and investigates its impact on hedging

effectiveness using data from stock, currency, and commodity markets. The findings reveal that stock futures contracts show

strong asymptotic dependence, while currency futures have weak asymptotic dependence and most commodity futures lack

asymptotic dependence with the underlyingspots. Further, stock futures have thehighest hedging effectiveness, whilecommodity

and currency futures show low hedgingeffectiveness for downside risk.Results also suggest that asymptotic dependence iscritical

for minimum‐variance hedging. Asymptotic dependence increases with the hedging horizon, leading to a better hedging

performance of the futures. It also appears that the hedging strategiessensitive to asymptotic dependenceperform better than the

competing models. The results for the entire period and the subsample periods offer similar conclusions.

JEL Classification: G11, G15, G17

1 | Introduction

Hedgers use futures contracts to safeguard their portfolios

against spot fluctuations by selling an appropriate amount of

futures contracts,against market crash. The hedged portfoliorisk

can be minimized when the spot and futures prices are perfectly

synchronized. The cost of carry model suggests that spot and

futures prices should move in lockstep. The model works

perfectly only under efficient and frictionless capital markets

when spot and futures price changes are perfectly correlated

(Chan 1992;Abhyankar1995; Bekiros and Diks 2008;Jiang

et al. 2019). However, empirical research reveals the contrary to

be true across many markets. The traded futures price typically

differs fromthe theoretical futures price dueto the market‐driven

supply and demand dynamics. Subsequent research confirms

the documented lead–lag relationship between spot and futures

prices across various studies (see Grünbichler et al. 1994;

Tse 1995;MinandNajand1999;Wangetal.2017). The

lead–lag relationship between pricemovements demonstrates the

speed at which one market reflects new information relative to

the other. Many reasons, such as infrequent trading, discreteness

of stock prices, and differences in liquidity in spot and futures

markets, give rise to market frictions that cause the lead–lag

relationship in the prices of the two markets. Furthermore, due

to transaction costs, arbitrage interactions, and long memory

volatility process, futures prices frequently breach bounds,

particularly for high‐frequency data (See MacKinlay and

Ramaswamy 1988; Daigler 1990; Lien and Tse 1999;Dark2007;

Andreou and Pierides 2008).

The aforementioned studies show that various factors govern

the dynamics between spot and futures prices, and their

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https://doi.org/10.1002/fut.22546

comovement often deviates from the theoretically established

framework. To the best of our knowledge, none of the research

has explicitly analyzed this deviation in terms of the depen-

dence structure between spot and futures markets. The

dependence structure of assets in a portfolio may alter

depending on the market conditions (Babaei, Sepehri, and

Babaei 2015; Karmakar 2017; Cerrato et al. 2017; Chang and

Li 2022). The variation in dependence structure at different

quantiles of the joint distribution could have a variety of effects

on hedged portfolio risk. For example, if a spot–futures pair has

a high dependence on average, the hedged portfolio's variance,

which is a two‐sided risk measure, may be low.

1

Similarly, if the

tail of the distribution shows high dependence, tail‐risk

measures, such as value at risk (VaR) and conditional value

at risk (CVaR), of the portfolio might be low. Because investors

are more concerned about the portfolio's downside risk (Veld

and Veld‐Merkoulova 2008), asymptotic dependence, that is,

dependence between assets in extreme situations, becomes

critical for hedging (Kellner and Gatzert 2013). Therefore,

hedgers should be cognizant of extremal or asymptotic

dependence when utilizing futures contracts to protect their

portfolio from downside risk. Furthermore, it is intuitive that

the degree of asymptotic dependence between spot and futures

is proportionate to the hedging effectiveness against the tail

risk. Compared to a weak asymptotic dependent or asymptotic

independent pair, a strong asymptotic dependent spot–futures

pair may form a highly efficient hedged portfolio.

Since the asymptoticdependence seems to have a bearing on the

downside risk minimization hedging effectiveness, this study

estimates asymptotic dependence for losses of spot–futures pairs

for different asset classes and investigates its impact on the

downside risk profileof the hedged portfolio. More specifically, it

seeks to address the following questions. First, do different asset

classes have different levels of asymptotic dependence for losses?

If yes, what are the economic explanations for the varying levels

of asymptotic dependence for different asset classes? Second,

does the increase in asymptotic dependence between spot and

futures losses enhance the hedging effectiveness in terms of

downside risk reduction? Third, are asymptotic dependence and

hedging effectiveness dependent on the hedging horizon? If yes,

what is the plausible economic explanation for this relationship?

Fourth, does asymptotic dependencehave any implication for the

minimum variance hedging strategy? Finally, is any specific

model‐based hedging strategy sensitive to asymptotic depen-

dence highly effective against extreme price fluctuations? We

measure the dependence structure using paired dependence

measures and investigate the impact of asymptotic dependence

on the hedging effectiveness using different competing models

across 19 futures contracts, whose underlying assets include

stock indices, currencies, and commodities.

The literature on hedging has gradually progressed from

minimum variance hedge ratios (MVHRs) (see Johnson 1960;

Ederington 1979; Myers and Thompson 1989; Kroner and

Sultan 1993; among others) to hedge ratios explicitly deter-

mined to minimize the portfolio's downside risk. As investors

do not have quadratic preferences, and returns rarely follow

elliptical probability distributions, variance is not a reasonable

risk metric. Harris and Shen (2006) suggest utilizing historical

simulation to compute the minimum‐VaR and minimum‐CVaR

hedge ratios, confirming that this strategy can potentially

deliver small but consistent improvements over minimum

variance hedging. Other studies construct downside risk‐based

hedged portfolios using semiparametric and parametric tech-

niques (see Cao, Harris, and Shen 2010; Hilal, Poon, and

Tawn 2011; Barbi and Romagnoli 2014). Lien and Tse

(1998,2000), Demirer and Lien (2003), and Mattos, Garcia,

and Nelson (2008) calculate hedge ratios to minimize the

hedged portfolio's lower partial moment (LPM), another

indicator of downside risk. Ubukata (2018) considers four

downside risk measures, VaR, CVaR, exponential risk measure

(ERM), and LPM, to form a hedged portfolio. Considering the

various downside risk measures analyzed in the literature, this

study focuses on two generic downside risk measures, LPM and

semivariance (SV), and three quantile‐related risk measures,

VaR, CVaR, and ERM, to examine the impact of asymptotic

dependence on hedging effectiveness.

One central problem associated with downside risk measures in

hedging is appropriately capturing the assets' distribution. In

most cases, the true probability distribution of the underlying

risk process is unknown (Kellner and Gatzert 2013). Traditional

methods either use the assets' empirical distribution or estimate

the entire marginal distribution using sample data. However,

using these methods carries the inherent risk of misestimating

the distribution's tails, potentially resulting in errors in

predicting hedge ratios, as accurately modeling the tails of the

distribution is critical for factoring the asymptotic dependence

in the hedging strategy. Misestimating the tails could have

serious consequences because downside risk measures rely

heavily on the precise prediction of extreme (tail) occurrences

like catastrophic losses during a market crash; therefore, it is

critical to measure tail risk appropriately.

In this study, to model the occurrence of extreme losses, we use

the extreme value theory (EVT) to better assess downside risk

measures. The EVT may be used to model marginal distribu-

tion, allowing for more generic fat‐tailed asset return distribu-

tions. Specifically, the threshold exceedances method in the

EVT allows the occurrence of extreme events and captures the

heavy‐tailed distribution through the generalized Pareto distri-

bution (GPD). McNeil and Frey (2000) propose a two‐stage

approach to apply EVT. A generalized autoregressive condition-

ally heteroscedastic (GARCH)‐type model may be applied in

the first stage to obtain (nearly) independently and identically

distributed (i.i.d.) residuals, andthe EVT can be used on the i.i.d.

residuals in thesecond step. Thus, while modelingthe margins of

individual return series, we use the GARCH–EVT method,

which efficiently incorporates the well‐known volatility cluster-

ing phenomenon and, at the same time, captures extreme and

fat‐tail behavior of the asset returns.

Financial data often show nonlinear dependence structures and

asymmetric dependency in the tails of portfolio assets (Ang and

Chen 2002; Erb et al. 1994; Longin and Solnik 1995,2001).

Copula models can capture nonnormal distributions with

asymmetric dependencies between assets in portfolio construc-

tion. Therefore, we adopt a GARCH–EVT–copula combined

approach to capture the stylized properties of data, which

include volatility clustering, fat tails, and nonlinear as well as

asymmetric tail dependence.

2

Since the combined approach can

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capture the stylized properties of the data appropriately, it is

expected that the asymptotic dependence can be measured

accurately, which is critical in estimating its impact on hedging

effectiveness.

The GARCH–EVT–copula model is sensitive to asymptotic

dependence. Therefore,to examine whether any strategy sensitive

to asymptotic dependence outperforms one insensitive to it in

downside risk hedging, we use three other models, namely,

GARCH–copula, DCC, and ADDC. While GARCH–copula is also

sensitive to asymptotic dependence, DCC and ADCC are

insensitive to asymptotic dependence. Since DCC and ADCC

models assume that the joint distribution is normal, they cannot

capture the varying dependence structure often observed in

financial return series.

The findings of this study reveal that stock index futures have

strong asymptotic dependence with their underlying spots.

Several currency futures also show asymptotic dependence with

the underlying spots, though in a weak form. On the other

hand, the majority of commodity futures lack asymptotic

dependence with the underlying assets. As a result, the hedged

portfolio of stocks, created using stock futures, is the most

effective in mitigating the downside portfolio risk. However, the

hedged portfolios using currency and commodity futures are

relatively less efficient and vulnerable to large losses. The

results show that a hedging strategy accounting for asymptotic

dependence between spot and futures performs well in reducing

downside risk. Further, the degree of asymptotic dependence

impacts the portfolio's hedging effectiveness. Futures contracts

with weak or no asymptotic dependence with underlying assets

reduce risk less than asymptotically dependent futures con-

tracts. We also observe that the variance of the hedged portfolio

reduces in the presence of strong asymptotic dependence. When

switching from daily to weekly returns, we observe an increase

in asymptotic dependence between spot and futures, conse-

quently improving hedging effectiveness. Finally, on investigat-

ing the impact of asymptotic dependence on the downside risk

minimization hedging effectiveness during the normal and

turbulent periods separately, the findings align closely with

those observed for the entire period.

The contributions of this paper are manifold. Numerous studies

discuss the impact of asymptotic dependence on hedging, and

yet, none have empirically examinedthe dependence structure of

spot and futures at quantiles of the joint distribution to link it

with the risk profile of the hedged portfolio. For example, Keller

and Gatzert (2013) analyze the asymptotic dependence between

the S&P 500 index and 21 companies, aiming to quantify the

basis risk of an index‐linked hedging strategy,such as using a put

option on an index to hedge against a company's losses.

Essentially, in their study, the underlying of the derivative

instrument is not the asset being hedged. In contrast, our

research focuses on assessing the effectiveness of futures in

mitigating losses of their own underlying assets (spot prices) by

analyzing the asymptotic dependence between them. This

represents a direct hedging approach using spot and correspond-

ing futures, making our study fundamentally different from that

of Keller and Gatzert (2013). Other studies, like Poon et al.

(2004), merely suggest that an effective hedging instrument

should show a high Pearson correlation and be asymptotically

dependent on the underlying asset without providing any

empirical evidence. Similarly, Poon et al. (2003) recommend in

their conclusion that future research could explore the gains in

hedging efficiency and improvements in option pricing through

extreme dependence measures. Therefore, ours is the first study

to explicitly measure the asymptotic dependence between spot

and futures losses for different asset classes and investigate its

impact on hedging effectiveness. Thus, this is the first contribu-

tion of the present study. Second, the study uses a robust

methodology and a comprehensive approach to investigate the

impact of asymptotic dependence on hedging effectiveness.

Third, to the best of our knowledge, this is the first study to

provide a plausible economic rationale for the distinct depen-

dence structures observed for various asset classes. Finally, for

the robustness check, the study also examines the role of

asymptotic dependence in the risk profile of thehedged portfolio

during normal and turbulent market periods separately.

The remainder of the paper is structured as follows. Section 2

presents the various risk measures and techniques for estimat-

ing minimum downside risk hedge ratios. Section 3introduces

the data set and reports the empirical findings. Finally, section 4

brings the study to a conclusion.

2 | Methodology

Let

R

tSand

R

t

F

be the log returns of the spot and futures prices

at time t.

R

t

h

is the hedged portfolio return at time tconstructed

with a long spot position and a fraction (h) of short position in

futures, such that

R

RhR=−

,

thtStF(1)

where hindicates the risk‐specific hedge ratio. The optimal

hedge ratio (OHR) is estimated to reduce a specific risk measure

in the hedged portfolio.

In this section, we first introduce the concepts of various downside

risk measures used in this study. Then,weexplaintheprocedureto

estimate the hedge ratio using the GARCH–EVT–Copula method.

2.1 | Downside Risk Measures

Here, first, we introduce generic downside risk measures, which

include LPM and SV, and then the quantile risk measures, such

as VaR, CVaR, and ERM.

2.1.1 | Generic Downside Risk Measures

2.1.1.1 | LPM. Fishburn (1977) developed a model, which

forms the basis of the LPM. The LPM is a family of downside

risk measures defined as:

∞



LPM τRE τR

τRdFR

(; )= {(max[0, −]) }

=(−)(),

nhhn

τ

hn h

−

(2)

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