Variance minimizing strategies for stochastic processes with applications to tracking stock indices

• Published date: 01 June 2021
• Author: David B. Colwell,Nadima El‐Hassan,Oh Kang Kwon
• Date: 01 June 2021
• DOI: http://doi.org/10.1111/irfi.12285

ORIGINAL ARTICLE

Variance minimizing strategies for stochastic

processes with applications to tracking stock

indices

David B. Colwell

1

| Nadima El-Hassan

2

| Oh Kang Kwon

3

1

School of Banking and Finance, Australian

School of Business, University of New South

Wales, Kensington, New South Wales,

Australia

2

School of Mathematical and Physical

Sciences, University of Technology Sydney,

Broadway, New South Wales, Australia

3

Discipline of Finance, The University of

Sydney, Sydney, New South Wales, Australia

Correspondence

Nadima El-Hassan, School of Mathematical

and Physical Sciences,

University of Technology Sydney, PO Box

123, Broadway, NSW 2007, Australia.

Email: nadima.el-hassan@uts.edu.au

Abstract

This paper extends the notion of variance optimal hedging

of contingent claims under the incomplete market setting to

the hedging of entire processes and applies the results to

the problem of tracking stock indices. Sufficient conditions

under which this is possible are given, along with the

corresponding variance minimizing strategy. The perfor-

mances of tracking error variance (TEV) minimizing, locally

risk minimizing, and variance minimizing strategies in track-

ing stock indices are investigated using both simulated and

historical market data. In particular, it is shown using

S&P500 data over the period 2000 and 2015 that the TEV

of the variance minimizing strategy is statistically lower than

other strategies at the 95%confidence level for 6-month

holding periods.

KEYWORDS

incomplete markets, index tracking, portfolio selection, variance

minimizing strategy, variance optimal hedging

JEL CLASSIFICATION

D52; D81; G11

1|INTRODUCTION

Providing investors with the return on a benchmark index, such as those published by Standard & Poors, Bloomberg,

or MSCI, is the primary goal of tracker funds. Given that the target index often comprises of a significant number of

constituent assets, and some with limited liquidity, it is impractical to hold a portfolio that exactly mirrors the index

Received: 30 January 2017 Revised: 26 June 2019 Accepted: 22 August 2019

DOI: 10.1111/irfi.12285

© 2019 International Review of Finance Ltd. 2019

430 International Review of Finance. 2021;21:430–446.wileyonlinelibrary.com/journal/irfi

itself. Consequently, tracker funds usually construct portfolios consisting of only a subset of the assets that make up

a given index and apply various criteria to try and minimize the subsequent difference between the index and the

tracker portfolio returns.

A popular approach, based on the mean–variance analysis of Roll (1992) involves choosing the portfolio weights

by minimizing the tracking error variance (TEV) over a given investment horizon. Alternative strategies for con-

structing tracker portfolios include heuristic-based strategies in which each asset, or a cluster or highly correlated

assets, are equally weighted, diversity weighting strategies that allow a tuning parameter to slide between the

extremes of equally weighted and capital weighted portfolios, and fundamental weighting strategies that determine

the portfolio weights according to the underlying assets' accounting variables such as the book value. For further

details on these index tracking strategies, and an empirical comparison of their relative performances, refer to Chow,

Hsu, Kalesnik, and Little (2011) and the references therein. It should be noted that these approaches are somewhat

static in the sense that they do not take into account the dynamic nature in which the target index, and the assets in

the tracker portfolio, evolve over time, and the impact this has on the optimality of the tracker portfolio weights.

In this paper, we introduce an alternative dynamic approach to index tracking based on the contingent claims val-

uation theory from the incomplete markets literature. The standard approach for valuing contingent claims in the

usual complete market setting is via hedging strategies consisting of underlying assets that replicate the given claim

in a self-financing manner. See for example Harrison and Kreps (1979) and Harrison and Pliska (1981). However,

given a contingent claim in an incomplete market setting, it is not possible, in general, to find a replicating strategy

that is also self-financing, and it becomes necessary to sacrifice one of these two conditions. A hedging criterion that

sacrifices the self-financing condition, called risk minimization, was introduced in Föllmer and Sondermann (1986))

and extended to local risk minimization in Schweizer (1988). The notion of local risk minimization was further

extended to the multidimensional case in Schweizer (2008) where it was shown that the general results from the

one-dimensional case carry over naturally to the higher dimensional setting. An alternative notion, called variance

minimization, is a criterion that sacrifices perfect replication, and was considered, for example, in Duffie and Richard-

son (1991) and Schweizer (1992).

Local risk minimization has been applied in a variety of contexts including, for example, in insurance models in

Vandaele and Vanmaele (2008) and Henriksen and Møller (2015)), and models of defaultable claims in Biagini and

Cretarola (2012), energy derivatives in Leoni, Vandaele, and Vanmaele (2014), Lévy processes in Arai and Suzuki

(2015), and stochastic volatility regime switching processes in Goutte (2013). Mean–variance hedging has been used,

for example, with stochastic volatility models in Biagini, Gausoni, and Pratelli (2000) and Černý and Kallsen (2008)),

insurance models in Wong, Chiu, and Wong (2014), models with partial information in Mania, Tevzadze, and Tor-

onjadze (2008) and Kohlmann, Xiong, and Ye (2007), and insider trading models in Biagini and Øksendal (2006).

Although the two hedging criteria have been studied extensively in the literature, this has primarily been in the

context of hedging a single contingent claim with fixed maturity. However, in certain financial situations, such as

those faced by fund managers who are required to track the return on a benchmark index, or total return swap pro-

viders with the need to hedge their exposure to the return on the underlying asset, there arises the need to hedge a

continuum of contingent claims with different maturities, viz. the future values of the target process over a given

investment period. An extension of the notion of local risk minimization to the hedging of entire processes was con-

sidered in Colwell, El-Hassan, and Kwon (2007), and this paper examines the corresponding extension for the notion

of variance minimization.

In this paper, a sufficient condition for the existence of the variance minimizing strategy for a given process is

obtained, and, when the required conditions are satisfied, we provide an expression for the variance minimizing

strategy in feedback form. The results are then applied to the problem of tracking stock indices in which a small sub-

set of constituent assets is used to form tracker, or hedge, portfolios for the target index. Using both simulated and

historical market data, three approaches to index tracking, namely the TEV minimization, local risk minimization, and

variance minimization are compared. It is found that the variance minimizing strategy outperforms the other strate-

gies by simultaneously achieving the minimum tracking error mean and variance when applied to simulated and

COLWELL ET AL.431