Cross‐Hedging Ambiguous Exchange Rate Risk

• DOI: http://doi.org/10.1002/fut.21793
• Published date: 01 February 2017
• Author: Kit Pong Wong
• Date: 01 February 2017

Cross-Hedging Ambiguous Exchange

Rate Risk

Kit Pong Wong*

This paper examines the behavior of an exporting firm that sells its output to two foreign coun-

tries, only one of which has futures and options available for its currency. The firm possesses

smooth ambiguity preferences and faces multiple sources of ambiguous exchange rate risk.

We show that the separation theorem fails to hold in that the firm’sproduction and export de-

cisions depend on the firm’s attitude toward ambiguity and on the incident to the underlying

ambiguity.Given that the random spot exchange rates are first-order independent with respect

to each plausible subjective distribution, we derive necessary and sufficient conditions un-

der which the full-hedging theorem applies to the firm’s cross-hedging decisions. When these

conditions are violated, we show that the firm includes options in its optimal hedge position.

This paper as such offers a rationale for the hedging role of options under smooth ambiguity

preferences and cross-hedging of ambiguous exchange rate risk. ©2016 Wiley Periodicals,

Inc. Jrl Fut Mark 37:132–147, 2017

1. INTRODUCTION

According to the 2013 triennial central bank survey of turnover in foreign exchange markets

coordinated by the Bank for International Settlements (BIS), the U.S. dollar remained the

dominant vehicle currency that accounted for 87% of the total daily global turnover in April

2013. The euro and the Japanese yen were the second and third most traded currencies,

respectively. Trading activities increased strongly in currency forwards and options. Trading

volumes of currency forwards reached $680 billion in 2013 from $475 billion in 2010, a

43% increase, while those of currency options increased the most, by more than 60%. The

rise in turnover of currency forwards and options together accounted for almost a quarter of

global foreign exchange turnover growth between 2010 and 2013.

The vast majority of trading of foreign exchange instruments continues to be conducted

over the counter and not traded on organized exchanges, rendering the prevailing use of non-

standardized contractual terms. The minor currencies of the bottom 29 out of 53 countries

included in the BIS triennial survey had an average of 0.134% of the total daily global foreign

exchange turnover in April 2013, which consisted of 0.043% in spot transactions, 0.041%

in outright forwards, 0.042% in foreign exchange swaps, 0.001% in currency swaps, and

0.007% in foreign exchange options. These figures suggest that trading of foreign exchange

instruments of these minor currencies is likely to be thin and illiquid, thereby resulting in

prohibitively high transaction costs. Firms that have positions in these minor currencies

Kit Pong Wong is a Professor of Finance and the Director of the School of Economics and Finance at the

University of Hong Kong, Hong Kong. The author thanks Bob Webb (the editor) and an anonymous referee

for their helpful comments and suggestions. The usual disclaimer applies.

JEL Classification: D21, D81, F31

*Correspondence author,School of Economics and Finance, University of Hong Kong, Pokfulam Road, Hong Kong.

Tel: +852-2859-1044, Fax:+852-2548-1152, e-mail: kpwong@econ.hku.hk

Received August 2015; Accepted April 2016

The Journal of Futures Markets, Vol. 37, No.2, 132–147 (2017)

©2016 Wiley Periodicals, Inc.

Published online 2 June 2016 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21793

Cross-Hedging Ambiguous Exchange Rate Risk 133

are as such induced to look for foreign exchange instruments of a related currency for

hedging purposes. Such an exchange rate risk management technique is referred to as “cross-

hedging.”

The purpose of this paper is to provide theoretical insights into the decision making of

an exporting firm that sells its output to two foreign countries, only one of which has futures

and options available for its currency.To this end, we depart from the extant literature that is

developed within the standard von Neumann–Morgenstern expected utility paradigm (Adam-

M¨

uller, 2000; Broll & Zilcha, 1992; Change & Wong, 2003; Viaene & Zilcha, 1998; Wong,

2013a, 2013b). Takinginto account the possibility that the firm cannot unambiguously assign

a probability distribution that uniquely describes the exchange rate risk, we define uncertainty

in the sense of Knight (1921) to be made up of two components, risk and ambiguity.While risk

aversion is the aversion to a set of outcomes with a known probability distribution, ambiguity

aversion is the additional aversion to being unsure about the probabilities of outcomes.1The

distinction between the known–unknown and the unknown–unknown is relevant, which is

justifiable by ample experiments (Chow & Sarin, 2001; Einhorn & Hogarth, 1986; Sarin

& Weber, 1993) and surveys (Chesson & Viscusi, 2003; Viscusi & Chesson, 1999) that

document convincing evidence that individuals appear to prefer gambles with known rather

than unknown probabilities.

In this paper, we adopt the approach of Klibanoff, Marinacci, and Mukerji (2005)

(hereafter, referred to as the KMM model) to characterize the firm’s preferences by “smooth

ambiguity aversion.” The KMM model has the following recursive structure. First, ambi-

guity is represented by a second-order probability distribution that captures the firm’s un-

certainty about which of the subjective beliefs govern the exchange rate risk. Second, the

firm’s expected utility under ambiguity is measured by taking the second-order expectation

of a concave transformation of the first-order expected utility of profit conditional on each

plausible subjective distribution of the exchange rate risk. This recursive structure creates

a crisp separation between ambiguity and ambiguity aversion, that is, between beliefs and

tastes, which allows the conventional techniques in the decision making under uncertainty

to be applicable in the context of ambiguity (Alary, Gollier, & Treich, 2013; Broll & Wong,

2015; Cherbonnier & Gollier, 2015; Gollier, 2011, 2014; Iwaki & Osaki, 2014; Snow, 2010,

2011; Taboga, 2005; Treich, 2010; Wong, 2015a, 2016a).

As a benchmark, we examine first the case of perfect hedging wherein futures and

options are available for both foreign currencies. In this benchmark case, we show that

the celebrated separation and full-hedging theorems of Danthine (1978), Feder, Just, and

Schmitz (1980), and Holthausen (1979) hold. Specifically, the separation theorem states

that the firm’s production and export decisions are independent of the firm’s preferences

and the underlying exchange rate uncertainty. The full-hedging theorem states that the firm

optimally opts for a double full-hedge via the fairly priced futures for both foreign currencies

so as to completely eliminate its exposure to the exchange rate risk.2As there is no residual

exchange rate risk that is hedgeable, the fairly priced options for both foreign currencies play

no role as a hedging instrument and are not used by the firm.

When the firm can only cross-hedge its exposure to the exchange rate risk, the separa-

tion theorem no longer holds. We show that there is a wedge between the expected marginal

1Dated back to the Ellsberg’s(1961) paradox, ambiguity has been alluded to the violation of the independence axiom,

which is responsible for the decision criterion being linear in the outcome probabilities. See also Dillenberger and

Segal (2015) and Machina (2014).

2The full-hedging theorem is analogous to a well-known result in the insurance literature that a risk-averse individual

fully insures at an actuarially fair price (Mossin, 1968).