Semivariance and semiskew risk premiums in currency markets

• Published date: 01 March 2021
• Author: José Da Fonseca,Edem Dawui
• Date: 01 March 2021
• DOI: http://doi.org/10.1002/fut.22160

J Futures Markets. 2021;41:290–324.wileyonlinelibrary.com/journal/fut290

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© 2020 Wiley Periodicals LLC

Received: 10 August 2020

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Accepted: 11 August 2020

DOI: 10.1002/fut.22160

RESEARCH ARTICLE

Semivariance and semiskew risk premiums in currency

markets

José Da Fonseca

1,2

|Edem Dawui

2,3

1

Department of Finance, Business School,

Auckland University of Technology,

Auckland, New Zealand

2

PRISM Sorbonne, University of Paris 1

Pantheon ‐Sorbonne, Paris, France

3

The World Bank, Washington,

District of Columbia

Correspondence

José Da Fonseca, Department of Finance,

Business School, Auckland University of

Technology, Private Bag 92006, 1142

Auckland, New Zealand.

Email: jose.dafonseca@aut.ac.nz

Abstract

Using a model‐free methodology, variance, and skew swaps are extracted from

currency options for several foreign exchange rates. These swaps are decom-

posed into semivariance and semiskew swaps and can also be used to define

the variance‐skew swap. The decomposed “up”and “down”semivariance

swaps, the “down”semiskew swap and the variance‐skew swap explain well

the currency excess return. These properties remain valid when considering

the prediction of the currency excess return. Lastly, trimming these variables

does not affect the results implying that extreme values of the distribution

convey little information regarding the evolution of the currency excess return.

KEYWORDS

currency risk premium, semimeasures, skew risk premium, variance risk premium, variance‐skew

risk premium

JEL CLASSIFICATION

G11; G12; G13

1|INTRODUCTION

The explanation of the currency risk premium is an important problem in finance as it synthesizes how two economies

interact, Engel (2016). Beyond the usual macroeconomic variables used as determinant of the currency risk premium, the

option market offers an interesting route to investigate this problem. Since the seminal work of Bates (1991), many studies

have shown that options could be used to extract market participants' expectations for the currency risk premium.

Using a model‐free approach that has been extensively used during the recent years, a variance swap contract is extracted

from currency options and used to build factor models for the foreign exchange excess return. This contract enables the

computation of the variance risk premium that is a key quantity in asset pricing. Going one step farther than the literature, we

decompose the variance swap contract into up and down components, the former depending on the upper tail of the currency

distribution while the latter depends on the lower tail of the currency distribution. What is more, these semivariance swap

contracts allow us to compute the variance‐skew swap that captures the skewness of the currency distribution. To further

investigate the importance of the currency skewness distribution we follow Kozhan, Neuberger, and Schneider (2013)and

compute a skew swap value from foreign exchange rate options and, similarly to the variance swap contract, it is decomposed

into up and down semiskew swap contracts. This large set of option related variables allows us to build several factor models

for the currency excess return and assess to which extent they can explain and predict its evolution.

We show that decomposed variance and skew variables have higher explanatory power for the currency excess

return than undecomposed ones; that the variance‐skew swap is highly informative for the currency excess return; that

the down semiskew swap is as informative as the skew swap regarding the currency excess return; a combination of

down semivariance and semiskew swaps have higher explanatory power for the currency excess return than an up one;

for certain currencies third order moment related quantities, mainly down quantities, provide additional information to

second order moment related quantities. Trimming these explanatory variables, by removing extreme movements, and

re‐estimating the same factor models allow us to show that the information extracted from the far ends of the tails

contain no information on the evolution of the currency excess return. Lastly, the predictability of the 1‐,3‐, and

6‐month currency excess return is improved when semivariance and semiskew swaps are used in place of

undecomposed variance and skew swaps, it further illustrates the importance of the decomposition.

The paper is organized as follows. We present the key ingredients to obtain the quantities from option prices in

Section 2. A description of the empirical data used in our analysis is provided in Section 3. Regression tests and analysis

are performed in Section 4and Section 5concludes the paper.

2|ANALYTICAL RESULTS

The main purpose of this study is to analyze the variance, the variance‐skew and the skew risk premiums for the foreign

exchange option market using a model‐free methodology based on call and put foreign exchange rate options. To this

end,

ck()

tT,and pk()

tT,denote the European call and put option prices at time

t

with maturity

T

and strike

k

on the spot

foreign exchange rate denoted

s

t, which represents the value at time

t

in the domestic currency of one unit of the foreign

currency. It is often more convenient to use the forward foreign exchange rate (or forward price), so

f

t

T

,stands for the

forward price value at time

t

with maturity

T

that is related to the spot value through the standard equality

f

se=

tT trrTt

,(−)( −)

df , where

r

d

and

r

f

represent the risk free domestic and foreign rates, respectively. Lastly, it is con-

venient to define rf f=ln −ln

tT TT tT

,,,

, the log return of a position in the forward contract of maturity

T

for the period

tT

[

;]

. The availability of these derivative products enables the computation of a variance swap contract, a variance‐

skew swap contract as well as a skew swap contract along with the risk premiums associated with those contracts in a

model‐free way. In this study, we use the approach proposed by Kozhan et al. (2013). Extracting distribution

information from option prices, like higher moments, has a long history and has been performed in many works.

2.1 |The variance and semivariance risk premiums

A variance swap contract, payer of the fixed leg and receiver of the floating leg, is a contract between two counterparties

that involves the payment at maturity

tτ

+of an amount specified at the initiation date

t

, this amount is called the

variance swap rate, and receiving at maturity (i.e.,

tτ

+) of the swap contract the realized variance of a given asset

computed over the interval

tt τ

[

;+]

. The amount specified at the initiation date

t

is called the fixed leg as it is known

during the life of the contract while the amount received at maturity is called the floating leg of the swap as it is known

only at the end of the contract, so during the life of the contract that quantity fluctuates. Following the literature, it is

known that the variance swap rate is given by

var b

ck

kdk b

pk

kdk=2() +2()

,

tt τ

tt τf

tt τ

tt τ

ftt τ

,+

,+

+,+

2,+ 0

,+

2

tt τ

tt τ

,+

,+

∫∫

∞(1)

var var

=

+

,

tt τ

utt τ

d

,+ ,+(2)

with

b

tt

τ

,+

the zero‐coupon value at time

t

with maturity

tτ

+that is expressed in the domestic currency. The quantity

vartt

τ

u

,+ captures the second moment of the upper tail distribution of the underlying asset while vartt

τ

d

,+ captures the

second moment of the lower tail distribution. More precisely, vartt

τ

u

,+ involves only call options with strikes larger or

equal to the forward price

f

tt

τ

,+ and, as such, it depends on the second moment of the log return of the forward price

(i.e., rtt

τ

,+)conditional on the event that it is positive, thatis to say, conditional on

1

r{>0

}

tt τ,+ . Similarly, vartt

τ

d

,+ depends on

the second moment of the log return of the forward price (i.e., rtt

τ

,+)conditional on the event that itis negative, that is to

say, conditional on

1

r{<0

}

tt τ,+ . Also, as option prices are computed under the risk neutral probability, the variance swap

rate is also called the risk neutral variance and hereafter we use interchangeably these names.

Notice that Equation (1) involves a continuum of options and as in the market only a finite number of options are

available the two integrals are approximated by sums, see Equation (23) in Kozhan et al. (2013) for details.

DA FONSECA AND DAWUI

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291

The floating leg of the variance swap is the realized variance of the underlying currency computed over

tt τ

[

;+]

using end‐of‐day values as it is the standard practice in the market and it is given by

rvarg r=(

¯)

,

tt τ

it

tτ

vii,+

=

+−1

,+1

∑(3)

with rf f

¯=ln −ln

ii itτit

τ

,+1 +1,+,+ with

i

ttt τ{,+1,…, +−1}∈the set of days covering the interval

tt τ

[

;+]

and

gx ex()=2( −1−

)

vx

. Performing a Taylor expansion of the exponential function the terms in the sum turn out to be

r

¯

ii,+1

2

, hence justifying the expression of realized variance for rvartt

τ

,+.The fixed and floating legs being defined, the

realization of a variance swap, payer of the fixed leg and receiver of the floating leg, is given by

vs rvar var=−

.

tt τtt τtt τ,+,+ ,+(4)

In practice this product is used to hedge against an increase of the currency's volatility. Indeed, at time

t

the amount

vartt

τ

,+ is specified (and paid at time

tτ

+) and if over the period

tt τ

[

;+]

the currency's volatility increases sub-

stantially, the quantity rvartt

τ

,+ received at time

tτ

+is larger than the amount specified at time

t

, the net value is

positive. Conversely, if the currency's volatility remains low or decreases over the period

tt τrvar

[

;+],

tt

τ

,+

is smaller

than vartt

τ

,+, overall the investormakes a loss. In this example, the point of view of a volatility protection buyer is taken,

the counterparty in that contract acts as a volatility protection seller and the cash‐flows are exactly the opposite.

Following the literature such as Carr and Wu (2009) or Ammann and Buesser (2013), taking the expectation (under

the historical probability measure) of Equation (4) leads to the variance risk premium, it is denoted

vs vs=[ ]

.

τtt τ,+

This quantity is negative in practice, it is interpreted as the amount an investor is willing to pay to hedge against volatility risk.

It is known that variance risk premiums for equity options and equity index options are negative, see Carr and Wu (2009).

It is of interest to compare an investment made in the variance swap contract with the return of an investment made

in a risky asset and therefore it is convenient to introduce the excess return of an investment made in the variance swap

contract, it is denoted by

xv rvar

var

=−1.

tt τ

tt τ

tt τ

,+

,+

,+

(5)

The qualifier “excess”follows from the fact that va

r

in Equation (1) involves at the denominator a zero‐coupon

bond, it makes that quantity a forward price, see Carr and Wu (2009) for details.

Similarly to the decomposition performed for the variance swap rate vartt

τ

,+ in Equation (2), the realized variance

can be decomposed into two components conditional on the return of the forward price

f

tt

τ

,+, it leads to

rvar grgr11=(

¯)+(

¯)

,

tt τ

it

tτ

vii r

it

tτ

vii r,+

=

+−1

,+1 {>0}

=

+−1

,+1 {<0}

tt τtt τ,+ ,+

∑∑ (6)

rvar rvar

=

+

.

tt τ

utt τ

d

,+ ,+(7)

Combining the decompositions for the risk neutral variance vartt

τ

,+ and realized variance rvartt

τ

,+,it is natural to

define the semivariance swap contracts

vs rvar var=−

,

tt τ

utt τ

utt τ

u

,+,+ ,+(8)

vs rvar var=−

,

tt τ

dtt τ

dtt τ

d

,+,+ ,+(9)

as these swaps involve “half”(roughly speaking) of the underlying currency distribution. Moreover, as vstt

τ

u

,+ depends

on the upper part of the underlying currency distribution, it is tempting to name it the up semivariance swap contract

while for obvious reasons vstt

τ

d

,+ is named the down semivariance swap contract.

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