Optimal cross-hedging under futures mispricing: A note

Original Article

Optimal cross-hedging under futures mispricing:

A noteReceived (in revised form): 21st October 2013

Juan Angel Lafuente

is Associate Professor of Finance at the University Jaume I Faculty of Law and Economics. He received his PhDdegree in economic analysis and quantitative economics from the University Complutense of Madrid in 1999. Hehas published extensively on topics related to financial markets and derivatives in refereed journals.

Correspondence: Juan Angel Lafuente, Department of Finance and Accounting, University Jaume I, Campus delRiu Sec, Castellón 12080, SpainE-mail: [email protected]

ABSTRACT This note provides a theoretical discussion on the optimal hedge ratio whena cross-hedging is performed under futures mispricing. Assuming that the investor’s riskmeasure is the conditional variance, it is showed that the optimal hedge ratio depends on thecorrelation structure and the relative volatility of the three assets involved: the futuresinstrument, the underlying asset of the futures contract and the portfolio to be hedged.

Journal of Derivatives & Hedge Funds (2013) 19, 181–188. doi:10.1057/jdhf.2013.11

Keywords: futures; cross-hedging; mispricing

INTRODUCTION

Futures contracts are essential instruments infinancial risk management. When tradersimplement futures hedging strategies, it is themost often case that the underlying asset of thefutures contract does not match the portfolioto be hedged. Transaction costs and transitoryilliquidity encourages price deviations in thefutures market from the theoretical no-arbitragevalues. This note attempts to provide atheoretical discussion on the optimal hedge ratio,taking into account these two aspects: (i)mispricing fluctuates randomly around 0 and (ii)the asset to be hedged is not a replicatingportfolio of the underlying asset of the futurescontract. Assuming a variance-minimizing

strategy, we extend the model proposed inLafuente and Novales (2003) by introducingmismatching between the asset to be hedged andthe underlying asset of the futures contract toshow how the nature of the correlations and therelative volatility affect the optimal hedge ratio.The nature of the futures position and thenumber of futures contract required for hedgingare discussed.

THE CROSS-HEDGING MODEL

Let Wt denote the price of the asset to be hedgedand St the spot price concerning the underlyingasset of the futures contract at period t,respectively, and assume that the instantaneous

© 2013 Macmillan Publishers Ltd. 1753-9641 Journal of Derivatives & Hedge Funds Vol. 19, 3, 181–188www.palgrave-journals.com/jdhf/

return of the both assets evolve according to ageometric Brownian motion:

dWt ¼ μW ; tWtdt + σW ; tWtdz1; t (1)

dSt ¼ μS; tStdt + σS; tStdz2; t (2)

where dz1, t= ξ1(dt)1/2 and dz2, t= ξ(dt)1/2 are

Wiener processes. The non-arbitrage cost-of-carry valuation of the futures contract leads to thefollowing relationship between spot and futuresprices:

Ft;T ¼ Ster - dð Þ T - tð Þ (3)

where r-d is the net cost of carry of theunderlying stock, and T is the maturity period.This relationship implies that spot and futuresmarkets share a common source of risk.However, traded futures prices depart from thistheoretical valuation (see Kawaller and Koch,1984; Yadav and Pope, 1990, 1994; Draper andFung, 2003, among others) and spot and futuresprices do not share a common ARCH feature(see, for example, Arshanapalli and Doukas,1994). Following Lafuente and Novales (2003),we model both aspects including a specific noisein the dynamics of the futures prices, whichare derived by applying the Ito’s lemma from(2) and (3):

dFt ¼ μF; tFtdt + σS; tFtdz2; t + σM ; tFtdz3; t (4)

where σM reflects another source of basis risk anddz3, t= ξ3(dt)

1/2 is a Wiener process. Let Σt

denote the variance covariance matrix of theGaussian noise vector ξ≡(ξ1, ξ2, ξ3)′, that is:

Xt

¼ E ξtξ′t

� � ¼ 1 ρ12; t ρ13; t1 ρ23; t

1

0@

1A

Consider an investor who is long ct units of theasset W at time t and decides to hedge returnvariability by using a futures contract whichunderlying asset does not match the previous

one. Assuming that the investor’s risk measure isthe conditional variance, the optimizationproblem can be stated as follows:

Min Vart ctdWt

Wtdt- ht

dFtFtdt

� �htf g

s:t:

dWt ¼ μW ; tWtdt + σW ; tWtdz1; t

dFt ¼ μF; tFtdt + σS; tFtdz2; t + σM ; tFtdz3; t

which amounts to:

Min c2t σ2W ; t + h

2t σ2S; t + σ

2M ; t + 2ρ23; tσS; tσM ; t

� �- 2htct ρ12; tσW ; tσS; t + ρ13; tσW ; tσM ; t

� �htf g

By taking the first-order necessary conditionand solving, the minimum variance hedge ratio isfound to be:

htct¼ ρ12; t σW ; t σS; t + ρ13; t σW ; t σM ; t

σ2S; t + σ2M ; t + 2ρ23; t σS; t σM ; t(5)

The optimal hedge ratio is undeterminedonly when the futures market returns arenon-stochastic ( ρ23, t=−1,σS, t= σM, t). In thiscase, risk cannot be hedged regardless of thechoice of the ratio to be applied. When the assetto be hedged and the underlying asset of thefutures contract share a common noise(dz1, t= dz2, t), which implies ρ12, t= 1 andσW, t= σS, t, the model becomes equivalent tothe model originally proposed in Lafuente andNovales (2003), with perfect matching betweenthe asset to be hedged and the underlying assetof the futures contract.

Proposition 1: The optimal hedge ratio is just afunction of (i) the correlation betweennoises, (ii) the relative standard deviationbetween the specific and common noise of

Lafuente

182 © 2013 Macmillan Publishers Ltd. 1753-9641 Journal of Derivatives & Hedge Funds Vol. 19, 3, 181–188

the futures market and (iii) the relativestandard deviation between the commonnoise in spot and futures markets and thedisturbance of the asset to be hedged.

Proof : Equation (5) can be rearranged asfollows: □

htct¼

ρ12; tσW ; t

σS; t+ ρ13; t

σW ; t

σS; t

σM ; t

σS; tσ2S; tσ2S; t

+σ2M ; t

σ2S; t+ 2ρ23; t

σM ; t

σS; t

¼ θt ρ12; t + ρ13; tϕt

� �1 +ϕ2

t + 2ρ23; tϕt(6)

where: θt= (σw, t)/(σs, t), ϕt= (σM, t)/(σs, t)The conditional variance of the hedged

position can be written as follows:

c2t σ2W ; t 1 +

htct

� �2 1

θ2t+

ϕt

θt

� �2

+ 2ρ23; tϕt

θ2t

!"

- 2htct

� �ρ12; t

σS; tσW ; t

+ ρ13; tσM ; t

σW ; t

� ��:

Using the above expression, and equation (6),the minimized conditional variance can beexpressed as follows:

c2t σ2W ; t 1 -

ρ12; t + ρ13; tϕt

� �21 +ϕ2

t + 2ρ23; tϕt

" #

¼ c2t σ2W ; t

1 - ρ2

12; t

� �+ϕ2

t 1 - ρ2

13; t

� �+ 2ϕt ρ23; t - ρ12; tρ13; t

� �1 +ϕ2

t + 2ρ23; tϕt

2666664

3777775 ð7Þ

where (1−ρ12, t2 )+ϕt

2(1−ρ13, t2 )+2ϕt( ρ23, t−ρ12, t

ρ13, t)/(1+ϕt2+2ρ23, tϕt) is the ratio between the

conditional variances of the hedged and

unhedged position. Alternatively, (ρ12, t+ρ13,tϕt)

2/(1+ϕt2+ρ23, tϕt) is the difference between

both these conditional variances but expressed inpercentage of the conditional variance of theunhedged position. Both ratios can be used as ameasure of hedging effectiveness (HE). Let usdenote the second ratio for further discussion asHE= (ρ12, t+ρ13, tϕt)

2/(1+ϕt2+2ρ23, tϕt).

Proposition 2: If ϕt= ((σM, t)/(σs, t))→0the optimal hedge ratio converges to(σWS, t)/(σS, t

2 ) and the variance reductionfactor approaches to (1−ρ12, t

2 ).

Proof : From equation (6), substituting ϕt= 0the optimal hedge ratio becomes θt ρ12, t=(σW, t)/(σS, t) (σWS, t)/(σW, t σS, t)=(σWS, t)/(σS, t

2 ). Relative to the variancereduction factor, see equation (7). □

As the common noise becomes morerepresentative of the uncertainty of the futuresreturns, spot and futures market innovations tendto be increasingly similar. In the limit case, nobasis risk arises and, consequently, the optimalhedge ratio and the HE also depend on thenature of the relationship between the returns ofthe asset to be hedged and the returns of theunderlying asset of the futures contract. In thiscase, and denoting mean returns byμ, the optimalhedge ratio can be computed from the followingregression:

lnWt

Wt - 1

� �- μW ¼ α + β ln

StSt - 1

� �- μS

�+ εt

and the variance reduction factor is simply thevariance of the W asset returns that remainsunexplained by the variance of S asset returns.

Proposition 3: When Σt equals to the identitymatrix, the optimal hedge ratio is θt/1+ϕt

and full HE is achieved.

Optimal cross-hedging under futures mispricing: A note

183© 2013 Macmillan Publishers Ltd. 1753-9641 Journal of Derivatives & Hedge Funds Vol. 19, 3, 181–188

Proof : From equations (6) and (7), substitutingρi, j, t= 1∀i≠j, i , j= 1, 2, 3 leads to ht/ct= θt/1+ϕt and HE= 1, respectively. □

When the underlying asset returns are perfectand positively correlated with both the derivativeasset returns (as implied in the theoreticalvaluation according to the cash-and-carry model)and the returns of the asset to be hedged, thehedged position becomes a riskless portfolio,regardless of the relative variance betweenmarket noises.

Proposition 4: With positive correlations,the optimal hedge ratio increases notonly as the common and the specificnoise of futures returns become morecorrelated with the noise of the asset tobe hedged, but also as the asset to behedged becomes more volatile relativeto the underlying asset of the futurescontract.

Proof : From equation (6), we have: □

∂ htct

� �∂p12; t

¼ θt1 +ϕ2

t + 2p23; tϕt>0 if p23; t>0

∂ htct

� �∂p13; t

¼θt∂ ht

ct

� �∂p12; t

¼ θtϕt

1 +ϕ2t + 2p23; tϕt

>0 if p23; t>0

∂ htct

� �∂θt

¼ p12; t + p13; tϕt

1 +ϕ2t + 2p23; tϕt

>0 if p12; t; p13; t>0

The above partial derivatives confirm that asthe asset to be hedged and the derivativeinstrument tend to have common fluctuations,the required number of the futures contract

increases. Moreover, when fluctuations of theunderlying asset of the futures returns show lessintense amplitude relative to the asset to behedged, a greater number of futures contract isrequired to immunize the value of the spotposition.Although the above propositions deal with

partial derivatives, the optimal hedge ratiodepends on five parameters (three correlationsand two variance ratios), leading to a continuousrange of possibilities. To provide additionalinsights, we show how the optimal hedge ratioevolves for alternative bivariate parameterstructures once the other three parameters takespecific values.Arbitrage trading tends to produce positive

correlations between the derivative and itsunderlying asset. Moreover, in a context ofincreasing globalization of the world economy,shocks in financial markets disseminate in asimilar way, and assets tend to share commontrends. This way, let us assume not only thatthe conditional variances of the three assetsobserved arises from joint dependence on acommon factor, leading to common volatilitymovements across returns, but also that therelative importance of the mispricing fluctuationsis negligible in terms of the volatility of thecommon noise between the derivative and itsunderlying asset. The next two figures displaythe optimal ratio as a function of ρ12,t and ρ23,tunder alternative scenarios for the relativevolatility between the asset to be hedgedand the underlying asset of the futurescontract.As expected, when returns of the asset to be

hedged are relatively more volatile than returnsof the underlying asset of the futures contract,the number of derivative contracts required isclearly higher (compare Figures 1 and 2).

Lafuente


00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

rho12

rho23

hedg

e ra

tio

Figure 1: Optimal hedge ratio with θt=1, ϕt= 0.1, ρ13, t=0.9.

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

rho12

rho23

hedg

e ra

tio

Figure 2: Optimal hedge ratio with θt=1.5, ϕt=0.1, ρ13, t= 0.9.



When the price discovery process of thefutures market is not in the way expectedunder rational expectations, spot and futuresreturns could be negatively correlated. In thiscase, there are combinations for the threecorrelations involved that lead to a long ratherthan short position in the futures market toperform the hedging strategy. Observe fromequation (6) that, under positive correlationbetween the common and the specific noiseof the futures returns, the optimal hedge ratiois negative when ρ12, t<−ρ13, tϕt. Figure 3provides insights into this aspect by allowingnegative values of parameter ρ12,t inparameterization benchmark used in Figure 1.Finally, we show the joint effect of the

relative noise variance ratios on the optimal

hedge ratio under positive and non-perfectcorrelation for all the assets involved in theproblem (Figure 4).Given the correlations, the optimal ratio

depends on the relative variance noiseratios in a nonlinear way. Consistently withFigure 2, as the relative volatility of the assetto be hedged in terms of the volatility of thecommon noise increases a higher numberof derivative contracts is needed. However,higher volatility of the specific noise relativeto the volatility of the common noiserebalances the intensity of fluctuations betweenthe asset to be hedged and the derivativeasset, reducing the number of contractsinvolved in the hedging operation and itscost.

-0.9-0.8

-0.7-0.6

-0.5-0.4

-0.3-0.2

-0.10

0.1

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

rho12rho23

hedg

e ra

tio

Figure 3: Optimal hedge ratio with θt= 1, 0, ϕt=0.1, ρ13, t= 0.9.

Lafuente


CONCLUSIONS

This note provides a theoretical discussion for theoptimal cross-hedge ratio under futuresmispricing. Assuming that assets evolve accordingto a geometric Brownian motion, it is showedthat the optimal ratio that minimizes theconditional variance of the hedged positiondepends on the correlation matrix and therelative volatility of the asset to be hedged andthe hedging instrument in terms of theunderlying asset of the futures contract. Weanalyze bivariate joint effects of correlations andnoise variance ratios for the assets involved in thehedging operation. It is showed that, underpositive correlations, the cost of hedging aportfolio that mismatches the underlying asset ofthe futures contract increases as the relative

volatility of the underlying asset of the futuresmarket decreases. However, the relative volatilityof the specific noise in the futures market couldbe exploited to reduce the number of futurescontracts required.

ACKNOWLEDGEMENT

I would like to thank Alfonso Novales for theirhelpful comments and suggestions. Financialsupport the Spanish Ministry of Educationthrough grant ECO2012-31941, the GeneralitatValenciana through grant Prometeo II/2013/015and the University Jaume I through grantP1.1A2012-09 is gratefully acknowledged. Theusual caveat applies.

0.20.4

0.60.8

11.2

1.41.6

1.82

2.2

0.20.4

0.60.8

11.2

1.41.6

1.82

2.2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

phi

theta

hedg

e ra

tio

Figure 4: Optimal hedge ratio with positive correlations ρ12, t= ρ13, t= ρ23, t=0.5.



ReferencesArshanapalli, B. and Doukas, J. (1994) Common volatility in

S&P 500 stock index and S&P 500 index futures pricesduring October 1987. Journal of Futures Markets 14(8):915–925.

Draper, P. and Fung, J.K.W. (2003) Discretionarygovernment intervention and the mispricing of indexfutures. Journal of Futures Markets 23(12): 1159–1189.

Kawaller, I. and Koch, T.W. (1984) Cash and carry tradingand the pricing of treasury bill futures. Journal of FuturesMarkets 4(2): 115–123.

Lafuente, J.A. and Novales, A. (2003) Optimal hedgingunder departures from the cost of carry valuation:Evidence from the Spanish stock index futuresmarket. Journal of Banking and Finance 27(6):1053–1078.

Yadav, P.K. and Pope, P.F. (1990) Stock index futuresarbitrage: International evidence. Journal of Futures Markets10(6): 573–603.

Yadav, P.K. and Pope, P.F. (1994) Stock index futuresmispricing: Profit opportunities or risk premia. Journal ofBanking and Finance 18(5): 921–953.

Lafuente


Optimal cross-hedging under futures mispricing: A note

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