Credit-implied forward volatility and volatility expectations

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Contents lists available at ScienceDirect

Finance Research Letters

journal homepage: www.elsevier.com/locate/frl

Cre

dit-implie

d forward volatility and volatility

expectations

Hans Byström

∗

DepartmentofEconomics,LundUniversity,Box7082,22007Lund,Sweden

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Articlehistory:

Received16September2015 Accepted26October2015 Availableonline31October2015 JELclassiﬁcation: G1 G10 G17 G53 Keywords: CDS

Impliedvolatilitytermstructure Forwardvolatility

Forwardstartoptions

a

b

s

t

r

a

c

t

Weshowhowonecanbackoutimpliedforwardvolatilityterm struc-turesfromcreditdefaultswapspreads.Suchforwardstockvolatility termstructuresareusefulforinstanceinforwardstartoptionpricing. Weﬁndthetermstructuretobedownward-sloping,andthecredit market’svolatilityforecaststendtovarymoreacrosstimethanacross maturities.Long-termvolatilityexpectations,inturn,arefoundtobe lowandstablewhileshort-termexpectationsarehigherandmore volatile.Thevolatilityexpectation’smean-reversionrate,ﬁnally, in-dicatesthatthecreditmarketexpectsvolatilityshocksintheequity markettolastforseveralyears.

1. Introduction

WefollowByström(2013)andbackoutimpliedstockvolatilitiesfromcreditdefaultswapspreads. While Byström(2013)backsout ordinaryimplied volatilities,weinstead focusonso-calledforward volatilities,i.e.volatilitiesfortime-periodsthatstartinthefuture.Suchvolatilitiesareneededwhen pricingexoticderivativessuchasforwardstartoptions,cliquetsorforwardstarting“goldenhandcuff” employeestockoptions(Alford&Boatsman,1995;Glasserman&Wu,2011).Wethenproceedto estimat-ingvolatilityexpectations,fortheﬁrsttime(webelieve)explicitlyestimatingthecreditmarket’s short-andlong-termequityvolatilityexpectations.

∗ _{Corresponding}_author._Tel:₊₄₆_2229478;_fax:₊₄₆_2224118. E-mailaddress:hans.bystrom@nek.lu.se

http://dx.doi.org/10.1016/j.frl.2015.10.027

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Inadditiontorepresentingthecreditmarket,ratherthantheequitymarket,thecreditdefaultswap (CDS)markethascertainfeaturesthatmakeitparticularlyusefulforthepurposeofforecastingforward volatilities.First,thematuritiesofcreditdefaultswapsaremuchlongerthanthoseofordinaryequity options.Thismakesitpossibletoforecaststockvolatilitiesstartingseveralyearsintothefuture.Second, theavailabilityofconstant-maturityCDScontractswitharangeofmaturities,n=1,2,3,… ,10years makeitpossibletobackoutforwardvolatilitiesforanycalendaryear1to10yearsintothefuture.This featuresetsthisstudy/marketapartfrompreviousstudies/marketsrelyingonbothfewerand/orchanging maturities.Third,theCDSmarkethasbecomeamaturemarketcoveringmanyregions,countriesand ﬁrms(Byström,2015).ThismakestheCDSmarketapromisingnewcandidateforanyonewhowantsto estimateimpliedforwardstockvolatilities.

Thereareseveralstudiesthatlookatthetermstructureofimpliedvolatilities.1 Inthispaperwebuild, ﬁrstandforemost,onByström(2013)backingoutimpliedstockvolatilitiesusingcreditdefaultswaps.We thenusethesevolatilitiestocalculateforwardvolatilitiesandshort-andlong-termvolatilityexpectations (inthecreditmarket)forthestocksintheDowJonesIndustrialAverage(DJIA)index.

2. Impliedvolatility

Impliedstockvolatilitiesaretypicallyinferredfromequityoptions.AsdescribedbyByström(2013, 2015),however,impliedstockvolatilitiescanalsobeinferredfromcreditdefaultswapsusingCDS pric-ingmodelssuchastheindustrybenchmarkCreditGradesmodel(CreditGrades,2002).InCreditGrades,a firmdefaultswhenitsassetvalue,Vt,fallsbelowitsstochasticdefaultthresholdLtD.Disthefirm’sdebt levelandLtistheglobalrecoveryrateonthefirm’sliabilities.TheCreditGradesCDSspreadforacertain maturity,T,isthen

spreadCDST =r

(

1−R

)

1−P

(

0

)

+erξ

₍

_G

₍

_T₊

_ξ)

₋_G

_(ξ))

P

(

0

)

−P

(

T

)

e−rT₋_erξ

(

_G

(

_T₊

ξ)

₋_G

(ξ))

(1) wherethesurvivalprobabilityP(t)is

P

(

t

)

=N

−At 2 + ln

(

d

)

At

−dN

−At 2 − ln

(

d

)

At

and d= V0 LmeanD eλ2, At2=

σ

2t+

λ

2, G

(

t

)

=dz+1 2N

−ln

(

d

)

σ

√t −z

σ

√ t

+d−z+1 2N

−ln

(

d

)

σ

√t +z

σ

√ t

,

ξ

=

λ

2

σ

2, z=

1 4+ 2r

σ

2.

σ

istheassetvolatilityand

σ

₌(

σ

EEt)/(Et+LmeanD)whereEtistheequityvalueand

σ

Eisthestock volatility(thatwebackout).ristherisk-freeinterestrate,RistheCDS-speciﬁcrecoveryrateandLmean and

λ

isthemeanandstandarddeviationoftheglobalrecoveryrate,Lt.WesetLmean=Randfollowthe recommendationsinCreditGrades(2002)choosingR=0.5and

λ

=0.3.

1_Stein₍₁₉₈₉₎_looks_at_the_{mean-reverting}_properties_of_implied_volatility._Xu_and_Taylor₍₁₉₉₄₎_estimates_forward_volatilities

andestimatesshort-andlong-termvolatilityexpectations.Heynenetal.(1994)testsrestrictionsontheimpliedvolatilityterm structureandCampaandChang(1995)teststheexpectationshypothesisintheimpliedvolatilitytermstructure.Mixon(2007)also testswhetherthetermstructureofimpliedvolatilitiesisconsistentwiththeexpectationshypothesisandEgelkrautetal.(2005) studiesforwardvolatilitiesandvolatilityprediction.

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3. Impliedforwardvolatility

VolatilitiesimpliedbyoptionpricesorCDSspreadsareexpectationsofvolatilitiesovertime-periods startingtoday.Forcertainpurposes,such aspricingforwardstart optionsoremployee optionswith “goldenhandcuffs”,therequiredvolatilitiesareexpectedvolatilitiesacrosstime-periodsstartinginthe future.Thesevolatilitiesareknownasforwardvolatilitiesand,inoursetup,impliedforwardvolatilities (variances)fT,tforeachfutureyearT,2≤T≤10,canbecomputedattimetfromimpliedvolatilitiesyT,t (yT,t=

σ

E)withmaturitiesT,1≤T≤10,as:

fT,t=Ty2T,t−

(

T−1

)

y2T−1,t. (2) 4. Avolatilitytermstructuremodel

Inourvolatilitytermstructuremodel,whichissimilartothemodelbyXuandTaylor(1994),CDS mar-ketparticipantsformvolatilityexpectationsthatarefunctionsofthreeparameters;theshort-term ex-pectation

α

,thelong-termexpectation

μ

andtherateofmeanreversion,

φ

,whichdeterminesthespeed withwhichthevolatilityexpectationrevertstowards

μ

.XuandTaylor(1994)showsthattheexpected forwardvariancegT,tisalinearcombinationof

α

2and

μ

2:

gT,t=

μ

2t +xT,t

α

2 t −

μ

2t

(3) with xT,t=

φ

T−1₋

φ

T 1−

φ

, (4)

assuming

φ

<1.ByminimizingthedifferencebetweenfT,tandgT,t,followingthethree-step regression-basedestimationproceduredescribedbyXuandTaylor(1994),wegetdaily

α

t,

μ

t and

φ

ˆt estimates. ComparedtoXuandTaylor(1994)wehavetheadvantageof(i)havingconstantmaturitiesacrossthe sample,(ii)havingmaturitiesthatarecountedinyearsand(iii)eachdayhavingthesamenumberof maturities(10).

5. Data

OurdatasetcontainsCDSspreads,stockpricesandleverageratiosfortheﬁrmsintheDJIAindexfrom January1,2006toSeptember30,2010.2 ,3 _The_CDS_data_is_available_for_constant_maturities_of_n_years_with n₌1,2,3,… ,10,andallCDScontractsareseniorUSDcontracts.Thestockpricesarealsodenominated inUSD,andasaproxyfortherisk-freeinterestrateweusetheaverageUS3monthTreasurybillrate.4 6. Results

Descriptivestatisticsonforecastedone-yeartonine-yearaheadforwardvolatilitiesispresentedin

Table1(year2startsoneyearfromnowandyear10startsnineyearsfromnowetc.).5 ,6 _The_near-term

volatilityforecastsareobviouslyhigherthanthedistant-termforecasts.Thenear-termforecastsarealso morevolatile.7 InFig.1weshowtheforwardvolatilitiesfortheaverageﬁrminthesampleanditisclear

2_Only_{non-financial}_firms_are_included_in_the_study_and_since_some_firms_lack_available_data_our_sample_is_reduced_to₂₂_firms. 3_The_start_date_is_dictated_by_data_{availability;}_before₂₀₀₆_we_do_not_find_CDS_data_for_all_the₍₁₀₎_maturities._The_end_date,

inturn,isdictatedbytheneedforareasonablylongout-of-sampleperiodfortheforwardvolatilityforecasts(thecalendaryears 2011–2014).

4_Leverage_ratios_are_downloaded_from_the_web-page_of_A._Damodaran_and_the_rest_of_the_data_is_downloaded_from_Datastream. 5_While_we_only_estimate_one-year_forward_{volatilities,}_our_method_and_data_allow_for_the_estimation_of_n_-year₍_n₌_2,_3,_…)

volatilitiesaswell.

6_We_proxy_the_January_1,₂₀₁₁_forecast_with_the_September_30,₂₀₁₀_forecast.

7_It_should_be_noted,_though,_that_although_the_nine_forward_volatility_forecasts_are_made_{simultaneously}_they_are_forecasts_for

differenttime-periods.Forexample,theone-yearaheadforecastsareforthecalendaryears2007to2012andthenine-yearahead forecastsareforthecalendaryears2015to2020.

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Table1

Descriptivestatisticsforannualizedone-yeartonine-yearaheadcredit-implied forwardvolatilitiesaveragedacrossthe22non-ﬁnancialDJIAﬁrmsforthe time-periodJanuary1,2006toSeptember30,2010.

Mean(%) Stdev(%) Skewness Kurtosis

Year2 41.4 5.7 –0.10 –1.22 Year3 38.8 4.6 0.41 –0.40 Year4 39.3 4.2 0.14 –1.13 Year5 38.0 3.7 0.54 –0.21 Year6 35.4 3.1 0.74 0.88 Year7 34.2 3.1 0.89 1.36 Year8 34.0 3.3 0.71 0.42 Year9 33.5 3.3 0.72 0.50 Year10 33.1 3.3 0.73 0.55

Fig.1. One-yeartonine-yearaheadcredit-impliedforwardvolatilitiesaveragedacrossthe22non-ﬁnancialDJIAﬁrmsforthe time-periodJanuary1,2006toSeptember30,2010.

that,regardlessofhorizon,theforecastedvolatilityincreasesduringthecrisis.Fig.2showsa3D-plotof theforwardvolatilitytermstructuresampledonaquarterlybasisanditisclearnotonlythattheterm structureisdownward-slopingbutalsothatbothitslevelandslopevaryovertime.

Thefinancialcrisismakesitdifficulttoevaluatetheaccuracyofvolatilityforecastsandinthispaperwe thereforechooseadifferentpath.InFigs.3–4wepresentsnapshotsfromthesetofforecastsinFig.1.We limittheforecaststothoseforwardvolatilityforecaststhataredonethefirsttradingdayeachcalendar yearfrom2006to2011.Theresultingforwardvolatilityforecastsforthecalendaryears2007to2014are plottedbothinFig.3,withthedependentvariablebeingtheyear,

τ

,fortheforecast,andinFig.4,with thedependentvariableinsteadbeingtheyear,t,whentheforecastismade.

Inanidealworld,wherethemarketalwayspredictsfuturevolatilitiescorrectly,thesevencurvesin

Fig.3shouldoverlap.Inreality,theforecastsdifferfromeachother,however;thecreditmarket over-estimatedtheforwardvolatilityfortheyearsafterthecrisisandunder-estimateditforthecrisisyears. ThecurvesinFig.4,inturn,shouldbehorizontalifthemarkethadperfectforesight.Instead,theactual

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Fig.2. The(average)credit-impliedforwardvolatilitytermstructuresampledonaquarterlybasisoverthetime-periodJanuary1, 2006toSeptember30,2010.

Fig.3. Credit-impliedforwardvolatilityforecastsmadetheﬁrsttradingdayinJanuaryeachyearfrom2006to2011.Thevariable onthex-axisistheyear,τ,fortheforecast(thecalendaryears2007to2014).

forecastsdemonstrateanupward-slopingpatternwheretheforecastoftheforwardvolatilityincreases overtimeregardlessoftheyear,

τ

,fortheforecast.Ingeneral,theforecastvariationissmalleramongthe various

τ

thanamongthevarioust;i.e.thecreditmarket’sequityvolatilityforecaststendtovarymore acrosstimethanacrossmaturities.

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Fig.4. Credit-impliedforwardvolatilityforecastsforthecalendaryears2007to2014.Thevariableonthex-axisistheyear,t,when theforecastismade(theﬁrsttradingdayinJanuaryeachyearfrom2006to2011).

Fig.5. Short-andlong-termvolatilityexpectations(αandμ)averagedacrossthe22non-ﬁnancialDJIAﬁrmsforthetime-period January1,2006toSeptember30,2010.

Themodel-impliedshort-andlong-termvolatilityexpectations

α

and

μ

arepresentedinFig.5(mean valuestogetherwith25%-and75%-percentiles).Thecreditmarket’slong-termexpectationsarequite stablearound30%whiletheshort-termexpectationsﬂuctuatebetween40and75%.Theaveragevolatility expectationmean-reversionrate

φ

isfoundtobe0.9andvariesbetweenaround0.8in2006and0.95in 2010;i.e.,a“half-life” of3–13yearswithanaverageof7years(0.97_≈_0.5).

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4. Conclusion

Inthispaperweshowhowonecanbackout impliedforwardstockvolatilitiesfromCDSspreads andthenusethesevolatilitiestoestimateshort-andlong-termvolatilityexpectations.Wefindthe for-wardvolatilitytermstructuretobedownward-sloping,andthecreditmarket’svolatilityforecaststend tovarymoreacrosstimethanacrossmaturities.Thecreditmarket’slong-termvolatilityexpectationfor thestocksintheDJIAindexisaround30%whiletheshort-termexpectationfluctuatesbetween40and 75%.Meanwhile,theaveragemean-reversionrateinthevolatilityexpectationcorrespondstoa“half-life” ofaround7years.I.e.,volatilityshocksintheequitymarketareexpectedtolastforlong.Webelievethis tobethefirstattemptatestimatingthecreditmarket’sshort-andlong-termexpectationsaboutfuture stockvolatility.

Acknowledgment

Financial assistancefromTheMarianneandMarcusWallenberg Foundation[MMW2012.0019]is gratefullyacknowledged.

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