Option-Implied Libor Rate Expectations across Currencies

K.7 Option-Implied Libor Rate Expectations across Currencies Gebbia, Nick Please cite paper as: Gebbia, Nick (2016). Option-Implied Libor Rate Expectations across Currencies. International Finance Discussion Papers 1182. http://dx.doi.org/10.17016/IFDP.2016.1182 International Finance Discussion Papers Board of Governors of the Federal Reserve System Number 1182 October 2016

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1182 October 2016 Option-Implied Libor Rate Expectations across Currencies Nick Gebbia NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

∗ Option-Implied Libor Rate Expectations across Currencies Nick Gebbia Federal Reserve Board of Governors October 13, 2016 Abstract Inthispaper,Istudyrisk-neutralprobabilitydensitiesregardingfutureLiborratesdenominatedin Britishpounds,euros,andUSdollarsasimpliedbyoptionprices. IapplyBreedenandLitzenberger’s (1978) result regarding the relationship between option prices and implied probabilities for the underlying to estimate full probability density functions for future Libor rates. I use these estimates in casestudies,detailingtheevolutionofprobabalisticexpectationsforfutureLiborratesoverthecourse of several important market events. Next, I compute distributional moments from density functions estimated for fixed horizons and test for Granger causality across the three Libor rate distributions considering their mean, standard deviation, skewness, and kurtosis. I further break these relationshipsdownbyvariousfixedhorizonlengths,aswellastheslopeandcurvatureinthetermstructure of moments over different horizons. The results show a rich interconnectedness among these three Liborratesthatextendswellbeyondlevelsoffuturemeanexpectations. JELClassifications: C14,E43,G13 Keywords: options,futures,Libor,pdf,distribution,moments,Grangercausality ∗The views presented in this paper are solely the responsibility of the author and should not be interpreted as reflecting theviewsoftheBoardofGovernorsoftheFederalReserveSystemorofanyotherpersonassociatedwiththeFederalReserve System. I would especially like to thank Stephanie Curcuru, Michiel De Pooter, Rob Martin, Marius Rodriguez, and Charles Thomas, whose insights and guidance were necessary conditions to this paper’s existence, and are very much appreciated. I also would like to acknowledge the Global Monetary and Sovereign Markets (GMSM) and Global Capital Markets (GCM) sectionsforthoughtfulfeedbackonanearlierversionofthisproject.Allerrorsaremyown.Email:nick.j.gebbia@gmail.com

1 Introduction Option prices provide a unique insight into the probabilities assigned by markets to various future outcomesforaparticulareconomicvariable. Anoptioncontractallowsitsownertobuyorsellanunderlying securityorcommodityatapre-determinedprice. Assuch,optionscanbeusedtohedgeagainstdownside orupsiderisks,oralternativelytospeculateonthefuturevalueofaparticularsecurityorcommodity. In thehedgingcase,optionsprovideaneffectiveceilingorfloortothepriceoftheunderlying. Ontheother hand, astheprice ofanoption’sunderlyingexceeds (fallsbelow)thepriceat whichaspeculatoragreed to buy (sell), the speculator earns a return by exercising the option to buy (buying at the market price) and selling at the market price (exercising the option to sell). In either case, the price an individual is willingtopayforaparticularoptionisbasedintimatelyuponthelikelihoodassignedtovariousoutcomes fortheoption’sunderlying. CoxandRoss(1976)formalizethisrelationshipbyexpressingthepriceofaEuropeanoption1 asthe discounted expected payout of the option at maturity. Breeden and Litzenberger (1978) directly relate thesecondpartialderivativeofcalloptionpricewithrespecttostriketotheprobabilitydensityfunction (pdf) assigned to the underlying. Since these results were established, option prices have been applied invariouscasestoderiveestimatesofmarket-assignedprobabilitiesfordifferentunderlyingsecuritiesor commodities(forexample,seeMalz(1997)inthecaseofforeignexchangerates,orMelickandThomas (1997)andDatta,Londono,andRoss(2014)inthecaseofoilprices). Optionpriceshavebeenappliedtoestimatemarket-assignedprobabilitiesforfutureinterestrates,as well. Clews, Panigirtzoglou, and Proudman (2000) estimate pdf’s for future British pound-denominated Libor interest rates and use distributional characteristics to shed light on the impact of important events on market expectations. For example, they show that the advent of operational independence for the Bank of England in May, 1997 had little contemporaneous effect on market uncertainty around future GBP Libor rates. Ivanova and Gutiérrez (2014) use options to study probabilities assigned to future euro-denominatedLibor(Euribor)rates,investigatingtheforecastingperformanceofsuchmeasuresand usingforecastbiasestoinferriskaversion. Two general approaches to estimating option-implied probabilities have emerged in the literature as particularly useful. One involves assuming a functional form for the terminal distribution of the underlying(forexample, aweightedmixtureoftwolognormalortwonormaldistributions), anddetermining the parameters of the distribution by minimizing the resulting option pricing errors (see Bahra (1997); Melick and Thomas (1997)). The other works by interpolating a fine set of option prices and using the 1 A European option is one that can only be exercised on its maturity date. Contrastingly, American options allow the optionholdertoexerciseatanytimepriortoexpiration. 1

result of Breeden and Litzenberger (1978) to produce a pdf by taking the second derivative of option prices numerically. This paper does not take a stance on whether one method outperforms the other. However, previous research comparing these two methodologies in Monte Carlo simulation settings using similar data has favored the method of interpolating option prices and taking numerical derivatives (Cooper(1999);BlissandPanigirtzoglou(2002)). In this paper, I use options on Libor futures for interest rates denominated in euros (EUR), British pounds (GBP), and US dollars (USD) in order to study market-assigned probabilities for the three rates. This paper contributes to the existing literature on option-implied pdf’s in a few ways. First, I detail the evolution of pdf estimates for Libor rates denominated in EUR, GBP, and USD over important case studies,includingeventscriticaltothefinancialcrisisof2007-08. Thesecasestudiesserveasmeticulous overviews of the ways in which market-assigned probabilities for future interest rates were affected by major events. I then move to a more thorough investigation of the time series dynamics among distributionalcharacteristicsforthethreeLiborrates. IperformtestsofGrangercausalityacrossthethree Liborratedistributionsconsideringtheoption-impliedmean,standarddeviation,skewness,andkurtosis relatedtopdf’satfixedfuturehorizons;thesetestsarecarriedoutforthelevelsofthefourdistributional moments at different horizons, as well as their slope and curvature across horizons. By comparing estimates of distributional characteristics across the three separate Libor rates, this paper offers unique insightsintotheirinterrelationshipsextendingbeyondcomparisonsofspotorevenexpectedfuturelevels. Finally,asamethodologicalcontribution,ItakeuptheissueoftheAmericanoptionpremiumasitrelates to option-implied pdf’s using a Monte Carlo simulation approach; while previous research has focused on the level of the American option premium (see Tian (2011)), I tie the premium directly to its impact onprobabilityfunctionestimates. Theremainderofthepaperisstructuredasfollows: Section2reviews the data and methodology; Section 3 presents empirical case studies of the evolution of future Libor rate pdf’s over important market events; Section 4 details the results of Granger causality tests over distributionalmomentsacrossthethreeLiborrates;andSection5concludes,alongwithsuggestionsfor furtherresearch. 2 Estimating Probability Density Functions and Distributional Moments from Option Prices2 2.1 Data Futures contracts for 3-month Libor rates denominated in GBP and EUR trade on the London International Financial Futures and Options Exchange (LIFFE), and similar contracts for Libor denominated in 2 AppendixAcontainsfurthertechnicaldetailregardingoptionpricingmodelscoveredinthissection. 2

USD trade on the Chicago Mercantile Exchange (CME). In order, these futures contracts are typically referred to as Short Sterling, Euribor, and Eurodollar futures. The futures contracts are cash settled at expiration based on the prevailing 3-month Libor rate in the relevant currency at that time, as well as the notional value of the contract. Prices are quoted as 100 minus the annualized 3-month rate – for example,aneffectiverateof2.79%wouldimplyacontractpriceof100−2.79=97.21. Option contracts written on 3-month Libor futures are available on the same exchanges. For futures on GBP and EUR Libor, options exist with maturities in the nearest eight March-quarterly months (i.e. March, June, September, December); for options on USD Libor futures, available option maturities span the nearest sixteen March-quarterly months. All sets of options also offer serial maturities in the four nearest months that do not follow the March-quarterly cycle, so that there is always an available option maturity in each of the six upcoming months. However, when an option is exercised, it delivers the futurescontractexpiringintheupcomingMarch-quarterlymonth. Therefore,thevalueofaserialoption atexpirationwillbebasedonafuturesrate(forexample,aFebruaryoption’svalueatexpirationisbased on the concurrent price of the March futures contract). On the other hand, March-quarterly options expireonthesamedateastheunderlyingfuturescontract,sothatthevalueofaMarch-quarterlyoption atexpirationisbasedonaspotLiborrate.3 Forthisreason,Irestrictanalysistooptioncontractsmaturing in the March-quarterly cycle. Futures and options data for EUR and GBP Libor comes from Thomson Reuters,andUSDLibordataisfromCME. FiguresB1andB2inAppendixBdetailoptionliquiditytrendsforMarch-quarterlyEurodollaroptions from May 4th, 1998 through March 31st, 2016. Figure B1 shows open interest, and Figure B2 shows volume. Not surprisingly, option liquidity peaked in the early stages of the 2007-08 financial crisis as investorslikelysoughtinsuranceagainstanuncertainmonetarypolicyresponse. Asinterestratessettled to historically low levels following the crisis, trading of Eurodollar options dipped significantly through 2013. More recently, Eurodollar options have seen a secular rise in liquidity alongside the possibility and realization of monetary policy “liftoff” from near the Zero Lower Bound (ZLB) in the United States. Figures B3 and B4 show open interest and volume as they evolve with an option’s time to maturity, aggregated over the same period. Trading is heaviest between 3 and 6 months prior to an option’s expiration. FiguresB1-B4alsodisaggregatetrendsinmarketliquiditytodetailout-of-the-moneyoption tradingindependently. Itisapparentthatout-of-the-moneyoptionscapturethesolidmajorityoftrading action in the Eurodollar market. In total, out-of-the-money Eurodollar options account for 73% of open interestand82%ofvolumeoverthefull period. Finally, FigureB5separatesshareoftotalopeninterest and volume by month of option maturity. Trading tends to be heaviest for December options, which accountfor30%ofbothopeninterestandvolumeovertheperiod. 3 Table B1 in Appendix B reports the maturity date conventions for Short Sterling, Euribor, and Eurodollar futures and optionsintheMarch-quarterlycycle. 3

Thepaymentandexercisestructuresoftheoptioncontractsproveimportanttothefollowinganalysis. All three Libor-based options are American, meaning the buyer has the right to exercise at any time prior to the option’s maturity. This is in contrast to European-style options, which only allow exercise at contract maturity. Eurodollar option contracts require payment of the option premium at the time of purchase; however, Short Sterling and Euribor options use a futures-style margining system. For Short Sterling and Euribor options, a premium is agreed at the time of purchase, but it is not paid at that time. Instead, the net value of the option is marked-to-market daily, and the payout of the contract at exercise becomes the value of the option net of the agreed premium. As a result, pricing of Short Sterling and Euribor options involves no discounting.4 Additionally, the margining structure resultsinzeroopportunitycostofholdingtheoption,meaningtheAmericanearlyexercisepremiumcan beignoredandShortSterlingandEuriboroptionscanbepricedasthoughtheywereEuropeanoptions.5 TheanalysisinthispaperassumesoptionsareEuropean. AppendixCtakesuptheissueoftheAmerican early exercise premium included in Eurodollar option prices, concluding that there is little accuracy lost inthispaper’sanalysisbytreatingEurodollaroptionsasthoughtheywereEuropean. Finally, it is important to note that Libor rates may include a risk and/or term premium component in comparison to the interest rates targeted by central banks in policy decisions; therefore, estimated probability distributions for future Libor rates necessarily blur together market views of future central bank policy along with anticipated risk or term premia. These factors may reinforce or counteract one another: for example, if stresses on the financial sector increase, markets may expect monetary policy easing to lower rates, while simultaneously expecting a greater risk premium between Libor and the corresponding policy rate to push Libor upward. The following analysis does not formally attempt to distinguish between implied expectations for central bank policy rates and other factors contributing to aspreadbetweenLiborandthecorrespondingpolicyrate,butinsteadtreatsLiborastheprimaryfocus. 2.2 Basic Methodology In an efficient (and risk-neutral) market, the price of a European option is equal to the discounted expectation of its value at maturity.6 In the case of a put option on the futures price F struck at K and T lastingfromtime t toamaturitydate T, P(K,t,T) = e −rτ E [max(K−F ,0)] t T (1) (cid:90) K = e −rτ (K− f)π (f)df , F ,t T −∞ 4 SeeChenandScott(1993). 5 SeeLieu(1990)andChenandScott(1993). 6 ThisresultisoriginallyduetoCoxandRoss(1976). 4

where r is the risk-free rate used for discounting,7 τ ≡ T −t, and π (f) is the pdf for F from time F ,t T T t evaluated at a terminal futures contract price of f. It is important to note that F is considered as a T randomvariable. Using the insights of Breeden and Litzenberger (1978), it is possible to extract the market-assigned cumulative distribution function (cdf) and pdf for F directly from Equation (1). Taking the first deriva- T tiveofEquation(1)withrespectto K yields dP(K,t,T) (cid:90) K = e −rτ π (f)df . F ,t dK T −∞ (cid:90) K Wherethecdfof F evaluatedat K isdefinedbyΠ (K)≡ π (f)df,itisthentruethat T F ,t F ,t T T −∞ dP(K,t,T) Π (K) = erτ . (2) F ,t T dK Equation(2)statesthatthecdfof F isequaltothe(futurevalueofthe)firstderivativeoftheputpricing T functionwithrespecttostrike. TakingthederivativeofEquation(2)withrespectto K yields d2P(K,t,T) π (K) = erτ , (3) F T ,t dK2 showing that the pdf for F is equal to the (future value of the) second derivative of the put pricing T functionwithrespecttostrike. After estimating π (·), it is possible to evaluate the overall methodology using Equation (1). Ob- F ,t T served options can be re-priced using the estimated pdf, and these prices can then be compared against actualmarketprices. Smallermagnitudesofre-pricingerrorsgenerallyindicatethattheestimatedpdfis increasingly reflective of market expectations. Pricing errors must also be balanced with the plausibility oftheshapeoftheresultingpdf,apointfurtherelaboratedbelow. 2.3 Interpolation of Option Prices InordertoapplyEquations(2)and(3),theinstantaneousfirstandsecondderivativesoftheputpricing function with respect to strike are required. Of course, we do not observe the latent option pricing 7 I set r = 0 for Short Sterling and Euribor options. Eurodollar discounting is based on a curve constructed using an overnightrateandTreasuryyieldsatvarioustenors. 5

function, but only prices at discrete intervals. As a result, derivatives are computed numerically after interpolatingputoptionpricestoasufficientlyhighgranularity. The question arises of how best to produce a stable and accurate interpolation function. The choice of interpolation specification is not trivial as we are directly interested in the slope and curvature of the estimated pricing function, which of course depend on the interpolation approach. Further, it is clear thattheactualpricingfunctionwillexhibitahighdegreeofcurvature,especiallysonearthemodeofthe pdffortheunderlying(Equation(3)directlylinksthepdfwithcurvatureintheputpricingfunction). In order to replicate this curvature in a robust way I first transform prices to implied volatilities and strikes to deltas (specifically, put deltas), both of which are derived via the Black (1976) model for pricing options on futures.8 Only out-of-the-money put and call options are used as inputs, as these tend to be more liquid than in-the-money options and therefore embed more reliable information.9 Transforming option prices to implied volatilities was introduced by Shimko (1993). The implied volatility function tends to entail a lower degree of curvature than the price function, yielding a simpler space overwhichtointerpolate. Anoption’sdeltaexpressestherateofchangeintheoptionpricerelativetoa changeinthepriceoftheunderlyingsecurity. Asaresult,convertingstrikestodeltasprovidestwomain benefits.10 First, deltas compress the domain under consideration: far out-of-the-money options have deltas approaching zero, while far in-the-money put options have deltas approaching −e −rτ . This eases extrapolation beyond the range of observed strikes as well as interpolation across time (both discussed further below). Second, because deltas change more rapidly near-the-money but group together for far in-the-money or out-of-the-money options, interpolating over deltas increases interpolation granularity near-the-moneytobettercapturetheinherentcurvatureofthepricingfunction. A cubic smoothing spline is used to perform interpolation of implied volatilities over deltas. The cubic smoothing spline is a piecewise cubic polynomial function that has continuous first and second derivatives at observed data points, where the function is segmented (so-called “knots” of the spline). In addition to providing a very flexible functional form, the cubic smoothing spline offers an explicit balancingbetweensmoothnessoftheinterpolatedfunctionandfiterrorsthatresultfromthesmoothing process. Specifically,thecubicsmoothingspline g(·)minimizesthefollowingobjectivefunction: n (cid:90) (1−λ) (cid:88) w(i)(cid:2) y(i)−g(x(i))(cid:3)2 + λ (cid:2) g (cid:48)(cid:48)(t)(cid:3)2 dt , i=1 x 8 NotethattheBlack(1976)assumptionofalognormallydistributedunderlyingfuturespriceatoptionexpirationdoes notcarrythroughtoestimatedpdf’s. Rather,theBlack(1976)modelsimplyprovidesamappingbetweenpricesandimplied volatilities,aswellasstrikesanddeltas. 9 IvanovaandGutiérrez(2014)verifythatout-of-the-moneyliquidityisgreaterforEuriborfuturesoptions,whileClews, Panigirtzoglou,andProudman(2000)arguethesameholdsforShortSterling. FiguresB1-B4inAppendixBdemonstratethe relativelyhighliquidityofout-of-the-moneyoptionsonEurodollarfutures. 10 InterpolatingimpliedvolatilitiesoverdeltaswasfirstperformedbyMalz(1997). 6

where x represents n observed deltas and y represents n implied volatilities, w is a set of weights, and λ is known as the “smoothing parameter”. The first term in the above objective function is simply the weighted sum of squared residuals at the spline’s knots, while the second term is meant to reflect the overalldegreeofcurvatureinthespline. Thesmoothingparameterfullydeterminesthebalancebetween fit residuals and spline smoothness. In fact, as λ → 1, the spline approaches a weighted least squares (WLS) regression estimate. Weights w are imputed as each option’s vega, equal to the rate of change in pricerelativetoimpliedvolatility. AspointedoutbyBlissandPanigirtzoglou(2002),becausevegaisthe change in price relative to implied volatility, weighting by vega over implied volatilities is equivalent to constantweightingoverprices. Thesmoothingparameterλpresentsachallenge,asitisnotreadilyapparentthataparticularvalue willproveidealinanysense. Oneapproachtodeterminingλautomaticallybasedoninputdatainvolves cross validation. This approach selects λ so as to minimize the average leave-one-out residual (i.e. for each i of the n observations, g(·) is estimated while ignoring data point i, after which the residual for datapointi iscalculated). InthecaseofoptionsonLiborfutures,testingofthecrossvalidationapproach yields clearly undesirable results. In a number of trials, cross validation selects a smoothing parameter atornearzero,whichinturnproducesimplausiblychoppyimpliedpdf’s. Examplesofsuchpdf’scanbe seen in Figures B6, B7, and B8 in Appendix B. These figures show estimates based on options maturing in December, 2010,11 which were priced on January 4th, 2010, for each of Short Sterling, Euribor, and Eurodollar in turn. Respectively, cross validation selected smoothing parameters of 0.00000, 0.00000, and0.00002onagranularityof0.00001between0and1. Alternatively, one might think to implement a similar approach which would instead be based on pricingerrors. Asmentionedpreviously,Equation(1)canbeusedtore-priceasetofoptionsbasedonan estimatedpdffortheunderlyingsecurity. Fromtheseestimatedprices,pricingerrorscanbecomputedas thedifferencebetweenestimatedandactualoptionprices. Insteadofselectingλtominimizeleave-oneoutimpliedvolatilityresiduals,onecouldalternativelyattempttominimizeleave-one-outpricingerrors. However,thisturnsouttobeprohibitivelyexpensivecomputationally.12 Duetotheundesirableperformanceofcrossvalidationinselectingthesmoothingparameter,Iinstead follow the alternative approach in the literature, which is to select a smoothing parameter based on subjective balancing between the plausibility of resulting pdf estimates and the magnitude of pricing 11 MaturitydatesareDecember13,2010forEURandUSDLibor,andDecember15,2010forGBPLibor. 12 Aselaboratedbelow,thesplineisevaluatedatalargenumberofpoints,whichquicklyaddstocomputingtimerequired foracrossvalidationapproachbasedonpricingerrors. 7

errors.13 I find λ=.03 tends to produce reasonable pdf’s while keeping option pricing errors generally within one or two ticks.14 Figures B9 through B14 in Appendix B show example pdf’s alongside the pricingerrorsthatresultafterre-pricingoptionsusingtheestimatedpdf. Pricingerrorsareshownforthe set of options used as inputs to each estimation, which includes out-of-the-money options meeting basic pricingassumptionrestrictions(asdetailedfurtherbelow). InFiguresB9throughB14,thevastmajority ofpricingerrorsarewithintwoticks,whilemanyarewithinonetick. In order to ensure the estimated pdf integrates to 1, the fitted spline function is extrapolated so that the full delta range between −e −rτ and 0 is covered.15 Extrapolation is performed with a quadratic polynomial function on either end of the observed delta range, as in Ivanova and Gutiérrez (2014). The full resulting function of implied volatilities over deltas is then evaluated at 50,000 evenly spaced delta values between the minimum and maximum deltas.16 Minimum and maximum deltas used for evaluationareoffsetbyasmallvaluefromtheactualboundsof−e −rτ and0,becausethesedeltasimply strikes approaching ∞ and 0, respectively (see Appendix A). In the case that any implied volatilities are evaluatedasnon-positive,theestimationprocedureisquit. Finally,theevaluatedimpliedvolatilitiesand deltasareconvertedbacktopricesandstrikesviaaninverseoftheBlack(1976)model. Because the estimated put option prices remain discrete – as opposed to a continuous function – derivatives and integrals must be taken numerically. The formal definition of a derivative or an integral provides the basis for approximating continuous derivatives and integrals with discrete (but highly granular)datapoints. Thefollowingequationssummarizetheserelationshipsforanyfunction f(x): df(x) f(x)− f(x−h) ≡ lim , (4) dx h→0 h and   (cid:90) b n i (cid:88) (cid:88) f(x)dx ≡ lim f a+ (∆x )(∆x ) , (5) a max 2≤i≤n (∆x i )→0 i=2  j=2 j  i 13 Forexample,Datta,Londono,andRoss(2014)demonstrategraphicallythetradeoffsinvolvedinincreasingordecreasing thesmoothingparameterwhenestimatingoption-impliedpdf’sforfutureoilprices.Theysettleonaconstantvalueforλ,which keepspricingerrorslowwhileproducingreasonablepdfestimates. 14 Theticksizeistheminimumpriceincrement. ForallthreeLibor-basedcontracts,theminimumincrementis.005. An exceptionismadeforEurodollaroptioncontractsthatdeliverthenearestexpiringfuturescontract, aswellasforlow-priced optionsexpiringinthetwoupcomingMarch-quarterlymonths,inwhichcasestheminimumticksizeis.0025. Forsimplicity andconsistency,Iwillrefertotheminimumticksizeas.005,thoughobservedpriceswithticksof.0025arenotalteredinany way. 15 Resultingpdf’sareonlyreportediftheirareaintegratestobetween0.99and1.01 16 Iuseaverylargenumberofpointsforevaluationinordertocloselyapproachacontinuoussetting, whileaccounting for some balancing of the computation time required. As we are interested in approximating instantaneous first and second derivatives,itisimportantthattheoptionpricegranularityisveryfineinordertoavoidbias.Further,applicationsthatrequire integrationovertheestimatedpdf-forexample,re-pricingoptionsorcomputingmomentsofthedistribution-gainaccuracy asthepdfapproachesanear-continuoussetting. 8

whereintheseconddefinitionn−1isthenumberofdiscretesub-intervalson(a,b),and∆x i ≡ x i −x i−1 . Note that, while the second equation simplifies in the case where ∆x is a constant value for all i, this i does not hold for our purposes. Discrete observations are evaluated in implied volatility - delta space at a constant delta interval. However, as previously mentioned, an option’s delta does not change at a constant rate with respect to strike – in fact, the gamma of an option specifies this rate of change across strikes. Therefore, the interval between strikes is not constant, so the above formula is applied when integrating. The process of smoothing implied volatilities over put option deltas does introduce the possibility of negative estimated probabilities, even after cleaning input option prices as described further below. In rare cases, the estimated set of implied volatilities over deltas results in a subset of implied volatilities exhibiting large swings when placed over strikes, a pattern carried through to the relationship between pricesandstrikes. Itisapparentinthesecasesthatthesetofimpliedvolatilitiesrepresentsasituationin which implied volatility – and, thereby, price – is not a proper function of strike (i.e. at least one strike exists that corresponds to more than one implied volatility), which contradicts Black’s (1976) option pricingmodel. Thisisanalogoustoconsideringthepoints(0,0),(1,-1),(4,2),and(9,-3): atsomedegree of granularity, it becomes apparent that the function being traced out is neither piecewise linear nor (cid:112) sinusoidal, but rather displays y = ± x. Unfortunately, there is no clear restriction to impose for the discrete set of implied volatilities over strikes (for example, monotonicity). However, because such a set of implied volatilities tends to produce swings in prices over strikes and thereby negative estimated probabilities (and because negative probabilities are not sensible), any estimated pdf’s are only used in furtheranalysisiftheyproducenonegativeprobabilities. 2.4 Converting Futures Prices to Interest Rates The preceding methodology is used to obtain a highly granular set of put option prices over strikes. These prices apply to options written on the relevant futures contract price, which is based on a simple transformationoftheLiborratebeingreferenced. Specifically,where F istherandomvariabledenoting T thefuturescontractpriceattime T, thenR =100−F isthecorrespondingannualized3-monthLibor T T rateinpercentagepoints. Ofcourse,wearedirectlyinterestedinprobabilityassignmentsforfutureLibor interestraterealizations,ratherthanforfuturescontractprices. Becausepricesareinterpolatedandextrapolatedinthesettingofoptionsonthefuturesprice,Equations (2) and (3) are applied initially in the same setting. Probabilities are transformed to the interest ratesettingafterbeingestimatedinafuturespricesetting,as 9

Prob (R ≤ a) t T = Prob (100−R ≥ 100−a) t T (6) = 1−Prob (100−R ≤ 100−a) t T = 1−Prob (F ≤ 100−a), t T whichgivesthedesiredcdfoverinterestrates.17 Similarly,whenconsideringpdf’s, π R ,t (a) = π 100−F ,t (a) = π F ,t (100−a). (7) T T T Interpolatingandextrapolatingoptionpricesinthesettingofthefuturescontractprice(F )provides T ausefultoolforcircumventinglimitationsembeddedintheBlack(1976)optionpricingmodel. TheBlack (1976) model assumes a lognormal probability distribution for the underlying security, which in turn implies no probability mass assigned below zero (see Appendix A). As noted previously, this assumption does not carry through to probability functions estimated as in this paper; the Black (1976) model is employed only to provide a mapping between prices and implied volatilities, as well as between strikes and deltas. However, these mappings break down for strikes below zero. As a result, while the above methodology could be applied directly to the interest rate setting by subtracting option strikes from 100 and reversing call and put options (e.g. 97.25 put → 2.75 call), this approach would break down for options with converted strikes below zero (or, original strikes above 100). Deriving probabilities for the futures contract price before converting to the interest rate setting then provides the advantage of retaininginformationembeddedinoptionswithstrikesabove100. Inthisway,onecanactuallyviewthe futurespriceasausefultransformationoftheLiborrateasrelatestoestimatingpdf’s. 2.5 Cleaning Input Prices Inadditiontorestrictinginputpricestoout-of-the-moneyoptions,inputsarescreenedsotheymeetbasic option pricing assumptions. These assumptions can be thought of in terms of the price level, slope, and curvatureoverstrikes. First,alloptionpricesofzeroorlowerareimmediatelyremoved. The next two restrictions regarding slope and curvature of option prices are simple results of Equations (2) and (3), combined with the fact that probabilities are defined as non-negative. Equations (2) 17 NotetheimplicitassumptionthattheprobabilityofarandomvariableZ beingequaltoanyparticularvalueisassumed to be zero. Or, lim Prob(k−ε < Z < k+ε) = 0 for any constant k. This allows equalities and inequalities to be used ε→0 interchangeably. 10

and (3) reveal that the first and second derivatives of the put pricing function must be non-negative, as the corresponding cdf and pdf are non-negative functions. Setting up an equation for the call pricing function C(K,t,T)analogoustoEquation(1)andtakingderivativesyieldssimilarlythat18 dC(K,t,T) Π (K) = erτ + 1 (8) F ,t T dK and d2C(K,t,T) π (K) = erτ . (9) F T .t dK2 Equation(8)impliesthat dC(K,t,T) −e −rτ ≤ ≤ 0. dK Therefore, the call pricing function must be non-increasing over strikes, and from Equation (9) it must havenon-negativecurvature. When option prices violate either the slope or curvature assumption, it is not obvious which option is “in the wrong” and should be removed. In any subset of prices that violates the slope or curvature assumption,Iremovetheoptionfurthestfromthemoney. Therearetworeasonsforthisselection. First, deeper out-of-the-money options are likely to be less liquid, so their prices are more likely to be stale. Second, the minimum tick size imposed on prices can be seen as introducing an error to the prices that would be observed in a continuous setting, which bears relatively more importance to deeper out-ofthe-money options with already low prices. In practice, I allow for slight deviations from the slope and curvature assumptions. This accommodates the possibility that prices are not perfectly arbitraged, and accountsforasmalldegreeofpriceinaccuracyduetodiscretepricingticks. After option prices are cleaned, the estimation procedure is only carried out if there remain at least threeoptionpricescomprisingatleastonecallandoneput. 2.6 Fixed Horizon Estimates The procedure outlined above lends itself naturally to estimating the put option pricing function for fixedhorizons,whichinturncanbeusedtogeneratefixedhorizonprobabilities. Optioncontractsallow 18 AfullderivationcanbefoundinAppendixA. 11

directlyforestimationofpdf’scorrespondingtoaspecificoptionmaturitydate. However,wemightalso be interested in estimates for a fixed horizon, for example 180 days ahead from the option pricing date. While the former type of estimate can show how expectations for interest rates at a particular point in the future evolve over time, fixed horizon estimates control for the impact of time to maturity. Pdf’s for a particular maturity date will tend to narrow with the passing of time, as more information is garnered on the likely price of the underlying futures contract at expiration. On the other hand, fixed horizon estimatesdemonstratetheevolutionofmarketexpectationsunaffectedbytimetomaturity. Fixed horizon estimates are computed by interpolating implied volatility - put delta functions across time, similar to Datta, Londono, and Ross (2014). For a given hypothetical fixed horizon maturity date, dataiscollectedfortheMarch-quarterlycontractswithexpirationsthatmostcloselyprecedeandfollow that date. Each set of options is then used to estimate a function of implied volatilities over put option deltas. Foreachof25,000evenlyspaceddeltasoverthedeltarangecorrespondingtothefixedmaturity date,thecorrespondingimpliedvolatilityisthelinearinterpolationbetweenthetwofunctions,weighted based on the fixed maturity date’s proximity to each contract’s actual maturity date. Implied volatilities anddeltasareagaintransformedbacktopricesandstrikes,fromwhichprobabilitiescanbeestimated. Fitting implied volatilities over deltas instead of strikes makes for more consistent interpolation of implied volatilities across time, as the domain covered is always (−e −rτ ,0). For a hypothetical fixed maturitydate,therangeofstrikesthatshouldbeusedforinterpolationwouldnotbeclear,asthesupport ofthepdfπ (·)overstrikesvarieswith t and T –evenforaconstantτ. F ,t T 2.7 Calculating Distributional Moments Onceapdfhasbeenestimatedfromoptionprices,computationofdistributionalmomentsprovidesuseful summary statistics, particularly when these moments are viewed over a span of time. Central moments of the distribution are computed using the traditional formulas, which are applied to the discrete case using the definition found in Equation (5) to compute integrals. Appendix A contains the continuouscase formulas referenced for calculating moments, with skewness and kurtosis being normalized by the standarddeviation(squarerootofvariance). Intheremainderofthispaper, referencestoskewnessand kurtosis are in terms of the normalized computations. Additionally, I focus on the standard deviation ratherthanvariance,asithasamorenaturalinterpretationwithunitsbeinginpercentagepoints. While distributional moments are discussed in terms of implied pdf’s on future interest rates rather than futures contact prices, moments are initially computed in regard to the futures price and are then transformed to the interest rate setting. It is straightforward to show that the following relationships hold: 12

Mean(R ) = 100 − Mean(F ) T T Variance(R ) = Variance(F ) T T Skewness(R ) = −1 × Skewness(F ) T T Kurtosis(R ) = Kurtosis(F ) T T It is worth briefly noting the potential influence that the risk-neutral assumption of this paper’s methodologymighthaveonestimateddistributionalmoments. Ifinvestorsaremoreriskaversetohigher (lower) relative to lower (higher) interest rates, then option payouts in the case of higher (lower) rates will be priced above what the actual market-assigned probabilities would imply, resulting in a higher (lower) estimated pdf mean and skewness. Similarly, if investors are relatively more risk averse to a more (less) extreme interest rate environment, estimates of kurtosis might be overstated (understated). Variance(andtherebystandarddeviation)arelikelytobeoverstatedtotheextentthatinvestorsarerisk averse generally, as this would increase the price of out-of-the-money options used as inputs, implying a wider spread of probability mass. Ivanova and Gutiérrez (2014) provide evidence that risk-neutral pdf estimatesforfutureEuriborrateslikelyoverstatethetailsofthedistributioningeneral,andparticularly theright-handtailasrelatestointerestratelevels(i.e. investorsaremoreriskaversetolargerdeviations fromtheexpectedfuturerate,andareparticularlyriskaversetohigherinterestrates). 3 Empirical Case Studies The methodology laid out above is now applied to four sample case studies. These case studies demonstratetheefficacyofsuchanapproachinilluminatingtheevolutionofmarketsentimentregardingfuture interestrates. Particularlyvolatilehistoricalcaseswithrespecttofinancialmarketconditionsareselected forstudy,asthesecasesentailimmediateandapparentchangesininterestrateexpectations. 3.1 Case 1: BNP Paribas Freezes Funds19 Paramount in triggering the 2007-08 financial crisis was heavy exposure throughout financial markets – especially the exposure carried by large and systemically important banks – to the performance of subprimehomemortgages. Asderivativeproductsincludingasset-backedsecurities(ABS)(inparticular, mortgage-backed securities (MBS)) and collateralized debt obligations (CDO) spread, the potential for 19 Newssources:“BNPParibassuspendsfunds”(2007,August9);Kingsley(2012,August6). 13

a significant market crash on the back of a spike in mortgage defaults grew. So markets were on edge when French bank BNP Paribas announced on August 9th, 2007 that it would suspend valuation for three of its ABS-based funds, claiming it was “impossible to value certain assets fairly” in those funds alongside disappearing liquidity. Indeed, the BNP Paribas announcement was a watershed moment in the development of the financial crisis; as Adam Applegarth – former CEO of the nationalized British bankNorthernRock–putit,August9th was“thedaytheworldchanged.” Figure 1: PDF Evolution, Case 1 EURLibor,Maturity=Dec. 17,2007 Figure1showsestimatedoption-impliedpdf’sforthe3-monthEuriborrateonDecember17th,2007, using options priced on August 8th and 9th, 2007. While actual and expected interest rates would continue to slide downward as the financial crisis unfolded, it is clear that the BNP Paribas announcement madeanearlycontributiontoinvestorexpectationsofmonetarypolicyeasing. Figure1revealsthatmuch of the change in interest rate expectations following the BNP Paribas announcement occurred through a movementofprobabilitymasstowardthelefttailofthedistributionandlargelyoutofthecenter,rather than a simple shift of the entire distribution. In fact, though the spot 3-month Euribor rate had been steadilytrendingupwardsincelate2005toreach4.35%byAugust8th,2007,theassignedprobabilityof falling back below 4.25% by mid-December nearly doubled from 6.84% to 12.53% in the course of one day. UncertaintyforwhereinterestrateswouldlieinDecemberincreasedaswell: thestandarddeviation widenedfrom16basispointsto18basispoints. AsimilartrendcanbeseeninexpectationsfortheDecember,2007USDLiborrateinFigure2below. Such similarity is likely reflective of the extent of financial market exposure to asset-backed securities whose returns were highly correlated. Normalized skewness and kurtosis decreased as the standard deviation jumped from 42 to 58 basis points. The mean of the distribution fell by 16 basis points from 5.12%to4.96% 14

Figure 2: PDF Evolution, Case 1 USDLibor,Maturity=Dec. 17,2007 3.2 Case 2: Lehman Brothers Files for Bankruptcy20 Of course, the bankruptcy of Lehman Brothers was a further watershed moment in the unfolding of the financial crisis. On September 15th, 2008, Lehman Brothers filed for bankruptcy following the decision by Federal Reserve and United States Treasury officials not to bail out the bank. As broader economic conditions continued to sour, the Federal Reserve would hold an unscheduled meeting weeks later and decide to resume interest rate cuts, which had seen a respite during the preceding six months. The collapseofLehmanBrothersmarksanimportantturningpointinthedepthofthefinancialcrisis. Figure 3: PDF Evolution, Case 2 USDLibor,Maturity=Dec. 15,2008 Figure 3 shows estimated pdf’s for USD Libor on December 15th, 2008, using options priced on September 12th and 15th of the same year. Though the Federal Reserve had left its target interest rate 20 Newssource:Sorkin(2008,September14). 15

unchangedatthemostrecentthreeFederalOpenMarketCommittee(FOMC)meetings,Lehman’sfailure appears to have immediately set off expectations for further monetary policy easing. Similar to the BNP Paribas case, the change in interest rate expectations resulting from Lehman was not a straightforward shift of the entire distribution, but rather movement of probability from the center of the distribution toward the left tail. The standard deviation of the distribution concomitantly increased from 43 to 57 basispoints. 3.3 Case 3: European Markets Deteriorate; ECB Responds21 Late summer 2011 was a tumultuous period for financial markets, particularly so in Europe. Broadly, global growth concerns were on the rise as composite purchasing managers indexes (PMI’s) dipped for boththeUnitedStatesandtheEuroarea. EspeciallysalientinEuropewasburgeoningskepticismoverthe ability of so-called European periphery countries – including Greece, Ireland, Italy, Portugal, and Spain – to meet creeping debt obligations alongside rising bond yields. On August 4th tensions boiled over: an ECB intervention in bond markets ignoring Spanish and Italian bonds was interpreted as signalling economic conditions in those countries so dire as to be unworthy of attempted rescue, while a letter written by European Commission president José Manuel Barroso indicated that risks were not confined to the periphery; stock markets tumbled, with the Dow Jones Euro Stoxx index down 3.75% on the day andtheS&P500down4.78%. On August 4th, the ECB announced that it would provide loans to banks at fixed rates through early 2012, while simultaneously announcing a longer-term refinancing operation. Days later, on August 7th, the ECB decided to resume bond purchases under the Securities Markets Programme (SMP), which had not been active since March 2011. The reinstatement of SMP purchases was intended to relieve the pressuresofhighinterestratesongovernmentborrowing. Figure 4 details the evolution of pdf’s for the 3-month Euribor rate on December 19th, 2011 over the most volatile few days of this episode. The evolution was not a linear one. As market pessimism came to a head on August 4th, skewness dropped from near-zero into more negative territory, reflecting greatertailprobabilityfordownwardinterestratemoves. Interestingly,uncertaintydecreasedslightlyas thestandarddeviationdroppedfrom42to39basispoints,possiblyreflectiveofastabilizingeffectfrom the ECB’s August 4th policy measures. Much of the remainder of the distribution caught up with the left tailonAugust5th asskewnessreturnedtonear-zero,whilethestandarddeviationbouncedbackfrom39 to 45 basis points. Following the revival of SMP, the mean of the distribution pushed further downward on August 8th (with a net decline of 37 basis points from August 3rd). Skewness turned positive as the 21 Newssources:Bowley(2011,August4);EuropeanCentralBank(2011,September);Rettman(2011,August4). 16

left tail of the distribution became increasingly constrained by the ZLB while the right tail remained significant. Figure 4: PDF Evolution, Case 3 EURLibor,Maturity=Dec. 19,2011 3.4 Case 4: BOE Governor Carney Says Rates May Remain Low “for some time”22 Afewmonthsinto2014,marketparticipantshadcometoexpectthattheBankofEngland(BOE)might begin increasing its target interest rate in the coming year – which would mark the first change to the BOE’s policy rate since it had bottomed out at 0.50% in March, 2009. However, when Governor Mark CarneydeliveredtheBank’sregularInflationReportonMay14th,2014,theseexpectationsweresubdued with language considerably more dovish than anticipated. Figure 5 below shows pdf’s for the 3-month GBP Libor rate on December 15th, 2014, based on option prices from May 13th and 14th. Following GovernorCarney’sremarks,thestandarddeviationfellasthedistributioncompressedbacktowardzero, while normalized skewness and kurtosis increased. In the process, the mean rate expectation dropped from80to72basispoints. 22 Newssource:“BankofEnglanddampensinterestraterisespeculation”(2014,May14). 17

Figure 5: PDF Evolution, Case 4 GBPLibor,Maturity=Dec. 15,2014 4 Cross-Rate Relationships in the Term Structure of Distributional Moments23 Usingfixedhorizonpdfestimatesatvaryinghorizons,itispossibletoviewelementsofthetermstructure ofdistributionalmomentsforfutureinterestratesacrosstime. Muchattentionhasbeenpaidtotheterm structure of interest rates, especially concerning Treasury yield curves. Examining the term structure of distributional moments around future interest rate expectations complements traditional yield curve analysisbypaintingafullerpictureofmarketviews. Further,with3-monthLiborratesastheunderlying, emphasishereisgiventoshortratesviewedoverthenearterm. Thissectioncomparessimplemeasures of the level, slope, and curvature of distributional moments on future 3-month Libor rates denominated in US dollars, euros, and UK pounds. Analysis focuses on more recent history, which is due in part to greaterconsistencyofdataavailability,aswellasanefforttofocusattentionontheperiodofhistorically lowinterestratesfollowingthefinancialcrisisof2007-08. FiguresB15throughB18inAppendixBshowtheestimatedmean,standarddeviation,(normalized) skewness, and (normalized) kurtosis for the three Libor rates at fixed horizons of 180, 360, and 540 days ahead. The series span from the beginning of 2009 through March 31st, 2016. A cursory view of these figures yields some insight into market expectations over this period, which is made more rich by considering the term structure of multiple moments in tandem. For example, an important trend in financial markets over recent years has been the divergence in interest rates between the Euro area andotheradvancedeconomiesincludingtheUnitedKingdomandtheUnitedStates. FigureB15reflects that markets have consistently expected an increasing near-future path for GBP and USD Libor rates 23 The “term structure” here is considered across varying forward horizons with the same forward terms (3 months), as opposedtobeingacrossspotrateswithvaryingterms. 18

since mid-2013. This timing coincides with the “taper tantrum”, when in May, 2013 Federal Reserve Board Chairman Ben Bernanke stated the Fed might begin slowing the pace of bond purchases under its quantitative easing program. The “taper tantrum” period did see an increased slope for projected Euribor rates, though this quickly faded, and the difference between 540-day-ahead and 180-day-ahead Euribor expectations has remained near zero since early 2014. As the slopes of mean rate expectations for GBP and USD Libor have become elevated, so too has uncertainty increased for these rates: Figure B16 shows that the standard deviation for each rate 540 days ahead has remained higher since mid- 2013. AsEuribormeanexpectationsapproached–andeventuallybreached–zerooverthesameperiod, the standard deviation generally fell as the distribution compressed, with a slight trend upward since mid-2014. Figures B17 and B18 show that skewness and kurtosis both jumped for GBP Libor in mid- 2013, especially for rates 180 days ahead, while both moments have broadly decreased across the term structure since then for both GBP and USD Libor. Overall, as mean expectations for GBP and USD Libor rateshaveincreased,thebulkofeachdistributionhasgenerallymovedupward,whiletheright-handtail has not seen a comparable shift. Skewness and kurtosis for Euribor rates have not seen a clear secular trendoverthepastfewyears. 4.1 Granger Causality across Rates ItisusefultounderstandthetimeseriesdynamicsofdistributionalmomentsacrossLiborratesforvarious components of the term structure. I examine these cross-rate dynamics by testing for Granger causality inavectorautoregression(VAR).ThegeneralVARisexpressedasfollows: Y t = C + Z 1 Y t−1 + Z 2 Y t−2 + ... + Z k Y t−k + U t , (10) whereeachY isa3-by-1vectorofthegivenmomentandtermstructurecomponentpairing(forexample, i standard deviation level 360 days ahead) for each Libor rate at time i, C is a 3-by-1 vector of constants, eachZ isa3-by-3matrixofcoefficients,U isa3-by-1vectoroferrortermsattime t,andkisthenumber i t oflagsincludedintheVAR.24 Intotal,24VAR’softhisformareestimated,correspondingtothenumber of combinations between distributional moments and term structure components being considered. The moments include mean, standard deviation, skewness, and kurtosis. The term structure components are levels at 180, 360, and 540 days ahead; the difference between levels at 360 and 180 days ahead (referredtoas“Slope1”);thedifferencebetweenlevelsat540and360daysahead(referredtoas“Slope 2”); and the difference between Slope 2 and Slope 1 (referred to as “Curvature”). Slope and curvature measurescaptureinformationregardingthetrajectoryofmomentsabstractedfromtheirlevels. 24 LaglengthisdeterminedbyminimizationofSchwarz’sBayesianInformationCriterion. 19

After a VAR is estimated, Granger causality is determined by Wald tests. Let each Y retain the i same ordering with respect to the three Libor rates. For a particular VAR, consider the test that the relevantpairingofmomentandtermstructurecomponentforthesecond-positionedrateGrangercauses that for the first-positioned rate (e.g. 540-day-ahead kurtosis for Euribor Granger causes that for GBP Libor). This is equivalent to testing whether the entry in the first row, second column of each Z matrix i of coefficients is equal to zero, with the null hypothesis being that all such entries are equal to zero. A rejectionofthenullhypothesisimpliesGrangercausality,meaningpreviousvaluesoftherelevantpairing forthesecond-positionedratecontainexplanatorypowerforthecurrentvalueofthepairingforthefirstpositionedrate,controllingforpreviousvaluesofthepairingforthefirst-andthird-positionedrates(e.g. past values of 540-day-ahead kurtosis for Euribor contain explanatory power for the current value of 540-day-aheadkurtosisforGBPLibor,controllingforpreviousvaluesof540-day-aheadkurtosisforGBP andUSDLibor). SuchtestsarerevealingofthedirectionsinwhichexpectationsfordifferentLiborrates mayaffectoneanotheracrossvariousdistributionalandtermstructurecharacteristics. Itisworthnoting thatGrangercausalityisnotthesameasformalcausality,butshouldbeinterpretedsimplyaspastvalues ofonevariablecontainingresidualexplanatorypowerforthecurrentvalueofanothervariable. WhentestingforGrangercausalityinaVARwithpossiblyintegratedtimeseries,TodaandYamamoto (1995) demonstrate that the distribution under the null hypothesis is non-standard. However, they introduce a simple solution: after determining the lag length k of the VAR by use of an information criterion, one estimates a VAR using lag length k+d with d being the maximum order of integration, then performs Wald tests for Granger causality using k lags of explanatory variables. This approach resolves asymptotic inference. Relevant to this paper, augmented Dickey-Fuller (ADF) tests fail to reject thenullhypothesisofnonstationarityinmanyofthetimeseriesconsidered. Asaresult,testsforGranger causalityareperformedbothinthetraditionalmanner(theVARstillbeingestimatedwithklags)aswell asusingTodaandYamamoto’s(1995)methodwhereapplicable. Tables B2 and B3 in Appendix B synthesize the results of all Granger causality tests. The sample period begins on May 7th, 2009 (the date on which the European Central Bank cut its main policy rate target to 1%, rounding out the series of rate cuts in the United States, United Kingdom, and the Euro area following the financial crisis onset) and continues through March 31st, 2016. The results in Table B2 use the traditional approach to testing for Granger causality. In Table B3, Toda and Yamamoto’s (1995) method (with d =1) is used for testing Granger causality in all VAR’s in which an ADF test fails to reject the null hypothesis of nonstationarity in at least one of the time series at the 5% confidence level, and uses the traditional approach in remaining VAR’s. The two tables serve as a robustness check on one another, as well as a basic indication of the extent to which nonstationarity might be affecting the results of traditional Granger causality tests in this case. VAR’s are set up as in Equation (10), and 20

resultingbivariateGrangercausalityresultsarereorganizedinbothtablestoeasereadingofrelationships betweenpairsofLiborrates. An“x”denotesGrangercausalitysignificantatthe5%confidencelevel. As a concrete example, the “x” under the GBP-180 column and the EUR-Mean row in either table denotes thatinaVARof180-day-aheadmeanratesamongallthreeLiborrates,theEuribormeanGrangercauses the GBP Libor mean at a 5% significance level under the relevant testing procedure. The sections of each table corresponding to tests of autocorrelation are left blank, as all series are autocorrelated. The exception is the EUR-EUR section in Table B3, which is used as a marker to denote which results come from VAR’s using Toda and Yamamoto’s (1995) method; a blue “O” signifies that Granger causality tests for the corresponding pairing of distributional moment and term structure component are performed usingTodaandYamamoto’s(1995)approach. InoteheresomebroadtakeawaysfromTablesB2andB3inanattempttosummarizetheresults;this isnotmeanttobecomprehensive,butrathertodrawattentiontosomeimportantpatternsthatemerge. The most basic trend to note in Tables B2 and B3 is the striking overall degree of interconnectedness among the three Libor rates, present across varying distributional moments and term structure components. It is immediately clear that – in addition to levels of mean expectations – combinations of higher distributionalmomentsandsimplemeasuresofthebroadertermstructurebearrelationshipsacrossrates as well. In other words, the cross-rate dynamics are not thoroughly captured by simply examining the levels of future mean expectations; there is a rich interplay among rates involving standard deviation, skewness,andkurtosis,aswellasmeasuresoftheslopeandcurvatureoffuturepaths. USD Libor expectations contain the most consistent explanatory power across moments and term structure components, particularly so in relation to GBP Libor rates. USD Libor expectations are found to Granger cause GBP Libor expectations across both Tables B2 and B3 in 20 of 24 combinations of distributionalmomentandtermstructurecomponent,withtheonlyexceptionsbeinglevelsofskewness and kurtosis 540 days ahead, Slope 2 for kurtosis, and Curvature for skewness. Similarly, the mean and standarddeviationofUSDLiborexpectationsGrangercausethoseofEuriborexpectationsacrossallterm structure components except the Curvature of standard deviation; USD Libor contains little explanatory powerforEuriborintermsofskewnessandkurtosis,however. Thereareacouplepotentialexplanations for the relatively great importance of USD Libor rate expectations to those for EUR and GBP Libor. It is certainly possible that USD-denominated interest rates hold greater causal sway relative to alternative interest rates; for example, Brusa, Savor, and Wilson (2016) find that Federal Reserve policy decision announcements bear an outsized influence across global stock market risk premia, which is not true for any other central bank they examine. It may also be true, however, that USD Libor expectations simply serveasastrongerproxyforomittedmacrovariables,whichinturnarelinkedcausallytoEURandGBP Libor expectations. Distinguishing between these explanations is beyond the scope of this paper, though 21

theevidenceofBrusa,Savor,andWilson(2016)seemstoimplytheformerexplanationholdswater. WhileUSDLiborrateexpectationsholdmuchexplanatorypowerforEURandGBPLiborexpectations, the converse is not true. In fact, Euribor expectations are statistically important precursors of USD Libor expectations in only 2 out of 24 total pairings of moment and term structure component when applying Toda and Yamamoto’s (1995) testing procedure. Similarly, while the levels of GBP Libor mean expectations consistently Granger cause their USD Libor counterparts, higher distributional moments of GBPLiborexpectationsbearlittleimportancetoUSDLiborwhenfollowingTodaandYamamoto(1995). On the whole, higher moments than the mean of USD Libor expectations are consistently not found to beGrangercausedbythoseofeitherEURorGBPLiborrates,whileitisofcourseimportanttonotethat levelsofmeanexpectationsforGBPLibordoGrangercauseUSDLibormeanexpectations. EURandGBPLiborexpectationsshareclearrelationshipsrunninginbothdirections,especiallysofor means and standard deviations. The levels of mean expectations for GBP Libor Granger cause those for Euribor across fixed horizons of 180, 360, and 540 days ahead. Additionally, while levels of GBP Libor standard deviations are not very consistently related to those for Euribor, the slopes and curvature of standard deviations over time are; so, while the actual levels of uncertainty in GBP Libor rates are less reliably linked to levels of uncertainty in Euribor, the path at which GBP Libor uncertainty is evolving does help predict the path of Euribor uncertainty. In the opposite direction, the standard deviation of Euribor expectations Granger causes that of GBP Libor expectations across term structure components, withtheexceptionofthe180-day-aheadlevel. MeanexpectationsforEuriborhavelittlepredictivepower forGBPLiborexpectationsacrosstermstructurecomponents. 5 Conclusion ThispaperusesoptionsonLiborfuturesforratesdenominatedinEUR,GBP,andUSDinordertoestimate risk-neutral market-assigned pdf’s for future interest rates. The methodology is reviewed, and estimates areappliedtoproducecasestudiesofprobabilityestimatesastheyevolveoverimportantmarketevents. DistributionalmomentsarecomparedacrossratestostudytheirinterrelationshipsinaGrangercausality framework. Thecasestudiescarriedoutinthispaperelucidatethesignificantimpactofspecificmarketeventson market expectations for future interest rates by detailing precisely how expectations changed along the full probability distribution. For example, while the mean expectation for USD-denominated Libor fell upon news of the collapse of Lehman Brothers in September, 2008, the probability distribution did not undergoasimplelevelshift;rather,probabilitymovedtowardthelefttailandprimarilyoutofthecenter of the distribution, resulting not only in a lower mean expectation but also a higher standard deviation. 22

The evolution of expectations for GBP-denominated Libor rates as BOE Governor Mark Carney stated interest rates might remain low “for some time” provides evidence on the critical role of statements by centralbankauthoritiesinshapingmarketbeliefs. Granger causality tests considering the future path of distributional moments across the three Libor ratesdemonstratethelargeextenttowhichexpectationsforthethreeratesareinterrelated. Thedynamics among mean expectations for future interest rates do not fully capture the rich interconnectedness among broader probabilistic expectations for these rates. I find generally that expectations for USDdenominatedLiborratesbearthemostconsistentpredictivepoweracrosscombinationsofdistributional moment and term structure component for EUR and GBP Libor, while EUR and GBP Libor expectations do not provide much consistent explanatory power for USD Libor. USD Libor holds consistent explanatory power for GBP Libor across mean, standard deviation, skewness, and kurtosis, while being linked to Euribor primarily in mean and standard deviation. GBP Libor holds predictive power for mean levels as wellas theslope andcurvature ofstandard deviationin Euribor expectations, while Euriborstandard deviation Granger causes that of GBP Libor across levels (excluding the shortest horizon of 180 days), slope,andcurvature. FurtherresearchmightbuildupontheGrangercausalityresultsestablishedherewithestimatesofthe magnitude of various relationships, for example in a multivariate vector error correction model among cointegrated series. Additionally, it would be useful to study the relationships of the distributional momenttimeseriesusedinthispaperwithothereconomicvariablesofinterest,includingequityindexesor exchange rates. Time series of distributional moments might be derived from calibrated real-world (i.e. not risk-neutral) probability densities as well, as detailed in Ivanova and Gutiérrez (2014); it would be instructive to test whether results in this paper are sensitive to the assumption of risk-neutrality. Finally, furtheranalysismightviewtherelationshipsamongLiborrateexpectations(andpossiblyothereconomic variables)astheyhaveevolvedovertime. 23

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Appendix A: Technical Appendix Black (1976) Model Conversion between prices and implied volatilities occurs through the Black (1976) model for pricing optionsonfuturescontracts. Foraputoption,theBlack(1976)priceisequalto P =e −rτ(cid:0) KΦ(−d )−FΦ(−d )(cid:1) , 2 1 where ln (cid:128)F(cid:138) + σ2τ d = K (cid:112) 2 1 σ τ and ln (cid:128)F(cid:138) − σ2τ (cid:112) d = K (cid:112) 2 =d −σ τ. 2 σ τ 1 Adding to prior notation, Φ(·) is the cumulative distribution of the standard normal, F is the futures contractprice,andσ istheimpliedvolatility. Relating Deltas and Strikes Aputoption’sdelta,∆ ,denotesthesensitivityoftheoptionpricetoachangeintheunderlyingfutures P rate,or ∂P ∆ = =e −rτ(cid:0)Φ(cid:0) d (cid:1)−1 (cid:1) . P ∂F 1 Fromthis,onecansolveforthestrikeofaputoptiongivenitsdeltaasthefollowing: K =Fe α , where σ2 (cid:112) α= τ−(σ τ)Φ−1(erτ∆ +1). P 2 Theaboveformulamakescleartheboundson∆ . Theinverseofthestandardnormalcdf,Φ(·),can P only be evaluated on (0,1). Therefore, it must be that 0 < (erτ∆ + 1) < 1, or −e −rτ < ∆ < 0. P P Additionally,itisthecasethat 26

lim K =∞, ∆ →−e−rτ+ P and lim K =0. ∆ →0− P Because of this, when evaluating implied volatilities as a function of deltas I place the end points for evaluationasmallvalueabove−e −rτ andasmallvaluebelow0. Vega An option’s vega, v, is the sensitivity of the option price to a change in implied volatility. Specifically, in thecaseofaputoption, ∂P (cid:112) v = =Fe −rτφ(d ) τ, P ∂σ 1 whereφ(·)isthestandardnormaldensityfunction. Proof of Equations (8) and (9) Asinthecaseofputoptions,webeginbydescribingthecalloptionpricingfunctionas C(K,t,T) = e −rτ E [max(F −K,0)] t T (cid:90) ∞ = e −rτ (f −K)π (f)df . F ,t T K Then, taking the derivative with respect to K using Leibniz’s rule for differentiating under an integral yields dC(K,t,T) (cid:90) ∞ = −e −rτ π (f)df F ,t dK T K = −e −rτ(1−Π (K)) F ,t T = e −rτ(Π (K)−1). F ,t T SolvingforΠ (K), F ,t T 27

dC(K,t,T) Π (K) = erτ +1, F ,t T dK which is Equation (8). It is then straightforward that taking the derivative of Equation (8) with respect tostrike K yieldsEquation(9). Distributional Moment Formulas Continuous-case references used for calculating distributional moments are listed below. In practice, integrals are computed using the discrete-case equivalent via Equation (5) applied over all option strikes evaluatedinthecourseofinterpolatingandextrapolatingoptionstrikesandprices. Below,µ isequiva- X lenttoMean(X),andσ isthestandarddeviationofX,orthesquarerootofVariance(X). X (cid:90) Mean(X): E[X] = t π (t)dt X X (cid:90) (cid:148) (cid:151) Variance(X): E (X −µ )2 = (t−µ )2π (t)dt X X X X (cid:150)(cid:18)X −µ (cid:19)3(cid:153) 1 (cid:90) Skewness(X): E X = (t−µ )3π (t)dt σ σ3 X X X X X (cid:150)(cid:18)X −µ (cid:19)4(cid:153) 1 (cid:90) Kurtosis(X): E X = (t−µ )4π (t)dt σ σ4 X X X X X 28

Appendix B: Figures and Tables Figure B1: Eurodollar Options Aggregate Open Interest Figure B2: Eurodollar Options Aggregate Volume 30-daylaggedmovingaverage 29

Figure B3: Eurodollar Options Aggregate Open Interest Bytimetomaturity Figure B4: Eurodollar Options Aggregate Volume Bytimetomaturity 30

Figure B5: Eurodollar Options Trading Bymonthofmaturity 31

Figure B6: PDF using Cross Validation for Smoothing Parameter - Short Sterling Maturity=December15,2010 Figure B7: PDF using Cross Validation for Smoothing Parameter - Euribor Maturity=December13,2010 Figure B8: PDF using Cross Validation for Smoothing Parameter - Eurodollar Maturity=December13,2010 32

Figure B9: PDF and Pricing Errors - Short Sterling Maturity=December15,2010 Figure B10: PDF and Pricing Errors - Euribor Maturity=December13,2010 Figure B11: PDF and Pricing Errors - Eurodollar Maturity=December13,2010 33

Figure B12: PDF and Pricing Errors - Short Sterling Maturity=December17,2014 Figure B13: PDF and Pricing Errors - Euribor Maturity=December15,2014 Figure B14: PDF and Pricing Errors - Eurodollar Maturity=December15,2014 34

Figure B15: Option-Implied Mean, by Days-to-Maturity 35

Figure B16: Option-Implied Standard Deviation, by Days-to-Maturity 36

Figure B17: Option-Implied Skewness, by Days-to-Maturity 37

Figure B18: Option-Implied Kurtosis, by Days-to-Maturity 38

Table B1: Maturity Date Conventions for March-Quarterly Futures and Options ShortSterling ThirdWednesdayofthemonth Euribor TwobusinessdaysbeforethethirdWednesdayofthemonth Eurodollar TwoLondonbankbusinessdaysbeforethethirdWednesdayofthemonth Sources:IntercontinentalExchangeandCMEGroup.ContractspecificationsforShortSterlingandEuribor futuresandoptionscanbefoundonwww.theice.com,andforEurodollarfuturesandoptionson www.cmegroup.com. 39

ecnacfiingiS ytilasuaC regnarG ycnerruC-ssorC robiL M3 :2B elbaT DSU PBG RUE :etaRtnednepeD evruC 2epolS 1epolS 045 063 081 evruC 2epolS 1epolS 045 063 081 evruC 2epolS 1epolS 045 063 081 x . . x x . x . . . . x naeM RUE . . . . . x x x x x x . veDdtS . . . . . . x . x . . . ssenwekS . . . . . . . . x . . x sisotruK . x . x x x x x . x x x naeM PBG x x . x x x x x x . x x veDdtS . . . x x . x . x . . . ssenwekS . . . x x . x . x . x . sisotruK x x x x x x x x x x x x naeM DSU x x x x x x . x x x x x veDdtS . x x x x x x . x . . . ssenwekS x . x x x x . . . . . . sisotruK eerhttsrfieht,gnipuorgetartnednepedhcaenihtiW.6102,13hcraMsdnedna9002,7yaMsnigebdoirepelpmaS.noretircnoitamrofninaiseyaBs’zrawhcSybdetcelesshtgnelgalhtiw,sRAVmorfdetamitsE:setoN 2epolSsi”evruC“dna;ylevitcepser,daehasyad063.sv045dna,daehasyad081.sv063taslevelnisecnereffidehtetoned”2epolS“dna”1epolS“;daehasyad045dna,063,081taslevelotdnopserrocsnmuloc ,)daehasyad081slevelssenweks,elpmaxerof(tnenopmocerutcurtsmretdnatnemomlanoitubirtsidfogniriaphcaerofnurerasRAV.levelecnedfinoc%5ehttaecnacfiingislacitsitatsskram”x“nA.1epolSsunim .gnidaeresaeotknalbtfelerasnoitcesgnidnopserroceht;detalerrocotuaeraseiresllA.noitatneserprofelbatsihtnidezinagroerspihsnoitaleretairavibgnitluserhtiw 40

dohtem )5991( s’otomamaY dna adoT gnisu ,ecnacfiingiS ytilasuaC regnarG ycnerruC-ssorC robiL M3 :3B elbaT DSU PBG RUE :etaRtnednepeD evruC 2epolS 1epolS 045 063 081 evruC 2epolS 1epolS 045 063 081 evruC 2epolS 1epolS 045 063 081 x . . . . . x . . . . x O O O O O naeM RUE . . . . . x x x x x x . O O O O O veDdtS . . . . . . x . x . x x O O O ssenwekS . . . . . . . . x . . x O O sisotruK . . . x x x x x . x x x naeM PBG x . . . . x x x x . . x veDdtS . . . . . . x . x x . . ssenwekS . . . . . . x . x . . . sisotruK x x x x x x x x x x x x naeM DSU x x x x x x . x x x x x veDdtS . x x . x x x . x . . . ssenwekS x . x . x x . . . . . . sisotruK eerhttsrfieht,gnipuorgetartnednepedhcaenihtiW.6102,13hcraMsdnedna9002,7yaMsnigebdoirepelpmaS.noretircnoitamrofninaiseyaBs’zrawhcSybdetcelesshtgnelgalhtiw,sRAVmorfdetamitsE:setoN 2epolSsi”evruC“dna;ylevitcepser,daehasyad063.sv045dna,daehasyad081.sv063taslevelnisecnereffidehtetoned”2epolS“dna”1epolS“;daehasyad045dna,063,081taslevelotdnopserrocsnmuloc ,)daehasyad081slevelssenweks,elpmaxerof(tnenopmocerutcurtsmretdnatnemomlanoitubirtsidfogniriaphcaerofnurerasRAV.levelecnedfinoc%5ehttaecnacfiingislacitsitatsskram”x“nA.1epolSsunim otdesusinoitcesRUE-RUEehtni”O“eulbA.gnidaeresaeotknalbtfelerasnoitcesgnidnopserroceht;detalerrocotuaeraseiresllA.noitatneserprofelbatsihtnidezinagroerspihsnoitaleretairavibgnitluserhtiw ytilasuacregnarGeht,sesacesehtfohcaeni;ycnerrucenotsaeltaroflevelecnedfinoc%5ehttatoortinuafosisehtopyhllunehttcejerotdeliaftsetrelluF-yekciDdetnemguAnahcihwniRAVayfingis .enofonoitargetniforedromumixamagnimussa,)5991(otomamaYdnaadoTfohcaorppaehtyolpmeelbatsihtnidetroperstluserecnacfiingis 41

Appendix C: The American Early Exercise Premium in Eurodollar Option Prices The American option early exercise premium is the premium paid for an American option compared against an equivalent European option. The only difference in value between the American and the European option in this case is the fact that the American option can be exercised at any time prior to theexpirationdate,whiletheEuropeanoptioncanonlybeexerciseduponexpiration. Barone-Adesiand Whaley(1987)deriveamethodwhichcanbeusedtoestimatetheearlyexercisepremiumembeddedin the price of an American option. This method, however, relies on the assumptions of the Black-Scholes option pricing model, which include the assumption of geometric Brownian motion in the underlying security and thereby a lognormal terminal distribution. The approach outlined in this paper does not makesuchanassumption. MelickandThomas(1997)demonstratethattheupperboundonanyAmericanoptionearlyexercise premium (considered multiplicatively) is equal to erτ . The intuition is that the greatest early exercise premium comes for an option which will be exercised as soon as possible with near certainty; and, for suchanoption,thepriceissimplytheundiscountedexpectedpayout(comparethisagainstEquation(1), in which the price of a European option is the discounted expected payout). Considering the analysis in Section4,themaximumUSD-denominatedrisk-freerateusedfordiscountingat540daysaheadoverthe sampleperiodfromMay7th,2009toMarch31st,2016isjustover1%. Then,themaximumupperbound onthedifferencebetweenaEuropeanandAmericanoptionpriceisabout1.5%. Themultiplicativeearly exercisepremiumwillbesmallerasoptionsarelessin-the-money,orinfactareout-of-the-money. Inthis paper, I use only out-of-the-money option prices as model inputs. As the value of an out-of-the-money option if exercised today is zero, the American price will certainly involve some degree of discounting based on likely time to exercise. Therefore, the maximum daily upper bound on the early exercise premiumforEurodollaroptionsusedintheanalysisofSection4iscertainlylessthan1.5%. While the relatively small bound on the early exercise premium points to a small impact on pdf’s estimated from Eurodollar option prices, it ignores the interaction of option prices with the actual estimation procedure. I therefore use Monte Carlo simulation to better understand the potential effect of the American early exercise premium in Eurodollar option prices on estimated pdf’s. A set of European optionpricesforagivenpricingandmaturitydatecanbeestimatedbyassumingabasicfunctionalform for the early exercise premium and shocking observed American option prices based on the premium. Characteristics of the pdf’s estimated from each set of option prices can then be compared in order to inferthedegreetowhichtheearlyexercisepremiummightbeaffectingimpliedpdf’s. In estimating European option prices, I work with the generous assumption that the daily upper 42

bound for the early exercise premium of erτ is effective at-the-money. From here, the multiplicative early exercise premium is linearly phased out over option strikes to reach 1 where the estimated cdf reaches2.5%forputsor97.5%forcalls(i.e. theearlyexercisepremiumisassumedtobezeroforout-ofthe-moneyoptionswherelessthan2.5%probabilityisassignedtoapositiveoptionvalueatexpiration). EuropeanoptionpricesarethenestimatedbymultiplyingobservedAmericanoptionpricesbytheinverse ofthecorrespondingpremium. I select all Wednesdays over the sample period from May 7th, 2009 to March 31st, 2016 for the simulationoptionpricingdates,andIestimatefixedhorizonpdf’s540daysaheadasdescribedinSection 2.6 using both observed option prices and simulated European option prices. I then calculate the mean, standard deviation, (normalized) skewness, and (normalized) kurtosis of each pdf. Figure C1 shows histograms of the percent differences in pdf moments, where positive (negative) values imply a higher (lower)valueusingestimatedEuropeanoptionprices. Figure C1: Percent Differences in PDF Moments after Option Price Shocks Figure C1 shows that the overall effects of even a generous specification for the early exercise premiumareminor. Differencesinthepdfmeanarenegligible,whiledifferencesinhighermomentsareall 43

within small margins. It is not surprising that the standard deviation is consistently lower when using estimated European option prices, or that the normalized skewness and kurtosis are consistently higher. Because European option prices will be lower than American prices, and because out-of-the-money options are used as inputs, estimated probabilities will be less dispersed in order to match lower out-ofthe-money option prices. Further, as the early exercise premium falls for increasingly out-of-the-money options, simulated price shocks will push prices increasingly downward nearer the money, implying less probability mass near the center of the distribution relative to the tails in the case of hypothetical European options. Finally, as skewness is normalized, a lower standard deviation will result in higher skewness even if there is little to no change in non-normalized skewness. Table C1 below shows the averageandmedianpercentdifferencesbetweenmomentsestimatedfromhypotheticalEuropeanprices andobservedAmericanprices,complementingFigureC1. Table C1: Percent Differences in PDF Moments after Option Price Shocks Mean StdDev Skewness Kurtosis AverageDifference 7.24e-06 -0.20 0.35 0.63 MedianDifference 0.00 -0.17 0.29 0.53 Note:“Difference”isthepercentdifferencebetweenagivendistributionalmomentasestimatedfrom hypotheticalEuropeanoptionpricesorobservedAmericanoptionprices. 44