QUANT PERSPECTIVES

Interest Rate Variance Swaps and the Pricing of B Y ANTONIO MELE A ND Y OSHIKI O BAYA SHI

ne of the pillars supporting the recent movement toward standardized measurement and trading of volatility (see http://www.garp.org/ and explicitly account for the distinct characteristics of each risk-news-and-resources/2013/december/a-push- Oto-standardize-interest-rate-volatility-trading.aspx) - - atility pricing. The meaning of this mouthful is best understood plied volatility of forward rates. This article provides an overview of how volatility pricing and indexing methodologies - - - Government Bonds of spanning variance swap payoffs with those of options on the same underlying. challenging, mainly because of the high dimensionality of The price of volatility derived in this framework carries a - clean and intuitive interpretation as the fair market value of ward volatility of a three-month future on the 10-year Treasury note, and that available for trading are American-style options naturally lends itself as the basis of a benchmark index for desired volatility in a strictly model-free fashion? income market and serves as the underlying for standardized The practical answer depends on the magnitude of two is- futures and options contracts for volatility trading. A model-free options-based volatility pricing methodology American prices. A second relates to the mismatch in maturity between the options and the underlying futures — i.e., prices and exchange rates) was branded and popularized as the one-month options are used to span risks generated by three- - - - Fortunately, in practice, situations arise in which the model- ogy has been carried over to other markets, such as those for dependent components may be presumed small enough, such gold, oil and single stocks. - In contrast, to create analogous volatility indexes for the -

www.garp.org MARCH 2014 RISK PROFESSIONAL 1 valuevalue of of a a variance variance swap, swap,value andvalue and to to of of know know a a variance variance when when swap,the swap,the approximation approximation and and to to know know iswhen iswhen of of a a the thetolerable tolerable approximation approximation magnitude magnitude is is givenof givenof a a tolerable tolerable the the magnitude magnitude given given the the context.context. context.context. Let Ft (S, T) be the forwardLet priceF (S, at )t, be for the delivery forward at priceS, of ata coupont, for delivery bearing at bondS, of expiring a coupon at bearingT, bond expiring at , Let Ft (S, T) be the forwardLet priceFtt(S, atTT)t, be for the delivery forward at priceS, of ata coupont, for delivery bearing at bondS, of expiring a coupon at bearingT, bond expiring at TT, Bt(T) BBt((T)) with t S T,i.e.valueF oft (S, awith varianceT)=t Bt(ST swap,) , where,i.e. andP toFtt(( knowSS,) is)= thewhent priceT the, where at approximationt ofP at ( zeroS) is coupon theis of price a bond tolerable at expiringt of magnitude a zero coupon given bond the expiring QUANTwith t S PERSPECTIVEST,i.e. Ft (S,withT)=t PSt(S), whereTT,i.e.PFtt((SS,)T isT)= thePP pricet((SS)), where at t ofP at ( zeroS) is coupon the price bond at expiringt of a zero coupon bond expiring ≤≤ ≤≤ ≤≤Pt(S)≤≤ t at S t, and Btcontext.(T) is theat priceS t at, andt ofB the( underlying) is the price bond. at t of the underlying bond. at S ≥t, and Bt (T) is theat priceS t at, andt ofB thett(TT underlying) is the price bond. at t of the underlying bond. ≥ value of a variance≥ swap,≥ and to know when the approximation is of a tolerable magnitude given the Let rt be the instantaneousLet Ft (S,LetT) short-termr bebe the the forward instantaneous rateprocess price at andt short-term, for let deliveryQ be rate the at processS risk-neutral, of a coupon and let probability. bearingQ be the bond It risk-neutral expiring probability. at T, It Let rt be the instantaneousLet short-termrtt be the instantaneous rate process and short-term let Q be rate the process risk-neutral and let probability.Q be the It risk-neutral probability. It context. Bt(T) is well-known (see,with e.g.,t Mele,isS well-knownT 2013,,i.e. ChapterFt (see,(S, T e.g.,)= 12)P that Mele,(S) , in where 2013, a diff ChapterPusiont (S) setting, is 12) the that priceFt ( inS, at aTt) diof satisfiesffusion a zero setting, couponF bondt (S, expiring) satisfies is well-known (see,Let e.g.,Ft (≤S, Mele,isT) well-known be≤ 2013,the forward Chapter (see, price at e.g., 12)t, for that Mele,t delivery in 2013, a at diSff, Chapterusion of a coupon setting, 12) bearing thatFt bond( inS, aT expiring) di satisfiesffusion at T, setting, Ft (S, TT) satisfies at S t, and B ( ) is theBt(T) price at t of the underlying bond. with t S T,i.e. Ft (S,TT)= , where Pt (S) is the price at t of a zero coupon bond expiring ≤≥ ≤ dF (S, ) Pt(S) at S t, and BdF( )τ( isS, theT) price at t of the underlyingdFdFττ((S,S, bond.TT)) Let rt bet Tτ the instantaneousT = vτ (S, T) short-termdWF S (τ) rate,=τ processvτ((t,S, S) and), dW let SQ(τbe) , theτ risk-neutral(t, S)(1), probability. It (1) ≥ = vτ (S, T) dWF S (τ) ,=τvτ((t,S, ST)T), dWFFS (τ) ,τ (t, S)(1), (1) Let rt be theF instantaneousτ (S, T) short-term rate· processFFτ((S,S, andT)) let Q∈be the risk-neutral·· probability.∈∈ It is well-knownFτ ( (see,S, T) e.g., Mele, 2013,· Chapterτ T 12)∈ that in a diffusion setting, Ft (S, T) satisfies is well-known (see, e.g., Mele, 2013, Chapter 12) that in a diffusion setting, Ft (S, T) satisfies ter 4) provide further details on how onewhere mayv τestimate(S, T) is these the instantaneous towhere WFs (v␶(S, volatility) is the process instantaneous adapted volatility to WF S ( processτ), a multidimensional adaptedS- to W S Brow-(τ), a multidimensional Brow- where vτ (S, ) is the instantaneouswhere vττ(S, volatilityTT) is the process instantaneous adapted volatility to W S ( processτ), a multidimensional adapted to WFFS Brow-(τ), a multidimensional Brow- T dFτ (S, T)dFτ (S, T) F approximation errors. For example, a numerical experiment forward probability, =Qvτs(S,S ) dW= vSτ((τS,) ,τ) dW(t, S) S, (τ) ,τ (t, S) , (1) (1) nian motion under the Snian-forwardnian motion motion probability, under under the theFQSFS-forward-forwardT, definedF probability, probability, throughT theFQQ Radon-NikodymFSS,, defined defined through through , the the Radon-Nikodym Radon-Nikodym derivative, derivative, nian motion under the S-forward probability,Fτ (S, T) FτQ(FS,S T, defined)· through∈· the Radon-NikodymF ∈ derivative, based on Vasicek (1977) model of short-termas follows: interest rate dy- derivative, as follows: asas follows: follows: S as follows: SS where vτ (S, T) is the instantaneous volatility processSt adaptedrτ dτ to WF S (τ), a multidimensionalrτ dτ Brow- dQ S e− rτ dτ S tt rτ dτS where vτ (S, T) is the instantaneousFS volatilityt processdQdQFS adaptedee−− to WF (τ), a multidimensional Brow- nian motion under the S-forwarddQ probability,F =Qe−S , defined, throughF the Radon-Nikodym derivative, (2) that (1) the early premium embedded in American op- = F ! , == ! (2) ,, (2) (2)(2) dQ P!t (S) SdQ P! (S) asnian follows: motion under the SdQ-forwardS probability,P (S) QFdQ, definedS throughPtt(S) the Radon-Nikodym derivative, !GS t S ! S ! rτ dτ !GG !G S t ! as follows: !dQ! F e− ! and S denotes the informationand S setdenotes at time theS information. = ! set at, time S!!. S (2) and S denotes the informationand S setdenotesdenotes at time the the Sinformation! information.!dQ setPt set( Sat) attime time S. S.! rτ dτ G GG ! ! S dQ S e ! t maturity mismatch is negligible when the differenceConsiderG in the maturi following- payoConsiderffof a variance the following swap:!G payoffFof a variance− swap: Consider theand followingdenotes payo theConsider informationffof a variance the set following at time swap:S!. payo ffof a variance= ! swap:, (2) S ! dQ Pt (S) ties is as small as two months, as in the example above. G ! S Consider the following payoffof a variance swap: !G and S denotesπ (T,T the) informationVt (T,S,T) setPππ at((t,(T,T, time T,T S,)) ST!.)VV,Tt((T,S,T,S,TS,)) P((t,t, T, T, S, S, T)),T,TS,S, π (T,T) ≡Vt (T,S,T) −P (t, T,T S, T≡!) ,Tt ≤TS, −P T ≤ G ≡π (T,T) Vt (T,S,−T) P (t, T,≡ S, T) ,T≤S, − ≤ Maturity and Numéraires Mismatches Consider the following≡ payoffof a− variance swap:≤ T 2 T To illustrate the maturity mismatch issue,where supposeV (thatT,S, available) T v (S, ) 2 dτ is the percentageT integrated22 variance, and the fair value of the t T t wherewhereτ TV TV t t ( ( T,S,T,S, T )2 ) t v v ττ ( (S, S, T ) ) d d τ τ is is theis the the percentage percentage percentage integrated integrated in- variance, variance, and and the the fair fair value value of of the the where Vt (T,S,Twhere) ≡Vt (T,S,∥vτT()S, Tt) ∥vτd(τS,isTT) thedτ percentageist the percentageT integrated integrated variance, variance, and and the the fair value fair ofvalue the of the for trading are European-style options expiring at T on a≡ 10-t ∥ tegrated≡ ∥ ∥variance,∥ ≡and≡ the∥∥ fair value∥∥ of the strike (t,T,S, ) is strikestrike P((t,t, T, T, T)isstrike)is P"(t, T,strikeTstrike)is PP((t,t, T, T,TT)is)isπ (T,T) Vt (T,S,T) P (t, T, S, T) ,T S, year Treasury note forward expiring at S T. PThe optionT span" " "" ≡ − ≤ T 1 T Q TT operates under the so-called T-forward probability, whereas the 1 T T t 2rτ dτ 1 QF TT Q where V (PT,S,(t, T, S,1)T)= v E(tS,er−τ )dτ dτVist (T,S,1 theT percentage) = EQt F (Vrt integratedτ(dT,S,τ T)) , (3) variance,(3)Q andFTT the fair value of the (t, T, S,t )= T ttPt (eT−()τt, T,rτT S,dτ V)=t (T,S, ) =e− FttT r(τVdtτ(VT,S,(T,S,)) , ) = F ((3)V (T,S, )) , (3) P(t, T, S, T)= ≡ E Pe∥P−(t,t T,# S, T∥TV)=(T,S, T)EEt$t=e−Et (V (VT,S,tt(T,S,T))T,T) = EEtt ((3)Vtt(T,S,TT)) , (3) P - T Pt (T )Et " t PPt((TT)) Et t T strike P (t, T, PTt)is(T ) # " t $ ## " $$ ability, which generates risks in the that wherecannot the second equality follows" # by" a change of probability, E$t denotes" conditional expectation under Q where the secondthe equality risk-neutral follows probability, by a and changeF T ofunder probability, the T -forwarddenotes probability. conditional That is, the expectation fair value of under be spanned by the set of all available options.where theUnless second T=S equality, we wherewhere follows denotes the the by secondconditional second a changeEt equality equality of expectation probability, follows follows byunder byET at adenotes change change the risk-neutral of conditional of probability, probability, prob expectation-Ettdenotesdenotes under conditional conditional expectation expectation under under t Q 1 Et QE T the variance swap is the expectedF T realized variance underQ theQFrTTτ-forwarddτ probability. It isF a notable (t,Q T,TQ S,FT )= e− tF T V (T,S, ) = (V (T,S, )) , (3) the risk-neutral probability,theability,the risk-neutral and risk-neutral PandEt F underTunder probability, probability, thethe TT-forwardE-forward andt and Et probability.probability.underundert the the That ThatTT-forward-forward is, is, theEt the probability. probability. fairt valueT of That That is, is, the the fair fair value value of of the risk-neutralpoint probability, of departure and fromE thet standardt under equity thePt (T caseT)-forward in whichEt probability. the fair value of That a variance is, the swap fair is the value of This issue is reminiscent of convexitythe problems variance arising swap is in the expected realized variance under# the" T -forward probability.$ It is a notable the variance swaprisk-neutral is the expectation, expectedthefairthe value variance variance realizedofassuming the swap swap variance interest variance is is the the rates swap expected expected under are is constant. the the realized realizedexpectedT -forward variancevariance realized probability. undervariance under the the ItTT is-forward-forward a notable probability. probability. It It is is a a notable notable where theunder second the equality T-forward follows probability. by a change It is a of notable probability, point of denotesdepar- conditional expectation under pointpoint of of departure departureFor from from the case thepoint thepoint at standard standard hand, of of departure departure we obviouslyequity equity from from case case cannot the thein in assume which standardwhich standard constant the the equity fairequity fair interest value value case case rates. of of inE in at a Moreover,which which variance variance the the we swap swap fair stillfair value valueis is the the of of a a variance variance swap swap is is the the Q T needthe to risk-neutral evaluate the RHS probability, of Eq. (3). and FollowingF theunder VIX Methodology, the T -forward one might probability. try to link the That is, the fair value of risk-neutralrisk-neutral expectation, expectation,risk-neutral assumingrisk-neutral assuming interest interest expectation, expectation, rates rates are are assumingE assumingt constant. constant. interest interest rates rates are are constant. constant. spirit of the standard VIX Methodology cannot be calculatedexpectation invariance Eq. (3) to swap the value is the of arisk-neutral log-contract (Neuberger, expectation, 1994)—i.e., assuming a contract interest with a payoff For the casethe at variancehand, weFor swap obviously the is casethe cannot expected at hand, assume realized we obviously constant variance cannot interest under assume rates.the T -forward constantMoreover, probability. interest we still rates. It is Moreover,a notable we still For the case at hand,FT (S, weT)For obviously the case cannot at hand, assume we obviously constant cannot interest assume rates. constantMoreover, interest we still rates. Moreover, we still at the strictest level of theoretical rigor. equal to ln F rates(S, ) at are time constant.T . In the standard equity case, the value of a log-contract is indeed minus need to evaluatepoint the RHS oft departureneedT of Eq. to evaluate (3). from Following the the standard RHS the of VIX equityEq. Methodology, (3). case Following in which one the the might VIX fair valueMethodology, try to of link a variance the one might swap tryis the to link the The silver lining is that, as long as Tneed and S to-T evaluateare small, the the RHS need ofFor Eq. tothe evaluate (3).case at Following thehand, RHS we the ofobviously VIX Eq. Methodology, (3). cannot Following assume one the constant might VIX Methodology, try to link the one might try to link the expectation in Eq.risk-neutral (3) toexpectationexpectation the expectation, value of in in a Eq.log-contract Eq. assuming (3) (3) to to the the interest (Neuberger, value value ratesof of a a log-contract 1994)—i.e.,log-contract are constant. a(Neuberger, (Neuberger, contract with 1994)—i.e., 1994)—i.e., a payoff a a contract contract with with a a payo payoffff numerical impact of both the maturity expectationmismatch and in early Eq. (3)ex- to the value of a log-contract (Neuberger,3 1994)—i.e., a contract with a payoff FT (S,T) FFT((S,S,T)) equal to lnFT (S,T) atFor time theequalT . case In to the at ln hand,standardT T weat equity time obviouslyT case,. In cannot the the standard value assume of a equity log-contract constant case, interest the is indeedvalue rates. of minus a log-contract Moreover, we is indeed still minus ercise premium is likely to be small enoughequal to be ln ignored,Ft(S,T) at as time equalT . In to the ln standardFt(S,T) at equity time T case,. In the the standard value of a equity log-contract case, the is indeedvalue of minus a log-contract is indeed minus Ft(S,T) Ft(S,T) need to evaluate the RHS of Eq. (3). Following the VIX Methodology, one might try to link the expectation in Eq. (3) to the value of a log-contract (Neuberger, 1994)—i.e., a contract with a payoff to realize that a VIX-like implementation leads to an approxiequal- to ln F T ( S, T ) at time T. In the33 standard equity case,3 the3 value of a log-contract is indeed minus Ft(S,T) mation of the true fair value of a variance swap, and to know a log-contract is indeed minus one half the expected realized when the approximation is of a tolerableone magnitude half the expectedgiven the realized variance. In our case, it follows by Eq. (1) and Itˆo’s lemma that value of a variance swap, and to know when the approximation is of a tolerable magnitude given the context. one half the expected realized variance.one half In the ourone expected case, half the it follows realizedexpected3 by variance. realizedEq. (1) andvariance. In our Itˆo’s case, lemma In itour follows case,that itby follows Eq. (1) by and Eq. Itˆo’s (1) andlemma Itˆo’s that lemma that Q Q F (S, ) valuecontext. of a variance swap, and to know when the approximation is of a tolerable magnitude givenF S the F S T T Let Ft (S,T) be the forward price at t, for deliveryone half at the S, expectedof Et realized(Vt (T,S, variance.T)) = In our2Et case,ln it follows by Eq.. (1) and(4) Itˆo’s lemma(4) that − Ft (S, T) Let Ft (S, ) be the forward price at t, for delivery at S, of aT coupon bearingT bond expiringQ at S , Q S FQT (SS, T) Q S Q S FQT (SS, T) FT (S, T) context. T a coupon bearing bond expiring at , with tS , i.e. F T(V (T,S, )) = 2 ! F ln F "(V (T,S,.F (V))( =T,S,2 ))F = ln2 (4)F ln . . (4) (4) Bt(T) Et t T Et Et t Et T t ETt Et Q S Q S FF(TS,(S,) ) F (S, ) F (S, ) withLett Ft S(S, T),i.e. be theFt forwardF(tS, (ST,T)=)= price P ( S ) at , where t, for deliveryPtt ((SS) )is isthe at the Sprice, price of aat coupon att oft ofa zeroa bearing zero coupon coupon bond expiring bond expiring at F, − F t T T − − t T t T ≤ ≤T t However, Eq. (4) does not linkETt to( Eq.Vt (T,S, (3).T The)) = expectation2Et ! ln in Eq." (3) is. the fair value! of a (4)! " " bond expiringBt(T) at St, and Bt (T) is the price at t of the underly- − Ft (S, T) withat St t,S andTB,i.e.t (T)F ist ( theS, T price)= at t,of where the underlyingPt (S) is the bond. price at t of a zero coupon bond expiring ! " T, ≥≤ ≤ Pt(S) varianceHowever, contract expiring Eq. (4) at doesT , whereasnot link the toHowever, Eq. expectation (3).However, Eq. The (4)on expectation the does Eq. LHS not (4) of doeslink Eq.in Eq. to not (4) Eq.(3) islink the (3). is to fairthe Eq. The valuefair (3). expectation value of The a of expectation a in Eq. (3) in is Eq. the (3) fair is value the fair of a value of a Let r be the instantaneousing bond. short-term rate process and let Q be the risk-neutral probability. It at S t, andt Bt (T) is the price at t of the underlying bond. variance contractHowever, expiring Eq. (4) at S does. In not other link words, to Eq. the (3).market The num´eraire expectation for government in Eq. (3) bond is the contracts fair value of a ≥ Let rt be the instantaneous short-term rate varianceprocess and contract let Q expiringof a variance at T , whereascontractvariance theexpiring contractvariance expectation at expiringS contract. In other on theat expiring words,T LHS, whereas ofthe at Eq. Tmarket, the whereas (4) expectation is the the fair expectation value on the of LHS a on of the Eq. LHS (4) of is Eq. the (4) fair is value the fair of a value of a is well-knownLet r be the (see, instantaneous e.g., Mele, 2013, short-term Chapter rate 12) process that in and a di letffusionQ be setting, the risk-neutralFt (S, T) probability.satisfies It t be the risk-neutral probability. It is well-knownexpiringvariance variance(see, at ae.g., given contract contractMele, maturity expiringnuméraire expiringis at theatS for.T In price, variancegovernment whereas other of zeros words, contract thevariance expiringbond expectation the expiring marketcontracts contract at , num´eraireon at where expiringexpiring theS. In LHS other=at forat ofT Sa government. Eq.in words,given In Eq. other (4) (3) the is words, the and bondmarket fair contracts the= value num´eraireS market of a num´eraire for government for government bond contracts bond contracts is well-known (see, e.g., Mele, 2013, Chapter 12) that in a diffusion setting, F (S, ) satisfiesY Y Y Y in Eq.t variance (4). t contractT expiring at S. In other words, the market num´eraire for government bond contracts dFτ (S, T) Fexpiring (S,T at a given maturitymaturity isis thetheexpiring price atofexpiring zerosazeros given expiringexpiring at maturity a given atat maturity ,,is wherewhere the price =isT ofT the inin zeros price Eq. expiring (3)of zeros and at expiring=,S where at ,= whereT in Eq.= (3)T in and Eq. (3)= S and = S = vτ (S, T) dW S (τ) ,τ (t, S) , Y (1) Y Y Y Y Y Y Y Y F (S, ) · F ∈Toinexpiring evaluate Eq. (4). atthe a RHS given of maturity Eq. (3), weisinS need the Eq. price something (4).in of Eq. zeros (4).more expiring than atItˆo’s, lemma. where We= T havein Eq. to acknowl- (3) and = S dFττ (S, TT) Y Y Y Y = v (S, ) dW S (τ) ,τ (t, S) , (1) (1) τ T F edge thatin Eq. the dynamics(4). of FT (S, T) under the “variance swap” pricing measure are those under Q T , Fτ (S, T) · ∈ To evaluate the RHS of Eq. (3), weTo need evaluate somethingTo the evaluate RHS more of the thanEq. RHS (3), Itˆo’s of we Eq. lemma. need (3), something weWe need have something to more acknowl-F than more Itˆo’s than lemma. Itˆo’s We lemma. have to We acknowl- have to acknowl- where v (S, ) is the instantaneous volatility process adapted to W S (τ), a multidimensional Brow- τ T not underF Q S , as in Eq. (1). Now, by Girsanov’s theorem, we have the following: edge thatToF evaluate the dynamics the RHS of F ofT Eq.(S, T (3),edge) under we that need theedge the “variance something dynamics that the swap” dynamics moreof FT pricing than(S, T of) Itˆo’sF under measureT ( lemma.S, theT) are under “variance We those havethe under “variance swap” to acknowl-Q pricingF T swap”, measure pricing are measure those areunder thoseQF T under, QF T , where v (S, ) is the instantaneousS volatility process adapted of Ft (S,T wherenian motionvτ (S, T under) is the the instantaneousS-forward␶ probability, volatility processQF , defined adapted through to WF S the(τ), Radon-Nikodym a multidimensional derivative, Brow- notedge under thatQ theF S , dynamics as in Eq. of (1).FT Now,(S,notT) by under Girsanov’snot theQF underS “variance, as theorem, inQ Eq.F S swap”, as (1). we in have Now,pricing Eq. the (1). by measure following: Girsanov’s Now, by are Girsanov’s thosetheorem, under wetheorem,Q haveF T , the we following: have the following: as follows: dFτ (S, T) nian motion under the S-forward probability, QF S , definedS through thenot Radon-Nikodym under Q=S , asvτ ( inS, derivative, Eq.)(v (1).τ (T, Now,) byvτ ( Girsanov’sS, )) dτ + theorem,vτ (S, ) wedW haveT ( theτ) , following:τ (t, T ) , (5) rτ dτ F T T T T F dQ S e− t Fτ (S, T) − − · ∈ as follows: F = , dFτ (S, T) (2) dFτ (S, T)dFτ (S, T) !S = vτ (S, T)(vτ (T,T) vτ (S, T))=dτ +vτv(τS,(S,=T)(T)vτ dW(T,S,FTT)()(τvτ)v,(τT,(S,τTT) ))(t,dvτ T()+S,, vTτ ))(S,d(5)τT+) vdWτ (S,F TT()τ)dW,FτT (τ(t,) , T )τ, (t,(5) T ) , (5) dQ Ptt (rSτ d)τ dQF S S e− FdFτ (S,τ (TS,)T) − −Fτ (S, T) Fτ (−S, T) − · − ∈− · · ∈ ∈ !G where W T (τ) is a multidimensional= v (S, )( Brownianv (T, ) motionv (S, under)) dτQ+ Tv.( ThatS, ) is,dW theT forward(τ) ,τ price(t, is T not) , (5) ! = ! , F τ (2)T τ T τ T F τ T F and denotes2 the RISK information PROFESSIONAL set at timedQ MARCHS!. Pt (2014S) Fτ (S, T) − − www.garp.org· ∈ S ! S a martingale under Q T , such that by Itˆo’s lemma, G !G! where WF T (τ) isF a multidimensionalwhere BrownianWFwhereT (τ) motion isW aF T multidimensional( underτ) is aQ multidimensionalF T . That Brownian is, the forward Brownian motion price under motion isQ notF underT . ThatQF is,T . the That forward is, the price forward is not price is not and ConsiderS denotes the the following information payoff setof at a timevarianceS!. swap: ! a martingalewhere WF T under(τ) is aQF multidimensionalT , such thata martingaleby Itˆo’s Browniana lemma, martingale under motionQF underT under, suchQQ thatFFTT,. such by That Itˆo’s that is, lemma, the by Itˆo’s forward lemma, price is not G Q Q Q F (S, ) Consider the following payoffof a variance swap: F T F T ˜ F T T T a martingaleEt under(Vt (QT,S,F T ,T such)) = that 2Et by(ℓ Itˆo’s(t, T, lemma, S, T)) 2Et ln , (6) π (T,T) Vt (T,S,T) P (t, T, S, T) ,T S, Q Q Q − Q Q Ft (S,FTQ)(S, ) Q Q FQ (S, ) F (S, ) F T F T ˜ F T F!TF T T F"T T˜ F T ˜ F T TF T T T T ≡ − ≤ E (Vt (T,S,T)) = 2E (ℓ (t, T,E S, T))(Vt (ET,S,2E T(V))t (ln =T,S, 2E T))( =ℓ ( 2t,E, T, S,(Tℓ))(t, T,2 S,E(6)T)) ln2E ln , , (6) (6) t Q t Q t t t Q Ft (S, ) t t t π (T,T) Vt (T,S,T) P (t, T, S, T) ,T S, F T F T ˜ − F T Ft (TS, T)T − − Ft (S, T) Ft (S, T) T 2≡ − where≤ we have defined E (Vt (T,S,T)) = 2E (ℓ (t, T, S, T)) 2E ! ln " , (6)! ! " " where Vt (T,S,T) vτ (S, T) dτ is the percentage integrated variance, and the fairt value of the t − t F (S, ) ≡ t ∥ ∥ ! t T " wherestrikeVP (t,T,S, T, T)is) T v (S, ) 2 dτ is the percentage integrated variance,where we and have the defined fair value of the whereT wewhere have defined we have defined t T "t τ T ˜ ≡ ∥ ∥ where we have definedℓ (t, T, S, T) vτ (S, T)(vτ (T,T) vτ (S, T)) dτ. (7) strike P (t, T, T)is ≡− t T − T T " 1 T Q ˜ # ˜ ˜ t rτ dτ F T ℓ (t, T, S, T) vTτ (S, T)(vτ (T,ℓ (t,T) T, S,vτTℓ()(S,t,T T,)) S,dτT.)vτ (S, T)(vτ (T,S, T)() v(7)τv(τT,(S,TT) )) dvτ .(S, T)) dτ. (7) (7) P (t, T, S, T)= Et e− Vt (T,S,T) = Et (Vt (T,S,T)) , (3) P (T ) T ˜ ≡− t − ≡− t ≡− t − − t1 TheQ T two expressions in Eqs.ℓ ( (4)t, T, and S, T (6)) di#ffer byvτ the(S, “tilting”T)(vτ (T, termT) ℓ˜v(τt,(S, T,T S,#)) d)τ in. addition# to the (7) # "t rτ dτ $ F T P (t, T, S, T)= Et e− Vt (T,S,T) = Et (Vt (T,S,T)) , (3) ≡− t − Pt (T ) fact that the expectations on the RHS are taken# under two different probabilities.˜ The tilting term ˜ ˜ where the second equality follows by a change# " of probability, E$t denotesThe conditional two expressions expectation in Eqs. under (4) andThe (6) two diff expressionserThe by two the expressions “tilting” in Eqs. term (4) in and Eqs.ℓ (t, (6) T, (4) S,di andffTer) in by(6) addition the diff “tilting”er by to the the term “tilting”ℓ (t, T,term S, Tℓ)(t, in T, addition S, T) in to addition the to the Q ˜ F T encapsulatesfact thatThe the the two impact expectations expressions of thein maturity on Eqs. thefact (4)RHS mismatch: and that are (6) thefacttaken di the expectationsff thater under forward by the the two expectations “tilting” price dionfferent the is a term RHS martingale probabilities. onℓ arethe(t, T, taken RHS S, underT are) under The inQ taken additionF tiltingS twoand under di term tofferent the two probabilities. different probabilities. The tilting The term tilting term wherethe risk-neutral the second equalityprobability, follows and byEt a changeunder of theprobability,T -forwardEt denotes probability. conditional That is, expectation the fair value under of Q T yet wefact are insisting that the in expectations evaluating the on theexpectation RHS are of taken its realized under twovariance different under probabilities.Q T , motivated The as tilting we term the variance swap is the expected realizedF variance under the T -forwardencapsulates probability. the impact It is a of notable the maturityencapsulates mismatch:encapsulates the impact the forward the of impact the price maturity of is the a martingale mismatch:maturityF mismatch: under the forwardQF S theand price forward is a pricemartingale is a martingale under QF underS and QF S and the risk-neutral probability, and Et under the T -forward probability. That is, the fair value of are in ourencapsulates search for the the impact fair value of the of maturitythe original mismatch: variance the swap forward(t, T, price S, )inEq.(3).Weexpect is a martingale under Q S and thepoint variance of departure swap is from the expected the standard realized equity variance case in under which the theT -forward fairyet value we probability. are of insistinga variance It in is swap evaluating a notable is the theyet expectation we areyet insisting we of are its in insisting realized evaluatingP invariance evaluating theT expectation under theQF expectationT , of motivated its realized of asF its variance we realized under varianceQF T under, motivatedQF T , motivated as we as we ˜ ℓ (t, T, S,yet we) to are be insisting zero only in when evaluatingT = S the. expectation of its realized variance under Q T , motivated as we pointrisk-neutral of departure expectation, from the assuming standard interest equity rates case are in whichconstant. the fairare value inT our of a search variance for swapthe fair is thevalueare of in the our originalare search in our variance for search the fair swap for value theP (t, fair of T,the S, valueT original)inEq.(3).Weexpect of theF variance original swap varianceP (t, swap T, S, TP)inEq.(3).Weexpect(t, T, S, T)inEq.(3).Weexpect ˜ are in our search for the fair value˜ of the˜ original variance swap (t, T, S, )inEq.(3).Weexpect risk-neutralFor the expectation, case at hand, assuming we obviously interest cannot rates are assume constant. constant interestℓ (t, T, S, rates.T) to beMoreover, zero only we when still ℓT(t,= T,S. S, Tℓ)(t, to T, be S, zeroT) to only be zero when onlyPT = whenS. TT = S. Spanningℓ˜(t, T, S, ) to be zero only when T = S. needFor to the evaluate case at the hand, RHS we of obviously Eq. (3). cannotFollowing assume the VIX constant Methodology, interest rates. oneT might Moreover, try to we link still the needexpectation to evaluate in Eq. the (3) RHS to the of Eq.value (3). of a Following log-contract the (Neuberger, VIX Methodology, 1994)—i.e.,WeSpanning are onenot a mightdone contract yet, try with asto welink a payo still theff needSpanning to deriveSpanning the value of the log-contract in Eq. (6). It is equal to ln FT (S,T) at time T . In the standard equity case, thenatural value ofSpanning at a log-contract this juncture is to indeed rely on minus the “spanning” approach, led by Demeterfi, Derman, Kamal and expectation inFt( Eq.S,T) (3) to the value of a log-contract (Neuberger, 1994)—i.e.,We are a contract not done with yet, a as payo weff stillWe need are to not deriveWe done are the not yet, value done as weof yet, the still as log-contract need we still to derive need in Eq. to the derive (6).value It the of is the value log-contract of the log-contract in Eq. (6). in Eq. It is (6). It is equal to ln FT (S,T) at time T . In the standard equity case, the valuenatural of a log-contractWe at are this not juncture is done indeed yet, to minus rely as we onnatural still the “spanning” need atnatural this to derive juncture approach, at this the to juncture value rely led ofby on tothe Demeterfi, the rely log-contract “spanning” on the Derman, “spanning” approach,in Eq. Kamal (6). approach, led and It by is Demeterfi, led by Demeterfi, Derman, KamalDerman, and Kamal and Ft(S,T) natural at this juncture to rely on the “spanning”4 approach, led by Demeterfi, Derman, Kamal and 3 4 4 4 3 4 Zou (1999); Bakshi and Madan (2000); Britten-Jones and Neuberger (2000); and Carr and Madan (2001), among others. This approach links the value of the log- (and other) contracts to that of a one half the expected realized variance. In our case, it follows by Eq. (1) and Itˆo’s lemma that portfolio ofZou out-of-the-moneyZou (1999); (1999); Bakshi Bakshi and (OTM) and Madan Madan European-style (2000); (2000); Britten-Jones Britten-Jones options. and and Neuberger Neuberger (2000); (2000); and and Carr Carr and and Madan Madan Zou (1999); Bakshi and Madan (2000); Britten-Jones and Neuberger (2000); and Carr and Madan To apply(2001), these spanning among others. arguments This approach to our context, links the consider value of the a Taylor’s log- (and expansion other) contracts with remainder, to that of a (2001),(2001), among among others. others. This This approach approach links linksthe value the ofvalue the of log- the (and log- other) (and contracts other) contracts to that of to a that of a QF S ZouQ (1999);F S BakshiFT (S, andT Madan) (2000); Britten-Jones and Neuberger (2000); and Carr and Madan Et (Vt (T,S,T)) = 2Eportfoliotportfolioportfolio ofln out-of-the-money of ofout-of-the-money out-of-the-money. (OTM) (OTM) (OTM) European-style European-style European-style options. options. options.(4) one half the expected realized variance. In our case, it follows by Eq. (1) and Itˆo’s lemma that− (2001),FT (S, amongT) F others.tF(TS,(S,T This)T) approachFt (S, T links) the value of the log- (and other) contracts to that of a ln ToTo apply!To apply apply= these these spanningthese spanning spanning"− arguments arguments arguments to our to context, toour our context, context,consider consider a consider Taylor’s a Taylor’s a expansion Taylor’s expansion withexpansion remainder, with with remainder, remainder, one half the expected realized variance. In our case, it follows by Eq. (1)portfolio andFt (S, Itˆo’s ofT) out-of-the-money lemma thatFt (S, (OTM)T) European-style options. To applyF theseF(S,F(TS,) spanning(S,)F ) (FS, argumentsF()TS,(S,F) (S,) F tot)F(S, ourt (S,) context,) consider a Taylor’s expansion with remainder, Q S However,Q S Eq.F (4)T (S, does) not link to Eq. (3). The expectationT T T TF intT(TS,T Eq.)TT (3)T tT isT theT fairT value of a F F T ln ln ln = = = − − − + 1 ∞ + 1 Et (Vt (T,S,T)) = 2Et ln QUANT. PERSPECTIVES(4) Ft (FS,t F(S,t)(S,)T) Ft (S,Ft F()S,t (S,)T) Q Q F (S, ) T T (K T FTT(S, T)) 2 dK + (FT (S, T) K) 2 dK , one half the expectedvariance− realizedF S contract variance. In expiringF ourt (S, case,T) it at followsT ,Fby whereasS Eq. (1) andT the Itˆo’s expectationT lemma that F onT (S, the−T) LHSFT ( ofS, T Eq.) −F (4)t (S, T is) the fairK value of a − K one half the expected realized variance.(V (T,S, In our case,)) = it follows2 by Eq.ln (1) and Itˆo’s lemma. that ! 0 Ft(S,TF)t((4)FS,tT(S,) T) Ft(S,T) # one half the expected realized variance.Et Int ! our case,T it follows"E byt Eq. (1) and Itˆo’s lemma that ln "= − + 1 +1 1 ∞" ∞ ∞ + 1 +1 1 Ft (S, ) Ft (S, ) + + variance contract expiring− at S. In otherFt words,(S, T) the market num´eraireT for government(KT (FKT(K(S,FTTF))( bondTS,(S,T))2TdK)) contracts+ dKdK+ +(FT (S, T(F)T(F(TKS,()S,T)T)2 dKK)K), dKdK, , Q S Q S FT (S, ) K 2 2 K 2 2 F F !T " − !−0− 0 − − − K KFt(S,T) F (S, ) − − − # K K Q SEt (Vt (T,S,T)) = 2EQt S ln FT (S, ) . (4) " F!t(S,!0T) " Ft(S,tT) T # # However, Eq. (4) does not link to Eq. (3).Q TheF expectationQ − inF Eq.F (S,F (3)(S,) is)T the fairand value take expectations of a under Q" T", + 1 ∞ " " + 1 expiringF SEt at(Vt a(T,S, givenT)) maturity = F2SEt !TlnistT theT price" . of zeros expiring at(4) , where F =(KT Fin( Eq.S, )) (3) anddK + = S (F (S, ) K) dK , Et (Vt (T,S,T)) = 2Et− ln Ft (S, . ) (4) T T 2 T T 2 T and take expectations− under Q T , − K − K However, Eq. (4) does not link to Eq.− (3). TheF!Yt (S, expectationT) " in Eq. (3) is theY fair! value0 Y ofF a T Y Ft(S,T) # variance contract expiring at T , whereasHowever, the Eq. expectation (4) does not link on to the Eq. (3). LHS The! of expectation Eq. (4)" in is Eq. the (3) fair is the value fair value ofand ofand a a take take expectations expectations" under underQFQTF, , " in Eq. (4). QF T Q T F (S, ) (F (S, )) variance contractHowever,variance Eq. expiring contract (4) does expiring at notT at, link whereasT , whereas to Eq. the the (3). expectation expectation The expectation on the LHS on in of the Eq. (4) LHS (3) is isthe of the fair Eq.fair valueF (4)value of a is of theT a fairT valueEt ofQF aT T T variance contract expiring atHowever,S. In Eq. other (4) words, does not the link market to Eq. (3). num´eraire The expectation for government in Eq. (3)is bond the fair contracts valueand take of a expectations under Q T , Q Q T Et lnQF T FT (S, T=) FEt (FFTT(FS, T)) 1 To evaluate the RHS of Eq. (3), we need something moreQ thanQ T Itˆo’sF F(S,( lemma.S,) ) We(F have(F(S,(S, to)) )) acknowl- variancevariance contract contract expiring expiring at T at, whereasS. In other the words, expectation the market on num´eraire the LHS for of government Eq. (4) is bond the faircontracts valueEt ofFtF a(TlnS,F T) T T T=T Ft (ES,t ETt ) T T −T1 T one halfvariancevariance the expected contract contract expiring realized expiring at variance.T at, whereasS. In the other In expectation our words, case, on theit the follows market LHS of by Eq. num´eraire Eq. (4) is(1) the and forfair valuegovernmentItˆo’s of$ lemma a Et Et bond thatFlnt (lnS,%T contracts) = F=t (S, T) − 1 1 expiring at a given maturity is the price of zeros expiring at , where = T in Eq. (3) and = S $ F %(S, )Q T F (S, ) − expiring at a given maturity is the price of zeros expiring at , where = T in Eq. (3) and = S Ft (S,t T)T F Ft((S,S,t T) )T − T varianceY contract expiringedge at thatS. In the other dynamics words,Y the of marketFTY( num´eraireS, T) under for government the “varianceY bond contracts swap”Q T pricingFT$(S, ) measure% (F areT (FS,t those(S,T))) under QF , variance contract expiring at S. In otherY words, the market num´eraireY for governmentY bond contractsY F $ T 1%Et 1 T 1 1 ∞ ∞ 1 1 expiring at ain given Eq. (4). maturity is the price of zeros expiring at , where = TEin Eq.ln (3) and = = S Ft(FS,t1(S,) T) in Eq. (4). t 1 PutPutTt (tK(K) ) dKdK+ 1+ Callt ∞(KCall) t (dKK) , 1dK(8) , (8) expiring at a given maturity is the priceS of zeros expiring at , where = T in Eq. (3) and = S Ft (S, T) Ft (1S, T) − 2 2 1 ∞ 2 12 expiring at a given maturitynot underis theY priceQ of, zeros as in expiring Eq. (1). at , Now, where by= YT Girsanov’sin Eq. (3)Y and theorem,= S we have− theYP− following:(PTt ()T ) 0 PutK K(K) Ft(dKS, ) + KCall (KK) dK , (8) To evaluateQ theF S RHSY of Eq. (3),F we need somethingQF moreS thanYF Itˆo’sT (S, lemma.YT) We have to acknowl-Y $ % t !!0" Putt (Kt ) 2"dK2 +TFt(S,T) Callt (Kt #) 2 dK2 #, (8) in Eq. (4). Y T S Y Y Y t (K− −P "P(Ttt ()(FTKt()S,) denote) the pricesK K of" European putsK K To evaluate the RHS of Eq.in Eq. (3), (4). weunder needE somethingQF (, V nott (T,S, under moreT)) thanQ =F 2 Itˆo’sE lemma. ln We have. to acknowl- (4) 1 t ! !T0 0 1 Ft(FS,tT(S,) T) 1 # # in Eq. (4). edge that the dynamicst of FT (S, T) under the “variancet swap” pricing measure are those under Q T , " " ∞ " " − Ft (S, ) F Putt (K) dK + Callt (K) dK , (8) T where Putandt (K) calls and Call struckt (K) denoteat K, theand prices we ofhave European relied2 puts on and the calls following struck at K ,pric and2 - we have To evaluateTo evaluatenot the undertheorem, the RHSQ RHSS , as of inof we Eq. Eq. Eq. have (1). (3), (3), Now, we the we need by following: need Girsanov’s something something theorem, more! we than more have Itˆo’s the than following: lemma." Itˆo’s We lemma.have to acknowl- We have to acknowl-− Pt (T ) K K edge that the dynamics of FToT evaluate(S, T) theunder RHS theF of Eq. “variance (3), wedF needτ ( swap”S, somethingT) pricing more measure than Itˆo’s arelemma. those Wewhere haveunder to Put acknowl-QtF(TK,) and Callt (K) denote the!" prices0 of European puts"F andt(S,T) calls struck at #K, and we have reliedwherewhere on thePut Put followingt (Kt ()K and) and pricing Call Callt (T equations:Kt ()K denote) denote the the prices prices of ofEuropean European puts puts and and calls calls struck struck at atK,K and, and we we have have edge that the dynamics of FT (S, T) under the= “variancevτ (S, swap”T)(v pricingτ (T,T measure) vτ are(S, thoseT)) underdτ +QvFτT(,S, T) dWF (τ) ,τ (t, T ) , (5) edge that the dynamics of F (S, ) under the “variance swap” pricing measure are those under Q T , T not under Q S , as inHowever,edge Eq.that (1). Eq. Now,the (4)dynamics by doesdF Girsanov’sτ (S, notT ofT) F linkTT( theorem,S,F toTτ )( Eq.S, underT) we (3). the have− The “variance the expectation following: swap” in pricing− Eq.relied measure(3) on is therelied thereliedF following are fairon onthose the valuethe following pricing following under· of equations:pricing aQ pricingF , equations: equations:∈ F not under Q S , as in Eq.= (1).vτ ( Now,S, )(v byτ (T, Girsanov’s) vτ (S, theorem,)) dτ + vτ we(S, have) dW theT ( following:τ) ,τ (t, Twhere) , Put(5) t (K) and Callt (K) denote the prices of European puts and calls struck at K, and we have F T T T T F Put (K) Q Call (K) Q not under QF S , as inF Eq.τ (S, (1).T) Now,− by Girsanov’s− theorem, we have the· following: ∈ t F T + t F T + variancenot contractunder QF expiringS , as in atEq.T , (1). whereas Now, the by Girsanov’s expectation theorem, on the LHS we have of Eq. the (4)relied following: is theon the fair following value= pricingE oft a ( equations:K FT (S, T)) , = Et (FT (S, T) K) . Put (K)PtPut(TPut)t Q(Kt ()K) Q QT −T CallP+t (TCall)(KCallt)(Kt ()K)Q Q QT T − + dF (S, ) where W T (τ) is a multidimensional Brownian motion undert Q T .F T ThatF is,F the forward+ + pricet is not F T F F + dFτ (S, T) whereτ WTF T (τ) is a multidimensionalF Brownian motion under QF T . That is, the forward price is not = F =(=KEt Et F(TK(KS,FT))F(TS,(S,T,))T)) , , = = =Et Et (F(TFT((FS,(TS,(S,T))T)K)K) .. . dFτ (S, T) = vτ (S, T)(vτ (T,T) vτ (S, T)) dτ + vτ (S, T) dWF T (τ) ,τ (t, T ) , (5) PEP(tTt ()T ) & −T ' P P(Tt ()T )Et& T ' − variance= vτ contract(S, )(v expiringτ (T, ) atvτS(.S, In other)) dτ + words,vτ (S, the) marketdW T ( num´eraireτ) ,T τ ( fort, T government) , (5) bondPutt (K contractst) Q T − Callt (K)t Q T − T aF martingale(S,=T ) vτ under(−S, TQ)(vTTτ,T( suchT,␶T that) v byτ (− Itˆo’sS, T)) lemma,Tdτ + vτ (S,F T) dWF· (τ) ,τ (t, T∈) , P(5)t (T) Ft F (S,−) is not a+ T martingalePt (T ) underF Q T +− dFτ (S,τ ) T wherea martingale WF F ( under Q T , such that by Itˆo’s lemma, Because Ft (S, T) is= not a martingale(K FT under(S, Q))F ,, the first term= on the RHS(FT ofF(S, Eq.) (8)K is) not. one, but Fτ (S, T) − Fτ (S, T)T −− − F · · ∈ ∈ Q Et & & T ' ' Et & &T ' ' = vτ (S, )(vτ (T, ) vτ (S, )) dτ + vτ (S, ) dW T (τ) ,FτPTt (Tℓ˜)(t,T,S,t, TT) ,& (5)˜− ' Pt (T ) & − ' expiring at a given maturityT is theT price ofT zeros expiringT at , whereT = TF inequals Eq.T Et (3)e and =, whereS ℓ (t, T, S, T) is as in Eq.T (7). Utilizing this expression and combining QF . ThatQ is, the forwardQ price is not aQ martingaleF (S, ) under QFBecause,Because Ft F(S,t (S,T )T is) not is not a martingale a martingale under under QFQT F, the, the first first term term on on the the RHS RHS of ofEq. Eq. (8) (8) is not is not one, one, but but whereFτ (WS,T (τ)) is a− multidimensionalF T BrownianF−T motion under QF TT . ThatT is,T the· forward price is not∈ & T' & ' F T Y (V (T,S, )) = 2 (ℓ˜(t, T, S, )) 2 FY ln BecauseY, Ft (S,(6)T) isQ notQT T a˜ Y martingale˜ under QF , the first term on the RHS of Eq. (8) is not one, but where WF T (τ) is a multidimensionalEt t BrownianT E motiont underT QF T E.t That is, the forward priceEq. is (8) not with(F Eq.F (6),ℓ(t,T,S,ℓ we(t,T,S,) T find) T) that the˜ fair˜ value of the variance swap in Eq. (3) is, T in Eq. (4). Q T T − Ft (QS, TT) BecauseQ equalsT equalsF˜tE(S,t ETt )Q ise notTe a martingaleF, where,( whereS, underℓ ()t,ℓ ( T,t,Q T,S,F S,T,)T the is) asis first as in in Eq. term Eq. (7). on (7). the Utilizing Utilizing RHS of this Eq. this expression (8) expression is not one,and and combining but combining where WF (τ) is a multidimensionala martingale Brownian under Q T , such motion that byunder Itˆo’s lemma,QFF . That is, the! forwardF " price isF not ℓ(t,T,S,T) F ˜T T F equals˜ eQ T ˜ , where ℓ (t, T, S, ) is as in Eq. (7). Utilizing this expression and combining a martingale under QF T , such that by Itˆo’s lemma,Et (Vt (T,S,T)) = 2Et (ℓ (t, T,Et S, T))F 2Eℓ(t,T,S,t T) ln ˜ T , (6) equalsEq.Eq.Et (8) (8) withe with( Eq.( Eq. (6),, (6), where we) we) findℓ find(t,Q thatF T,T that S, theTℓ˜( thet,T,S,) fair is fairas) value in value Eq. of (7). ofthe the Utilizingvariance variance thisswap swap expression in inEq. Eq. (3) (3)and is, is, combining a martingale under ToQwhere evaluateT , suchWF thatT the(τwhere)RHS by iswe a Itˆo’s multidimensionalhave of Eq. lemma,defined (3), we need Brownian something motion more under than Itˆo’sQF T . lemma. That is, We the have forward−P ( tot, T, acknowl- S, priceT)=2 isF1t not(ES, T) e T ℓ˜(t, T, S, T) F Eq. (8) with( Eq. (6), we) find! that− thet fair" value of− the variance swap in Eq. (3) is, QF T QF T QF T FT (S, T) Eq. (8) with( Eq. (6), we) find that the fair value of the variance swap in Eq. (3) is, Q Q T ˜ Q F (S, ) Q Q T ˜ ˜ a martingale underF TEQt T(,V sucht (T,S, thatT)) =F by 2TEt Itˆo’s(ℓ (t, lemma, T, S, T)) F2TEt Tln T , (6) (6) ( Ft(S,TF)T F ℓ(t,T,S,ℓ(t,T,S,T) T) )) edge that the dynamics of(VFFt (TT,S,(S, T))˜) = under 2 the(ℓ˜(t, “varianceT, S, )) 2 swap”ln pricing measure, are those(6) under(t,( T,t,Q T,S,F S,T)=2,2)=21 1 e e 1 ℓ˜(t,ℓ˜( T,t,∞ T,S, S,) ) 1 Et T ℓ (t, T, S,EtT) vτ (S,TT)(vτ (ET,t−T) vτ (S, T)) dFτt.(S, T) (7) P P T T Et Et T T where we have defined − F!t (S, ) " + Q Q−T− ˜˜ Putt (K) −dK−+ Callt (K) dK . (9) Q T Q T ≡− t Q T FT (−S, ) T F TF ℓ(ℓt,T,S,(t,T,S,T) ˜˜2 2 F F # F ! T " (t, T,( S,t, T,)=2 S, )=21Pt 1(T ) 0 ee ℓKℓ((t,t, T, T, S, S,F)t(S,) T) K not under Q S , as in Eq. (1).˜ Now, by Girsanov’s theorem, we have the following:P P T T Et(E!t"( Ft(FS,tT(S,) T) T" T )))) # Et (Vt (T,S,F T)) = 2Et (ℓ (t, T, S, T)) 2Et ln , (6) − − 2 2 − 1 1 ∞ ∞ 1(9)1 where we haveQ definedT Q T ˜ Q T FT (S, ) + Put (K) dK + Call (K) dK . (9) where we have definedThe twoF expressions in Eqs. (4) and− (6)F differ by the “tilting”Ft (S, termT) ℓ (t, T,TF S, T) in addition to theT (+ Ft(S,T) Putt (Kt ) dK2))+ Callt (Kt ) dK2 . (9) (V (T,S, )) = 2 (ℓ˜(t, T, S, )) 2 ln , ( 2 (6)PF(tT(S,) ) 1 2K ∞)) 1 2K Et t T Et ! T "Et Accordingly, a formulation of anPt index(Tt ) ofT!0 government0 bondK volatility indexFt(FS,tT( isS,) T) K # fact that the expectations on the RHST are taken˜ under two different probabilities. The tilting term +2 !" "Putt (K) 1dK + ∞" Call" t (K) dK1 . # (9) dFτ (S, ) T ℓ (t, T, S, T) − vτ (S,FTt)((S,vτ T(T,) T) v+τ (S,PT))(Td) τ. Putt (K)K2 dK +(7) Callt K(K2 ) dK . (9) T ˜ ! " t ! 0 2 Ft(S,T) #2 encapsulates= v ˜(S, theℓ impact)((t, T,v ofS,(T, theT) maturity) v mismatch:vτ(S,(S, T)))( thevdττ( forwardT,+Tv) price(vS,≡−τ ( isS, a)T martingale))tdWdτ. T under(τ) ,Q Sτand (t,(7) T−) , P ((5)T ) " K " K where we have defined τ ℓ (t, T,T S, T)τ T≡−vτ (S,τ T)(vTτ (T,T) vτ (τ−S, T))Tdτ. F F Accordingly,(7) a formulationt ! 0 of an index of government bondFt( volatilityS,T) index is # Fτ (S, ) − ≡− −t − #· (7)Accordingly,∈ a formulation of" an index of government1 bond" volatility index is T yet we are insisting in evaluatingt the# expectation of its realized variance under QF T , motivated as we GB-VI(t, T, S, T) P (t, T, S, T) where we have defined # Accordingly,Accordingly, a formulation ofa anformulation index of government of≡ anT bondindext volatility of government index is bond The twoare in expressions our searchT for in theEqs. fair (4) value and of (6) the di originalffer by variance the “tilting” swap P term(t, T, S,ℓ˜(Tt,)inEq.(3).Weexpect T,Accordingly, S, ) in addition a formulationto the of an˜ index of government* − bond volatility index is The two expressions in Eqs.The (4) and two (6) expressions differ by the “tilting” in Eqs. term (4)ℓ˜ and(t, T, (6) S, ) di inffT additioner by the to the “tilting” term ℓ (t, T, S, T) in addition to1 the1 where Wℓ˜(t,T T,(τ S,) is) aℓ˜(t, multidimensional T, S, T) tov be zero(S, only)( whenv Brownian(TT,= S). motionv (S, under)) dτ. Q T . ThatT is, the forward(7) volatility price isis not GB-VI(t, T, S, ) (t, T, S, ) F fact thatT the expectationsτ onT the RHSτ areT takenT τ underT two differentF probabilities. Thewhere tiltingP ( termt, T, S, T) is as in Eq. (9).GB-VI(t, T, S, T)T 1 P (Pt, T, S, T)T fact that the expectations≡− t fact on the that RHS the are expectations taken− under two on different the RHS probabilities. are taken The tilting under term two different probabilities.GB-VI(t, T, S, The) tilting≡ ≡ T term(Tt, T,t t S, ) term ℓ ˜ ( t, T, S, ) in additionvτ to(S, the )(factvτ that(T, the) expectationsvτ (S, )) dτ on. (7) T **P − − T a martingaleencapsulates under QF theT#, impact such ofthat the maturitybyT Itˆo’s mismatch: lemma, theT forward priceT is a martingaleT under QF S and ≡ T 1t encapsulates theSpanning impact of the maturity mismatch:≡− the forward price is a martingale− under Q S and * − encapsulates the impactt of the maturity mismatch: theF forward price isGB-VI( a martingalet, T, S, T under) 5 QF S Pand(t, T, S, T) yet we are insisting in evaluating the expectation# of its realized variance under QF T , motivatedwherewhere asP we(Pt,( T,t, T,S, S,T)T is) asis as in in Eq. Eq. (9). (9). yet we are insisting in evaluating the expectation of its realized˜ variance under Q T , motivated as we ≡ T t The two expressions in Eqs. (4) andWe are (6) not di doneffer yet, by as the we still“tilting” need to derive term theℓ ( valuet, T, of S, theT log-contract) in additionF in Eq.where to (6). the ItP is(t, T, S, T) is as in Eq. (9). * − are in ourQ searchT term foryet the encapsulates fairwe valueare insisting of Q the theT original impact in evaluating variance of the swap maturity thePQ(t, expectation T,T S, mismatch:T)inEq.(3).WeexpectFT (S, ofT its) the realized variance under QF T , motivated as we areThe in our two search expressionsnaturalF for at the this fair juncture in value Eqs. to of rely(4) the on original and theF “spanning” (6)˜ variance differ approach, swap by theP led(t, “tilting” byT, S,Demeterfi,FT)inEq.(3).Weexpect term Derman,ℓ˜( Kamalt, T, S, and ) in addition to the fact that the expectations on˜ theEt RHS(V aret (T,S, takenT)) under = 2E twot ( diℓff(t,erent T, S, probabilities.T)) 2Et Theln tilting term, T (6) 5 ˜ ℓ (t, T, S, T) toforward be zeroare only price in when our isT searcha= martingaleS. for the under fair value −Q S and of the yet original wewhereF are(S, insist variance()t, T,- S, where swap) is as in ( t, Eq. T, (9).S, )inEq.(3).Weexpect5 ℓ (t, T, S, T) to be zero only when T = S. F ! t PT " T P T 5 encapsulates the impactfact of that the the maturity expectations mismatch: on the the RHS forward are price taken4 is under a martingale two different under probabilities.Q S and The tilting term ing inℓ˜ (evaluatingt, T, S, T) the to beexpectation zero only of when its realizedT = S. variance Funder whereencapsulates we haveSpanning defined the impact of the maturity mismatch: the forwardT price is a martingale under QF S and yet we are insisting in evaluatingSpanning the expectationQFT, motivated of its as realized we are variance in our under search Q forF the, motivated fair value as of we Time Deposits 5 yet we areWe insisting are not done in evaluating yet, as we still the need expectation to derive the of value its realized of the log-contract variance in under Eq. (6).Q ItT , is motivated as we are in our search for the fairWe valueare not of done the yet, original as we still variance need toT swapderive theP (t, value T, S,Time ofT the)inEq.(3).Weexpect log-contract Deposits in Eq. (6). It isFTime deposit variance contracts share an interesting feature natural at this˜ junctureSpanning to rely on the “spanning” approach, led by Demeterfi, Derman, Kamal and ˜ arenatural in our at search this juncture for ℓ ( t, the to T, rely S, fair T on ) valueto the be “spanning” zero of thev τonly( originalS, approach, Twhen)(vτ variance ledT(T,= byST. ) Demeterfi, swapvτ (S, Derman,PT())t,d T,τ. KamalS, T)inEq.(3).Weexpect and with government(7) bonds: they can be priced based on the same ℓ (t, T, S, T) to be zero only when T = S. ≡− − ˜ We are not#t done yet, as we still need to derive the valuechange-of-numéraire of the log-contract set forth in Eq. in (6).the previous It is section. A point ℓ (t, T, S, T) to be zero only when T = S. 4 Time deposit variance contracts share an interesting feature with government bonds: they can be Spanningnatural at this juncture4 to rely on the “spanning” approach,of leddeparture by Demeterfi, arises when Derman, expressing Kamal time and deposit volatility in Spanning The two expressions in Eqs. (4) and (6) differ by the “tilting” term ℓ˜(t, T, S, T) in addition to the priced based on the same change-of-num´eraireterms of basis point volatility set forthof rates in, as the opposed previous to the section. more fa A- point of departure fact thatSpanning the expectations on the RHS are taken under two different probabilities. The tilting term We are not done yet, as we still need to derive the value of the log-contractarises when in Eq.expressing (6).4 It time is miliar deposit notion volatility of percentage in terms volatility of basis of prices. point volatilityTo accommo of- rates, as opposed to encapsulates the impact of the maturity mismatch: the forward price is a martingale under Q S and natural at this juncture toWe rely are on not the done “spanning” yet, as we approach, still need led to by derive Demeterfi,the the more value Derman, familiar of the Kamal log-contract notion and ofdate percentage in Eq.thisF practice, (6). volatility It we is need of to prices. consider To spanning accommodate arguments this dif practice,- we need to yet wenatural are insisting at this in juncture evaluating to rely the onexpectation the “spanning” of its realized approach, variance led by under Demeterfi,Q T , motivated Derman, Kamal as we and consider spanning argumentsF ferent diff fromerent those from in thosethe previous in the section. previous section. are in our search for the fair value4 of the original variance swap P (t, T, S, T)inEq.(3).Weexpect ℓ˜(t, T, S, T) to be zeroThis only approach when T = linksS. the value of4 the log- (and other) contracts The Underlying Risks The Underlying Risks Let lt Zou (1999); Bakshi and Madan (2000); Britten-Jones and Neuberger (2000); and Carr and Madanfor the time period from t to t- Spanning Zou(2001), (1999);style among Bakshioptions. others. and This Madan approach (2000); links Britten-Jones the value of and the Neuberger log- (and (2000); other) contractsand Carr toand that Madan of a Let lt (∆) be the simply compounded interest rate on a deposit for the time period from t to t+∆. (2001),portfolio amongTo of out-of-the-money apply others. these This approach (OTM) spanning European-style links thearguments value options. of the log-to our (and other)context, contracts consider to that of ation, we refer to lt We are not doneportfolioTo yet, apply of as out-of-the-money these we spanning still need arguments (OTM) to European-style to derive our context, the options. considerAs value a a non-limitative Taylor’s of the expansion log-contract with illustration,remainder, in Eq. (6). we Itrefer is to lt ( ∆) as the LIBOR. t, one To apply these spanning arguments to our context, consider a Taylor’s expansion with remainder, natural at this juncture toFT rely(S, T) onFT the(S, T) “spanning”Ft (S, T) approach, led by Demeterfi, Derman, Kamal and ln = − Define a as one where at time t, one party agrees to pay a counterparty a payoff FFTt ((S, T)) FT (S,FTt)(S,Ft)(S, T) ln T = − T 100×(1–lst (S,SS Ft (S, T) Ft(FS,tT()S, T) + 1 equal∞ to 100 (1+ 1 lS (∆)) Zt (S, S +∆) at time S. The forward LIBOR price, Zt (S, S +∆),is Ft(S,T) (K FT (S, T))4 dK + (FT (S, T) K) dK , Zt (S,S - 12 ∞ × 1−2 − − ! 0 − + K Ft(S,T) − + K # " (K FT (S, T)) 2 dKagreed+ " (F atT (S, timeT) K)t such2 dK that, in the absence of arbitrage − 0 − K F (S, ) − K trage !" " t T # and take expectations under QF T , and take expectations under QFT, and take expectations under QF T , QF T Q F T FT (S, T) Et (FT (S, T)) Zt (S, S +∆) = 100 (1 ft (S, S +∆)) , (10) (10) Et ln = QF T 1 QF T FFTt ((S, T)) Et Ft(F(S,T (TS,) T)) − × − Et $ln % = 1 Ft (S, ) Ft (S, F)t(S,T) − T 1 T 1 ∞ 1 (8) $ % QF S Ft(S,T) Putt (K) dK + Callt (K) dK , (8) where ft (S,S 1 12 ∞ 12 − Pt (T ) ! 0 whereK fFt(S,(TS,) S +∆K ) is# the forward LIBOR, which satisfies: ft (S, S +∆) = Et (lS (∆)). Because " Putt (K) 2 dK + " Callt (K) 2 dK , (8) Q Fs − Pt (T ) ! 0 K Ft(S,T) K # ft (S,S (lSlsfs (S,Sft (S,S " " t S where Putt (K) and Callt (K) denote the prices of EuropeanlS ( puts∆) and = callsfS struck(S, at SK+, and∆ we), haveft (S, S +∆) is a martingale under QF . Therefore, assuming that the whererelied onPut thet (K following) and Call pricingt (K) denote equations: the prices of European puts and calls struck at K, and we have relied on the following pricing equations: information in this market is driven by Brownian motions, the forward price, Zt (S, S +∆), satisfies

Putt (K) QF T + Callt (K) QF T + = Et (K FT (S, T)) , the= following:Et (FT (S, T) K) . PutPtt((TK) ) QF T − + CallPt t(T(K) ) QF T − + = Et (K FT (S, T)) , = Et (FT (S, T) K) . Pt (T ) & − ' Pt (T ) & − ' dZτ (S, S +∆) z Becausewww.garp.orgFt (S, T) is not a martingale& under QF'T , the first term on the& RHS of Eq. (8) is' not one, but = v (MARCHS, ∆) dW S2014(τ) , RISKτ ( PROFESSIONALt, S) , 3 (11) Q τ F F T ℓ˜(t,T,S,T) ˜ Becauseequals EtFt (S,eT) is not a, martingalewhere ℓ (t, T, under S, T)Q isF T as, the in Eq. first (7). term Utilizing on the RHS this expression of Eq. (8)and is not combining one, butZτ (S, S +∆) ∈ Q T ˜ equalsEq. (8)E withF Eq.eℓ(t,T,S, (6),T) we, findwhere thatℓ˜(t, the T, S,fairT) value is as of in the Eq. variance (7). Utilizing swap inthis Eq. expression (3) is, and combining t ( ) z Eq. (8) with( Eq. (6), we) find that the fair value of the variancewhere swapW inS Eq.(τ (3)) isis, a multidimensional Brownian motion under Q S , and v (S, ∆) is a vector of instan- Q ˜ F F τ F T ℓ(t,T,S,T) ˜ P (t, T, S, T)=2 1 Et e ℓ (t, T, S, T) − Q T ˜ − (t, T, S, )=2 1 F eℓ(t,T,S,T) ℓ˜(t, T,taneous S, ) volatilities, adapted to W S (τ). P T ( Et Ft(S,T) T )) F 2 − − 1 ∞ 1 + ( Ft(S,T) Putt (K) dK + )) Callt (K) dK . (9) P 2(T ) K12 ∞ K12 + t !"0 Put (K) dK + Because"Ft(S,T) Call (K we) shalldK#. deal(9) with variance swap designs referencing rates instead of prices,we P (T ) t K2 t K2 t !"0 "Ft(S,T) # Accordingly, a formulation of an index of government bondconsider volatility index the is forward LIBOR equivalent to Eq. (11), as follows: Accordingly, a formulation of an index of government bond volatility index is 1 GB-VI(t, T, S, T) P (t, T, S, T) ≡ T 1 t df τ (S, S +∆) f GB-VI(t, T, S, T) * − P (t, T, S, T) ≡ T t = vτ (S, ∆) dWF S (τ) ,τ (t, S) , (12) * − f (S, S +∆) ∈ where P (t, T, S, T) is as in Eq. (9). τ where P (t, T, S, T) is as in Eq. (9). 5 f 1 z where, by Itˆo’s lemma, vτ (S, ∆) 1 f (S, S +∆) v (S, ∆). 5 ≡ − τ− τ ! " LIBOR Variance Contracts and Volatility Indexes

6 Time Deposits Time Deposits TimeTime deposit Deposits variance contracts share an interesting feature with government bonds: they can be Timepriced deposit based on variance the same contracts change-of-num´eraire share an interesting set forth feature in the with previous government section. Abonds: point they of departure can be pricedarisesTime basedwhen deposit expressingon variancethe same time contracts change-of-num´eraire deposit share volatility an interesting inset terms forth ofin featurebasis the previous point with volatility government section. of A rates point bonds:, as of they opposed departure can to be priced based on the same change-of-num´eraire set forth in the previous section. AWe point define the of basisdeparture point LIBOR integrated rate-variance as, arisesthe more when familiar expressing notion time of depositpercentage volatility volatility in terms of prices. of basis To accommodate point volatility this ofWe rates practice, define, as the opposed basis we needpointLIBOR to to integrated rate-variance as, arises when expressing time deposit volatility in terms of basis point volatility of rates, as opposed to T 2 the more familiar notion of percentage volatility of prices. To accommodate this practice, we need tof,bp 2 f consider spanning arguments different from those in the previous section. Vt (T,S,∆) T fτ (S, S +∆) vτ (S, ∆2 ) dτ, f,bp ≡ t2 f considerthe more spanning familiar arguments notion of percentage different from volatility those inof theprices. previous To accommodate section. this practice,We define we theV need basist ( pointT,S, to ∆ LIBOR) integrated!fτ (S, S +rate∆) -variancevτ "(S, ∆) as,d"τ, We define the basis point LIBOR integrated≡ t rate-variance as," " such that, by arguments similar to those! leading to Eq." (3)" in Section" " 2, the fair value of the time consider spanning arguments different from those in the previous section. T " " 2 The Underlying Risks such that, by arguments similar tof,bp those leading to Eq.2 (3)" in Section"f 2, the fair value of the time deposit rate-variance swap generatedV (T,S, atT t∆,) and payingf (S, off Sat+T∆,) isv2 the(S, following:∆) dτ, f,bp t τ τ deposit rate-variance swapV generated(T,S,∆) at t, andf 2 ( paying≡S, St + o∆ff) atvfT(,S, is∆ the) following:dτ, The Underlying Risks t ≡ τ ! τ " " f,bp !t bp " " Let lt (∆) be the simply compounded interest rate on a deposit for the time period from t to t+∆V.t (T,S,∆) Pf (t, T," S, ∆) ,T"" S, " The Underlying Risks such that, by argumentsf,bp similar to those−bp leading to" Eq. (3)" in≤ Section 2, the fair value of the time such that, by arguments similarVt to those(T,S, leading∆) Pf to( Eq.t, T, S,(3)"∆ in) ,T Section"S, 2, the fair value of the time AsLet a non-limitativelt (∆) be the simply illustration, compounded we refer interest to lt (∆ rate) as on the a LIBOR. deposit for the time perioddeposit from rate-variancet to t+∆ swap. generated− at t, and paying offat≤T , is the following: QUANT PERSPECTIVESdepositis rate-variance swap generated at t, and paying offat T , is the following: Let lt (∆) be the simply compounded interest rate on a deposit for the time period from t to t+∆bp. QF T f,bp As aDefine non-limitative a forward illustration, contract as we one refer where to l att (∆ time) ast, the one LIBOR. party agrees to payis a counterparty a payoff (t,f, T,bp S, ∆) = bpV (T,S,∆) . (13) Pf V (T,S,Q∆E) t P (t,t T, S, ∆) ,T S, f,bpbp (t, T,t S, ∆) =bp F T− Vff,bp (T,S,∆) . ≤ (13) As a non-limitative illustration, we refer to l (∆) as the LIBOR. Vt Pf (T,S,∆) P E(tt, T, S,t∆# ) ,T S, $ equalDefine to 100 a forward(1 contractl (∆)) asZ one(S, where S +∆ at) at timet timet, oneS. The party forward agrees toLIBOR pay a price, counterpartyWeZ face(S, two S complications. a+ payo∆),isff The first− isf the same maturity≤ mismatch arising in the government S t t # $ × − − We faceis two complications. The first is the same maturity mismatch arising in the government Define a forward contract as one where at time t, one party agrees to pay abond counterparty case. The second a payo complicationff is that we areQ dealing with a notion of basis point variance, equalagreed to at 100 time(1t suchlS ( that∆)) inZ thet (S, absence S +∆) of at arbitrage time S. The forward LIBORis price, Zt (S, S +∆),is bp F T f,bp bond case. The second complicationP isf ( thatt, T, weS,Q ∆ are) = dealingEt V witht a(T,S, notion∆) of. basis point variance, (13) × − − which necessitates a differentbp treatment fromF T the percentagef,bp case. equal to 100 (1 lS (∆)) Zt (S, S +∆) at time S. The forward LIBOR price, Zt (S, S +∆Pf),is(t, T, S, ∆) = Et Vt (T,S,∆) . (13) agreed at time t such that in the absence of arbitrage which necessitatesIn the percentage a different case, treatment it is by now from well-understood the percentage that# case. its formulation$ relates to a log-contract. × − − We face two complications. The first is# the same maturity$ mismatch arising in the government WeIn face the percentagetwo complications. case, it is The by now first well-understood is the same maturity that its mismatch formulation arising relates in the to a government log-contract. agreed at time t such that inZ thet (S, absence S +∆) =of 100 arbitrage(1 ft (S, S +∆)) , Thebond contract case. we Theshall second link(10) the complication expectation onis that the RHS we are of Eq. dealing (13) with is, instead, a notion a “quadratic” of basis point contract variance, × − bondThe case. contract The we second shall link complication the expectation2 is that on we the are RHS2 dealing of Eq. with (13) a is, notion instead, of basis a “quadratic” point variance, contract deliveringwhich anecessitates payoffequal a di tofferentfT (S, treatment S +∆) fromft (S, the S + percentage∆). To see case. how this contract is useful, note Zt (S, S +∆) = 100 (1 ft (S, S +∆)) , which necessitates a different(10) treatment2 from the−2 percentage case. deliveringQthat by aIn Itˆo’s payo the lemmaff percentageequal and to thef case,T (S, Girsanov itS + is by∆) theorem,nowft well-understood(S, S +∆). To that see how its formulation this contract relates is useful, to a log-contract. note Z (S, S +∆) = 100× (1− f (S, S +∆)) , F S (10) − where ft (S, S +∆) is the forwardt LIBOR, which satisfies:t ft (S, S +∆) = EthatInt the by Itˆo’s(percentagelS ( lemma∆)). case, and Because itthe is Girsanov by now well-understood theorem, that its formulation relates to a log-contract. × − The contract we shall link the expectation on the RHS of Eq. (13) is, instead, a “quadratic” contract martingale under QF . Therefore, assuming that the informaQ S - QF T 2 2 QF T bp bp The contractF we shall linkf ( theS, S expectation+∆) 2f on(S,the S + RHS∆)=2 of2 Eq. (13)ℓ˜ is,(t, instead, T, S) + a “quadratic”(t, T, S, ∆ contract) , (14) where f (S, S +∆) is the forward LIBOR, which satisfies: f (S, SS +∆) = (deliveringl (∆Et)). a payo BecauseT ffequal to f t(S, S +∆) f E(S,t S +∆). To see howPf this contract is(14) useful, note lS (∆)t = fS (S, S +∆), ft (S, S +∆) is a martingale undert Q . Therefore,Et assumingS QF T 2 that the −2 T Qt F T bp bp F Q S f (S, S +∆2 ) f (S, S +2∆)=2− ℓ˜ (t, T, S) + (t, T, S, ∆) , (14) deliveringF aE payot ffequalT to fT (S, S +t ∆) ft (S, S +E∆t ). To# see how this$P contractf is useful, note where ft (S, S +∆) is the forward LIBOR, which satisfies: ft (S, S +∆) = Et that(lS by(∆ Itˆo’s)).% lemma Because and− the& Girsanov− theorem, lSinformation(∆) = fS ( inS, this S + market∆), ft ( isS, driven S +∆) by is Brownian a martingale motions, under theQ forwardF S . Therefore, price,thatZ bywheret assuming( Itˆo’sS, S lemma+∆ and), thatsatisfies the Girsanov the theorem, # $ price, Zt (S,S where% & T where QbpT 2 2 2 f Q fT bp f bp lS (∆) = fS (S, S +∆), ft (S, S +∆) is a martingale under QF S . Therefore, assumingℓ˜F ( thatt, T, S) theT f (S, S +∆) v (S, ∆) vF (T,˜∆) v (S, ∆) dτ. (15) informationthe following: in this market is driven by Brownian motions, the forward price, Zt (S, SQ +∆E),t satisfiesfT (S, S +∆) τ ft (S, S +Qτ∆)=2Et τ ℓ (t, T,τ S) + Pf (t, T, S, ∆) , (14) F T ˜2bp ≡ 2 t2 − f F T ˜bpf −f bp Et fℓT (S,(t, S T,+ S∆) ) ft!f(τS,(S, S S++∆)=2∆) vτE(tS, ∆) ℓ vτ(#(t,T, T,∆ S)) +vτP(fS, (∆t,) T,d S,τ$.∆) , (14)(15) information in this marketdZ is(S, driven S +∆ by) Brownian motions, the forward price, Zt (S, S +∆), satisfies≡ −t # − $ the following: τ z On the other hand,% by taking! expectations& under the T -forward probability of a Taylor’s expansion = v (S, ∆) dW S (τ) ,τ (t, S) , (11)where % (11)& ## $ $ τ F On the2 other hand, by taking expectationsT under the T -forward probability of a Taylor’s expansion the following: dZZ ((S,S, S S ++∆∆)) ∈ whereof f (S, S +∆) withbp remainder, we obtain2 the following:f f f ττ z T ℓ˜ (t, T,T S) f (S, S + ∆) v (S, ∆ ) v (T,∆) v ( S, ∆) dτ. T- (15) = v (S, ∆) dW S (τ) ,τ (t, S) , of f 2 (S, S +∆)bp with remainder,(11) we2 obtainτ thef following:τ f τ f τ dZ (S, S +∆) τ F T ℓ˜ (t, T, S) f (≡S, St +∆) v (S, ∆) v (T,∆) v (S, ∆−) dτ. (15) Z (S,τ S +∆) z z Q T τ ! τ τ τ 2 τ F ␶ ∈ F 2 ≡ t 2 #− $ whereWF S (τ) is a multidimensionalwhere W ( Brownian= vτ ( motionS, ∆) dW underF S (τQ)F,S ,τ and(vt,τ S(S,) ,∆) is aE vectortOn thefT other(S, of S instan-hand,+∆!)(11) byf takingt (S, S expectations+∆ ) under the T -forward probability fT (S, ofS a Taylor’s expansion QF T 2 −2 # $ Zτ (S,z S +∆) ∈ f (S, S +∆) Qf (S, S˜ +∆) z On theEt other2 T hand,2 by taking expectationstF T ℓf (t,T,S) under the T -forward probability of a Taylor’s expansion whereW S (τ) is a multidimensionalQF , and v S Brownian motion under Q S , and v (S, ∆) is aof- vector=2fT remainder,(fS,%( SS,+ ofS∆+) instan-∆ with) − we remainder,& obtaine we the obtain 1following: the following: taneousF volatilities, adapted to WF (τ). F τ 2 t Q Et ˜ of f (S, S +∆2 ) with remainder,F T weℓf obtain(t,T,S) the following:− z T =2ft%(S, S +∆) E&t e 1 whereW S (τ) is a multidimensionaled to WF (). Brownian motion under Q S , and v (S, ∆) is a vectorQ offt(S,S instan-#+∆) # $ $ taneous volatilities,F adapted to W S (τ). F τ 2 F T 2 f 2 − ∞ f Because we shall deal with varianceF swap security designs referencing rates insteadE of pricesfT (S, S,we+∆) ft (S, S +∆) Q T + 2 t ft(S,S#+∆) 2 #Putt (Kf$,T,S,$∆) dKf + Callt (Kf ,T,S,∆) dKf , (16) F 2 f − ∞ f taneous volatilities, adapted to W S (τ). Et fTP(tS,(T S) '+∆0 ) ft (S, SQ +∆)˜ ft(S,S+∆) ( considerBecause the we forward shall deal LIBOR with equivalent varianceF swap to Eq. security (11), as designs follows: referencing rates instead+ =2 offprices2%(!S, S −+,we∆Put) t &(KF fT ,T,S,eℓf (t,T,S∆) dK) f +1 ! Callt (Kf ,T,S,∆) dKf , (16) Pt (T ) 0t Q ˜ Et f (S,S+∆) (16) 2 ' F T ℓf (t,T,S) − t ( referencing rates instead of prices, we consider the forward LI=2-ft%(S, S +!∆) E&t e 1 ! Because we shall deal with variance swap security designs referencing rateswhere,instead of pricesft(S,S,we#+∆) # − $ $ consider the forward LIBOR equivalent to Eq. (11), as follows: 2 f T ∞ f where, + ft(S,S#+∆) # Put$t (K$ff,T,S,∆) dKf f + f Callt (Kf ,T,S,∆) dKf , (16) df τ (S, S +∆) 2 P (T ) ℓ˜ (t,f T, S) T v (S, ∆) ∞v (T,∆) vf (S, ∆) dτ, (17) consider the forward LIBOR equivalent tof Eq. (11), as follows: + t ' 0Putf (Kf ,T,S,∆) dKτ f + τ Callft(S,Sτ(+K∆f) ,T,S,∆) dKf , (16)( = vτ (S, ∆) dWF S (τ) ,τ (t, S) , P (T )and: ˜! (12)t ≡ tf f ! −f t t ' 0 ℓf (t, T, S) !vτ (S, ∆) vτ#f(tT,(S,S∆+)∆) vτ (S, ∆) dτ$, ( (17) dffττ ((S,S, S S++∆∆)) f ∈ ! ≡ t ! − = v (S, ∆) dW S (τ) ,τ (t, S) , (12)where, (12) ! # $ df (S, S +∆) τ F where, zT 7 z fτ (τS, S +∆) f ∈ where, Call (100 (1 K ) ,T,S,∆) Put (100 (1 K ) ,T,S,∆) = v (S, ∆) dW S (τ) ,τ (t, S) , f (12)˜ T t f f f f f t f f 1 τ F z Put (K ,T,ℓf∆(t,) T, = S) v7τ (S, ∆) −vτ (T,∆) vτ (S, ∆) ,dτ, Call (K ,T,∆(17)) = − . f (S, S +∆) t ˜ f f ≡ t f f − t f where, by Itˆo’s lemma, vτ (S,τ ∆) 1 fτ− (S, S +∆) vτ (S, ∆). ∈ ℓf (t, T, S) vτ (S,!∆) vτ (T,∆)100vτ (S, ∆) dτ, (17)(17) 100 ≡ − ≡ t #− $ f 1 z ! # $ where, by Itˆo’s lemma, vτ (S, ∆) 1 fτ− (S, S +∆) vτ (S, ∆). z z 7 f ≡ ! − 1 " z and:Put (Kz,T,S,∆) and Call (Kz,T,S,∆) are the prices of OTM puts and calls written on Zt (S, S +∆) where, by Itˆo’s lemma, vτ (S, ∆) 1 fτ− (S, S +∆) vτ (S, ∆). and: t 7 t LIBOR Variance Contracts and≡! Volatility− Indexes" and: f f with strike pricez Kz and maturity T ;Put (Kf ,T,S,z ∆) and Call (Kf ,T,S,∆) are hypothetical OTM and:f Callt (100 (1 Kf ) ,T,S,∆) f t Putt (100 (1 Kf ) ,T,S,∆) t LIBOR Variance Contracts and! Volatility Indexes" Putz t (Kf ,T,∆) = − , Callt (Kf ,T,∆)z = − . f CallEuropean-stylet (100 (1 Kf ) ,T,S, options∆)100 onf the forwardPut LIBORt (100 (1 rate.K100f ) ,T,S,∆) LIBOR Variance Contracts and Volatility Indexes Putt (Kf ,T,∆) = z − z, Callz t (Kf ,T,∆) = − z . Putand:z (K f,T,S,100∆) and CallzCall(K ,T,S,t (100∆) (1 are theKf prices) ,T,S, of OTM∆) puts andf calls written100 on ZPut(S, St +(100∆) (1 Kf ) ,T,S,∆) LIBOR Variance Contracts and Volatility Indexes tPutt z (Kz,fT,T,,S∆) = t z t − z,T,S , Call (K f ,T,∆) = t − . We define the basis point LIBOR integrated rate-variance as, Finally,t by matchingf Eq. (16) to Eq.f (14),t we are able to use options on Z to price the basis with Kz and maturityz T ;Put (100Kf ,T,S,∆) and Call (Kf ,T,S,∆z ) are hypothetical OTM 100 WeWe define define the the basis basis point point LIBOR LIBOR integrated integratedraterate-variance-variance as, as, z fz Call (100 (1 Kf ) ,T,S,t ∆) f t Put (100 (1 Kf ) ,T,S,∆) ratePut-variancet (Kz,T,S, as,∆) andputs CallPutt ((KandK,T,z,T,S, ∆calls) =∆) written aret the− prices on ofZ OTMt (S,,S putsCall( andK ,T, calls∆) = writtent on Z−t (S,z S and+∆) . European-stylepointt z f volatility options on of thef forward,100 asz follows: LIBOR rate. t f 100 6 Putt (Kz,T,S,∆f)z and Callt (Kz,T,S,∆)f are thez prices of OTM puts and calls written on Zt (S, S +∆) T with2 strike price Kz andmaturity maturityFinally, by T matching;Putt t( Eq.Kf ,T,S, (16)z,T to,S∆ Eq.) and (14), Call we aret (K abletf,T,S, to usez,T∆ options,)S are hypothetical on Z to price the OTM basis- f,bp TT Putz (K ,T,S,∆) and Callz (K ,T,S,∆) are the pricesf of OTM puts and calls writtenf on Z (S, S +∆) 2 f 22 pointwitht volatilityz strike of pricef, asK follows:z tandz maturity T ;Putt (Kf ,T,S,∆) and Callt (Kf ,T,S,t ∆) are hypothetical OTM f,Vf,bpbp (T,S,∆) 22f (6S, S +∆) ffv (S, ∆European-style) dτ, options on the forward LIBOR rate. f f t τ τ withbp strike price Kz and maturity T ;Put2 (Kf ,T,S,∆) andQ CallT(Kf ,T,S,ℓ˜ (t,T,S∆) are) hypothetical OTMQ T bp VVtt ((T,S,T,S,∆∆)) ≡ tffττ((S,S, S S++∆∆)) vvττ((S,S,∆∆)) ddττ,, European-style(t, T, S, ∆ options) = 2 onf the(S, forwardt S +∆ LIBOR) rate.F t e f 1 F ℓ˜ (t, T, S) Finally, by matchingP Eq. (16) to Eq. (14), wet Q areT able˜ to useE optionst Q T on Z to price the basisEt ≡≡ tt! 6 European-stylebpf options2 on the forwardF LIBORℓf (t,T,S rate.) F ˜bp " " Pf (t, T,Finally, S, ∆) =by 2 f matchingt (S, S +∆) Eq.Et (16)e to Eq.1 (14),Et we areℓ ( ablet, T, S to) use− options− on Z to price the basis !! "" " ""point" volatility of f, as follows:Finally, by matching Eq. (16) to Eq. (14), we are− able− to use options on Z to price the basis ! ft(S,S!+!∆) ! ft(S,S" +!∆" ) ! "" " " ! "" such that, by arguments similar to those leading to Eq." (3)" " in Section"" " 2, the fair valueoptionspointpoint of volatility the volatility on time of Zf, 2to as of follows:fprice, as follows:2 the basisf point volatility∞f of ff, as follows: ∞ f suchsuch that, that, by by arguments arguments similar similar to to those those leading leading to to Eq. Eq. (3) (3)"" in in Section Section"" 2, 2, the the fair fair value value of of the the time time+ Putt (Kf ,T,S,∆) dKf + Callt (Kf ,T,S,∆) dKf , PtQ(T ) #+0˜ Q Putft(S,S(+K∆) f ,T,S,∆) dKf%+ Call (Kf ,T,S,∆) dKf , bp 2 F T $ ℓf (t,T,S) F T ˜bp $ t t previous section, the fair value of the time deposit rate-variance(t, T, S, ∆) = 2 f (bpS, S +∆) e P (T )Q T 1 ˜ Q ˜ ℓ (Qt,T T, S) Q deposit rate-variance swap generated at t, and paying offat T , is theP following:f t bp Et 2 2t F# 0ℓf (t,T,SEFt)T ℓf (t,T,SF) ˜bp F T ˜bp (18) ft(S,S+∆) % Pf P(t, T,(t, S, T,∆) S, =∆ 2 )f =t ( 2S, Sf+(∆S,) SEt+−∆)e −E e1 Et ℓ 1(t, T,E S) ℓ (t, T, S) depositdepositWe define rate-variance rate-variance the basis point swap swap LIBOR generated generated integrated at attt,, andrate and-variance paying paying o oas,ffffatatTT,, is is the the following: following: f t $ t − − − − t $ swap generated at t, and paying off at T, is the following: ! ft(S,S+!∆) ! ! ft(S,S+!∆") !" " !" ! "" "" 2 ˜ 2˜bpf ! ft(S,Sf +!∆) !∞ "∞ f " f ! "" (18) where ℓf (t, T, S)+ and ℓ (t, T,2 S) are definedPut (K in,T,S, Eqs. (17)∆) dK and+ (15). Call (K∞,T,S,∆) dK , f,bp T bp + PPut(T )t (Kf ,T,S,∆)t dKff + f f Callt (Kf ,T,S,t f ∆) dKf f f, 2 P (T ) t + # 0 Putt (Kf ,T,S,ft(S,S∆)+dK∆) f + Call%t (Kf ,T,S,∆) dKf , f,bp f,Vf,bptbp (T,S,∆) bpbpP (t, T, S, ∆) ,T S, t An# index0 of basis point$ time deposit rate-volatilityf is,t(S,S+∆$) % 2 f f $ Pt (T ) 0 $ ft(S,S+∆) V (T,S,VVtt ∆)((T,S,T,S,∆f∆))(S,P S−Pf+f (∆(t,t,) T, T,v S, S,(S,∆∆∆)),T),Tdτ, ≤S,S, #$ $ (18) % t ≡ τ −− τ ≤≤ ˜ ˜bp (18)(18) t where ℓf (t, T, S) and ℓ (t, T, S) are definedbp in Eqs. (17) and (15). (18) ! P (t, T, S, ∆) is " " where ℓ˜ (t, T, S) and ℓ˜bp (t, T, S)bp are defined in Eqs.2 (17) andf (15). is " " f TD-VIf (t, T, S, ∆) 100 bp An index of basis point time≡ deposit× & T rate-volatilityt is, suchisis that, by arguments similar to those leading to Eq. (3)" in Section" 2, the fairwhere valueℓ˜ (t, of T, the S) and timeℓ˜ (Ant, T,index S˜) of are basis defined point time in˜bp Eqs.deposit (17) rate-volatility and (15). is, − bp QF T f,bp f where ℓf (t, T, S) and ℓ (t, T, S) are defined in Eqs. (17) and (15). QQ TT bpbpP (t, T, S, ∆) = EFFt f,Vf,bptbp (T,S,∆) . (13) bp (13) deposit rate-variance swap generatedf( att,t T,, and S, ∆ paying) = offat TV, is the(T,S, following:∆) . An index of basis pointwhere timeAn(t, index(13) T, deposit S, ∆ of) is basis rate-volatilityas in Eq.point (18). time is, deposit rate-volatility is, PPff (t, T, S, ∆) =EEtt Vtt (T,S,∆) . AnPf index(13) of basis point time depositbp rate-volatility is, Pf (t, T, S, ∆) bp # $ Note that the above formulaTD-VI involvesbp (t, T, S,an∆ equally-weighted) 1002 portfolio of OTM options. This is in f ≡ bp× & T t Pf (t, T, S, ∆) We face two complications.f,bp The firstbp is the same## maturity mismatch$$ arising in thecontrast government to the percentage volatility case where thebp weight of each option− is inverselybp 2 proportional to WeWe face face two two complications. complications.V (T,S, The The∆ first first) is is the( thet, T, same sameS, ∆) ,T maturity maturityS, mismatch mismatch arising arising in in the the government government TD-VIP (t,f T,(t, S, T,∆) S, ∆) P 100(t, T, S, ∆) t Pf the square of itsbp strike. MO (2012; 2013d)2 providebp f additional intuition2 about≡f this feature× by& showing T t bond case. The second complication− is that we are dealing≤ with a notion of basis pointwhereTD-VIbp variance,(t, T,f S,(t,∆) T, is S, as∆ in) Eq.100 (18).TD-VIf (t, T, S, ∆) 100 bondbond case. case. The The second second complication complication is is that that we we are are dealing dealing with with a a notion notion of of basis basis point pointhow thevariance, variance,P hedgingf portfolio of a basis≡ point× volatility& indexT ditff≡ers from× that& of a percentageT t index, and − mismatch arising in the government bond case. The second Note that the above formula involves an equally-weighted− portfolio of OTM options.− This is in is which necessitates a different treatment from the percentage case. provide further analytical details on the behavior of the two indexes. whichwhich necessitates necessitates a a di difffferenterent treatment treatment from from the the percentage percentage case. case. bp contrast to thebp percentage volatility case where the weight of each option is inversely proportional to complicationbp is thatQ weT aref,bp dealing with a notion of basis point wherewhereMO (2012) contains(t,(t, T, T, S, the∆ S,) first∆ is) as formulation is in as Eq. in (18). of Eq. a variance (18). contract design cast in basis point terms F where Pf (t, T, S, ∆) isthe as square in Eq.P off its (18). strike. MO (2012; 2013d) provide additional intuition about this feature by showing In the percentage case,Pf it(t, is T, by S, ∆ now) = E well-understoodt Vt (T,S, that∆) . its formulation relates(13) to athat log-contract. applies to fixed income markets–namely, to interest rate swaps. In earlier work, Carr and Corso InIn the the percentage percentage case, case,variance, it it is is by by nowwhich now well-understood well-understood necessitates a different that that its its treatment formulation formulation fromNote relates relates the that per the to to- above a a log-contract. log-contract.how formula theNote hedging thatinvolves portfolio the abovean of aequally-weighted basis formula point volatility involves portfolioindex an diequally-weightedffers of from OTM that of options. a percentageportfolio This index, of isOTMin and options. This is in (2001)Note explain how that to the the above variance formula of price changes involves in markets an withequally-weighted constant interest rates. MO portfolio of OTM options. This is in The contract we shall link the expectation on# the RHS of Eq.$ (13) is, instead, a “quadratic”providecontrast further contract to analytical the percentage details on volatility the behavior case of wherethe two the indexes. weight of each option is inversely proportional to TheTheWe contract contract face two we we complications. shall shall link linkcentage the the The expectation expectation firstcase. is the on sameon the the maturity RHS RHS of of mismatch Eq. Eq. (13) (13) arising is, is, instead, instead,contrast in the a to a government “quadratic” the“quadratic” percentage(2013d, contractvolatility contract Chapter 2) case explain where that the the elegant weight replication of each optionarguments is in inversely Carr and proportional Corso (2001) break to- 2 2 contrastMO (2012) to contains the percentagethe first formulation volatility of a variance case contract where design castthe in weight basis point of terms each option is inversely proportional to delivering a payoffequal to22f (S, S +∆) 22ft (S, S +∆). To see howthe square this contract of its strike.down is MOthe useful, once square(2012; interest note of2013d) its rates strike. areprovide random, MO additional (2012; but that 2013d) the intuition random provide num´eraireabout additional this inherent feature intuition in each by about showing market this of feature by showing bonddeliveringdelivering case. a The a payo payo secondffffequalequal complication to toInff the(T(S,S, is Spercentage S that++∆∆ we)) aref f case,( dealing(S,S, S S it+ + is with∆ ∆ by).). a Tonow To notion see see well-understood how ofhow basis this this point contract contract variance, that is is useful,agethat useful, appliesvolatility note tonote fixed case income where markets–namely, the weight to interest rateof swaps.each Inoption earlier work, is inversely Carr and Corso TT −tt interestthehow square in the the hedging fixed of income its portfolio strike. space can of bea MO basis incorporated (2012; point volatilityinto 2013d) the replicating index provide di portfoliosffers additionalfrom and that the variance of a intuition percentage index, about and this feature by showing whichthat necessitates by Itˆo’s lemma a different and treatment the Girsanov from the theorem,−− percentage case. how the hedging portfolio(2001) of explain a basis how point to hedge volatility the variance index of price differs changes from in marketsthat of with a percentage constant interest index, rates. and MO thatthat by by Itˆo’s Itˆo’s lemma lemma and and theits the formulation Girsanov Girsanov theorem, theorem, relates to a log-contract. The contract we shall contract design. - provide further analyticalhow(2013d,provide details the Chapter further hedging on 2) the explain analytical behavior portfolio that the details of elegant the of twoona replication the basis indexes. behavior pointarguments of volatility the in Carr two andindexes. Corsoindex (2001) diff breakers from that of a percentage index, and In the percentage case, it is by now well-understood that its formulation relates to a log-contract. videdown onceMOadditional interest (2012) rates contains intuition are random, the first about but formulation that this the random feature of a num´eraire variance by showing inherent contract in each designhow market the cast of in basis point terms Q T Q T MO (2012)bp contains the first formulation of a variance contract design cast in basis point terms QQ F 2 2 QQ F ˜bp provide further analytical details on the behavior of the two indexes. The contract weEFFtT shallT 22 linkfT ( theS, S expectation+∆) 22f ont ( theS, S RHS+∆ of)=2 Eq.E (13)FFtTT is,˜˜bpbp instead,ℓ (t, T, a S “quadratic”) + bpbpPf (t, contract T, S, ∆hedginginterest)that, inapplies the portfolio fixed(14) to income fixed incomespaceof a can basis markets–namely, be incorporated point8 intovolatility to the interest replicating index rate portfolios swaps. differs and In the earlierfrom variance work, Carr and Corso EEtt ffTT((S,S, S S++∆∆)) f−ftt ((S,S, S S++∆∆)=2)=2EEtt ℓℓ ((t,t, T, T, S S)) that++P appliesPff ((t,t, toT, T, fixed S, S,∆∆ income)),, markets–namely,(14)(14) to interest rate swaps. In earlier work, Carr and Corso delivering a payoffequal to f 2 (S, S−+−∆) f 2 (S, S +∆). To see how this contract is useful, notecontract(2001)MO design. explain (2012) how contains to hedge the the variance first of formulation price changes in of markets a variance with constant contract interest design rates. castMO in basis point terms % T & t . To see how# this contract(2001)$ explain is useful, how to hedgethat of the variancea percentage of price changesindex, inand markets provide with constantfurther interest analytical rates. de MO- − ## $$ that(2013d, applies Chapter to 2) fixed explain income that the markets–namely, elegant replication arguments to interest in Carr rate and swaps. Corso (2001) In earlier break work, Carr and Corso thatwhere by Itˆo’s lemma%% and the Girsanov&& theorem, (2013d, Chapter 2) explaintails on that the the behavior elegant replication of the two arguments indexes. in Carr and Corso (2001) break wherewhere T down once interest rates are random,8 but that the random num´eraire inherent in each market of ˜bp TT 2 f f downf once interest rates(2001) are random, explain but how that to the hedge random the num´eraire variance inherent of price in changes each market in markets of- with constant interest rates. MO Q T bpbpℓ (t, T, S) 22f (S, S +∆Q )fTfv (S, ∆) ffv (T,∆) ffv (S, ∆) dτ. interest in the(15) fixed income space can be incorporated into the replicating portfolios and the variance F ℓ˜2ℓ˜ ((t,t, T, T, S S)) 2 ff ((τS,S, S S++∆∆))vFv ((τS,S,˜bp∆∆)) vv ((τT,T,∆∆)bp) vv ((τS,S,∆∆)) ddττ.. (15)(15) Et fT (S, S +∆) ≡ft (S,t τ Sτ +∆)=2Et ττ ℓ (t, T,τ Sτ ) + Pf (t,−interestτ T,τ S, ∆ in) , the fixed(14) income(2013d, space Chapter can be incorporated 2) explain into that the replicating the elegant portfolios replication and the variance arguments in Carr and Corso (2001) break ≡−≡ tt! −− contract design. !! # contract design.$ On the other% hand, by& taking expectations under# the##T -forward$ probability$$ of a Taylor’sdown expansion once interest rates are random, but that the random num´eraire inherent in each market of whereOnOn the the other other hand, hand, by by taking taking expectations expectations under under the theTT-forward-forward probability probability of of a a Taylor’s Taylor’s expansion expansion 2 T interest in the fixed income space can be incorporated into the replicating portfolios and the variance of22fT (S, S +˜bp∆) with remainder,2 we obtainf the following:f f 8 ofofffTT((S,S, S S++∆∆ℓ)) with with(t, T, remainder, S remainder,) fτ ( weS, we S obtain obtain+∆) vτ the the(S, following:∆ following:) vτ (T,∆) vτ (S, ∆) dτ. (15) 8 4 RISK≡ PROFESSIONALt MARCH 2014− contract design. www.garp.org Q T ! # $ QQ F 2 2 On theEFF othertTT 2 hand,2fT (S, by S taking+∆) expectations22ft (S, S + under∆) the T -forward probability of a Taylor’s expansion EE ffTT((S,S, S S++∆∆)) f−ftt ((S,S, S S++∆∆)) 2 tt of f (S, S +∆) with remainder,−−Q weT obtain˜ the following: T 2 QQ F ˜˜ ℓf (t,T,S) =222ft%(S, S +∆) EF&FtTT ℓℓfef((t,T,St,T,S)) 1 8 =2=2fft%t%((S,S, S S++∆∆)) EE&t&t ee 1−1 Q −− F T 2 f (S,S#+2∆) # $ $ Et fT 2(S, S +∆)t## ft (S,## S +∆) $$ $$ 22 fftt((S,SS,S−++∆∆)) f ∞ f + Q ˜ Putff t (Kf ,T,S,∆) dKf + ∞∞ Callff t (Kf ,T,S,∆) dKf , (16) ++ 2 F T PutPutℓf (t,T,S((K)Kff,T,S,,T,S,∆∆))dKdKff++ CallCall ((KKff,T,S,,T,S,∆∆))dKdKff ,, (16)(16) =2ft%(PS,t ( ST+) '∆) 0E&t e tt 1 ft(S,S+∆) tt ( PPtt((TT))'' 00! − fft!t((S,SS,S++∆∆)) (( !! # # $ $ !! 2 ft(S,S+∆) where, f ∞ f where,where,+ Putt (Kf ,T,S,∆) dKf + Callt (Kf ,T,S,∆) dKf , (16) Pt (T ) ' 0 T ft(S,S+∆) ( ! ˜ TT f ! f f ˜˜ ℓf (t, T, S) ffvτ (S, ∆) ffvτ (T,∆) ffvτ (S, ∆) dτ, (17) ℓℓff((t,t, T, T, S S)) ≡ vvττ((S,S,∆∆)) vvττ((T,T,∆∆)) v−vττ((S,S,∆∆)) ddττ,, (17)(17) where, ≡≡ !t −− T !!tt # $ ˜ f f ## f $$ ℓf (t, T, S) vτ (S, ∆) vτ (T,∆) vτ (S, ∆) dτ, (17) ≡ t 7 − ! # 77 $ 7 The next section illustrates how a variance swap design is affected by the num´eraire when deriving model-free indexes in markets.

Interest Rate Swaps

How is the variance swap security design affected by the presence of more complex num´eraires than the price of a zero? We now consider the pricing of variance swaps in the context of interest rate swap markets. For the sake of brevity, we focus on the basis point variance derivation to mirror market practice, which is also the formulation used for the CBOE SRVX Index. Let R (T , ,T ) be the forward swap rate prevailing at t, – i.e., the fixed rate such that the t 1 ··· n value of a forward starting swap (at T T0, with reset dates T0, ,Tn 1, tenor length Tn T , and ≡ ··· − − payment periods T1 T0, ,Tn Tn 1) is zero at t. It is well-known (e.g., Mele, 2013, Chapter 12) − ··· − − that R (T , ,T ) is a martingale under the so-called annuity probability Q defined through the t 1 ··· n A Radon-Nikodym derivative, as follows:

T dQA r ds PVBPT (T1, ,Tn) = e− t s ··· , dQ PVBPt (T1, ,Tn) ! T !G " ··· ! and PVBP (T , ,T ) is the “price! value of the basis point – ” i.e., the value at t of annuity paid t 1 ··· n over the swap tenor. We assume that R (T , ,T ) is a diffusion process, as follows: QUANT PERSPECTIVESt 1 ··· n dR (T , ,T )=R (T , ,T ) σ (T , ,T ) dW (τ) ,τ [t, T ] , (19) τ 1 ··· n τ 1 ··· n τ 1 ··· n · A ∈ where W (τ) is a Brownian Motion under Q , and σ (T , ,T ) is adapted to W (τ), and define A A τ 1 ··· n A the basis point realized variance of the forward swap rate arithmetic changes in the time interval [t, T ],1 T V bp (t, T ) R2 (T , ,T ) σ (T , ,T ) 2 dτ. n ≡ τ 1 ··· n ∥ τ 1 ··· n ∥ #t How do we design a variance contract in this case that allows model-free valuation? Consider the value of a payer swap withK fixed: rate, K: SwapT (K; T1, ,Tn) PVBPT (T1, ,Tn)[RT (T1, ,Tn) K] . ··· ≡ ··· ··· − The payoffof a payer is max Swap (K; ) , 0 and that. of a receiver is { T · } T ),0} and max Swap (K; ) , 0 . Notice that swaption prices contain. information about both interest rate {− T · } T ),0}. Notice that swaption - prices contain information about both interest rate volatility dom, but that the random numéraire inherent in each market and the value of an annuity. 9This reveals yet another funda- the replicating portfolios and the variance contractvolatility design. and the value of an annuity. This reveals yet another fundamental difference between equity The next section illustrates how a varianceand swap fixed design income is markets.price. Instead, Options swaps on equities and relate to aare single affected source by oftwo risk: sources the stock price. In- volatility andstead,volatility the value swaps and of and an the annuity. swaptions value of This an are annuity. reveals affected yet This by another two reveals sources fundamental yet another of risk: the fundamentaldifference swap rate, between diRfference( equity), and between the annuity equity affected by the numéraire when deriving model-free indexes in of risk: the swap rate, RT (.T T (.). and fixed income markets. Options on equities relate to a single source of risk: the stock price.· In- interest rate swap markets. factor,and fixed PVBP incomeT ( ). markets. Options on equities relate to a single source of risk: the- stock price. In- stead, swaps and swaptions are· affected by two sources of risk: the swap rate, R ( ), and the annuity stead,MO swaps (2012) and show swaptions howtility to component insulate are affected the in pure by a twomodel-free interest sources rate offashion. volatility risk: the They componentT swap· show rate, that inRT a ( model-freethe), and the fashion. annuity The next section illustrates how a variance swap design isfactor, affected PVBP by the num´eraire( ). when deriving · Interest Rate Swaps Theyfactor,T show PVBPvolatility thatT ( the and). annuitytheannuity value offactor factor an annuity. entering entering This into revealsinto the the yet payo anotherpayoffffof fundamental swaptionsof swaptions needsdifference needs to between be to worked equity into the · · model-free indexes in interest rate swap markets. MO (2012)variance showMO how (2012)and swapto fixed design insulate show- income howbe to the markets.worked allow to pure insulate for interest Optionsinto model-free thethe on ratepurevariance equities volatility interestpricing relateswap of rate component to thedesign avolatility singlevariance to sourceallow in component a contract model-free offor risk: model-free inthe in both fashion. stocka model-free percentage price. In- fashion. and stead, swaps and swaptions are affected by two sources of risk: the swap rate, R ( ), and the annuity They show thatbasisThey the point show annuity variance that factor the terms. annuitypricing entering More of factor intothe generally, entering thevariance payo MO intoff contractof (2013d, theswaptions payo in Chapter ffbothof needs swaptionspercentage 2) to develop be worked needs aandT framework· to intobasis be the worked that handles into the The next section illustrates how a variance swap design is affected by the num´erairefactor, when PVBP derivingT ( ). Interest RateThe next Swaps sectionnow illustratesconsider howthe apricing variance of swap variance designvariance isswaps aff swapected anyinvariance by designthe market the context num´eraire to swap andallow designof num´eraire when for model-free toderiving· allow of interest for pricing model-free in ofthe the fixed pricing variance income of contractthe space. variance Let in both us contract illustrate percentage in both how and this percentage framework and model-free indexes in interest rate swap markets. MO (2012) show how to insulate the pure interest rate volatility component in a model-free fashion. model-free indexesinterest in interest rate swap rateswap markets. markets. For thebasis sake point of brevity, variancespecializesbasis we point terms.They focus in variance theshow Moreon interest that generally,terms.develop the rate annuity More a swapMO framework factor generally, (2013d, case. entering Chapter MOthat into (2013d,handles the 2) payo develop Chapteranyffof swaptionsmarket a framework 2) develop and needs numéraire tothata framework be handlesworked into that the handles How is the variance swap security design affected by the presence of more complex num´eraires than the basis point variance derivationany to market mirror and anymarketConsider num´eraire market variancepractice, and a of variance swap num´eraireinterest design swap in to ofthe allowstarting interest fixed for income model-free atin thet with fixedspace. pricing the income followingLet of the us space.illustrate variance payo Letff contract: how us illustrate this in both framework percentage how this frameworkand the price of a zero?Interest We now Rate consider Swaps the pricing of variance swaps in the context of interest ratevolatility swapframework and the value specializes of an annuity. in Thisthe reveals interest yet another rate fundamentalswap case. difference between equity Interest Rate Swaps specializes inspecializes the interestbasis in pointrate the swapinterestvarianceand fixed case. terms. rate income swap More markets. generally,case. Options on MO equities (2013d, relate Chapter to a single 2) source develop of risk: a framework the stock price. that In- handles markets. For the sake of brevity, we... focus on the basis point variance derivation to mirror market bp How is the varianceLet R swapt (T1 security, ,Tn) designbe the aff forwardected by the swap presence rate ofprevailing more complex atany t num´eraires, market– i.e., andstead, than num´eraire swaps and of swaptions interest are in aff theected fixed by two income sources space. of risk: the Let swap us illustrate rate, RT ( ), how and thethis annuity- framework Consider a varianceConsider swap a variance startingπn ( swapt, at T )t with startingVn the( att, following Tt)withPn the payo(t, T following)ff: PVBP payoT ff(T: 1, ,T· n) practice, which is also the formulation used for the CBOE SRVX Index. factor, PVBPT≡( ). − × ··· Howthe is price the variance of a zero? swap We now security consider design the pricing affected of variance by the presence swaps in the of contextmore complex ofspecializes interest num´eraires rate in the swapoff: interest than rate· swap case. MO (2012) show! how to insulate the pure interest" rate volatility component in a model-free fashion. Let Rt (theT1,markets. price,T ofn) a For zero?beT theϭ theT We sake0 forward, with now of brevity, consider reset swap wedates rate the focus pricing prevailingT on0,... the of,T basis variancen-1, at tenor pointt, – swaps i.e.,variancelength the in the derivation fixedTn context-T, rateand toConsider of such mirrorpay interest- that market a rate variance thebp swap swap startingbp at t with the following payoff: ··· at T , whereπn (Pt,n T()t, T )They isVπn theshow(t,(t, Tfair that T)) the valueP annuityVn (t, of( Tfactort, the) T ) enteringcontractPVBP( intot, T such the)(T payo1, thatPVBPffof,T swaptionsn) (T needs, to,T be worked) into the practice, which is also the formulation used for the CBOE SRVX Index. ≡ n − n × Pn ··· T 1 n value of a forwardmarkets. starting For the swap sake of(at brevity,T TT1 0T we,1 with... focusT resetn onTn-1 the dates basisT0, point,T variancen 1, tenor derivation length T ton mirrorTvariance, and market swap design≡ to allow for model-free− pricing of× the variance contract··· in both percentage and ment periods≡ - , , - ) is zero ···at t. It −is well-known (e.g.,− ! bp " Let Rt (T1, ,Tn) be the forward swap rate prevailing at t, – i.e., the fixed rate such that the πn (t, T )! Vn (t, T ) Pn (t, T )" PVBPT (T1, ,Tn) payment periodspractice,T1 whichT0, is also,Tn theT formulationn 1) is zero used at t for. It the is well-known CBOE... SRVX (e.g., Index. Mele, 2013, Chapterbasis 12) point variance terms.≡ More generally,− A MO (2013d,bp × Chapter 2) develop··· a framework that handles ··· Rtat (TT1,, where,Tn) is Pan martingale(t, T ) is the un fair- valueat ofT, the where contract P nn (t(,t,T such T) is)= the thatE tfairV valuen (t, Tof) the, contract such that (20) value of− a forward··· starting− swap− (at T T0, with reset dates T0, ,Tatn 1T, tenor, where lengthPnT(nt, TTany), and is market the and fair num´eraire value! of interest the contract in the fixed suchincome" that space. Let us illustrate how this framework Let Rt (T1, ,Tn) be the forward swap rate prevailing at t, – i.e.,− the fixed rate such that the that Rt (T1, ,Tn) is ader martingale the so-called under theannuity so-called≡ probability annuity probability QA···QA defined through− the ···payment periods··· T1 T0, ,Tn Tn 1) is zero at t. It is well-known (e.g., Mele,at 2013,T , where ChapterPspecializesn ( 12)t, T ) is in the the interest fair value rate of swap the case. contract! such that" value of a forward starting− ··· swap (at− T− T0, with reset dates T0, ,Tn 1, tenor length Tn T , and A bp Radon-Nikodym derivative, as follows: − A Consider(t, a T variance)= swapV starting(t, at Tt)withA, thebp following payoff: (20)(20) that Rt (T1, ,Tn) is a martingale under≡ the so-called annuity probability···whereQEA defineddenotes through conditional− thePn expectationEt (t,n Tunder)= theV annuity(t, T probability) , QA . Moreover, by Eq. (19)(20) payment periods···T T , ,T T ) is zero at t. It is well-known (e.g.,t Mele, 2013, Chapter 12) Pn EtA nbp 1 0 n n 1 Pn (t, T )=Et Vn (t, T ) , (20) Radon-Nikodym derivative,− ··· as follows:− T − and Itˆo’s lemma, π (t,! T ) V bp (t, T )" (t, T ) PVBP (T , ,T ) dQ PVBPT (T1, ,Tn) n n !Pn " T 1 n that Rt (T1, ,Tn) isA a martingale underrsds the so-called annuity probability QA defined through the A ≡ − ! × " ··· ··· = e− t ··· A , A t T where Et denoteswhere conditionaldenotesA expectation conditionalwhere underE expectation denotes the annuity conditional! under probability the annuityexpectation" Q probabilityA . Moreover, under theQ by .annuity Moreover, Eq. (19) by Eq. (19) Radon-Nikodym derivative,dQ asdQ follows:A PVBPrsds PVBPt (T1, T (T,T1, n),Tn) Ewheret denotes conditional expectation under the annuity probability Q . Moreover,A by Eq. (19) ! T = e− t ··· , Et A at2T , where Pn (t, T ) is the fair2 value of the contract suchA that bp A G dQ " PVBPand···(T Itˆo’s, ,T lemma,) Rprobability(T , ,T Q)AR (T , ,T )= V (t, T ) = (t, T ) . (21) ! T t 1 n and Itˆo’s lemma,Et T 1 n t 1 n Et n Pn ! !G " ··· and Itˆo’s lemma, ! ... T ··· − ···(t, T )= A V bp (t, T ) , (20) and PVBPt (T1, ,Tn) is the “price! dQ valuet (AT1, of, theTn basisr ds pointPVBP –T ”( i.e.,T1, the,T valuen) at t of annuity paid Pn Et n ! = e− t s , ! " and··· PVBPt (T1, ,Tn) is the “price! value of the basis point – ”··· i.e., the value at t of annuity# paid $ dQ PVBPt (T1, ,Tn) A 2 A 2A 2 2 2 2 A bp ! AA "bpbp over the swap tenor. i.e.,··· the value at !t ofT annuity paid over the swapwhere Etenor.t theRT second(T1, equality,TnE)t RA RfollowsT (t T ( 1T , 1 , by ,T n ,T Eq. ) n )= (20).R t ( ET 1t , V n,T n ()= t, TE )t = V n P (n t, ( Tt, ) T )= . P n ( t, T ) ,.(21)(21) (21) G " ··· E whereR (ETt1,denotes,T conditionaln) R expectation(T1, under,Tn the)= annuityE probabilityV (t, TQ)A . Moreover,= Pn (t, by T Eq.) . (19) (21) over the swap tenor. ! ... ···t T − ··· ······ − −t ······ t n We assume thatWeR assumet (T1, that,TR n()T is, a di,Tff! )usion isRt a ( diT process,ff1,usion,Tn process,) is as a follows: diffusion as follows: process,Similar as follows: to our derivationand Itˆo’s# lemma, of Eq. (16),$ we now! take the expectation"! " under the annuity probability and PVBPt (T1, ···,Tn)t is1 the “price!n value of the basis point – ” i.e., the# value at t of annuity$ paid ! " ··· ··· where the second# equality follows2 $ by Eq. (20). where the secondQwhereof equality a the Taylor’s second follows expansion equality by Eq. follows of (20).RA (2T by, Eq.,T (20).) with2 remainder,A obtaining:bp over the swap tenor. A E TR (1T1, ,Tn)n R (T1, ,Tn)=E V (t, T ) = Pn (t, T ) . (21) Similar to our derivationt ofT Eq.······ (16), we− nowt take··· the expectationt n under the annuity probability dRτ (T1, ,TdRnτ )=(T1, Rτ ,T(Tn1)=, Rτ,T(Tn1), στ (,TTn1), στ (T,T1,n) Similar,TdWn) AdW(τ to)A,(τ our) ,τ derivationτ[t,[ Tt,] T,] , (19) of Eq.(19) (16), we now take the expectation under the annuity probability- We assume··· that Rt (···T1, ,T···n) is a··· diffusion··· process,···· as follows:· Similar∈ ∈Q of to a our Taylor’s derivation expansion of Eq.of# R2 (16),(T , we,T$ now) with take remainder, the expectation! obtaining:" under the annuity probability Q T A where the second equalityT follows1 by Eq.n (20). ··· F 2 2 tation under2 2 the annuity··· probability QA QA of a Taylor’sEt expansionRT (T1 of, RT,T(nT)1, R,Tt (nT)1 with, ,T remainder,n) obtaining: where WA (τ) is a Brownian Motion under QA , and στ (T1, ,Tn) is adaptedQA of a to Taylor’sWA (τ), and expansion defineSimilar to of ourR derivationT (T1, of Eq.,Tn (16),) with we now remainder, take the expectation obtaining: under the annuity probability where WA (τ) is a Brownian Motion under Q , and στ (T1, ,T···n) is adapted to WA (Qτ...),T and··· define ···−2 ... ··· dR where(T , W,TA)= (␶R (AT , ,T ) σ (T , ,T ) dWQA, and(τ) , ␴␶τ(T1,[Ft,, TT]n,)2 of R(19) (T1, ,T2 n) with··· 2remainder, obtaining: the basis pointτ realized1 variancen ofτ the1 forwardn swapτ ···1 rate arithmeticn A changes inE thet timeRT interval(QT1,of a Taylor’s,TT n)Rt( expansionTR1,t (T,T1n, of) R,T(Tn1), ,Tn) with remainder, obtaining: ··· ··· ···Q · # ∈ 2 A ···$ − ··· ··· T ··· ∞ p the basis point realized1 variance of the forward swap rate arithmeticF T 2 changesQ inT the time2 interval r is adapted to WA (␶= F 2 - 2 Rt(T1, ,TSwpnn) (K, T) dK + Swpn (K, T) dK , (22) [t, T ], Et RT (T1, ,Tn) Rt (T1Q, T ,Tn) n,t n,t 1 Et PVBPRT ((TT1#,, ,T,T2 nF)) R2 (RT$ ,t (T,T1,) ···R,T2 (nT ), ,Tr ) ∞ p (22) [t, T ], where WA (τ) is a Brownian Motion underT Q , and στ (T1, ,Tn) is adapted··· to=t W−A1 (···τ), andEtn··· define%−T 0 1 n ··· t Swpn1 n (K, T) dK + Rt(T1, ,TSwpnn) (K, T) dK , (22)' ance of thebp forward swap2 A rate arithmetic changes2 in the time···R (T , ,T )& ··· − ··· n,t & ··· n,t V (Tt, T ) R (T1, ,Tn) στ (T···1, ,Tn) dτ. PVBPt (T11, n,Tn) 0 Rt(T1, ,Tn) Rt(T1, ,Tn) n τ # 2 $ ··· # 2R%t(T1, $,Tn) ··· ∞ ∞ ' the basis pointbp realized variance1 2≡ of thet forward··· swap∥ rate arithmetic··· 2 ∥ changes# in the2 time= interval···$ & r ··· r Swpnr (K, T) dK +& ∞···p Swpnp (K, Tp) dK , (22) Vinterval(t, T ) [t,T], R (T#1, ,Tn) στ (T1,= ,Tn) dτ.= r PVBPSwpn(T , pn,t,T()K, TSwpn) dK + (K,n,t T) dK +Swpnn,t (K, TSwpnn,t) dK ,(K,(22) T) dK , (22) 1 n τ t 1 n % 0 n,t Rt(T1, ,Tn) n,t ' [t, T ], How do we design a≡ variancet contract··· in this∥ case that··· allowsPVBP model-free∥twhere(T1, valuation? Swpn,Tn)n,t Consider(K,0 Tr ) the and Swpn···n,t (K,p & T) denoteR thet(T1, prices,Tn) of& receiver··· and payer swaptions at t, PVBPwheret%(T Swpn1, ,T(K,n) T%) and0 Swpn (K, T) denote the prices ofR receivert(T1, ,T andn) payer' swaptions at 't, # T ··· & ···n,t & n,t & ··· & ··· value of a payer swap with fixed rate, K: where Swpnr (K, T) and Swpnp (K, T) denote the prices of receiver and payer swaptions at t, How do we design a variance contractbp in this case2 that allows model-freereferencing valuation?2 referencing aConsider swap a with swap the tenor withn,t tenorTn rTnT andT andn,t strike strikeKK,, andp and expiring at atT .T . Vn (t, T ) Rτ (T1, ,Tn) στ (T1, r,Tn) dτ. p−n.t(K−,T n.t(K,T) denote the prices of re- ≡ t ··· where∥ Swpn··· (K,∥ T) andr Swpnreferencing(K, T a) swap denote withp tenor theTn pricesT and strike of receiverK, and expiring and at payerT . swaptions at t, value of a payer swap with fixed rate, K: # n,twhere Swpnn,t (K, Tn,t) and Swpn (K, T)− denote the prices of receiver and payer swaptions at t, Swap (K; T , ,T ) PVBP (T , ,T )[R (T , ,T ) K] . ceiver and payern,t swaptions at t, referencing a swap with tenor How do we design aT variance1 contractn in thisT case1 thatn allowsT model-free1 n valuation? Consider the ··· ≡ ···referencing a···referencing swap with− tenor a swapTn withT tenorand strikeTn TK,and and strike expiringK, and at T expiring. at T . −Tn–T and strike K, and expiring at T. value of a payer swap with fixed rate, K: − TheSwap payoT (ffK;ofT1, a payer,Tn) swaptionPVBPT is(T1 max, ,TSwapn)[R(KT ;(T) 1, 0, and,Tn) thatK] of. a receiver is ··· ≡ ···{ T · }··· − max Swap (K; ) , 0 . Notice that swaption prices contain information about both interest rate {− T · } 10 10 The payoffof a payerSwap swaptionT (K; T1, is,Tn max) PVBPSwapT ((TK1,; ) ,,T0 n)[andRT (T1 that, ,T ofn) aK] receiver. is ··· ≡ { T ···· } ··· − 10 9 max SwapT (K; ) , 0 . Notice that swaption prices contain information about both interest rate {− The payo· ff} of a payer swaption is max SwapT (K; ) , 0 and that of a receiver is { · } 10 max SwapT (K; ) , 0 . Notice that swaption prices contain information about both interest rate 10 {− www.garp.org· } 9 MARCH 2014 RISK PROFESSIONAL 5 9 QUANT PERSPECTIVES

Matching Eq. (21) to Eq. (22) leaves:

Rt(T1, ,Tn) 2 ··· r ∞ p ␶ Pn (t, T )= Swpnn,t (K, T) dK + Swpnn,t (K, T) dK . PVBPt (T1, ,Tn) ! 0 Rt(T1, ,Tn) # ··· " " ··· Matching Eq. (21) to Eq. (22) leaves: (23) ␭ adapted to origination. We assume that (1) loss-given-default (LGD) is constant; (2) the short-term rate rτ is a Eq. (23) provides the expression for the value of the variance swap in a model-free fashion. It(23) origination.r. Let n be the We initial assume number that of (1) names loss-given-default in the index decided (LGD) isat constant; (2) the short-term rate rτ is a is a portfolio of equally weighted swaptions,origination. as in the case for Wetime deposits assume (see Eq. that 18); however, (1) loss-given-default (LGD) is constant; (2) the short-term rate rτ is a diffusionR (T , process;,T ) and (3) default arrives as a Cox process with intensity λ adapted to r. Let n be the this portfolio is rescaledt 1 by then inverse of the annuity factor (theorigination. num´eraire in the We interest assume rate - thattime (1) tϭ loss-given-defaultT0, and let each constituent (LGD) is have constant; a notional (2) the value short-term 1/n, rate rτ is a 2 ··· diffrusion process; and∞ (3) default arrivesp diff asusion a Cox process; process and with (3) intensity default arrivesλ adapted as a to Coxr. Letprocessn be with the intensity λ adapted to r. Let n be the Pn (t, T )= swap market),initial whereas number the portfolioSwpn ofin (18) names isn,t rescaled(K, in by T the the) dK pricedi indexff+ ofusion a zero decided (the process; num´eraire atSwpn inand time the n,t (3)t(K, defaultT 0),dK and arrives let. each as constituenta Cox process have with a notional intensity␭. λ adapted to r. Let n be the PVBPt (T1, government,Tn) ! bond0 market).2 Rt(T1, ,Tn) ≡initial number# of names in the index decided at time t T , and let each constituent have a notional volatility and the value of an··· annuity." This1 reveals yetinitial another number fundamental of names" diff···erence in the betweenindex decided equity at time t T0, and let each constituent have a notional0 Intuitively,value tilting a variance, the swap same by the LGD, market num´eraire and the at T (which same is one intensity, in the cases dealtλ. ≡ n 1 initial number of names in theThe index(23)1 number decided≡ of at names time t havingT0, and survived let each up constituent to Ti is have a notional and fixed income markets. Optionswith in sections on on equities time deposits relate and interest to rate a single swaps) causes source its fair valueof risk: to be defined the under stock price.value In- , then same LGD, and the same intensity, λ. value n , the sameorigination.origination. LGD,1 and We We the assume assume samethat intensity, that (1)n (1) loss-given-defaultλ loss-given-default. (LGD) (LGD)≡ is isconstant; constant; (2) (2) the the short-term short-term rate raterτ risτ is a a however, thisThe portfolio number is rescaled of names by having the inverse survived of the up annuity to Ti is ( T i) j=1 (1 I τj T i ) where where τ␶jj is thethe time at which Eq. (23) provides thea market expression space where for all the the relevant value information of the is given variance by thevalue price swap of availablen , the in derivatives—an a same model-free LGD, andfashion.S the≡ same It intensity,− { λ≤. } n n stead, swaps and swaptions areexpectation affected under theby num´eraire two sources probability of ofThe interest. risk:3 number the swapdiorigination.ff ofusion rate, namesR process;T ( having We), and assume and the survived (3) annuity that defaultThe up (1) numberto loss-given-defaultarrivesT is as of( a namesT Cox1) process having(LGD)(1 with survivedis constant; intensity up) where (2) toλ adaptedT thei isτ short-termis(T thetoi)r. Let ratejn=1ber(1 is the aI τj Ti ) where τj is the factor (thetime numéraire at which in namethe interestj defaults, rate swap anddiff market),usion the outstanding process;· whereas and notionalname (3) default j defaults, is arrivesi (τ)= and asi athe Cox(τ )withoutstanding processj=1 with(It) τnotionalj intensityTni1. isλ Nadapted(␶j)=1/nS to≡r. Let nτbe− the{ ≤ } is a portfolio of equally weightedFinally, an indexswaptions, of interest rate as swap in volatility the case is for timeThe deposits number (see of Eq. names 18);having however, survivedS upn to≡Ti is (Ti−) { ≤ } (1 τ T ) where τj is the 1 factor, PVBPT ( ). time at which!N name jSdefaults, andN the1 ≡j outstanding=1 I j notionali is (τ)= (τ)with (t) 1. the portfolio in (18) is rescaledtime atby whichthe priceinitialdi nameinitialff usionof numberaj numberzerodefaults, process; (the of of names andnu and names- (3) the1S in(␶ default in) the outstanding the with index index arrivesN decided(t)ϭ decided notional1. as a The atCox at time isprocess time indext S(t τT)= loss with0T, and ≡, andat intensity let(τ␶ letj)with eachshould each−λ constituentadapted{ constituent (≤obligort) } to1. haver. have Let a notionaln a notionalbe then · The index loss at τj should obligor j default is LGD I t τ T ,whereasthepremiumat! 0 n Ti is 1 !N S N ≡ this portfolio is rescaled by the inverse of the annuity factor1 time (the2 at num´eraire which name in thej interestdefaults,n j rate andbM the outstandingN notional≡ isS (τ)=N (τ1 )with≡ (t) 1. MO (2012) show how to insulate the pure interestbp rate volatility2 component1 1 in a model-free fashion.{ ≤ ≤ } 1 ≡ n méraire 1in the governmentIRS-VIn1(t, T )bond100 market).Pn (t, T ) The index loss at τj should obligor j default! is LGD TI i t τ T ,whereasthepremiumatTi is The≡ index× T losst valueinitial atvalueτjn numbershould, the, the same sameobligorof names LGD, LGD,jj in anddefaultdefault andthe the index the is sameis sameLGD decided intensity, n intensity, I t atτ j time Tλ bM. λ . t,whereasthepremiumat 1whereasT0, and the letN premium each constituent Sat n T iisis j haveNbM a notional≡ swap market), whereas the portfoliob CDX int ( (18)M) isn rescaled(Ti). byFinally,$ the− The price the index ofvalue an zero loss of protection (the at τ num´eraireshould leg minus obligorin the premiumj default{ leg≤ is is≤ LGD} ≡ ,whereasthepremiumat{ ≤ ≤ } T is They show that the annuity factor entering into× theS payo1 ffof swaptions1 needs1 to be workedj into1 the 1 n I t τj TbM n n i Intuitively,2 tilting a varianceCDX swap(M )by thevalue The(marketT )., number Finally,the numéraire same of the LGD, names value andCDX having of theprotection t ( M same survived ) intensity, leg ( Tup minusi). Finally, to Finally,λT. premiumis the the{ ≤(T valuevalue)≤ leg is } ofof protection protection(1 leg leg minus) where premiumτ is the leg is where Pn (t, T ) is as in Eq. (23). b t 1 n Thein number1 of namesb having survivedn up toiTi is (iTi) j=1 (1 I τIj τTi T ) wherej τj is the government bond market). × CDXS (M) (T ). Finally, the value× S of protection legS minus≡ premiumj=1 − leg{ is ≤j } i variance swap design to allowat T for (which model-free is one pricing in the of cases the variancedealt withb contract in tsections in both non percentage timei 1 minus and premium leg is S ≡ n −1 {1 ≤ } Intuitively, tilting a variance swap by the market num´eraire attimeTtimeThe(which at at whichnumber which× is oneS name of namein thenamesj casesjdefaults,defaults, having dealt and survived and the the outstanding up outstanding to Ti is notional notional(Ti) is isj=1(τ(1()=τ)=Inτj (Tτi()withτ))with where τ(tj)(ist) the1. 1. Credit DSXt = LGD v0t CDXt (M) v1t, S ≡ !N! − 1{ Sn≤ } N ≡ basis point variance terms.deposits More generally, and interest MO (2013d, rate swaps) Chapter causes 2) develop its fair avalue framework to· be −de that-b handles · 1 1 1 N 1 S N ≡ Thetime index at which loss name atDSXτj tjshould=defaults, LGD obligorv0 andt j theCDXdefault outstandingt (M is) LGD1v1t, DSX notionalt = LGD is ,whereasthepremiumat(vτ0)=t CDX(τ)witht (M) v1t(,t) Ti 1.is with in sections on time deposits and interest rate swaps) causesThe its fair index value loss to be at definedτj should under obligor j default is LGDn IntI τtj τjTbMTbM ,whereasthepremiumatn Ti is any market and num´eraire ofCredit interest volatility in can the be priced fixed through income credit default space. swaptions, Let although us illustrate the nature of how credit this risk frameworkDSX· t−=b LGD v0t · CDXt{(≤M{ ≤)≤ ≤v!1t},N}· − b S · N ≡ calls for a number of new features to take into account. For example,1 1 we need to consider1 credit1 1 a market space where all the relevantwhere v information0t is the value is given at t byof the $1Theb CDX price paidCDX indext ( ofoMtff( availableM)at loss) then at( timeT derivatives—anτ(ji).Tshouldi). Finally, of Finally, default obligor the the of value aj value representativedefault of· of protection protection is− LGDb firm, legn I legt minus providedτ minusj ·TbM premium premium,whereasthepremiumat leg leg is is Ti is specializes in the interest rateis variancegiven swap swaps by case. onthe loss-adjusted price forwardof available position in a CDSderivatives index, and web must— dealan with expectation the× survivalSn { ≤ ≤ } where 3v is the1 value at t of×1 $1S paidwhere offatv the0t is time the of value default at t ofof a $1 representative paid offat the firm, time provided of default of a representative firm, provided expectation under the num´erairecontingent probability probability and the defaultable of interest. annuity market0t num´eraire toCDX account3 for(M default) risk. (T ). Finally, the value of protection leg minus premium leg is under thedefault numéraire occurs probability before the of indexinterest.whereb expiry;vt andis the wheren valuevi1 att ist thewhereof $1 value paidv0t is at the offt valueofat an the at annuity timet of $1 of ofpaid default $1 off paid at of the at arepresentative time of defaultfirm, provided Consider a variance swap startingWe only present at thet with percentage the variance following contract formulation payoff: here. The risk0t we are dealing× withS 1 default occurs before the index expiry;default and where occursv1 beforet is the the value index at1 t expiry;of an annuity and where of $1v1t paidis the at value at t of an annuity of $1 paid at Finally, an index of interestisFinally, that of aT rate CDS1 an, index, swapindex,T forbM which volatilityof, until ainterest buyer pays either is periodicrate aswap premium default volatility (the CDS of index the is spread) representative and the firm orDSXDSX thet = expiry= LGD LGD ofv0 thetv indexCDXCDX (whichevert (M(M) )v1tv, , - default occurs before the index expiry;t and where· 0−vt1tb is the valuet · at 1tt of an annuity of $1 paid at seller insuresbp losses··· from defaults by any of the index’s constituents during the term of the contract. T1, ,TbM , until either· a default−1 b of the representative· firm or the expiry of the index (whichever T1, ,TbM , until either a default of the representativeDSX = LGD firmv1t or is the theCDX expiryvalue( Mat of )t theofv an, index annuity (whichever of $1 πn (t, T )If a constituentVoccursn ( defaults,t, T first)–the) theP defaultedn (t, T obligor value) ··· is removedPVBP of a fromdefaultableT T( theT1, index,, and,T,Tannuity then) index, until continues. either a default··· oft the representative0t firmt or the1 expiryt of the index (whichever ≡ − × 1 1 ··· bM occurs first)–the... value· of a−defaultableb annuity· . to be traded with abp prorated notional amount.occurs2 Options first)–the on a CDSwhere index···where value are European-style,v0 oftv0ist a isdefaultable the to the buy value value atpaid annuity att oft atof $1 .T $11 paid, paid,TbM off, o untilffatat the either the time time a of default of default default of of the of a arepresentative representative firm, firm, provided provided !IRS-VIAn CDS(t, T ) index100 payer" is anP optionn (t, T ) to enter a CDS index at T as a protection buyer with strike (payers) or sell (receivers) protection≡ at the strike× spreadT upont optionoccurs expiry. first)–the value of a defaultableA CDS index annuity payer. is an option to enter a CDS index at T as a protection buyer with strike at T , where Pn (t, T ) is the fairWe value assume of credit the events contract may occur oversuch aA sequence that CDS of regular indexdefaultwhere intervalsdefault payer (Tvi occurs01t,T occursisiis) with an the lengthbefore option before value the to at the indexentert indexof $1 aexpiry; paidCDSexpiry; o index andff andat where the at whereT timevas1tv ais of protectionis the default the value value of at buyer a att representativeoft of with an an annuity strike annuity firm, of of $1 provided$1 paid paid at at spread K. Upon exercise,$ the− protection− buyer would also receive a front-end protection1t arising from 1 A CDS index payer is an option to enter a CDS index at T as a protection buyer with strike b , for i =1, , bM, where M is the number of years the index runs, T0 is the time of the index ··· spread K. UponTdefault1T, exercise,, ,T occurs,TbM the, until,before protectionuntil either either thespreadof a indexbuyera defaulta defaultable defaultK expiry;would. Upon of ofannuity the also theand exercise, representative receive. representative where the av1front-endt protectionis firm the firm value or protection or buyer the the at expiryt wouldexpiryofarising an of also of annuity thefrom the receive index index of (whichever a$1 (whicheverfront-end paid at protection arising from losses occurringA beforebp the option matures.1 bM Accordingly, consider a loss-adjusted forward position at where Pn (t, T ) is as in Eq. (23). n spread······K. Upon exercise, the protection buyer would also receive a front-end protection arising from where P n ((t,t,T T)=Et Vn (t,11 T ) , losses(20) occurring before the option matures. Accordingly,T consider as a loss-adjusted forward position at t in a CDS indexlosses that starts occurring at occursT1 before,occurs, which,T first)–the the first)–thebM can, option until be value shownvalue either matures. of to of a a equaldefaultable a default Accordingly,defaultable the of following: the annuity annuity consider representative. . a loss-adjusted firm or the forward expiry position of the index at (whichever ! " losses··· occurring before theta optioninprotection a CDS matures. indexbuyer thatwith Accordingly, startsstrike spread at considerT , whichK. Upon a can loss-adjusted exercise, be shown the forwardto pro equal- position the following: at A t in a CDS indexoccurs thatAA CDS first)–the startsCDS index index at valueT payer, payerwhich of is ais canandefaultable an option be option shown to annuity to enter to enter equal. a aCDS the CDS following: index index at atT Tasas a aprotection protection buyer buyer with with strike strike where E denotes conditionalCredit expectation under the annuity probabilityt in aQ CDSA . Moreover, index that by starts Eq.tection (19) at T buyer, which would can also be shown receive to a equalfront-end the following:protection arising Creditt L spreadA CDS1K. Upon index exercise, payer is the an protection option to buyerenter a would CDS also index receive at T aasfront-end a protection protection buyerarising with strike from DSXt,T (τspread) K(.τ) Uponv1τ CDX exercise,τ (M the) CDX protectiont (M) buyer, would also receive1 a front-end protection arising from and Itˆo’s lemma, L 1from losses occurring before theL option matures. Accordingly, ≡ b NDSX (τ) (τ)−v CDX (M) DSXCDXt,T((Mτ)) , (τ) v1τ CDXτ (M) CDXt (M) , Credit volatility can bealthough priced through the nature credit of defaultcredit risk swaptions, calls lossesforspread althoughlosses a number occurringK occurring. the Upon oft,T nature beforenew exercise, before of the credit the the optionL option protectionrisk1τ matures. matures.1 τ buyer Accordingly, Accordingly, would alsot consider receive consider≡ b N a a loss-adjusted afront-end loss-adjusted protection forward forward− arising position position from at at " ≡DSXb N t,T (τ) (τ#) v1τ−CDXτ (M) CDXt (M) , A 2 2 A bp tlossesin a CDS occurring index before that starts the option at T ,≡ matures. whichb N can Accordingly, be shown to consider equal− the a loss-adjusted following: forward position at calls for a numberEt RT ( ofT1 new,features,T featuresn) where to takeR tot CDX(takeT into1, τ into( account.,TMn) account.)= is definedE t ForV Fornexample, as(t, example, the Tt) in value we= aP weCDSn need of(t, need TCDX index) to. to conτ (consider thatM- ),that startsset(21) credit suchstarts at thatTat", T which a forward can be position shown toat# equalτ in the the following:" # ··· − ··· where CDX (M) is defined" as the value of CDX# (M), set such that a forward position at τ in the sider creditindex variance is worthless, swapswhere viz on DSX loss-adjustedCDXLτ ((Mτt)in) = is a forward0, defined CDS index position as the that value starts of atCDXT ,τ whichτ(M), canset such be shown that a to forward equal the position following:τ at τ in the variance swaps# on loss-adjusted$ forward position in a CDS! index,τ,Twhere and" weCDX must(M deal) is with defined the survival as the value of CDX (ML), set such that a forward position at τ in the τ L indexL isL worthless,1 1 viz DSXτ (τ) = 0, wherecontingent the second probability equality andfollows the by defaultable Eq. (20). annuityindex market is worthless, num´eraire viz to account DSXτ,T for(τ) default = 0,DSXL risk.DSXt,T (τ()τ) (τ()τv)1τv1τCDXτCDX,T τ (τM(M) ) CDXCDXt (Mt (M) ), , index is worthless, viz DSX (τt,T) = 0,≡ ≡b1Nb N − − Similar to our derivationprobability of Eq. (16), and we the now defaultable take the expectation annuity 1market under thenuméraire annuity to probabilityv τ,T LvF We only present the percentage variance contract formulation here. The risk we are0 dealingτ DSXt,T withτ(τ) (τ) v1τ CDXτ (M) CDXt (M) , F 2 CDXτ (M)=LGD + ≡. b N F" " 1 − # #v v Q of a Taylor’s expansionaccount of R (T for, default,T ) withrisk. remainder, obtaining: 1 v␶0 τ(Mvτ ␶ (0Mτ ), set τ A is that of a CDS index, for whichT 1 a buyern pays periodic premiumwhereb where (the CDX CDS CDXτ index(M(M) is spread)) is definedv1 definedτ and as( asτthe the) v the1τ value value of ofCDXCDXτ (M(M),), setCDX set suchF suchτ (M that)=LGD that a forwarda forward+ position position at at.τ inτ in the the ··· τ CDXτ (MN)=LGD1 + " vτ0τ. vτ # b - suchL thatL CDX a forwardτv(1Mτ )=LGD position(τ) v1 τat +␶b in the index. is worthless,v1τ viz (τ) v1τ seller insures losses from defaults by any of the index’s constituentsindexwhere during is CDX worthless, theτ ( termM) is vizof defined the DSX contract. as(τ the) = value 0, of CDXτ (M), set such that a forward positionN at τ in the QF T 2 2 index is worthless, viz DSXτ,TLτ,Tb (τ) = 0, N v1τ (τ) v1τ E RT (T1, ,Tn) Rt (T1, Note,Tn) that (τ) v1τ is the natural num´eraire in- this market.L ␶,T(␶)=0, Indeed, CDXt (M) is aN martingale tIf a constituent··· defaults,− the defaulted··· obligorN is removed fromindex the index, is worthless, and the index viz DSX continues(τ) = 0, Note that (τ) v1τ is the natural num´eraireNoteτ,T that in this(τ market.) v1τ is the Indeed, natural CDX num´erairet (M) is a in martingale this market. Indeed, CDXt (M) is a martingale Rt(T1, under,Tn) the “survival contingent probability” Qsc, defined through theN Radon-Nikodym derivative,F F as to be# traded2 with a prorated$ notional··· amount.r Options on∞ a CDSN Note index that arep European-style,(τ) v1τ is the to natural buy1 1 num´eraire in thisv0τv market.0τ vτv Indeed, CDXt (M) is a martingale = Swpn (K, T) dK + Swpn (K, T) dK under, (22) theCDX “survival(M)=LGD contingent probability”+ τ .Qsc, defined through the Radon-Nikodym derivative, as spread) follows: and the sellern,t insuresunder losses the “survival from defaults contingentn,t byN any probability” of Qsc, definedCDXτ τ ( throughM)=LGD the Radon-Nikodym+ F . derivative, as (payers)PVBPt or(T1 sell, (receivers),Tn) % 0 protection at the strike spread uponRt(T1under, option,Tn) the expiry. “survivalT contingent' probability”b1 b Qsc, definedv10τvτ1 throughτ (τvτ()τ thev)1τv1 Radon-Nikodymτ derivative, as ··· & follows: & ··· dQ sc r (u) dufollows:(T ) vCDX1T τ (M)=LGD + NN . = e− τ N T , T We assume credit events may occur over a sequence of regularfollows: intervals (Ti 1dQ,Ti)withlengthsc b (T ) v1Tv1τ dQsc (τ) v1τ r (u) du (T ) v1T r constituentp defaults, the defaulted obligor isdQ removedr from− the Note(τ) thatvτ1τr N(u(␶)) duv1␶ is theT natural numéraireN = ine− thisτ market. NIn- , where1 Swpn (K, T) and Swpn (K, T) denote the prices of receiverNote andNote$FT that payer that swaptions(τ()τv)1τv1isτ atis the= thete,− naturaldQ naturalsc num´eraire num´eraireN in in this, this market.( market.T ) v1T Indeed, Indeed, CDX CDXt (Mt (M) is) is a martingalea martingale n,t n,t % N τ r (u) du dQ r (τ) v1τ b , for i =1, , bM, whereindex,M andis the numberindex continues of years theto be index traded runs, with$ T0 isa prorated theNN timedQ ofr the index t = e− (τ) v1τ N F , FT (M) is a martingale under $theT “survival% contingentN referencing a swap··· with tenor Tn T and striker K, and expiring at underT .underNote$ the the that “survival “survival(τ) contingentv1 contingentτ$ is the natural probability”%dQ probability”r num´eraireNQscQ,sc defined in, defined this market.through( throughτ)$v1τ theIndeed, the Radon-Nikodym Radon-Nikodym CDXt (M) is derivative, a derivative, martingale as as where FT denotes the information set$ at time T , which$ includes the$ pathFT of the short-term rate only. − r N $ wherer denotessc the% informationN set$ at time T , which includes- the path of the short-term rate only. where F denotesfollows:underfollows: the the information “survival set contingent at timeFT probability”T, which$ includesQsc, definedthe pathp through of the$ short-term the Radon-Nikodym rate only. derivative, as to buy (payers)The prices or sell of a(receivers) payer and protectionT receiver with at ther strike strikeK spreadexpiring $ at T , are, for$ any τ [t, T ], SWτ (K, T; M) 11 where FT denotes the informationtive, as follows:dQ setsc at time T , whichT includes(T the) v1T path ofp the short-term rate only. p The prices of a payer and receiver with strikeThe pricesK expiringdQ of$ asc payer at∈T and, are, receiverr for(ru() anyudu) du withτ ([ striket,T T) v],≡1T SWK expiring(K, T; atMT) , are, for any τ [t, T ], SWτ (K, T; M) upon option expiry. follows: = =e−e−τ τ NN , , τ p ∈ ≡ The prices of a payer and receiver withdQ striker K expiringT at T , are,∈(τ) v for any τ [t, T ], SW≡ (K, T; M) dQdQsc F r r (u) du (T()τv)11τvT1τ τ - $ T$FT= e−%τ% N N , ∈ ≡ dQ $ r$ (τ) v r r 12 $F 1τ lar intervals (Ti-1,Ti) with length 1/b, forwhere i=1,where...F,bMdenotes, denoteswhere theM the information information set set$ atT$ at time timeT%,T which, which includes includesN the the path path of of the the short-term short-term rate rate only. only. 10 TFT 12 $$ $ 12 r r $ p p is the number of years the index runs, ThewhereT0The is prices the pricesF timedenotes of of a payerof a payer the and information and receiverwhere receiver F with setT withdenotes atstrike strike time KtheTKexpiring 12,informationexpiring which includesat atT , Tset are,, are, theat for time for path any any Tτ of, τwhich the[t,[ Tt, short-term], Tin SW],- SWτ (K,(K, rate T; TM; only.M) ) T $ ∈ ∈ τ ≡ ≡ The prices of a payer andcludes receiver the with path strike of theK expiring short-term at T rate, are, only. for The any τ prices[t, T of], SW a p (K, T; M) ∈ τ ≡ 1212 12 6 RISK PROFESSIONAL MARCH 2014 www.garp.org QUANT PERSPECTIVES

payer and receiver with strike K expiring at T, are, for any indexes viable for serving as the underlying of tradable prod- p sc + r ␶⑀[t,T ␶ (K,TM)N(␶) v1␶ . E␶ T (M)-K) ␶ ucts such as volatility futures and options. + + (τ) v sc (CDX (M) K) andsc SWr (K, T; M) + (τ) v sc (K CDX (M)) . sc1τ Eτ (K,TT M)+N(␶) v1␶ . Er␶ [Kτ - T (M)) ]. 1τ scEτ T + (τ) vN1τ sc (CDX· T (M) K+)− and SWr (K, T; M) ≡(Nτ) v1τ · sc (K −CDXT (M)) +. (τ) v1τ E Eτ(CDXsc T (M) K) and+ SW τ (K,r T; M) (τ) v1τ EEτ (Ksc CDXT (M)) . + N (τ) v·1τWeτ E assume(CDX thatT (M− ) K) andτ SW (K, T≡; MN) (τ· ) v1ττ E− (K CDXT (M)) . N NWe· assume· τ that ! − − " τ ≡N≡N · ·! τ − − " We assume that! " ! " We assume! that dCDXτ "(M) sc j(τ;M) ! " ! =" E e 1 η (τ) dτ ! " dCDXCDXτ (M) (M) −sc τj(τ;M) − - dCDXτ (M)τ = Esc ej(τ;M) 1 η (τ) dτ CDXdCDX(Mτ) (=M−) τ #e sc# j(−τ;M1 ) η $(τ) d$τ τ =Eτ E e sc 1 ηj((ττ;M)) dτ sc CDXτCDX(M) (M−) #+ −σ (τ#; Mτ ) dW− $(τ−)+ $e 1 dJ (τ) , τ + σ#(τ; M#) dW·sc (τ)+$ ej(τ$;M) 1− dJ sc (τ) , income asset classes. sc +# #scr j#$(τ;M) $ sc$ sc + (τ) v1τ E (CDX+scT (σM()τ; KM) ) ·anddW SW ((τK,)+sc T; M)e (τj)−(vτ1τ;M1E) dJ(K (CDXτ)sc, T (scM)) . where W scN(τ) is awhere· multidimensionalτ W (␶+− σ ( Brownianτ·; M) dWτ motion(τ)+ and≡NJesc−(τ)isaCoxprocess(withintensity· τ 1 −dJ (τ )(␶,) From a methodological research perspective, the logical next We assume that · # $ − where W sc (τ) is a multidimensional! Brownian" motion␩ and(.#) andJ sc jump(τ)isaCoxprocess(withintensity size$ ! j(.)) under the " step in supporting the creation of a market for standardized scη ( ) and jump size j ( )) under the survival contingent probability.sc# $ where W (·τ)sc is a multidimensional· dCDX Brownianτ (M) motionsc j(τ;M) and J (τ)isaCoxprocess(withintensitysc η (where) and jumpWWe wish( sizeτ) is toj ( a price)) multidimensionalsurvival under a credit thecontingent variance survival Brownian= swapprobability. contingentE originatedτ e motion probability.1 atηt and(,τ and) dτJ paying(τ)isaCoxprocess(withintensity offat T , as follows: · · CDXτ (M) − − η ( ) and jump size j ( )) under the survival contingent# # probability.$ $ ηWe( ) andwish jump to price size aj credit( )) under variance the swap survival originated contingentsc at t, and probability.j(τ paying;M) offsc at T , ast, follows:and options on these indexes in a way that is consistent with the un- · · · · + σ (τ; M) dW (τ)+ e 1 dJ (τ) , We wish to price a creditpaying variance off at(V T swapM, (ast, Tfollows: originated) Pvar·,M (t, at Tt)), and( payingT )−v1T , offat T , as follows:derlying yield or credit curves, the indexes themselves and the We wish to price a credit variance swap− originated× at# Nt, and$ paying offat T , as follows: where W sc (τ) is a multidimensional Brownian motion and J sc (τ)isaCoxprocess(withintensity (VM (t, T ) Pvar,M (t, T )) (T ) v1T , term structure of volatility in order to facilitate risk manage- where η ( ()t, and T ) jump is the size fairj ( )) value under of the− the survival contract, contingent and×N probability. we have defined the percentage variance as, Pvar,M · (VM·(t, T ) Pvar,M (t, T )) (T ) v1T , ment and formulation of trading strategies by end users. We wish to price a credit(VM variance(−t, T ) swapPvar originated,M (t,× T atN))t, and paying(T ) v off1Tat, T , as follows: where Pvar,M (t, T ) is the fair value of theT contract,− and we haveT× definedN the percentage variance as, 2 2 sc where Pvar,M (t, T ) is the fairwhereV valueM (t, Tvar,M of) the (t,T(V contract,)M is(σ t,the( Tτ); fairMP)var and value,Md(τt, we+ T ))of have thej( T(contract, defined)τv;1MT , ) dJ the and(τ percentage) .we have variance as, where Pvar,M (t, T ) is the fair valueT≡ oft the∥ contract,− ∥ andT ×t weN have defined the percentage variance as, % % FOOTNOTES V(t, T ) σ (τ; M) 2 dτ + j2 (τ; M) dJ sc (τ) . where PvarM,M (t, T ) is≡ theT fair∥ value of the∥ contract, andT we have defined the percentage variance as, In MO (2013d, Chapter 5,% Appendixt T D),2 we show%t that2 T sc VM (t, T ) σ (τ; MT ) dτ +2 j T (τ; M2 ) dJ (τ)sc. VM (t, T ) σ (τ; M) 2 dτ + 2 j (τ; Msc ) dJ (τ) . ≡ VtM (t,∥ T ) σ∥(τ; M) dτt+ j (τ; M) dJ (τ) . In MO (2013d, Chapter 5, Appendix% CDX≡ tt( D),M≡)∥ we∥ r show∥∥ that% t p 2 % %SWt t (K, T; M) %t % ∞ SWt (K, T; M) showed the remarkable property that the fair value of a Pvar,M (t, T )= dK + dK v K2 K2 In MO (2013d, ChapterIn MO (2013d, 5, Appendix1t Chapter& 0 5, D),Appendix we show D), we showthat that CDXt(M) p ' variance swap remains the same in this case. In MO (2013d, Chapter2 CDX 5,% Appendixt(M) SWr (K, D), T we; M show) that% SW (K, T; M) (t, T )= T t dK + ∞ t dK Pvar,M sc CDXtj((Mτ;)M2) r 1 2 p 2 sc v 2 KSW (K, T; M) ∞ SW (K,pK T; M) 1t & (CDXt,0 T2)=Ett(M) er t 1 j (τ;dKM+)CDXt(jM)(τ; Mt ) dJ (τdK) . ' (24) 2Pvar,M%− CDXSWt(M) (K,r T−; M−2) % −∞2 SW (2 K,p T; M) 2. Merener (2012) considers the replication of variance 2 v1t t t SW (K,K T; M) Kt SW (K, T; M) Pvar,M (t, T )= T (%&%0 ) t dK + %CDXt(M∞) * t +' dK Pvar,M (t, T )= 2 dK1 + 2 dK v1t & 0sc j(τ;MT) K 2 CDX12 t(M) scK 2 ' Accordingly, a credit2 volatility%Ev1tt & 0 indexesc is j(1τ;MK) j (τ; M) % j (2τCDX; M)t(MdJ)sc (τ) K. (24)' − % 2Et −e − 1 j (τ; M−)2 j%(τ; M) dJ (τ) . (24) Tt − t − − − 2 approximation, and does not rescale for the relevant no- sc (% ) jT(τ(;M% ) ) 1 2 * sc+ + 2 sc e j(τ;M1 ) j (τ; M) j (τ1; M2 ) dJ (τ)sc. (24) Accordingly,E at credit2 volatility indexe is 1 j (τ; M) j (τ; M) dJ (τ) . tion of market(24) numéraire in the manner we suggest in this Accordingly, a credit volatility− indexEtt is − − 1 − 2 −(% C-VI) tM (t, T ) 100− − Pvar,M− (2t, T ) * + section, thereby providing neither the index formulae and Accordingly,( %a credit) volatility≡ × indexT ist * + , −1 Accordingly, a credit volatility index is C-VIM (t, T ) 100 Pvar,M (t, T ) pricing in this section nor the hedging details described in Accordingly, a credit volatility index is ≡ ×1 T t C-VIM (t, T ) 100 , P−var,M (t, T ) - where Pvar,M (t, T ) is as in Eq. (24). ≡ × T t , The firstwhere termPvar on,M the(t, T ) RHS is as inof Eq. Eq. (24). (24) is model-free,−1 once we estimate the CDX defaulters intensity variance swaps priced and hedged based on parametric The first termC-VI on the( RHSt, T ) of Eq.100 (24) is model-free, once1 we( estimatet, T ) the CDX default intensity incorporated by v .where The second Mvar,M term (t,T is small for all intentsPvar and,M purposes, and should not materiallyassumptions. where Pvar,M (t, T ) is as1t in Eq. (24).C-VIM (≡t, T ) ×100T t Pvar,M (t, T ) incorporated by v1t. The second term is≡ small, for× all− intents T andt purposes, and should not materially Theaff firstect the term valueaff onect theof the value RHS index of of the Eq. approximated index (24) approximated is model-free, by by only only retaing retaing, once− we the the estimatefirst first term. term. the CDX default intensity 1t. The 3. Interestingly, our model-free expression for the variance whereincorporatedPvar,M (t, by T )v is. as The in secondEq. (24). term is small for all intents and purposes, and should not materially where Pvar,M (t,1t T ) issecond as in Eq.term (24). is small for all intents and purposes, and should - aTheffect first the value term of on the the index RHS approximated of Eq. (24) is by model-free, only retaing once the we first estimate term. the CDX default intensity The first term onnot the materially RHS of Eq. affect (24) the is value model-free, of the onceindex we approximated estimate the by CDX defaultment of intensity the forward swap rate, as it turns out by compar- incorporated by v1t. The second term is small for all intents and purposes, and should not materially incorporated by v1t. The second term is small for all intents and purposes, and should not materially affecta theffect value the value of the of index the index approximated approximated by only by retaingonly13 retaing the first the term. first term. 13 Closing Thoughts that complements the asset pricing foundations laid down The viability of volatility indexing using the methodologies 13 - - describe the behavior of variance risk-premiums under both interpretations of n (t,T 13 13 options used in the various index formulas render the resulting

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REFERENCES - able from www.garp.org/risk-news-and-resources/2013/ Journal of Financial Eco- december/ nomics Interest Rate Finance Institute. - - Quantitative Finance 12, 249-261. Journal of Finance - Journal of Portfolio Management 20, 74- Energy & Power Risk Management 4, 80. - Quantitative Finance 1, 19-37. script. - Institute. - Mele, Antonio, 2013. Lectures on Financial Economics Journal of Financial Economics manuscript. Available fromhttp://www.antoniomele.org. Fixed Income Securities Index. Available from http://www.cboe.com/micro/srvx/ default.aspx. Antonio Mele is a Professor of Finance with the Swiss Finance Institute in Lugano. as a tenured faculty at the London School of Economics. His academic expertise covers - - - from its think tank of academic researchers. Previously, he managed U.S. and Asian Ibid, 2013d. The Price of Fixed Income Market Volatility. manuscript.

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