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Interest Rate Variance Swaps and the Pricing of Fixed Income Volatility B Y ANTONIO MELE a ND Y OSHIKI O BAYA SHI

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Interest Rate Variance Swaps and the Pricing of Fixed Income Volatility B Y ANTONIO MELE a ND Y OSHIKI O BAYA SHI QUANT PERSPECTIVES Interest Rate Variance Swaps and the Pricing of Fixed Income Volatility 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 interest rate 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 swap 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 option 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 derivative, 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 exercise 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.
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