QUANT PERSPECTIVES

Interest Rate Variance Swaps and the Pricing of Fixed Income Volatility BY ANTONIO MELE

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ne of the pillars supporting the recent movement toward standardized measurement and trading of interest rate volatility (see http://www.garp.org/ risk-news-and-resources/2013/december/a-pushto-standardize-interest-rate-volatility-trading.aspx) atility pricing. The meaning of this mouthful is best understood

Y O S H I K I O B AYA S H I

and explicitly account for the distinct characteristics of each 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. The price of volatility derived in this framework carries a clean and intuitive interpretation as the fair market value of

challenging, mainly because of the high dimensionality 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 income market and serves as the underlying for standardized futures and options contracts for volatility trading. A model-free options-based volatility pricing methodology prices and exchange rates) was branded and popularized as the ogy has been carried over to other markets, such as those for gold, oil and single stocks. In contrast, to create analogous volatility indexes for the

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desired volatility in a strictly model-free fashion? The practical answer depends on the magnitude of two isAmerican option prices. A second relates to the mismatch in maturity between the options and the underlying futures — i.e., one-month options are used to span risks generated by threeFortunately, in practice, situations arise in which the modeldependent components may be presumed small enough, such -

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value of of aa variance variance swap, swap,value and to to know whenswap, the approximation approximation is of aathe tolerable magnitude given value ofknow variance swap, and to to know know is when the approximation of aa tt of aa variance and when approximation isis given of value and when the of tolerable magnitude context. context. context. context. (S, T) be the forward price at forthe delivery atprice S, of ofat a coupon coupon bearingat bond expiring Let F (S, T) be the forward price for delivery delivery at S, of of coupoaa Letprice t T) forward t,t, for S, aa coupon Let FFtt(S, at t,t,be for delivery at S, aat bearing bond expiring Let Ft (S, T) be the forward Bt (T) B (T) t (T) tprice awith variance swap, and know when the where approximation is ofprice a bond tolerable BtS (T) T) = where Pto (S) is = the at ofPP zeroisis coupon with tt ≤ ≤ SS ≤ ≤ T, T,value i.e. FFof (S, T) = Bprice (S) the price atexpi t of of t≤ ≤ ≤, T, T, i.e. i.e. F Ftt(S, t (S, T) the at texp with t= ≤ PS (S) T) at tt of aatt(S) zero coupon bond with i.e. (S),, where t (S, tt(S) Ptt(S) , where Ptt(S) is the PP context. (T) isis the the price atand ofB the underlying bond. at SS ≥ ≥ t,t, and and B Bt(T) (T) the price price at tt of of the the underlying underlying bond. bond. atprice ≥t, t, and B isis the at at SS ≥ tt(T) at tt of the underlying bond. at t value ofLet a variance swap, and to forward knowrate whenprocess the approximation is ofbe a tolerable magnitude given T) be the price at t, for delivery at S, of a coupon bearing Ft (S, be the instantaneous short-term and let Q the risk-neutral probabilit Let r be the instantaneous short-term rate process and let let Qthe be th t Let r t short-term and Q be Letshort-term rtt be the instantaneous instantaneous rate process and let Q berate the process risk-neutral probability Let rt be the context. Bt (T) (S, T) = , where P (S) is the price at t of a zero co with t ≤ S ≤ T, i.e. F (S, satisfies well-known (see, (see, e.g., Mele, 2013, Chapter Chapter 12) that in2013, diffusion setting, t(see, e.g., t is well-known well-known Mele, 2013, Chapter 12) that in aT) aT) diffusion set t(S, P (S) in isT) (see, Mele, 12) that in diffusion tdelivery satisfies isis well-known e.g., Mele, 12) that aaatdiffusion setting, FFtbond be 2013, the forward price ate.g., t, for S, Chapter of a coupon bearing expiring at T,sett Let Ft (S, Bt (T)price at t of the underlying bond. (T) is the at S ≥ t, and B t with t ≤ S ≤ T, i.e. Ft (S, T) = Pt (S) , where Pt (S) is the price at t of a zero coupon bond expiring (S,T) T) dFτ (S, (S, T) process and let Q be the risk-ne dFττ (S, dF dF is the price t(S, of the underlying at S ≥Let t, and instantaneous rate rtBbe t (T) τthe = vat vτ (S, (τbond. )T) , = τ∈ (t, S) T)short-term ·dW dW vττ(t, (τ)),, ττ ∈∈(t, (t,SS (S, T) dWFFSS (τ FSS (τ v∈ (S, T) = ) , τ= S) ,, ··dW T) · τ F (S, T) F Fττ (S, (S,and T) let τ rate process be the risk-neutral probability. It rt be theFinstantaneous FChapter T) T)e.g.,short-term τ (S, isLet well-known (see, Mele, 2013, 12)Qthat in a diffusion setting, Ft (

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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 one may (S,T) T) isisthese the instantaneous instantaneous volatility process adaptedvolatility to W WFSS (τ (τ multidimensional where to WFs v(! S- to (S, T) isis the theprocess instantaneous volatility process adapted to W WFFSBr ( where vττ (S, SB T) instantaneous process adapted (τ where (S, the volatility adapted to ),), aa multidimensional where vvττestimate F dFτ (S, T)dFτ (S, T) approximation errors. For example, a nian numerical experiment = v (τ ) , τ ∈ (t, S) , (S, T) · dW s S = v (S, T) · dW (τ ) , τ ∈ (t, S) , (1) forward probability, Q through the the Radon-Nikodym deriva motion under under the the S-forward S-forward probability, S , defined τ S-forward FQ FQ th nian motion motion under the probability, QRadon-Nikodym F τ probability, S , defined through FS-forward the nian the , defined through derivat nian motion probability, Fτunder (S, T) F FFS , defined through τQ(S, F S T) based on Vasicek (1977) model of short-term interest rate dy- derivative, as follows: as follows:as follows: S

that (1) the early exercise premium

as follows:

! ! SS !!!), a multidimensional process to WF S (τ where vτ (S, T) is the instantaneous volatility −! StSadapted rτ dτ dτ Browrrττdτ dQFSS!!!! e−volatility dQFFSS!adapted e−−tott W process where vτ (S, T) is the instantaneous S (τ ), a mu dQ e t rτ dτ Fderivative, dQ e ! = F !! nian motion probability, QF S , defined ,through the! Radon-Nikodym = ,, = embedded in American op- under the S-forward dQ (2) , = ! ! P (S) dQ (S) the , defined through Radonmotion under the S-forward QFdQ S PPtt(S) as nian follows: dQ !GGSS ! probability, Ptt(S) GGSS !S dQF S !! e− t rτ dτ as follows: denotes the the information information set at time time S. and GGS denotes (2) = set denotes theS. information set at time S.! and GGSSset !, time denotes the information and denotes the information atat time S. S. at and dQ !GS Ptset (S) − tS rτ dτ S ! as follows:

!

!

dQ

e

F Considerin the following payoff payoff of aa variance variance swap: maturity mismatch is negligible when the difference maturiConsider the following payoff of of aa!! variance variance swap:, = Consider the swap: Consider the following of swap: information setfollowing at time S. payoffdQ and GS denotes the Pt (S) GS ties is as small as two months, as in the example above. Consider the following payoff of a variance swap: S

(T,T) T) ≡information (T,S, S,T) T)− −PPππat (t, T,T) S,≡ T)VV ≤T) S,− (T, S, T) −PP(t, (t,T, T,S, S,T) T),, TT ≤ ≤ (T, T) ≡ the≡ set time S. and GS denotes (T, (t, T, S, T) ,,tt(T, TTS, ≤ S, ππ(T, VVtt(T,

π (T, T) ≡ Vt (T, S, T) − P (t, T, S, T) , T ≤ S, Maturity and Numéraires Mismatches Consider the following payoff swap: ""TT of a variance 2 ""TT ∥vwhere 22 dτ is is " To illustrate the maturity mismatch issue,where suppose that available 2 where V (T, S, T) ≡ (S, T)∥ dτ is the percentage integrated variance, and the the fair value value o the percentage in-fair where V (T, S, T) ≡ ∥v (S, T)∥ the percentage integrated vari T t τ 2 t τ where V (T, S, T) ≡ ∥v (S, T)∥ dτ is the percentage integrated varia t t dτ is the percentage τ Vt (T, S, T) ≡ Vtt(T,∥vS,τT) (S, integrated variance, and of t where ≡ T)∥ t ∥vτ (S, T)∥ dτ is tthe percentage integrated variance, and the fair value of the for trading are European-style options expiring at T on a 10tegrated variance, of the strike (t,T,S, ) isT ≤ S, strike PP(t, (t,T, T,T) T) strike (t, T,T) T) isisand π (T,the T)fair ≡ Vvalue strike T) is PP(t, T, t (T, S, T) − P (t, T, S, T) , strike isis P (t, T,strike year Treasury note forward expiring at S T. The option span # $ $$ " T##1 ""TT − "tT 2rτ dτ ""TT $$ ## QQ 1 operates under the so-called T-forward probability, whereas the 1 − r dτ E e P (t, T, S, T) = (T, S, T) =− E−Q (T, S, T)) , (3) (3) Q V T 1 r dτ t t tintegrated τdτ Vdτ(T, t FtTT(V 1 ≡ EttPt ∥v τ where VtT) (T,=S, T) (S, T)∥ is the percentage variance, and r dτ t FFT τ τ − r e S, T) = E (V (T, S, T)) , P (t, T, S, (T ) (T,T)) S,T) T) = =EEQ (t,tT, T,S, T) τS,T) t= EEtt=eeEtt Ft (Vtt(T, S, VVtt(T, eP(t, (T,PS, T) S, , Vt= P (t, T, -S, T) = Pt (T )Et P tt (T ) Ptt(T ) strike P (t, T, P T)t (T is ) the second equality follows by a change of probability, Et denotes conditional expectation under ability, which generates risks in the forward price that where cannot Q $ the wherethe the second follows bysecond change of probability, denotes conditional expectation under the T# -forward probability. That is, fair of uc the equality risk-neutral probability, Etequality "EETatadenotes where the second equality follows by change of probability, denotes t denotes be spanned by the set of all available options. Unless T=S, we where the change of probability, EEvalue conditional expectation under the risk-neutral probwhere second equality follows by aaand change of probability, conditional expectation un t r 1 follows −by QFtT t denotes QF T dτ Q τ TT Q the variance swap is the expected realized variance under the -forward probability. It is a notable F Q t T T Q E e (T, S, T) = E (V (T, S V P (t, T, T) = T S, Fprobability. under the T -forward That is, the fair valu the risk-neutral probability, and E F t t t F under the T -forward probabili the risk-neutral probability, and E t ability, andtt t under under probability. is,is,the under theThat T -forward probabilit the risk-neutral probability, and Ett probability. the TT-forward the the risk-neutralpoint probability, )-forward Pthe t (Tcase of departureand fromEthe standard equity in which the fair value ofThat a variance swapfair is thevalu This issue is reminiscent of convexity problems arising in the variance swap is the expected realized variance under the T -forward probability. It is a not fair of the variance swap is the expected realized variance the value variance swap therates expected realized variance under the the -forwar the variance swap isis the expected realized variance under the variance swap is the expectation, expected realized variance under the T -forward probability. ItTTis-forwar a nota risk-neutral assuming interest are constant. under the T-forward probability. It is a notable point of deparwhere second equality follows by a change of probability, E denotes condition point of departure the standard equity case in which the fair value of a variance swap is t point of departure from the standard equity case in which the fair val va Forfrom thethe case at hand, we obviously cannot assume constant interest rates. Moreover, we still point of departure from the standard equity case in which the fair FT

FT

point of departure from the standard equity case inQ which the fair value of a variance swap is FT need evaluate the RHS of expectation, Eq. (3). Following theunder VIXinterest Methodology, one might try to link the the Trates -forward probability. That thetorisk-neutral probability, and Eassuming risk-neutral expectation, assuming interest rates are constant. risk-neutral rates are constant. t risk-neutral expectation, assuming interest are constant. risk-neutral assuming interest rates are constant. spirit of the standard VIX Methodology cannot be expectation, calculated expectation in Eq. (3) to the value of a log-contract (Neuberger, 1994)—i.e., a contract with a payoff variance swap is the risk-neutral expectation, assuming interest variance swap is case the cannot expected realized variance underassume the T -forward probab For the the case casethe at hand, hand,(S,T) we obviously assume constant interest rates. Moreover, we For the case at hand, hand, we obviously obviously cannot assume constant inter For the at we cannot intere For at we obviously cannot assume constant interest rates. Moreover, we time T . In the standard equity case, the value of a log-contract isconstant indeed minus equal to ln FFTtrates at the strictest level of theoretical rigor. are constant. (S,T) at point of departure from the standard equity case in which the fair value of a need to to evaluate the the RHS of to Eq. (3). Following Following the VIX Methodology, one might try to to link link need to evaluate evaluate the RHS RHS ofVIX Eq. Methodology, (3). Following Followingone themight VIX Methodolog Methodolo need the of Eq. (3). the VIX of try The silver lining is that, as long as Tneed and S-Tevaluate are small, theRHS ForEq. the (3). case at hand, wethe obviously cannot assume constant risk-neutral assuming interest are constant. expectation inearly Eq. (3) to toexpectation theexpectation, value of ofin a log-contract log-contract (Neuberger, 1994)—i.e., contract with with pa expectation Eq. (3) to to the the valuerates of aa1994)—i.e., log-contract (Neuberger, 1994)— Eq. (3) value of log-contract 1994)— numerical impact of both the maturity expectation mismatch and exin Eq. (3) the value ain (Neuberger, aa(Neuberger, contract aa pa 3 FT (S,T) F (S,T) T F (S,T) For the case at hand, we obviously cannot assume constant interest rates T (S,T) T . In equity thestandard value of of aaequity log-contract indeed m equalto tobe lnFignored, atequity time TTcase, . In In the the standard equity case, the the value of of equal tothe ln standard ercise premium is likely to be small enough time .case, case, value aa equal to ln FT (S,T) atastime the value log-contract isis indeed m equal to ln (S,T) at FFtt(S,T) Ftt(S,T) at time T . In the standard need to evaluate the RHS of Eq. (3). Following the VIX Methodology, one m expectation in Eq. (3) to the value of a log-contract (Neuberger, 1994)—i.e., a c (S,T) 3 value of a log-cont to realize that a VIX-like implementation leads to an approxiat time time T . In the at equal to ln FFTt (S,T) 33 standard equity case, 3the

mation of the true fair value of a variance swap, and to know a log-contract is indeed minus one half the expected realized half the expected when the approximation is of a tolerableone magnitude given the realized variance. In our case, it follows by Eq. (1) and Itˆo’s lemma that 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 Inthe our oneexpected case, half the it follows realized expected realized Eq. (1) and variance. In our Itˆo’s case, lemma In it ourfo 3 byvariance. FT (S, T) QF S QF S xt. f a variance swap, and when approximation of delivery aone tolerable magnitude the (4) Let to Ft know (S,T) be thethe forward price at t,isfor S, of halfatthe expected realized variance. In our case, it follows by Eq. (1) and Itˆ o ’s lemm Egiven (V (T, S, T)) = −2E ln . t t t " ! (S, T) F T) QF S at t, for delivery at S,atof aT,coupon atF ST, QF S t FQ QF S TF(S, couponprice bearing bond expiring with tbearing S T, bond i.e. expiringEQ S t.t Ft (S, T) be the aforward " (Vt (T, ! ln (V (T, S, T)) = −2E (V (T, = −2E S, T)) E E t t Q t = t Q t F (S,t T)t. S, T)) Bt (T) F (S, T) T S S F (S,T)= , where P (S) is the price at t of a zero coupon (S, T) = , where (S) is the price at t of a zero coupon bond expiring ≤ S ≤ T, i.e. F t t t F F t at S, of a coupon bearing bond expiring at T, Ft (S, T) be the tforward price Pt (S)at t, for delivery . the fair value lnin Eq. (3) is S, T)) = expectation −2Et Et to (V However, Eq. (4) does not link Eq. (3). The t (T, Ft (S, T) t (T) bond expiring at Sthet,underlying and Bt (T) isbond. the priceatatt tofofathe underlyT, (T)Fist (S, theT) price at t of ≥≤t,Sand Bti.e. = B zero coupon bond expiring ≤ T, contract expiring at T , whereas the expectation on the LHS of Eq. (4) is the value Pt (S) , where Pt (S) is the pricevariance However, Eq. (4) does not link to However, Eq. (3). However, Eq.The (4)expectation does Eq. not (4) does link in Eq. to notEq. (3) link (3). istofair the Eq. The fair(3) exv ing bond. be the instantaneous short-term rate process and let Q be the risk-neutral probability. It tt,rand t Bt (T) is the price at t of the underlying bond. variance contract expiring at S. In other words, the market num´ e raire for government bond cont However, Eq. (4) does not link to Eq. (3). The expectation in (3) is the fair variance contract expiring at T , whereas variance the contract variance expectation expiring contract on the at expiring T LHS , whereas of at Eq. T , the whereas (4) expectation is the the fair ex Let r be theChapter instantaneous short-term rate process and let T) Q satisfies of a variance contract expiring at S. In other words, the market -known (see, e.g., Mele, t2013, that inand a diffusion Ft (S,maturity instantaneous short-term rate12) process letexpiring Q besetting, the probability. ItTprice rt be the atrisk-neutral ae.g., given Y is of zeros expiring at Y,num´ where Y other = TS. in and Y variance contract expiring at , variance whereas the expectation on the LHS Eq. (4)(3) is words, the fai variance contract atthe S.for In other words, contract variance the expiring market contract at expiring S. eraire In at for words, InEq. other the bond mark be the risk-neutral probability. It is well-known (see, Mele, expiring numéraire government bond contracts expiring atof agovernment given (S, T) satisfies known (see, e.g., Mele, 2013, Chapter 12) that in a diffusion setting, F t in Eq. (4). atcontract variance expiring In other words, the market ewhere raire for Fexpiring t (S,T maturity at is the the price at ofexpiring expiring atnum´ ,iswhere =T inin a given maturity Y S. is expiring price of zeros azeros given expiring at maturity a given at Y, Y maturity the price Y= isgovernment of T the zeros price Eq.expirin (3) ofbon zea dFτ (S, T) = vτ (S, T) · dWF S (τ ) , τ ∈To(t,evaluate S) , (1) the RHS of Eq. (3), we need something more than Itˆ o ’s lemma. We have to ackn S expiring at a given maturity Y is the price of zeros expiring at Y, where Y = T in Eq. (3) Fττ (S, (S, T) T) in Eq. (4). in Eq. (4).in Eq. (4). dF = vτ (S, T) · dWF S (τ ) , edge τ ∈ (t, S)Eq. ,the dynamics under the “variance swap” pricing measure are those under Q that in (4). (1) of FT (S, T)(1)

To evaluate the RHS of Eq. (3), we Toneed evaluate something Tothe evaluate RHS more of thethan Eq. RHS (3), Itˆoof’swe Eq. lemma. need (3),something we Weneed havesom to m Fτ (S, T) volatility process adapted not to W ), aevaluate multidimensional Browvτ (S, T) is the instantaneous F S (τ To , as in Eq. (1). Now, by Girsanov’s theorem, we have the following: under Q S the RHS of Eq. (3), we need something more than Itˆ o ’s lemma. We have F the dynamics of FT (S, T) under (S, T) under measure T) are under “variance those theun “s edge that edge thatthe edge the“variance dynamics that theswap” dynamics of FT pricing of F T (S, the where v!(S, ) is the process instantaneous volatility process adapted of Fderivative, t (S,T through the motion S-forward probability, QF S , defined is thethe instantaneous volatility adapted to WFnot (τunder ),Radon-Nikodym athat multidimensional Browvτ (S, T)under S edge (S, T) under the “variance swap” pricing measure are those dynamics of F T , as in Eq. (1). Now, by Girsanov’s , as theorem, in Eq. , as (1). we in have Now, Eq. the (1). by following: Girsanov’s Now, by Girsano theore Qthe not not Q under Q FS FS FS dFRadon-Nikodym τ (S, T) ows: through the derivative, otion under the S-forward probability, !QF S , defined !S = (S, T) (v (T, T) − v (S, T)) dτ + v (S, T) we · dW ) , following: τ ∈ (t, T ) , , as in Eq. (1). Now, by Girsanov’s theorem, have the not under Q S−v τ τ τ τ F T (τ F Fτ (S, dQF S !! e− t rτ dτ dFT) dFτ (S, T)dFτ (S, T) τ (S, T) ws: (2) ! ! = − ! S rτ dτ , = −vτ (S, T) (vτ (T, T) − vτ (S, T))=dτ−v = −v +τv(S, T)T) (v·τ dW (T, (S,FT) (v −τ)v,(T, τT)T)) ∈−(t,dτ vτT()+ T (τ τ (S, τ (S, ! FdF Fτ (S, T) Fτ (S, T) T) T) dQdQ e Ptt (S) τ (S, τ (S, F S ! GS where W That is, the price Brownian motion under Q = −v (S, T) (v (T, T) − v (S, T)) dτ + T) · dW (τ ) , τ ∈ (t,is Tv.τ (S, T (τ ) is a multidimensional Tforward , (2) = τ τ τ F F F 2 RISK PROFESSIONAL 2014 Fτ (S, T) www.garp.org dQ M!GAS R C HPt (S) S denotes the information set at time S. , such that by Itˆ o ’s lemma, a martingale under Q T where WF T (τ ) isFa multidimensional where WFwhere is, the forward Brownian motion isWaF Tmultidimensional (τ under ) is aQmultidimensional Brownian Brownian motion pr u T (τ ) F T . That nsider thethe following payoffsetofata time variance denotes information S. swap: where WF T under . That is, the forward (τ ) is aQFmultidimensional Brownian motion under Q " ! T , such that by Itˆ o ’s lemma, , such that , such by Itˆ that o ’s lemma, by Itˆ o ’s lem a martingale a martingale a martingale under Q under Q T FT FFT

"This ! among To apply these spanning arguments to ourlinks context, consider alogTaylor’s expansion withtoremainder, (2001), among others. This approach links value of the log(and other) contracts to that of a (2001), others. approach thethe value of (and other) contracts (S, F (2001), among others. This approach the value of the Neuberger log-the (and other) contracts to and that Madan of a that of a Q(1999); Q S T Zou Bakshi andT) Madan (2000);links Britten-Jones and (2000); and Carr FS . (OTM) (4) ln Et F (Vt (T, S, T)) = −2E portfolio out-of-the-money (OTM) European-style options. of of out-of-the-money (OTM) European-style options. t portfolio portfolio of out-of-the-money European-style options. Fothers. T) among This links the value of the log- (and other) contracts to that of a T) (S, T) −approach Ft (S, T) FT (S, tF(S, ed variance. In our case, it follows by Eq. (1) and Itˆo’s lemma that(2001), Tthese To apply spanning arguments to context, aexpansion Taylor’s expansion with remainder, apply these spanning arguments ourour context, consider a Taylor’s expansion with remainder, = ln ToTo apply these spanning arguments to ourtocontext, consider aconsider Taylor’s with remainder, portfolio of out-of-the-money (OTM) European-style options. F F (S, T) (S, T) f the expected realized variance. In our case, it follows by Eq. (1) and Itˆ o ’s lemma that t t " ! ! # " ∞ (S, T) (S, −T) T) value F(S, apply spanning to our consider aaTaylor’s expansion with remainder, (S, T) T) − F (S, T)context, FTarguments F However, does to Eq. (3). TheToexpectation in Eq. (3) the fair of t (S, T" T− (S,FT) T) FtT) (S,is FT these tF T (S, FF (S,T) T) not link Q S Q S Eq. F(4) T (S, lnT = t= = " ! + 1 + 1 . (4)ln Flnt (S, ln Et F (Vt (T, S, T)) = −2Et F F F (S, T) F(K (S, T) T) (S, T) F F (S, T) (S, T) t t − − F (S, T)) (F (S, T) − K) dK + dK , t t t T is the fair2value of a T (S, T) Fand QF Scontract variance expiring TQ,Fby whereas expectation on the LHS of Eq. ! !T"(S, #K2 Tthe !"T) ## Four T)itat S Eq. (1) F(4) F the expected variance. In case, follows Itˆ o’s lemma that thatF t (S, K t (S, T) T (S, T) " ∞ Ft (S,T) "− " ∞ " ∞ one half one the half expected realized variance. InS, our case, follows by Eq. (1) and Itˆ o’s lemma F (S,T) Ft (S,T) F (4) (V T)) =it−2E Erealized t (S,T) 1 1 alf the expected realized variance. Int (T, our case, it follows byt Eq. (1)lnand Itˆo’s lemma. that ln F (S, T) =0 1 1 t 1 1 + + + ++ + ,+ Ft (S,(K T)−(K " F (S, T) − − for FT(K (S,F (FT (S, T) −(S, K)(S, dK t −T)) T) − K) dK, , − T))T)) dKdK ++ (FT(F T)K −2 dK K) variance contract expiring Qat !!S. FIn other eraire bond T (S, T TF(S, 2 !" 0 − government # 2KdK T) " twords, the market num´ Q K 2 contracts T (S, K 2K"F2 (S,T) K "is !Qt Eq. Ft (S,T) 0 0 Ft (S,T) . (4) (Vt (T, S, T)) = −2E ln F Ft (S,T) ∞ T)the Q T (S, SEt expectation S not link to Eq. expiring (3). The in (3) fair value of a F F 1 1 and take expectations under Q , T (S, FY FtT) (S, T)price + + Q F S Et . of zeros expiring at (4) Y, −where FY = (Vta(T, S, T))maturity =Q−2E T ln at given is the T in Eq. (3) and Y = S FS t (K − FT (S, T)) dK + (FT (S, T) − K) dK , (4) (Vt (T, S, T)) = −2Et Et Ft (S, .T) K2 K2 wever, Eq. (4) not link on to the Eq. LHS (3).ln in value Eq. and (3) the fair value of a QT , T , expectations under Q FEq. T) 0 Ft (S,T) F ,Q t (S,expectation t T , whereas thedoes ofThe (4) is the fair ofoftake ais and take expectations under expectations However, Eq. (4) does not link to Eq. (3). The expectation in Eq. (3) is the fair valueand atake % under $ QF T F F inexpectation Eq. (4). T) value Fthe QF T(4) %Et ofQ a(FT (S, T)) T$ eS. contract expiring at Tmarket ,to whereas the expectation oninofthe LHS of Eq. isof However, Eq. (4) does not link to Eq. (3). The for expectation Eq. (3)is is the fair value a(S, expiring at T , whereas the on the LHS Eq. (4) the fair value of a expectations Invariance other words, the num´ eexpectation raire government bond contracts $fair and Q% , (F owever, Eq. (4)contract does not link Eq. (3). The expectation in Eq. (3) is the fair value of aQ than = 1 acknowlln E QFTQ T) T)) − to FT$(S,under T(S, FET % F T have t take To evaluate the RHS Eq. (3), we need something more ’s lemma. We (F(S, (S, Q T) T)) EE FToF(S, =T)tFt (S, 1 T)) lnF TT)Itˆ Et of QtFa(S, T (S, T− F variance contract expiring at S. In other words, the of market num´ eraire for government bond contracts T t (FT variance contract expiring at T , whereas the expectation on the LHS of Eq. (4) is the fair value =t t(S,T) ln E eY contract expiring at S. In other words, the market num´ e raire for government bond contracts =F − 1− 1 ln E F (S, T) T) expected realized variance. In our case, it follows by Eq. (1) and Itˆ o ’s lemma that nce contract expiring at T , whereas the expectation on the LHS of Eq. (4) is the fair value of a t ! # t$ t is the price of zeros expiring at Y, where Y = T in Eq. (3) and Y = S % Q F F (S, T) (S, T) ! # " " F F (S, T) (S, T) T expiring at a given maturity Y is the price of zeros expiring at Y, where Y = T in Eq. (3) and Y = S t T) under the “variance swap” pricing are those edge that dynamics FT (S, " FFtt (S,T) variance contract expiring at S. Inthe other words, theof market num´ eraire for government bond contracts T)) Et 1F (F FT (S, tT)tmeasure (S,T) QF T !" under1 Q T (S, ! ## 1 1F T ," ∞ ∞ 1 nce contract expiring at S. In other words, the market num´ e raire for government bond contracts 1 " ∞ " " ∞ g at a in given maturity Y is the price of zeros expiring at Y, where Y = T Ein Y=− = S 1 1 ln (3) and Ft1(S,T) Eq. (4). FPut (S,T) " ! t− t Eq. − Put (K) dK + Call (K) dK , (8) (K) dK + Call (K) dK , (8) tt t t2 121 xpiring at a given maturity Y is the price of in zeros expiring at Y, where Y = T in Eq. (3) and Y = S t (S, T)the (S, T) 2 2 1 1 Eq. Now, Girsanov’s under Q(3), S ,zeros Pt− (T K KCall Ptfollowing: (T ) F−)t ! K K ng at a given Y is the price expiring at =lemma. T in Eq. (3) and Y = S we Fhave Put dK t+(S,T) Call dK, , (8)(8) (S, T) F (S,T) Put Q(1). Q Fof t (K) 2 dK t (K) # 2 dK To maturity evaluatenot the RHS of Eq. weas need something thanFItˆ oby ’s We have totheorem, acknowlTY t (K) t (K) 0"0 (K) denote SY, where 2of+FEuropean 2 FS Fmore " (K the prices puts T S t t (4). P (T ) K K under Q , not under Q P (T ) K K F (S,T) ∞ t t (4) 1 t ln We S, T))than =F −2E Etsomething F (Vt (T, more Ft (S,T) n Eq.(3), (4). we need 0 0 Ft (S,T) Eq. Itˆot’s lemma. have .to acknowl1 1 swap” pricing measure under QF , that the dynamics of FT (S, T) under the “variance (4). edge FtItˆ (S, T) are those − at K, the Put dKon + Callt (K) dKwe ,have(8) t (K) 2 and calls struck weofof have relied pricand to Call prices European andthe callsfollowing struck at K,K and where Put theorem, the following: To T) evaluate the RHS ofwe Eq.have (3), we need something morewe than o’sthan lemma. have to acknowlt (K) t (K) denote Pt (Tthe ) and K 2putsputs evaluate the RHS Eq. (3), we need something more oWe ’s lemma. We have acknowlasof in Eq. (1). by Girsanov’s theorem, the following: notthe under Qthe (S, T) dF 0 Fand t (S,T)calls struck at K, and we have (S, under “variance swap” pricing measure those under Q , Fof, Eq. τNow, T oT evaluate RHS (3), we need something more than Itˆ ohave ’s are lemma. WeItˆ have to acknowl(K) and Call (K) denote prices European where Put tF tand (K) and Call (K) denote the prices of European puts and calls struck and have where Put (K) Call (K) denote the prices of European puts and calls struck at at K,K, and wewe have where Put t t t t −vτ (S,swap” T) (vpricing − vτare (S,those T)) dτ on +Qthe vFτT (S, dWF Tequations: (τ ) , τ ∈ (t, T ) , (5) relied following τ (T, T) measure under , T) · pricing dge that the dynamics of FT (S, T) under the=“variance under the “variance swap” pricing measure are thosemeasure under Q , fair hat the dynamics of FT) T T) under the “variance swap” pricing are those under Qdenote at the dynamics of Flink T , equations: Fto T) T (S, relied on the pricing equations: Ffollowing TT)(S, dF τ (S, F ,not Eq. (4) does Eq. (3). The expectation in Eq. (3) is the value of a ). Now, by Girsanov’s theorem, we have the following: relied on the following pricing τ (S,not relied on the following pricing equations: (K) and Call (K) the prices of European puts and calls struck at K, and we have where Put t −vτ (S, T) (vby T) − vτ (S, theorem, T)) dτ + vτ we (S, T) · dW ) , τ ∈ (t, T ) , (5) t Now, Girsanov’s have the following: under QF S , as in Eq.=(1). τ (T, F (τ & ' & ' Callt (K) Putt (K) Q Q T) Now, by Girsanov’s theorem, we have the following: Eq. nder Q Q SS, ,asasin F τ (S, (1). in Eq. Now, the by Girsanov’s we have the(4) following: der = pricing Eof = Et − FT (S, T))+ , (FT (S, T) − K)+ . relied on the following equations: t tract FFexpiring at T ,(1). whereas expectationtheorem, on the LHS of Eq. isPut the fair value aQ(K & ' & & ' & ' ' P (T ) P (T ) (K) (K) Put Call & ' & Q Q (K) (K) Put Call Q t t t t (K) (K) Call T T t t + Q Q T T + + F the forward t F F + T))T)) + '+. . WF T (τ ) is is, a multidimensional Brownian motionis not under QFET .F TThat F TE dF = (K − (F(S, (S, T) − K) τ (S, =(K Et EF− E −F , , t is not T)− − K) K) where WT) )where is a multidimensional motion under QF . That is, the forward T (S, T T) TF(S, = E=t = , price . T)) t F(K F (τ t t (F(F T) dF(v T (S, T T(S, tt (T τ (S, = −v (S, T) (vdτ T)vBrownian −(S, vτ (S, T)) dτ + vTτ (τ (S,)T) ·raire dW (τ ) T , government τ, ∈price (t, T(5) )P, (Tbond (5)= T (t, P P ) (T ) τ(S, τ (T, P P (T ) (T ) expiring at S. In other words, the market num´ e for contracts & ' & ' F t ,tract T) (T, T) − v T)) + T) · dW , τ ∈ ) t t (K) (K) Put Call Q Q P ) (T ) τ τ τ t t = −v (S, T) (v (T, T) − v (S, T)) dτ + v (S, T) · dW (τ ) , τ ∈ (t, T ) , (5) T T T + one, but (S,(K) −isFTunder not t Ft (S, T) is= τunder τ Itˆ τ F (S, T)where F RHSQ W (! thatunder by o’s Q lemma, aFmartingale Q FT not QaF+ ,martingale onunder Eq.T)(8) is not τ T) (S, dF F τF, Tsuch o’s lemma, Because a martingale EF)tat Fmartingale , the firstt term = Ethe (FTofF(S, . (S, T)) − K) ( Fττ (S, T) t F T , such that by Itˆ Q ˜) Pt (7). (T (T ) Utilizing this expression and combining ℓ(t,T,S,T) T)the (v (T,ofT)zeros −price vτexpiring (S,isT)) (S, T)Y ·under dW (τBecause ) , Et (3) τPFt ∈ (t,T)TY ) not ,is, where (5) τ forward a given maturity Yτ (S, is the price Y, = TF Tin Eq. Sa )martingale " !vτwhere eand ℓ˜(t, T, S, T) isunder as Q in Q Eq. equals Q=T−v . That is, notdτat aQQ + martingale Q T) not , the first term RHS is not one, (S, is = first term onon thethe RHS of of Eq.Eq. (8)(8) is not one, butbut t((S, F T ,Because tF F T F, Tthe FT (S, Fτ (S, Q Q ) a martingale where W T) (τ ) isF a multidimensional under That is,T)the forward is not "under (S, T) is not a( under Q Because F T , the first term on the RHS of Eq. (8) is not one, but Q F T . ln tprice QFEq. ˜! , (6) (Vt (T,Brownian S, T))Brownian = 2Emotion (ℓ˜motion (t,under T, S, T)) 2E Et ˜ martingale F T F T) ( ℓ(t,T,S,T) WF T (τ F) Tis a multidimensional Q− . That is, the forward price is not Eq. (8) with (6), we find that the fair value of the variance swap in Eq. (3) is, ℓ(t,T,S,T) T t t ˜ ˜ F is as in Eq. (7). Utilizing this expression and combining e , where ℓ (t, T, S, equals E E e , where ℓ (t, T, S, T) is as in Eq. (7). Utilizing this expression and combining equals Ft (S, T) (S, T) is not a martingale under Q the first term on the RHS of Eq. (8) is not one, but Because F T , T) F (S, T) Q Q Q Q t ˜ t mensional Brownian motion under Q . That is, the forward price is not T T F t T T T T ( ) F FF F, where ℓ ˜(t, T, S,(T) ,is as in Eq. (7).(6) ’s lemma, a martingale under QF T , such that by ItˆoE eQℓ(t,T,S,T) Utilizing this inexpression and combining EtS, ˜ ln (Vt (T, S, T)) = 2Et F equals (ℓ˜(t, T, T)) T 2E )) o’s lemma, tingale under QF T , such that by Itˆ ℓ(t,T,S,T) F− t tEq. Eq. (8) Eq. that of variance swap ewith ,(6), where ℓ˜find (t,Q T, S,the T) is fair asvalue invalue Utilizing this expression and combining equals Et(8) Eq. with (6), we(is find that fair of (7). thethe variance swap in Eq.Eq. (3)(3) is, is, ˜ the ℓ(t,T,S,T) W That is, the forward price not (τwhere )RHS is we aItˆ multidimensional Brownian motion QoF’sT .lemma. Tthe ˜Eq. Fwe (S, T) defined ate of’s Eq. (3), we need something moreunder than We have to acknowlt e − ℓ (t, T, S, T) P (t, T, S, T) = 2 1 − E F that ! Itˆ " ch by ohave lemma, t with that the of the thevariance variance swap in (3) Eq.is,(3) is, "Eq. !Q T T) (8)Eq. FT (S, (8)Eq. with(6), Eq. we (6), find we find that the fairvalue value swap in)) Eq. Q T QF#T ˜ ( fair )) ( ( of ( !" # F F (S, T) " ∞ Q T Et F T (V Q ˜ T QF Q S,that T)) Q =Fby 2E (’s ℓ (t,lemma, T, S, T))swap” −F 2E , are (6)under Q T , T ln T ˜ ℓ(t,T,S,T) T the T t (6)those such o“variance under t (T, (S,T) ˜t Itˆ T) pricing measure engale dynamics ˜(t, F T ,(S, ℓ˜(t, T, S, e P (t, T) − EtT F eℓ(t,T,S,T) 2= 1 − ℓ− 1 ,T) (6) (VF S, T)) =under 2E T)) − 2E Et F ofQ " ! T,)) S,)) T)T) P (t, T, T, S,(FS, T)( = 2 21 −1F(E ˜(t, t (T, Ft (S, tT) ≡(ℓ−(t, T,vτS,(S, t T) − vln ( t T) (v (T, (S, T)) dτ. (7) ℓ T, S, τ τ + Callt (K) 2 dK . we have defined Ft (S, T) Q QT T ˜ ˜ Putt (K) ˜ 2 dK + FT (S, ! # Q Twhere # (9) t Q T (TE P (t, −)t EFt F0 !e"ℓ(t,T,S,T) eℓ(t,T,S,T) − T)FT) "t (S,T) " ∞ K t1 −ℓK ℓ˜(t, (t,T,T,S,S, P (t, T, S, T, T)S,=T)2 = 12P− (S,T)" ∞ " as in Now, byT)) Girsanov’s weT)have S ,T)) Ft (S,T) F , the !following: (6) ln ,FS, = Eq. 2Et F(1). (ℓ˜(t, T, S, − 2Et F theorem, 1 2 2 1 1 1(9) where we haveQdefined ! # (S, T) F # Q Q #. . T to the " dK Ft (S, term T) (t, T,TFS,TT) in addition T ˜by the “tilting” +" " Ft (S,T) Put Call dK +! Put Call (9)(9) we have defined t (K) 2 dK t (K) 2 dK F T expressions in Eqs. (4) and (6) F t (K) t (K) The "+∞+ , a formulation ln Accordingly, (Vt (T, S, T)) = #2Et differ (ℓ (t, T, S, T)) − ℓ˜2E Et two 11 K 1 P )of0government K2 ∞ t)(S,T) P(6) (TF K K2 t (T of2 an bond is t index 0 Ft (S,T) FCall t t (S,T) Put dKvolatility + (7) index dK 1 . (9) +2T)) t (K) t (K) on the two different probabilities. The F tilting term T) v (S, T) (v (T, T) − v (S, dτ. (t, T, S, T) ≡ − # TRHS Tare takenℓ˜under 2 2 t (S, τ τ τ (S, T)fact that the expectations + Put (K) dK + Call (K) dK . (9) P (T ) K K t t t ˜(t,(v 0 Ft (S,T) vτ(S, (S,T)) T) (v vτ (S, dτ. F T under ℓimpact T, S, T) T) ≡ −− vτmismatch: * K2 τ (T, = −vτ ℓ˜(S, T) (T, dτ +T) (S, · tdW (τ ) ,QFAccordingly, τand ∈ (t,(7)T ) a, formulation (5) Pt (T ) K2 encapsulates theT, the the forward aT)) martingale τ−price Ft (S,T) vτ t(S, T) (vτ (T, T) − vτv(S, T))isT) dτ. (7) (t, S, T)τof≡ − maturity Accordingly, a formulation index government bond volatility index of0of anan index of of government bond volatility index is is 1 (7) (S, T) yet we are insisting P (t, T, S, T) GB-VI(t, T, S, T) ≡ in evaluatingt the expectation of its realized variance under Q , motivated as we we have defined a formulation of index of government bond index is T− t volatility # T for the fair value of the original variance swap P (t, T, S,˜ T) in FEq. (3). Accordingly, Accordingly, a an formulation of an index of government bond ** are in our search We expect ˜ Accordingly, a formulation of an index of government bond volatility index is expressions (4) and (6) differ by the “tilting” term ℓ (t, T, S, T) in addition to1 the The two expressions inThe Eqs.two (4) and (6) differ byin theEqs. “tilting” term ℓ (t, T, S, T) in addition to the ˜ 1 he two expressions in Eqs. (4) and (6) differ by the “tilting” term ℓ (t, T, S, T) in addition to the *T) volatility is ℓ˜(t, T,−S, T) to vbe zero only when T =T) S.# − . That is, the forward price is not (τ ) is a multidimensional Brownian motion under Q T (S, T) (v (T, v (S, T)) dτ. (7) T, S, T) ≡ P (t, T, S, T) GB-VI(t, T, S, ≡ P (t, T, S, T) GB-VI(t, T, S, T) ≡ T τ τ τ F probabilities. Thewhere P (t, T, S, T) is as in Eq. (9). act that the expectations on thethe RHSexpectations are taken underon twothe different tilting term 1 Tterm T T, − t− tS, T) that RHS arev taken under two different probabilities. tilting hat the expectations the RHST) are≡taken under two T) different probabilities. The tilting term t fact P (t, GB-VI(t, T) ≡ * −mismatch: vτto (S, (vτ that (T, dτ. (7) T, S,The ℓ˜on(t,of T, term ino’s addition the fact the on τ (S, T)) such that by Itˆ lemma, encapsulates underSpanning QFthe T −1t T ,impact theS,maturity the forward priceT) is − a expectations martingale under QF S and sulates the impact ofencapsulates the maturity mismatch: is a martingale under Q t the forward F S and the impact of theprice maturity mismatch: the forward price isisGB-VI(t, aas martingale (t, T, S, T) T, S, T)under ≡5 QF S Pand where P (t, (9). where (t, T, T, S, S, T)T) is as in in Eq.Eq. (9). asP we yet we are insisting in evaluating the expectation of its realized variance under QF T , motivated T −t ˜(t, " !Qaddition asItPwe eqs. are insisting evaluating theasexpectation realized variance We in are notdiffer done yet, we still needoftoits derive the value ofS, theunder log-contract in Eq. where (6).the is(t, T, S, T) is as in Eq. (9). (4) and (6) by the “tilting” term ℓ T, T) in to F T , motivated yet we value are insisting in evaluating the expectation ofKamal its realized (S, T) F Q Qthe are in our for the fair of original variance swap PQ (t, T, T)Derman, inTEq. (3). We expect variance under QF T , motivated as we encapsulates the impact ofswap the maturity the T term T S, mismatch: Fsearch F T “spanning” FT) natural atthe this juncture toof onoriginal the approach, by Demeterfi, and for fair the P (t, T, S, inln Eq. (3).ℓ˜(t, Weterm expect e˜nour twosearch in Eqs. (4) and (6) by “tilting” term T, S, , T) in addition (6)to the (Vare (T, S,value T)) =rely 2E ℓ˜variance (t,differ T, S, T))the −led 2E Eexpressions taken under two different probabilities. The tilting t S. ( t 5 (t,the T, S,tRHS T) to forward bet zero only when T = T, (9). S, T) in Eq. (3).5 We5 expect are in our search for the fair value of the original variance swap F where (S, T) S where P (t, T, S, is asPin(t, Eq. price , S, T) to be zero only when T =isS.a martingale under Q F and yet we tare insist- T) at the expectations on the RHS are taken under two different probabilities. The tilting term 4 he maturitying mismatch: the forward price is a martingale under Q S and ˜ F inℓ (t, evaluating expectation its realized T, S, T) the to be zero only of when T = S. variance under Spanning ulates theexpectation impact of the maturity mismatch: the forward price a martingale under QF S and ve defined , motivated as of we Time ating the of its realized variance Qfor ning Deposits 5 Q FT, motivated as we are in our under search fair isvalue F T the We are not done yet, as we still need to derive the value of the log-contract in Eq. (6). It is , motivated as we are insisting in evaluating the expectation of its realized variance under Q T # F Time deposit variance contracts share an interesting feature P (t, T, S, T) in Eq. (3). We expect re value theyet, original variance are notof done as we still need toT swap derive the value of the log-contract in Eq. (6). It is Time Deposits Spanning natural at thisfor to rely the approach, led byswap and (3). We expect our fair of“spanning” the original PT)) (t,Derman, T, Kamal S, T)Kamal in al at search this to onvalue the “spanning” approach, led Demeterfi, andEq. T) ≡onbe − vτonly (S, T) (vτvariance (T,byT) − vDemeterfi, dτ. (7) bonds: they can be priced based on the same ℓ˜juncture (t,the T,rely S, with government to zero when T=S. τ (S,Derman, when T juncture = S. t done yet, as we still need to derive the value of the log-contract in Eq. (6). It is are Tnot change-of-numéraire set forth infeature the previous A pointbonds: they can S, T) to be zero onlyWe when = S. 4 Time deposit variance contracts share an interesting with section. government 4 natural at this juncture to rely on the “spanning” approach, led by Demeterfi, Derman, Kamal and of departure arises when expressing time deposit volatility Spanning expressions in Eqs. (4) and (6) differ by the “tilting” term ℓ˜on (t,the T, S,same T) in change-of-num´ addition to the eraire set forth in the previous section. in priced based A point of depart terms of basis point volatility of rates, as opposed to the more faeing expectations on the RHS are taken under two different probabilities. The tilting term we still need to derive the value of the log-contract in Eq. (6). is miliar arises when expressing deposit volatility in terms of basis point volatility of rates, as oppose notion of percentage volatility of prices. To accommo4 It time and sely theon impact ofyet, the maturity the forward priceofisthe a Kamal martingale under Q S (6). are not done as we stillmismatch: need the value log-contract in Eq. It is F the “spanning” approach, ledtobyderive Demeterfi, Derman, and this practice, we needoftoprices. consider arguments dif-practice, we nee the more familiar notion ofdate percentage volatility Tospanning accommodate this as those we and nsisting evaluating the on expectation of its realized variance QF T , motivated at this in juncture to rely the “spanning” approach, led byunder Demeterfi, Derman, Kamal ferent from in the previous section. consider spanning arguments different from those in the previous section. earch for the fair value4 of the original variance swap P (t, T, S, T) in Eq. (3). We expect to be zero This only approach when T =links S. the value of 4the log- (and other) contracts The Underlying Risks

QUANT PERSPECTIVES t

FS

FS

t

T

FT

FT

t

t

T

S

T

FT

T

FT

T

T

T

FT

FT

FT

FT

FT

t

t

S

T

The Underlying Risks Let lt for the time period from t to t style options. be the simplytion, compounded rate on a deposit for the time period from t to t+ Let l (∆) we refer to interest lt To apply these spanning arguments to our context, consider As a non-limitative illustration, we refer to lt (∆) as the LIBOR. t, one Define a forward contract as one where at time t, one party agrees to pay a counterparty a pa 100×(1–l S t (S,S at time S. The forward LIBOR price, Zt (S, S + ∆ equal to 100 × (1 − l (∆)) − Z (S, S + ∆) Z (S,S agreed at time t such that trage in the absence of arbitrage

Zou (1999); Bakshi and Madan (2000); Britten-Jones and Neuberger (2000); and Carr and Madan Zou (1999); Bakshi andThis Madan (2000);links Britten-Jones and and Carr to andthat Madan (2001), among others. approach the value of theNeuberger log- (and (2000); other) contracts of a (2001), others. This approach links the value options. of the log- (andt other) contracts to that of a portfolioamong of out-of-the-money (OTM) European-style not done yet, wespanning still need the valuea Taylor’s of theexpansion log-contract in Eq. (6). It is portfolio ofas out-of-the-money (OTM)to European-style options. To apply these arguments toderive our context, consider with remainder, To apply these spanning arguments to our context, consider a Taylor’s expansion with remainder, his juncture to rely on the “spanning” approach, led by Demeterfi, Derman, Kamal and FT (S, T) − Ft (S, T) FT (S, T) = ln −F FT (S,FT) FFTt (S, T) T)t (S, T) t (S, = !" ln s # " ∞ Ft (S, T) Ft (S, T) Ft (S,T) t S 1 1 − !" Ft (S,T) (K − FT (S, T)) 4++ K12 dK + " ∞ (FT (S, T) − K)++ K12 dK# , t 0 − (K − FT (S, T)) dK + Ft (S,T) (FT (S, T) − K) dK , K2 K2 0 Ft (S,T) and take expectations under QF T , FT and take expectations under QF T , % $ Q T Et F (FT (S, T)) FT (S, T) QF T t $ ln % = QF T −1 Et FFT (S, T) E Ft(F Q T (S, T) T)) T (S, − 1 Et F ln t = t !" # " ∞ Ft (S, T) (S, T) Ft (S,T) 1 Ft ! 1 1 " Ft (S,T) Putt (K) 2 dK + " ∞ − Callt (K) 2 dK# , (8) t 1 1 Pt 1(T ) K K 0 Q Fs − Putt (K) 2 dK + Ftt (S,T) Callt (K) 2 dK , (8) Pt (T ) K K t 0 Ft (S,T) S puts and calls S struck at K, and we havet where Putt (K) and Callt (K) denote the prices of European and Call the prices of European puts and calls struck at K, and we have where Putthe t (K) t (K) denote relied on following pricing equations: relied on the following pricing equations: ' ' Putt (K) Callt (K) Q T & Q T & = Et F & (K − FT (S, T))+' , = Et F & (FT (S, T) − K)+' . (K) (K) Put Call QF T QF T Pt t(T Pt t(T ) ) + + = Et = Et (K − FT (S, T)) , (FT (S, T) − K) . Pt (T ) Pt (T ) τ T) is not)a martingale under QF T , the first term on the RHS of Eq. (8) is not one, but Because Ft (S, ( QF T ˜ ℓ(t,T,S,T) ˜ under first(7). term on the RHS of Eq. (8) and is not one, but τ Because where ℓ (t, T, S, T)QisF Tas, the in Eq. Utilizing this expression combining equals EtFt (S, ( eT) is not)a, martingale QF T ˜ eℓ(t,T,S,T) where ℓ˜(t,the T, S, is as of in the Eq. variance (7). Utilizing thisEq. expression equals Eq. (8)Ewith Eq. (6), we, find that fairT)value swap in (3) is, and combining t

and take expectations under Q ,

(10)

Z (S, S + ∆) = 100 × (1 − ft (S, S + ∆)) ,

www.garp.org

(

(8) Q S where fLIBOR, (S,S where f (S, S + ∆) is the forward which satisfies: ft (S, S + ∆) = Et F (lS (∆)). Beca f (S,S (lS ls fs (S,S ft (S,S l (∆) = f (S, S + ∆), f (S, S + ∆)t is a martingale under QF S . Therefore, assuming that information in this market is driven by Brownian motions, the forward price, Zt (S, S + ∆), satis the following: dZ (S, S + ∆) M A∆) RC H F S2 (τ 0 1) ,4 RISK PROFESSIONAL 3 = vτz (S, dW τ ∈ (t, S) , ( Z (S, S + ∆) z

er spanning arguments different from those in the previous section. t We define the basis point LIBOR integrated rate-variance as, such that, by arguments similar to those leading Section 2, "the fair value of the time ! T to Eq. (3) in " ider spanning Risks arguments different from those in the previous section. 2 f,bp Underlying f 2,following: ! suchdeposit that, by arguments swap similar those leading Eq. (3) in Section the"fair value of the time " "2the T rate-variance generated at t, and paying offSat , is" Vto (T, S, ∆) ≡ to fτ2 (S, +T ∆) "v t " " τ (S, ∆)" dτ, f,bp 2 f Vt generated (T, S, ∆) at ≡ t, and fτ (S, St + off ∆) at (S, ∆) dτ, "v " deposit rate-variance swap paying T , is the following: τ Underlying Risks t f,bp simply compounded interest rate on a deposit for the time period to t+∆.Vsimilar etUnderlying lt (∆) be the Risks (T,toS,those ∆) − leading Pbp S,Eq. ∆) , (3)Tin≤Section S, t f (t, T,to such from that, byt arguments 2, the fair value of the time f,bp V (T, S, ∆) − Pbp (t, T, S, T ≤ S,2, the fair value of the time such that,deposit by arguments similarswap leading to Eq. (3)∆) in, Section tto those f rate-variance generated at t, and paying off at T , is the following: compounded rate deposit for the time period from t to t+∆. tnon-limitative lt (∆) be the simply as on theaLIBOR. illustration, we referinterest to lt (∆) is rate-variance swap generated at t, and paying off at deposit $ # T , is the following: Q be the simply compounded rate onLIBOR. a deposit for to thepay time period from tatopayoff t+∆. Let lt (∆) f,bp isa counterparty bp non-limitative illustration, tointerest lat T, S, ∆) (T,∆) S,$,∆) T . ≤ S, (13) Pbp efine a forward contract aswe onerefer where timeas t,the one party agrees t (∆) t #P V t T, S, f (t, Vtf,bp (T, S,= ∆)E− (t, Q f bp f,bp (t, T, =bpE(t, Vt∆) , (T,TS,≤∆) (13) Pf (T, Vtf,bp S, S, ∆)∆) −P S, . t T, S, (∆) as the LIBOR. non-limitative illustration, we refer to l t f fine a forward time t, one partyforward agrees to pay a price, counterparty a+payoff S + ∆)atat time S. The LIBOR ∆), is The first is the same maturity mismatch arising in the government to 100 × (1 − contract lS (∆)) −asZone We Z face twoScomplications. t (S,where t (S, is two complications. The first is the same maturity # mismatch $ ofinbasis We face arising the point government Define a×forward as(S, one where at time S. t, one party agrees to pay abond counterparty a∆), payoff case. The second complication is that we areQdealing with a notion variance, f,bp is (∆)) −Z S + ∆) at time The forward LIBOR price, Z (S, S + is Pisbp (1 − lS contract dtoat100 time t such that in the absence of arbitrage t tThe second complication $a(T, #=dealing T, we S, ∆) Et V S, ∆) of. basis point variance, (13) t case. f (t, bond case.necessitates that are with notion Q the bp treatment f,bp which a different from percentage (t, T, S, ∆) = E (T, S, ∆) . (13) V P − Zabsence ∆)arbitrage at time S. The forward LIBORwhich price, Zt (S,a S + ∆), is from the al at totime 100 × (1 − that lS (∆)) t t f t (S, S + of d t such in the necessitates different percentage case. In the case, treatment it is by now well-understood that its formulation relates to ainlog-contract. Wepercentage face two complications. The first is the same maturity mismatch arising the government We twocase. complications. The firstwell-understood is the on same mismatch ina the Inface the percentage case, it(10) isthe by now formulation toofagovernment log-contract. ed at time t such that inZthe of 100 arbitrage The contract we The shall link expectation the maturity RHS of its Eq. (13) with is, arising instead, “quadratic” contract S + ∆) = × (1 − ft (S, S + ∆)) , bond second complication is that wethat are dealing arelates notion basis point variance, t (S,absence 2 (S, bond case. Thewe second complication istreatment dealing with notion basis point variance, Thedelivering contract shall link the onwethe (13) instead, a “quadratic” contractnote Sthat + ∆) −are ftRHS Sof+Eq. ∆). Toa is, see howofthis contract is useful, anecessitates payoff equal toexpectation fT2 (S, which a different from the percentage case. (10) Zt (S, S + ∆) = 100 × (1 − ft (S, S + ∆)) , 2 2 which necessitates apercentage different percentage S+ ∆) theorem, −the ft well-understood (S, S + ∆). case. To that see how this contract is useful, note delivering payoff equaland to treatment fcase, byaIn Itˆ othe ’s lemma the Girsanov T (S, Qthat it isfrom by now its formulation relates to a log-contract. FS S + ∆) = 100 × (1 − ft (S, S +S∆)) , =E (10) Zt (S,LIBOR, e ft (S, S + ∆) is the forward which satisfies: ft (S, + ∆) (l Because In percentage it by now well-understood that its formulation relates to a log-contract. that by Itˆ oS’s (∆)). lemmacase, and theis Girsanov theorem, t the

QUANT PERSPECTIVES

FT

FT

FT

FT

shall link &the expectation on the QRHS#of Eq. (13) $is, instead, a “quadratic” contract % we Q Q S The contract martingale under QF . Therefore, assuming that the informabp 2 2 bp

˜ is,(t,instead, F we theS expectation on RHS Eq. a “quadratic” contract Eshall link fBecause + (S,Sthe S++∆) ∆)−=fof22E T,$S) + Pf (t, T, S, ∆) , (14)(14) T (S, t #S(13) + ∆) the fforward satisfies: ft (S, = The Et contract (ldelivering (∆)). + ℓ∆). To see how equal fT2ft(S, &∆)to − )ft=(S,fSS (S, S +is∆), ∆) is a which martingale under QFSS .+ ∆) Therefore, assuming that the Q t % a2 payoff Q t (S, t (S, S +LIBOR, bp this contract is useful, note ˜bpsee QF SaSE 2 (S,− 2∆) ℓ fequal +fT∆) f+t2 ∆) (S, − S+ = 2E (t, T, S) this + Pcontract , (14) S f (S, S + ∆). To how is ∆) useful, note delivering payoff to T (S, S t t f (t, T, S, t theorem, that by Itˆ o ’s lemma and the Girsanov re f (S, S + ∆) is the forward LIBOR, which satisfies: f (S, S + ∆) = E (l (∆)). Because t forward S+ ∆), tt assuming = tfS (S, S + market ∆),price, ft (S, S + ∆)byisBrownian a martingale under that the ! theorem, (S, satisfies mation in this isZdriven motions, theQ price, where F S . Therefore, thatZby Itˆ o’sS lemma and the Girsanov where # $ T t (S,S # $ % 2 !the &2 2 Qbp that Q f f ∆) = finS (S, + ∆), fist (S, S +by ∆)Brownian is a martingale QF S . Therefore, (t,fT, S) ≡T+ ∆)fτ −(S, +S ∆)+vτ∆) (S, ∆) vτfS) (S,$∆) dτ. (15) (14) # $ (t,−T, ftS(S, =#∆) 2Et vτ (T,ℓ˜bp + Pbp Etℓ˜ satisfies SQ +% ∆), mation thisSmarket driven motions,under the forward price, where Zt (S,assuming & S ollowing: T (S, f (t, T, S, ∆) , 2 f Q f f bp ˜bp ℓ fℓ˜T2bp(S, ∆)≡ − ft2ft(S, S + ∆) = 2E (t, T, S) + P (t, T,dτ. S, ∆) , (14) Et (t,ST,+S) (S, S + ∆) v (S, ∆) v (T, ∆) − v (S, ∆) (15) τ τ t τ τ f (S, S + ∆), satisfies rmation in this market is driven by Brownian motions, the forward price, Z dZ (S, S + ∆) t On the other hand, by taking τ t lowing: under the T -forward probability of a Taylor’s expansion (11) where = vτz (S, ∆) dWF S (τ ) , τ ∈ (t, S) , (11) expectations ! T $ expansion 2 Onfthe other hand, byremainder, taking expectations under the T -forward#probability of a Taylor’s whereof we obtain the following: Zττ (S, (S, S S+ + ∆) ∆) dZ following: 2 ! T (S, S + ∆) withℓ˜bp z T S) ≡ T- (15) (t, T, fτ (S, S + ∆)#vτf (S, ∆) vτf (T, ∆) −$vτf (S, ∆) dτ. = vτ (S, ∆) dWF S (τ ) , τ ∈ (t, S) , of fT2 (S, S + ∆)ℓ˜bpwith (11) 2 obtain f f remainder, we thev ffollowing: t (S, S + ∆) dZ (t, T, S) ≡ f (S, S + ∆) (S, ∆) v (T, ∆) − v (S, ∆) dτ. (15) τ S + ∆) % 2 & τ2 τ τ τ 2 Q Zτ (S, z S) where WF (! Brownian f (S,S t − ft (S, S + ∆) = vτz (S, ∆) dW (τ ) , τ ∈ (t, , (11) motion under Q , and v (S, ∆) is a vector of instaneWF S (τ ) is a multidimensional S E f (S, S + ∆) S T T F F τ taking expectations % 2the other hand, &# by Q tOn 2 # $ $ under the T -forward probability of a Taylor’s expansion zS + ∆) τ (S, E - other fremainder, S by + ∆) −Q fexpectations Sℓ˜ + ∆) the taking underfollowing: the T -forward probability of a Taylor’s expansion Thand, t (S, 2(S, (t,T,S) QF ,Zand obtain (S, S +S∆) withEwe remainder, fT22f +instan− = istheatof vector motion under QF S , and vτz (S, ∆)On W S (τ ) is a multidimensional #∆) $we obtain $ 1 the following: (τ ). usFvolatilities, adapted to vW F SBrownian t (S, of t# ˜ e 2 Q (t,T,S) the following: 2 with' ℓ obtain S+ ∆) remainder, we of fT (S, = z ( (S, S + ∆) E e − 1 2f ! ∞ t ed to W ().S (τBrownian a adapted multidimensional motiondesigns under Qreferencing (S, ∆)instead is at vector instanreW % !2of & f (S,S+∆) F S (τ F S , and vτ rates 2Qof ). swap security us volatilities, toFW ecause we) is shall deal with variance ( f f 2 (S, S + ∆) F f(S,S+∆) ∆) ! fprices, % 2 Et' & S +2we T (S, t f , T, S, ∆) dKf!+∞ + (16) Put− (K Callft (Kf , T, S, ∆) dKf , Q t #f S # $ $ ) + ∆) Et + f2TP(S, − fPut t (TS ). ous volatilities, adapted to WF S (τswap 0 f (S,S+∆) t (S, Q + ∆) (16) T,eℓ˜S,(t,T,S) ∆) dK− Callft (Kf , T, S, ∆) dKf , 2 # f + cause we shall deal withequivalent variance security designs referencing rates instead of2fprices, we tE(Kf ,$ der the forward LIBOR to Eq. (11), as follows: $ # (S, S + ∆) 1 (16) P2t (T )= t t Q ˜ (t,T,S) 0 f (S,S+∆) referencing rates instead of prices, designs we consider the forward LI∆)prices, E −1 eℓwe =instead 2ft (S, S +of 't! ( Because we shallLIBOR deal with variance security ! ∞ f (S,S+∆) er the forward equivalent toswap Eq. (11), as follows: referencing rateswhere, '! 2 ( f! T # ! f (S,S+∆) ∞ where, + Put (Kf f, T, S, ∆) dK Call$ft (Kf , T, S, ∆) dKf , (16) f f + f t 2 ˜ (S, S + ∆) df ! f f τ equivalent to fEq. (11), as follows: ider the forward LIBOR ℓf (t, (K T, S) ≡T vdK vτ(K (S,$ ∆) dτ, Pt (T ) #∆) vτ (T, ∆) τ (S, 0Put f− (S,S+∆) + (16) (17) Call f , T, S, ∆) f + f , T, S, ∆) dKf , t t = vτ (S, ∆) dWF S (τ ) , τ ∈ (t, S) , Pt (T )and: T, S) ≡ vtτf (S, ∆) vτff(T, ∆) − vτf (S, ∆) dτ, (17) ℓ˜f (t,(12) 0 (S,S+∆) (S,SS + + ∆) ∆) dffττ (S, f (12)where,where, (12) t ∆) dWF S (τ ) , τ ∈ (t, S) , ! S+ ∆)= vτ (S, 7 τ (S, where, f fτdf(S, S+ ∆) (1#− K ) , T, S, ∆)$ CallztT #(100 Pu ! T, f ! " F Sz (τ ) , τ ∈ (t, S) , = vτ (S, ∆) dW (12) T S) ≡ (t, v7τf (S, ∆) vτf (T,f∆) −$vτf (S, ∆) ,dτ, Callf (K , T, ∆) (17) = ℓ˜f∆) f (K , T, = Put −1 f f f ˜ t t (17)f f (S, S + ∆) t e, by Itˆo’s lemma, vτ (S, vτ (S, ∆) vτ (T, ∆)100 − vτf (S, ∆) dτ, ℓf (t, T, S) ≡ τ ∆) ≡ 1 − fτ (S, S + ∆) vτ (S, ∆). (17) ! " t , by Itˆo’s lemma, vτf (S, ∆) ≡ 1!− fτ−1 (S, S + ∆) v"τz (S, ∆). 7 z z f Put (Kz , T, S, ∆) and Call and: 7 t (Kz , T, S, ∆) are the prices of OTM puts and ca re, by Itˆ o’s lemma, vτ (S, ∆) ≡ Volatility 1 − fτ−1 (S, Indexes S + ∆) vτz (S, ∆). and: t OR Variance Contracts and and: f T ; Putft (KPut S,(1 ∆) with strike price Kz(1 and f , ztT, − Kf ) maturity , T, S, ∆) (100 − Kfand ) , T, S,Call ∆) t (Kf , T, S Callzt (100 f and: , Callft (Kf , T, ∆)z = . Put R Variance Contracts and Volatility Indexes z t (Kf , T, ∆) = S, ∆)100 on fthe forward Put − K100 Callt (100 (1 − Kf ) , T,options f ) , T, S, ∆) t (100 (1 rate. Call . (1 − K ) , T, = LIBOR Putft (Kf , T, ∆) = European-style z, OR Variance Contracts and Volatility Indexes z t−(K K f), ,T, T,∆) S, ∆) Putz (100 zCallt (100 (1 z z f f 100 100 and: Put (K S, ∆) and Call (Kz , T, S, ∆)tare thezfprices of OTM, puts andfcalls written on=Zt (S, St + ∆) t tPut LIBOR Variance Contracts and Volatility Indexes ,T,S Call (K , T, ∆) = (K , T, ∆) t z t, T, z,T,S f f we are t Finally, by matching (16) to Eq. (14), able to use optio fEq. f e define the basis point LIBOR integrated rate-variance as, 100 100 z and maturity T ; Put (K , T, S, ∆) and Call (K , T, S, ∆) are hypothetical OTM with strike price K z z f f t z define (1 − Kf ) ,t T, ∆) puts T,+ S, ∆) ∆) Callare Putt (100 define the the basis basis point point LIBOR LIBOR integrated integrated rate-variance rate-variance as, as, rate-variance f ) ,S t (100 (K , T,calls S, the prices ofZS,OTM and calls written on(1Z−t K(S, Putzt (Kz , T, S, Call as,∆) andEuropean-style puts and written on t (S,S z and , Callft (K . = ∆) on Putftt (K f ,zT, ∆) f , T, ∆) = options the forward LIBOR rate. point volatility , 100 aszt (K follows: 100 f of Putzt (K T, S,Call ∆) fare the and calls written on Zt (S, T,; Put S, ∆) zandfCall z prices of OTM puts z ,T z ,and ! T 6 " " and maturity (K , T, S, ∆) (K , T, S, ∆) are hypothetical OTM with strike price K z f f Finally, by matching Eq. (16) to Eq. (14), we are able to use options on Z to price the basis tt z maturity T 2 z,T,S t z,T,S ! ! ft f z " " " " T " " T f,bp Put (Kz! ,maturity T, S, ∆) areTthe prices of puts calls written Zt " (S, S + are ∆) hypothetica (Kzstrike , T, S, ∆) and K Call f ; Put , T, S, ∆) and Call S, ∆) with price 2 dτ, z tand f , T, !fOTM !and ! " on t (K t (K of f , asLIBOR follows: "" "v ""2European-style optionspoint ont volatility the forward rate. f V (T, S, ∆) ≡ fτ2 (S, S + ∆) ∆) f,bp f QF T ˜bp τ (S, t (T, bpstrike price Kz and maturity ;2Putforward ∆) and Q Call ∆) are hypothetical OTM with (t,T,S) ℓ˜fS, VVtf,bp ff22(S, ∆) F tT(Kf , T, (T,S, S,∆) ∆)≡≡ (S,6SS++∆) ∆)"v (S, ∆)"" "dτ, dτ, t (Kf , T, S, "vτfτf(S, European-style options onfT!the LIBOR rate. t "" ! ! " " ! (S, S + ∆) E (t, T, S, ∆) = 2 e ℓ (t − 1 − E P t ττ t Qare Finally, by matching to2 Eq. we able to use −options on ZT, S) to price the basis t t Q ˜ 6 bpfEq. (16) options 2 on(14), European-style the forward rate. tt S, ∆) = by fmatching Et LIBOR eℓ (t,T,S) Etwe are ℓ˜bp (t, P t (S, S + ∆)Eq. f (t, T, Finally, (16) to Eq.− 1(14), able to use options on Z to price th # % Finally, by matching#Eq. (16) to Eq. (14), we are able to use options on Z to price the basis $ $ point volatility of f , as follows: $ f (S,S+∆) ∞ ft (S,S+∆) $ ∞ point volatility of fprice , as follows: 2thePutbasis f as follows: f options on Z2to point!volatility of hat, by arguments similar to those leading to Eq. (3) in Section 2, the fair value ofvolatility the time f + Callf, f + f , T, S, ∆) dKf , t (K !ofPftQ,(Tas) follows: "" ! point !0 " t (K "f , T, S,Q∆) dK t, by the of at, by arguments arguments similar similar to to those those leading leading to to Eq. Eq. (3) (3) in in Section Section 2,2, the fair fair value value of the the time time + Put (K , T, S, ∆) dK + f f f (S,S+∆) ˜ t T T bp ! "" ! ! " " ! 2 (t,T,S) bp ℓ "F "ℓ˜ (t, ! "" f previous section,atthe fair value of off theat time rate-variance (T!E)Q − !1e0ℓ˜ − ST,+S,∆) E !F eSP Q T,˜S) QEFtT Q ˜ bp bp it rate-variance swap generated t, and paying T , deposit is the Pfollowing: (t,T,S) ft (S,S+∆) 2 t∆) f (t, T, S, ∆) = 2 ft P(S, −e1ℓf (t,T,S) − Et T,E S)t F T ℓ˜bp (t, T,(18) ∆) = 2t ft2 (S, 2 f+ −ℓbp1 (t,− f P(t, rate-variance swap generated at rate-variance swapLIBOR generated at t,t, and and paying paying off off at TT,, isis the the following: following: t (S, S t+ ∆) Et efine the basis point integrated rate-variance as, at #f$ (t, T, S, ∆) = %S) # % $ FT

FT

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swap generated at t, and paying off at T, is the following:

f,bp ! T " T, S, ∆)"2, T ≤ S, (T, S,2 ∆) −bp Pbp V f,bp f,bp "T,fS, " T≤ t (T, f(t,(t, −−PPbp ∆) S,∆) ∆) S, Vtf,bp (T,VV S, ≡ S, f∆) ∆) T, (S, ∆),," T dτ, ≤S, "vτ S, tt ∆) (T, f (t, τ (S, S f+

+

f

FT

$ ∞ $ ft (S,S+∆) #$ ∞ t (S,S+∆) $ ∞ 2 where ℓ˜ f(t, 2˜bp f ft (S,S+∆) T, S)+andPut ℓ (t,(K T, areS, defined inf ,Eqs. and (15). 2 fS), T, f , Put (K T, Call,fT, (KS, S,dK ∆) dK ∆)ft dK +S,(17) Callft (K ∆) f + f , T, f∆) dK f f f, (K f t P (T ) + time Puttfis, (K ∆) dKff+t Call 0 ft (S,S+∆) f , T, S, f , T, S, ∆) d Pt (T ) An index t of basist point deposit rate-volatility 0 (S,S+∆) t P (T )

& defined in Eqs. (17) (18) (18) (15). and where ℓ˜f (t, T, S) and ℓ˜bp (t, T, S) are Pbp T, S, ∆) bp f (t, bp ˜ ˜ 2× is T, S) are defined in Eqs. (17) and (15). where ℓf (t, T, S) and ℓ (t, TD-VI (t, T, S, ∆) ≡ 100 f point time depositT rate-volatility An index of basis is, −t $where # An T, index ofare basis deposit rate-volatility is, in Eqs. (17) and (15). , by arguments similar to those leading to Eq. Q (3) value ofT,the timeℓ˜bpwhere S) and (t, S) and (15). ℓ˜f (t, ˜bpEqs. #in Section T,defined S)point andtime ℓin (t, T, (17) S) are defined ℓ˜f (t, T f,bp 2, the$$fair F# QQ (t, T, S, ∆) = E (T, S, ∆) . (13) V Pbp T T bp f,bp (13) bp f,bp F F t bp & deposit rate-volatility is, T, S, ==EEtoff (T, S, (13) VV,ttist the PPff f(t, ate-variance swap generated at paying following: (t,t, T,and S,∆) ∆) (T, S,∆) ∆) .. An index of basis point (13) An of point timetime deposit rate-volatility is, where Ptime (t,index T, S, ∆) as in Eq. (18). f index t at T An ofisbasis basis point deposit rate-volatility is, & Pbp (t, T, S, ∆) f portfolio bp Note that the above formula an∆) equally-weighted of OTM options. ThisPisbp in TD-VIinvolves ≡ 1002 × f (t, T, S,& f to (t, T, S, ∆) −t & bp of eachT option e face two complications. The first is the same maturity mismatch arising in the government contrast to the percentage volatility case where the is≡ inversely bp weight(t, f,bp The bp bp1002proportional TD-VI T, S, ∆) × ace two first is the same maturity mismatch arising in the government face two complications. complications. The first is the same maturity mismatch arising in the government P (t, Pf (t, T, S, ∆) V (T, S, ∆) − P (t, T, S, ∆) , T ≤ S, f T, S, ∆) 2 bpstrike. MO (2012; 2013d) bp f additional 2 ×provide the square of its TD-VI T, S, ∆) ≡ 100 TD-VI S, ∆) ≡intuition 100 ×about this feature by showing T − t case. The second tcomplication isf that we are dealing with a notion of basis point variance, f (t, f (t, T, T where Pbp ∆) is as in Eq. (18). f (t, T, S, −differs t T − t index, and se. The variance, ase. The second second complication complication isis that that we we are are dealing dealing with with aa notion notion of of basis basis point point variance, how the hedging portfolio of a basis point volatility index from that of a percentage Note that the above formula involves an equally-weighted portfolio of OTM options. This is in mismatch arising from in thethe government bond further analytical details on the behavior of the two indexes. necessitates a different treatment percentage case. case. The second provide $ # bp contrast to the percentage volatility case where the weight of each option is inversely proportional to ecessitates treatment from the percentage case. ecessitates aa different different treatment from the percentage case. MO contains formulation ofEq. a variance contract design cast in basis point terms QFwe complication is=that are dealing∆) with a notionwhere of basis where P (t, S,first ∆) isin as (18). T where Pofbp (t, T,T, S,the ∆) is as Eq.in (18). (t,point T, S, ∆) as in(2012) Eq. (18). Pbp ffits f relates the square strike. MO (2012; 2013d) provide additional intuition about this feature by showing (t,is T,by S, ∆) E . its formulation (13) Vtf,bp (T, S, that Pbp the percentage case, it now well-understood to isathat log-contract. t f applies to fixed income markets–namely, to interest rate swaps. In earlier work, Carr and Corso e percentage ititisisby that formulation relates to aalog-contract. he percentagecase, case, bynow nowwell-understood well-understood thatits itstreatment formulation relates toabove log-contract. Note that thethe above formula involvesportfolio an differs equally-weighted portfolio of is OTM options. Th variance, which necessitates a different from the Note that perthe formula involves an equally-weighted offrom OTM options. This in how the hedging portfolio of aabove basis point volatility index that of a percentage index, andportfolio Note that formula involves an equally-weighted o (2001) explain how to hedge the variance of price changes in markets with constant interest rates. MO ontract we shall link the expectation on the RHS of Eq. (13) is, instead, a “quadratic” contract provide further analytical details on the behavior of the two indexes. contrast to the percentage volatility case where the weight of each option is inversely proporti ce two complications. The first is the same maturity mismatch arising in the government contrast to the percentage volatility case where the weight of each option is inversely proportional to tract we shall link the expectation on the RHS of Eq. (13) is, instead, a “quadratic” contract ntract we shall linkcentage the expectation on the RHS of Eq. (13) is, instead, a “quadratic” contract case. (2013d, Chapter 2)the explain that the elegant replication arguments in Carr and Corso (2001) of break contrast to percentage volatility case where the weight each option 2 MOsquare (2012) contains the firstMO formulation of a variance contract design cast in basis about point terms the of its strike. (2012; 2013d) provide additional intuition S +we ∆) −2case, fdealing S +by ∆).now To well-understood seeof how this contract is useful, note ring a payoff equal to22the fT2 (S, once interest rates are random, but that the random eraire inherent in each marketthis of feature by s the square ofvariance, its strike. MO (2012; 2013d) provide additional intuition about this feature by showing t (S, e. second complication notion basis point percentage that age volatility case where the (2012; weight of num´ each is inversely ++∆) fftt2(S, SSit++iswith ∆). see this contract isisdown useful, note g aThe equal to ffTT (S, (S,isS Sthat ∆)−−are (S, ∆). aTo To see how how this contract useful, note ng a payoff payoff equal toIn that applies to fixed income markets–namely, to interest2013d) rate swaps. Inoption earlier work, Carr and Corso the square of its strike. MO provide additional ab interest inthe thehedging fixed income space can be incorporated into the replicating portfolios and that the variance how portfolio of a basis point volatility index differs from ofand a intuition percentage inde how the hedging portfolio of a basis point volatility index differs from that of a percentage index, by Itˆ o ’s lemma and the Girsanov theorem, (2001) explain how to hedge the variance of price changes in markets with constant interest rates. MO cessitates a different treatment from the percentage case. its formulation relates to a log-contract. The contract we shall contract - from that design. Itˆ oo’s’s lemma and Girsanov theorem, Itˆ lemma and the the Girsanov theorem, provide further analytical details on the behavior of the two indexes. how the hedging portfolio of a basis point volatility index differs (2013d, Chapter 2) explain that the elegant replication arguments in Carr and Corso (2001) break provide further analytical details on the behavior of the two indexes. percentage %case, it is by now that its formulation relates$to a log-contract.vide # additional intuition about this feature how the MO (2012) contains the first anum´ variance design castofin basis point down rates are of random, butformulation that the random eby raireshowing inherent each market & well-understood QF%T% QF#T# ˜bp $$ MO (2012) contains theonce firstinterest formulation a variance contract design cast incontract basis point terms provide further analytical details on of the behavior ofinthe two indexes. 2 2 & & Q Q Q Q ℓ (t, T,a S) fT (S, Sexpectation + ∆) −22fon (S, SRHS + ∆) = 2E(13) +bp Pbp (t, T, S, ∆) , inapplies (14) EFFtTshall TT is, T 8 bp interest theportfolio fixed income space canbasis be incorporated into theinterest replicating portfolios and the variance ractEwe the the of Eq. instead, “quadratic” contract bp 22link bp that to fixed income markets–namely, to rate swaps. In earlier work, Carr and hedging of a point volatility index differs from FF t ˜ ˜ t f ℓℓ (t, ffTT (S, ++Papplies T, S, ,, MO (14) (S,SS++∆) ∆)==2E 2Ett (t,T, T,S) S) that Pff (t, (t,to T,fixed S,∆) ∆) (14) contains (S,SS++∆) ∆) −−fftt (S, Ett income markets–namely, to interest rate swaps. In earlier work, Carr and Corso contract design. (2012) the first formulation of a variance contract desig 2 2 (2001) explain how to hedge the variance of price changes in markets with constant interest rate To see see how how this is useful, note a payoff equal to fT (S, S + ∆) − ft (S, S + ∆).. To this contract contract is useful, that of a percentage index, and provide further analytical de(2001) explain how to hedge the variance of price changes in markets with constant interest rates. MO (2013d, Chapter 2) explain that the elegant replication arguments in Carr and Corso (2001) that applies to fixed income markets–namely, to interest rate swaps. In ea tˆ o’s lemma and the Girsanov! theorem, tails on the behavior of the two indexes. (2013d, Chapter that the elegant replication arguments in Carr and Corso (2001) break # $ 2) explain 8but that the random num´ T down once interest rates are random, e raire inherent in each ma ! ! ## TT (2001) explainbut how to the hedge the num´ variance of priceinchanges in markets with c f once$$ interest rates are random, that random eraire inherent each market of v#τf ˜(S, ∆) vτf$ (T, ∆) −down (15) ℓ˜bp (t, T, &S) ≡2 22fτ2 (S, S + ∆) ffvτ (S, ∆) dτ. bp Q T % 2˜ interest (15) in the fixed income space can be incorporated into the replicating portfolios and the v S) ffτ (S, SS+ vFvfTf(S, − v (S, ∆) dτ. (t, T,∆) S)≡ ≡ ft (S, (S,∆) +∆) ∆)Q (S,bp∆) ∆) vvfτf(T, (T,+∆) ∆) − v (S, ∆) dτ. (15) ℓbp (S,(t, S T, + − + = 2E Pbp (t, T, S, ∆) , (14) Et F fℓT˜ τ τ interest in the fixed income space can be incorporated into the replicating portfolios and the variance (2013d, Chapter 2) explain that the elegant replication arguments in Ca t τS t ττ ℓ (t, T,τS) f contract design. tt contract design. downexpansion once interest rates are random, but that the random num´eraire i n the other hand, by taking expectations under the T -forward probability of a Taylor’s he other the otherhand, hand,by bytaking taking expectationsunder underthe theTT-forward -forwardprobability probabilityof ofaaTaylor’s Taylor’sexpansion expansion ! T expectations # $ interest in the fixed income space can be8 incorporated into the replicating (S, S +˜bp ∆) with remainder, we obtain the following: we ,SS++∆) ∆) with remainder, weSobtain obtain the following: (t, T, remainder, S) ≡ fτ2 (S, + ∆) vτfthe (S, following: ∆) vτf (T, ∆) − vτf (S, ∆) dτ. (15) ℓ with 8 contract design. t 4 RISK PROFESSIONAL MARCH 2014 www.garp.org & QF%T% % 2 2 & & QQ Staking + ∆) expectations −22ft (S, S + ∆) the T -forward probability of a Taylor’s expansion 2fT (S,by eEFFother under tTT ff2hand, (S, S + ∆) − f (S, S + ∆) (S, S + ∆) − f (S, S + ∆) TT tt # t # $ $ ## Qwe $$following: $$ #obtain T S + ∆)2with remainder, the ℓ˜f (t,T,S) F#

t

t

0

ft (S,S+∆)

(18)

dQ

PVBPt (T1 , · · · , Tn )

GT

and PVBPt (T1 , · · · , Tn ) is the “price value of the basis point – ” i.e., the value at t of annuity paid over the swap tenor. We assume that Rt (T1 , · · · , Tn ) is a diffusion process, as follows:

QUANT PERSPECTIVES

dRτ (T1 , · · · , Tn ) = Rτ (T1 , · · · , Tn ) στ (T1 , · · · , Tn ) · dWA (τ ) ,

τ ∈ [t, T ] ,

(19)

where WA (τ ) is a Brownian Motion under QA , and στ (T1 , · · · , Tn ) is adapted to WA (τ ), and define the basis point realized variance of the forward swap rate arithmetic changes in the time interval [t, T ],1 # T Vnbp (t, T ) ≡ Rτ2 (T1 , · · · , Tn ) ∥στ (T1 , · · · , Tn )∥2 dτ. 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 payoff of a payer swaption is max {SwapT (K; ·) , 0} and T that. ),0} of aand receiver is .),0}.information about both interest rate max {−SwapT (K; ·) , 0}. Notice that swaption prices contain Notice that swaption T

dom, but that the random numéraire inherent in each market

prices contain information about both interest rate volatility and the value of an annuity. 9This reveals yet another funda-

volatility and the value of an annuity. This reveals yet another fundamental difference between equity

the replicating portfolios and the variance contract design. andswap fixeddesign incomeismarkets. Options swaps on equities relate to aare single source risk: the stock price. InThe next section illustrates how a variance price. Instead, and swaptions affected byoftwo sources volatility andstead, the value of and an annuity. This reveals yet another fundamental difference between equity (·), the annuity volatility and the value of an annuity. This reveals yet another fundamental difference between equity swaps swaptions are affected by two sources of risk: the swap rate, R . T T (. ). and affected by the numéraire when deriving model-free indexes in of risk: the swap rate, RT ( and fixed income markets. Options on equities relate to a single source of risk: the stock price. Infactor, PVBP and fixed income T (·). markets. Options on equities relate to a single source of risk: the- stock price. Ininterest rate swap markets. (·),show and stead, swaps and swaptions areswaptions affected by sources risk: therate swap rate,the Rcomponent Tswap (·), annuity MO (2012)and show how insulate the in pure interest volatility in model-free stead, swaps aretwo affected byaoftwo sources of risk: rate,the RTaannuity tilitytocomponent model-free fashion. They that theand thefashion. section illustrates how a variance swap design is affected by the num´ e raire when deriving (·). factor, PVBP volatility and the value of an annuity. This reveals yet another fundamental difference between equity T (·). factor, PVBP They show that the annuity factor entering into the payoff of swaptions needs to be worked into the Interest Rate Swaps annuity factor entering into the payoff of swaptions needs to T and fixed income markets. Options on equities relate to a single source of risk: the stock price. Inndexes in interest rate swap markets. MO (2012) show how to insulate the pure interest rate volatility component in a model-free fashion. MO (2012) show to insulate the interest rate volatility component a model-free fashion. variance swap design to worked allow forinto model-free pricing of the variance contract ininboth percentage and - how be thepure variance swap design to allow for model-free and the annuity stead, swaps andentering swaptionsinto are affected by two sources of risk: the swap rate, RT (·), into They show that the annuity the payoff of swaptions needs be worked pricing of the variance contract in both percentage basis They show thatfactor theterms. annuity factor entering into the payoff of swaptions needs to be the worked into the basis point variance More generally, MO (2013d, Chapter 2)to develop aand framework that handles PVBP T (·). The nextnow sectionconsider illustrates the how a variance of swap design is affected by the num´efactor, raire when deriving variance swap design to allow for model-free pricing of the variance in both percentage variance in the of Rate Swaps variance swap design to allow model-free pricing of contract the variance contract both percentage and any market num´ eshow raire of interest inthe the fixed income space. Let us illustrate howand this framework The next section illustrates how apricing variance swap design isswaps affected by thecontext num´ eand raire when deriving MO (2012) how tofor insulate pure interest rate volatility component in a in model-free fashion. model-free indexes in interest rate swap markets. basis variance terms. More generally, MO (2013d, Chapter 2) develop a framework that handles interest rate swap markets. For the sakepoint of brevity, we focus on develop a framework that handles any market and numéraire odel-free indexes in interest rate swap markets. basis point terms. More generally, MO (2013d, Chapter 2) develop a framework that specializes invariance the swap case. They showinterest that the rate annuity factor entering into the payoff of swaptions needs to be worked into thehandles ariance swapthe security affected byderivation the presence of more complex num´eof raires thanin the fixed income space. Let us illustrate how this framework basis design point variance mirror market practice, anytomarket and num´ evariance raire interest swap design to allow for model-free pricing of thespace. variance contract in both percentage and any market and num´ e raire of interest in the fixed income Let us illustrate how this framework Consider a variance swap starting at t with the following payoff: volatility and the value of an annuity. This reveals yet another fundamental difference between equity Interest Rate Swaps zero? We now consider the pricing of variance swaps in the in context of interest rate swap framework inMO the interest rate swap case. basis variance terms. More generally, (2013d, 2) source develop a framework that Inhandles specializes the interest rate case. and fixed incomeswap markets. Options on equities relate to a single of risk: the stock price. specializes inpoint the swap interest rate case. nterest Rate Swaps... !specializes "Chapter r How the sake of brevity, we focus on the basis point variance derivation to mirror market bp Let R (T , ,T ) be the forward swap rate prevailing at t, – i.e., (·),how and the stead, swaps and swaptions are affected bypayoff: two sources ofPVBP risk: the rate, RT, T anynum´ market and num´ eT raire of interest in the fixed income space. LetTswap us illustrate thisannuity framework t 1securityn design affected by the presence is the variance swap of more complex e raires than (t, ) ≡ V (t, T ) − P (t, T ) × (T , · · · ) π Consider a variance swap starting at t with the following n n 1 n n Consider a variance swap starting at t with the following payoff: PVBP chtheis isprice alsovariance formulation useddesign for CBOE SRVX Index. specializes inefactor, the rate swap case. ow the swap security affected by the presence of context more complex num´ raires thanT (·). ofthe a zero? We now consider thethe pricing of variance swaps in the of interest rate off:interest !swap " the pure interest"rate volatility component in a model-free fashion. MO (2012) show!how to insulate · · · , Tofn )aFor be theWe swapwe rate prevailing t, –swaps i.e., the fixed rate such the bpswapswap starting the sake of brevity, focus the point variance derivation mirror a rate variance at with the following e1 ,markets. price zero? now consider the pricing of variance inatthe of interest T"T , with reset dates Ton0,... ,Tbasis length T -T, and pay0forward n-1,attenor T n,context where P Tmarket )They is fair value of(t, the contract (t, ) that ≡ V (t,T T))≡ − P (t, T )T )t× , that ·PVBP · ·payoff: Tn )T (T1needs πntoConsider show that the annuity factor entering intoTsuch the of,swaptions into the nT(t, nnbp 1× nnthe π (t, V −PVBP Pn (t, )(Tpayoff , · · ·to, be Tnworked ) practice, which is alsoof the formulation used for the CBOE SRVX Index. ... ! " , with reset dates T , · · · , T , tenor length T − T , and rward starting swap (at T ≡ T arkets. For the sake brevity, we focus on the basis point variance derivation to mirror market 0 1, ,Tn-Tn-1) is zero 0 at t. It n−1 n variance swap design to allow for model-free pricing of the variance contract in both percentage and ment periods T1-T is well-known (e.g., bp " T (T1 , · · · , Tn ) ! T ) × PVBP ≡ VnMore (t,generally, T ) − PA (t, Tnthe ) be the )forward swap rate prevailing t, –(e.g., i.e., the fixed rate such that the Let n (t, T )terms. nMO 1 , · · ·, ,T basis pointπvariance (2013d, 2) develop a framework that handles bp Chapter ...atSRVX actice, CBOE Index. −RtT(T at t.for Itthe is well-known Mele, 2013, Chapter 12) iods T1which 0 , ·is· ·also n − Tformulation n−1 is zeroused R (T ,T a martingale unat T, where (t,T) is the fair value of the contract such that tat 1,, where n) is P n (t, T ) = E (t, T ) , (20) P V T (t, T ) is the fair value of the contract such that n the contract n T t fixednsuch , where PnT(t, TTany ), and ismarket the and fairnum´ value , tenor length value of a forward starting swap (at T ≡ T0 , with reset dates T0 , · · · , Tat eraire of of interest in the income that space. Let us illustrate how this framework n− (T1a, der ·martingale · · , Tthe the forward swap rate prevailing at t, – n−1 i.e., fixed rate such that the n ) be · ·Let , TnR)t is under the so-called annuity probability QAthedefined through the so-called annuity probability Q A at T , where P (t, T ) is the fair value of the contract such that specializes in the interest rate ! swap case. " ! n 12) payment periods T1 − T0 , · · · , Tn − Tn−1 ) is zero at t. It is well-known (e.g., Mele, 2013, Chapter " length Tn −Consider T , and lue ofderivative, a forward starting swap (at T ≡ T0 , with reset dates T0 , · · · , Tn−1 , tenor A starting bp dym A , !the bp at t)the with following"payoff: (20)(20) by Eq. (19) ) = Eswap (t, TE denotes conditional expectation QA . Moreover, whereQEA Tn )follows: is a martingale under the so-called annuity probability through thePn (t,aTvariance that Rt (T1 , · · · , as tPn V (t,n Tunder )= ) , (20) Vannuity tdefined t n (t, T probability

ayment periods T1 − T0!, · · · , Tn − Tn−1 ) is zero at t. It is well-known (e.g.,A Mele, 2013, Chapter 12) ! T ) = EA " T) , Pn (t, (20) Vnbp (t, t Radon-Nikodym derivative, as follows: "T and Itˆo’sQlemma, πn (t, T ) ≡ Vnbp (t, T ) − Pn (t, T ) × PVBPT (T1 , · · · , Tn ) PVBP · · , Tn ) probability T (T1 , · annuity is Aa !!martingale the so-called defined through the at Rt (T1 , · · · , Tn )dQ − t under r ds s A A A ! , = e " A where E denotes conditional expectation under the annuity t expectation under the annuity probability Qprobability by Eq. (19) by Eq. (19) T ! expectation under the Ewhere A . Moreover, ! PVBP Tn )whereconditional dQ !GT asdQ PVBP (Twhere , ·T ·(T · ,1 ,TE· n·t·) ,denotes t denotes A . Moreover, A! adon-Nikodym derivative, follows: conditional the annuity annuity probability Q"A .QMoreover, by Eq. (19) EA #conditional at2T , where Pn (t, T ) expectation is$ the fair2 valueunder of the contract such that t Adenotes , = e− t rs ds t 1 probability Q A − R (T1 , · · · , Tn ) = EA V bp (t, T ) = Pn (t, T ) . R (21) (T , · · · , T ) E and Itˆ PVBP dQ !GT · · ,lemma, Tn )and Itˆ t (T 1 ,o·’s 1 n t T t o’sand lemma, "n ! Itˆ o’s tlemma, ! ... " T ! Pn (t, T ) = EA (20) Vnbp (t, T ) , PVBP (T · · , Tvalue t t (T n− basis 1 , · the n) T T1 , · · · , Tn ) is the “price dQ value the point – ” i.e., at t of annuity paid r ds A !1,of ,T s ! " t !! " " ,value e of # at $# 22 and PVBPt (T1 , · · · , Tn ) is the “price= value the basis point – ,”· ·i.e., t of annuity # paid $ $ 2 2 A A bp PVBP · , where Tthe t (T 1 swap n )E i.e., the valuedQ at !tGof annuity paid over the tenor. ,.by Vprobability (t, T )TTQ)= RR ·,··by ,· T· ,Eq. )Tn−expectation R= , · ·V · ,n ,bp )) = follows RT2 second (t, TEE )At A = (21) (T1 , · E · ·Aequality , Twhere )2At (T − )(20). E p tenor. (21) T n (t, n (t, nE t (T n t· (T t the VPbp R (t, ).. PMoreover, = PTn )(t, TEq. ) . (19) (21) (21) ·(T ·1 ,,·1Tconditional (T TTnnthe =annuity EA1T,denotes 1 ,1t· · ·under n ) n− R over the swap tenor.

... Rta(T diffusion process, as follows: 1, ,Tn) is · · · ,RTn(T ) is, · a· · diffusion process, as afollows: me that and Itˆ o’s lemma, Similar to our derivation of Eq. (16), we now take the expectation under the annuity probability t (T1 , that , Tn ) is diffusion process, as follows: We R assume nd PVBP value of the basis point – ” i.e., the value at t of annuity paid t (T1 , · · · , Tn )t is 1the “price t

T

t

t

t

n

A

where follows the second follows by Eq. $(20). ! " where the second equality byequality Eq.follows Taylor’s of(20). RAt T2#R(T ···· ·· , ,T(20). Tn )− with secondexpansion equality Qwhere 2by bp 1 ,1 ,Eq. A of athe er the swap tenor. Vobtaining: , ·remainder, ·· ,T EA = Pn the (t, T )annuity . (21) (T Rt2 (T1take E n) = t n (t, T )under Similar to our derivation ofT Eq. (16),n )we now the expectation Tn1), = (Tn1), ·σ·τ· (T , Tn1), σ , dW Tn ) A · dW ) ,τ ∈derivation τ [t, ∈ [t, (19) dRτ (T1 , · · · ,dR Tnτ )(T=1 , ·R· ·τ ,(T · ·R · τ, T · ·τ ·(T, 1T, n· ·)· ·Similar (τ to )A,(τour T ]T,] , (19) (19) -probability of Eq. (16), we now take the expectation under the annuity probability to our derivation of Eq. (16), we now take the expectation under the annuity probability 2 We assume that Rt (T1 , · · · , Tn ) is a diffusion process, as follows: QSimilar , TEq. of RT (T #QA2 of a Taylor’s $the second 1 , · · · by n ) with whereexpansion follows (20). remainder, obtaining: T 2 equality F QA obtaining: , · ·expansion (T , · · ·,annuity expansion R· T2, Ttation (T · under ·R with QA of a Taylor’s n )1 , ·− n )21the T (T1of aR Taylor’s of, tTR (T ·,remainder, ·T·n,)T )probability withobtaining: remainder, QEt of

WA (τ ) is a Brownian Motion under QA , and στ (T1 , · · · , Tn ) is adapted WA (τ ),# and define 1 of Eq. n Similar2 to $our derivation (16), we now take the expectation under the annuity probability A to to T ) where is a Brownian Motion under Q(T , ,and σ (Tσ1 , (T · · · ,,· T· ·n ), Tis )adapted Wτ(T ),Tand define Q... % ' ...R A∈1(τ 2 n) with 2remainder, obtaining: F T ,T & ∞ QA,Achanges and ,[t, ,T dRτ where (T ,variance TnA)(!= Rof · · , Tτn ) swap (τ ) , #!in ]Rn,)T2 interval 1,− Ethe (T , of · ·R ,T(19) T&(T , ·of 1, · · · W τ Athe 1 ·forward τ 1rate arithmetic n · dW obtaining: a· Taylor’s expansion ,T1n ) · ·R,TT(T Q1Aof n)R n1), · · · , Tn ) with remainder, the basis point realized t (T t (T 1 ,··· t$ time # 2 Q p # $ % ' T r int realized variance of the forward swap rate arithmetic changes in the time interval 2 2 Q F & & T 1 2 2 is adapted to WA (! (K, T ) dK + ∞ Swpnn,t (K, T ) dK , (22) Et RT (T1= −R ,2TR [t, T ], n) n ) (T n ) (TRt,(T·$1·,··· E, · ·F· , TR , ·-t· (T · ,21TQ, · T)· #· − · ,T ,Swpn 2T ) n,t

p t PVBPTt (T r n) · ·),· ,and RT (T ETt n )define T 1, · · · , T 1 ,(τ 0 1 ,t· · · , Tn ) − Rt (TSwpn Rt (T1 ,··· ,TSwpn % ' T(22) )'dK , (22) =W a Brownian Motion #under QA , andarithmetic στ (T1 , · · · , Tn ) is 2adapted to here WA (τ ) isance n) A ' n,t&(K, T ) dK + n,t (K, & %& of Vthe swap time bp # Tforward &&t (T Rt (T )T& ∞ (T11,··· , · ·,T· n,% T) ≡ Rτ2 (T1 ,rate · · · , Tn ) ∥στ (T1 , · · · changes , Tn )∥2 dτ. in the PVBP n ) Rt (T ∞ 0 1 ,··· ,Tn R )Rt (T1 ,··· ,Tn ) ∞ 1 ,··· ,Tn ) n (t, 2 p p r 2 r e basis pointinterval realized [t,T], variance forward swap rate arithmetic changes in the time interval 1 2 of the 2 bp p t r Swpn (K, T ) dK + Swpn (K, T ) dK , (22) = T ) dK + (K, Swpn ) dK , (K, (22) n,t T ) dK + n,t Vn (t, T ) ≡ Rτ (T1 , · · · , Tn ) ∥στ (T1 ,=· · · , Tn )∥ dτ.= T ) dK , at(22) n,t (K, TSwpn PVBPSwpn ·n,t , Tn(K, ) t (T1 , · · p 0 Swpn Rt (T n,t the n,t 1 ,··· ,Tn ) (K, T· ),(K, Swpn ) denote prices receiver and payer swaptions t, (T1 , · valuation? ·PVBP ·Swpn , where Tn )trn,t PVBP p T t where Rt (T1 ,··· ,Tn ) ofreceiver T ],1 How do we design a variance (T Tand n,t (K, contract in this case that allows model-free Consider the t 10, · ·rn,t n )T ) and Swpn 0 Swpn t (T1 ,··· ,Tand n ) payer swaptions at t, n,t (K, T ) denote the prices of R # T p r of aapayer swap contract with fixed rate, K:case T−and strike atT .T .and payer swaptions at t, referencing a Consider swap awith Tn T−)rTand (K, Swpn (K, T )K, denote theexpiring prices of at receiver where Swpn 2 2 n,t wevalue design variance that allows model-free valuation? thetenor T and strike K,and expiring referencing swap withn,ttenor p and Vnbp (t, T ) in ≡ this R τ (T1 , · · · , Tn ) ∥στ (T1 , · · · r, Tn )∥ dτ. n.tn(K,T n.t(K,T) denote the prices of rep

1

n F

1

n

p tenor − T and strike K, and expiring T. with (K, TaSwpn )swap denote the receiver payer swaptions at t, where Swpn (K, T ) andr Swpnreferencing TTn)prices denoteof the pricesand of atreceiver n,t (K,swaptions and payer at t, referencing a swap and withpayer tenor swaptions at t, How do we design a variance contract in this casereferencing that allows model-free valuation? the strike − T tenor and K,and andstrike expiring at T expiring . a referencing swap with tenor nwith K, and at T . a swapTConsider Tn − T n,t ) and where Swpn (K, Tn,t ceiver SwapT (K; T1 , · · · , Tn ) ≡ PVBPT (T1 , · · · , Tn ) [RT (T1 , · · · , Tn ) − K] . n,t

yer swap with fixed rate, K:

t

Tn–T and strike K, and expiring at T.

lue of apayoff payer swap rate, K: T1 , a· ·with ·payer , Tnfixed ) ≡swaption PVBP (T1max , · · ·{Swap , Tn ) T[R(K; , · · · and , Tn )that − K]of. a receiver is Swap The ·) 1, 0} T is T (T T (K;of

max {−SwapT (K; ·) , 0}. Notice that swaption prices contain information about both interest rate

· , Tnmax ) ≡ {Swap PVBPT (T Swapswaption 1 , · · · , Tn ) [RT (T1 , · · · , Tn ) − K] . T (K; T1 , · · is of a payer T (K; ·) , 0} and that of a receiver is 9 T (K; ·) , 0}. Notice that swaption prices contain information about both interest rate

he payoff of a payer swaption is max {SwapT (K; ·) , 0} and that of a receiver is ax {−SwapT (K; ·) , 0}. Notice that swaption prices contain information about both interest rate

www.garp.org

9

9

10 10

10 10

10 M A R C H 2 0 1 4 RISK PROFESSIONAL

5

QUANT PERSPECTIVES

Matching Eq. (21) to Eq. (22) leaves: Pn (t, T ) =

2 PVBPt (T1 , · · · , Tn )

!"

Rt (T1 ,··· ,Tn )

Swpnrn,t (K, T ) dK +

"

∞

Swpnpn,t (K, T ) dK

#

!

.

$ adapted to Matching Eq. (21) to Eq. (22) leaves: We assume that (1) loss-given-default (23) (LGD) is constant; (2) the short-term rate r is a origination. τ (23) Eq. (23) provides the expression for the value of the variance swap in a model-free fashion. It r. Let n # be the initial inshort-term the index decided origination. We assume that of (1)names loss-given-default (LGD) isataconstant; ( origination. We arrives assume that (1) loss-given-default (LGD) isnumber constant; the rτ is !of"equally weighted is a portfolio swaptions, as in (3) the case for time deposits 18); however,process with intensity diffusion process; and default as a Cox λ adapted to r. (2) Let n be the (2) therate " ∞(see Eq. Rt (T1 ,··· ,T ) origination. We assume that (1) loss-given-default (LGD) is constant; short-term rate rτ time t"T each constituent have a notional value 1/n, 0, and letand by the ninverse of the annuity factor (the num´ e raire in the interest rate 2 this portfolio is rescaled diffusion process; (3) default arrives as a Cox process with p r diffusion and(the (3) default arrives as),dK aand Coxlet process with intensity λ aadapted to r. Let n be the intensity T) = (K, ) dK + Swpn T . each constituent have notional initial of names inbyTprocess; the index at ≡ T swap market), whereasnumber the portfolio Swpn in (18) isn,t rescaled the price of a zerodecided num´eraire in time the n,tt(K, 0 diffusion process; and (3) initial defaultnumber arrives as a Cox process with decided intensity λtime adapted Letletn eac be $. , · · government · , Tn ) bond PVBP 2 0market). Rt (T ) in let theeach index ≡ Tto0 ,r.and 1 ,··· and the valuet (T of1an annuity. This another fundamental difference between equity at time tof≡names 1 reveals yetinitial constituentathave a tnotional number ofsame names in,Tnthe index T0 , and , the swap same and intensity, λ. decided value Intuitively, tilting n a variance by theLGD, market num´ eraire the at T (which is one in the cases dealt The number of names having survived up to T i is , and let each constituent have a notio initial number of names in the index decided at time t ≡ T (23) 1 0 1 ! , loss-given-default the the same intensity, λ.thethe value d income markets. Options ononequities to rate a single source of risk: the stock price. In-(1) nsame LGD, and with in sections time depositsrelate and interest swaps) causes its fair value to be and defined under the same LGD, the same intensity, λ. value origination. We assume that (LGD) is is constant; (2)(2) short-term rate rτ n (1) origination. We assume that loss-given-default (LGD) constant; short-term rate n ,by 1 of the however, this portfolio isinformation rescaled the inverse annuity S (T (1 − I{τjλ. The number of of names having survived up to T ! i is and i ) ≡same ≤Ti! } ) where τ!jj is the time at which j=1 , the same LGD, the intensity, value q. (23) provides the expression the value the variance in a model-free fashion. It a market space wherefor all the relevant is given by the price swap of available derivatives—an n n nnames is S (T ) ≡ The number of names having survived up to T (·), and the annuity waps and swaptionsfactor are affected by two sources of risk: the swap rate, R 3 i i T ! 1 is S (T ) ≡ (1 − I ) where τ is the The number of having survived up to T diffusion process; and (3) default arrives as a Cox process with intensity λ adapted to r. Let n origination. We assume that (1) loss-given-default (LGD) is constant; (2) the short-term rate r( j=1b expectation under the eraire probability of interest. j i (τ ) and diffusion process; and notional (3) default arrives as iaS Cox process intensity λN(!)=1/n adapted to r. Let n (the numéraire in name the interest rate swap market), whereas ≤T } ni1. j=1 Nwith j≡ j defaults, the (τ (t){τnotional time atnum´ which defaults, and the outstanding is N = portfolio of equally weighted swaptions, asswap involatility the jcase for time deposits (seeofEq. 18);name however, iswith S (Tand (1 −isI{τ ) whereis τN The number names having survived up njto Ti)outstanding j is i ) ≡the Finally, an index of interest rate is j=1 1 j ≤T i} VBPT (·). time at which name defaults, outstanding notional (τ 1S(!) STlet (τ!let with N (t) ≡to1.have time at name defaults, and the outstanding notional is Nt ≡ (τ )loss a not initial ofisof names indefault the index at time T diffusion process; and (3) arrives as aThe Cox process with intensity λconstituent adapted r.have Let n b the portfolio in (18)loss is rescaled by$which the obligor price of jnumber ajdefault zero (the nuwith N(t)"1. index at should obligor , nand each an initial number names in the index decided at time t≡ T, 0and 0= j)each 1constituent LGD ,decided whereas the premium at The index at τj should i is portfolio is rescaled by the inverse of the annuity factor (the num´ erairename in the ≤T bM } the outstanding notional is N n I{t≤τjrate (τ1 )I with N (t) ≡ time at j interest defaults, and (τ ) is= LGD 2 1 component 1 τj should obligor j default 1which 2012) show how toméraire insulate 1in thethe pure interest rate volatility in a model-free fashion. n Sat 2 × market). 1 , w The index loss at government Pn (t,value IRS-VInbp1(t, T )bond ≡ 100 T ) j default is whereas the premium T is {t≤τ ≤T } should obligor j default is LGD I , whereas the premium at T is The index loss at τ , the same LGD, and the same intensity, λ.bMλ.}t1≡ T0 , and let each constituent initial of names in the index decided atjis time bMa not n i i jhave jnnumber ,ofthe same LGD, and the same value {t≤τ ≤T Tthe − t price nintensity, S (Ti ). by Finally, the value protection leg minus premium leg )× market), whereas the portfolio (18) is nrescaled of a nzero (the eraire in the t (M b CDXin obligor j×default isFinally, LGD ,! thelegpremium at T The loss τnum´ 1 the 1 j should {t≤τjvalue ≤TbM! }of ow that the annuity factor entering intoathe payoff of(M swaptions needs to beat worked into nwhereas 1 1 index 1 n Ithe nprotection Intuitively, tilting variance swap by the market numéraire CDX (M ) S (T ). the protection minus premi Finally, value of leg t i CDX ) × S (T ). Finally, the value of protection leg minus premium leg is is S (T ) ≡ (1 − I ) where τj τij The number of names having survived up to T , the same LGD, and the same intensity, λ. value 2 b n t i is S (T ) ≡ (1 − I ) where The number of names having survived up to T i i where Pn (t, T ) is as in Eq. (23). {τj{τ ≤T i i j=1 i} i} b 1n 1 ≤T n nment bond market). j=1 j !n Spercentage (T value of protection leg minus premium leg 1is1 CDX (Min) both × non i ).1 Finally, tsections swap design to allow model-free the variance and the at Tfor(which is onepricing in theofcases dealt with in time minus premium leg is b contract (τ )} )with time atat which name jCDX (τ(1 ) = Snotional (Ti ) ≡is isNj=1 − In{τSjn≤T τ(t) i The number of names having up to Ti is notional ) where withN N time which name jdefaults, defaults, andthetheoutstanding outstanding N (τ ) = j (t ntuitively, tilting adeposits variance swapinterest by the rate market num´ eraire at (which istoone the cases dealt DSX LGD · vbe − ) ·and vsurvived t = 0tin t (M 1t , 1 1 S i(τ Credit and swaps) causes its Tfair dent variance terms. More generally, MO (2013d, Chapter 2) develop avalue framework that handles 1 1 t = LGD · v0t − 1CDXt (M ) · v1t , bτjshould DSX obligor j default is LGD I , whereas the premium at The index loss at τ S (τ ) with N (t) time at which name defaults, and the outstanding notional is N (τ ) = DSX = LGD · v − CDX (M ) · v , 1 should obligor j default is LGD I , whereas the premium a The index loss at j tj t 1t n n {t≤τ } } in sections on time deposits andbeinterest rate swaps) causes its the fair value toriskbe defined under0t b j ≤T bMbM {t≤τ j ≤T bn volatility priced through credit default swaptions, nature ofhow credit this ket and num´eraire ofCredit interest incanthe fixed income space. Let although us framework DSXt = LGD · v0t − CDX 1 1illustrate 1credit 1 t (M ) · v1t , 1 (T calls for relevant a where number of new features to take into account. For example, we need to(M consider S ). Finally, the value of protection leg minus premium leg is is premium at CDX (M ) × should obligor j default is LGD I , whereas the The index loss at τ b v is the value at t of $1 paid off at the time of default of a representative firm, provided CDX ) × S (T ). Finally, the value of protection leg minus premium leg t i j rket space where all the information is given by the price of available derivatives—an {t≤τ ≤T } 0t t i j bM b n n is variance given by the price of available derivatives — an expectation n es in the interest rate swap case. swaps on loss-adjusted forward position in a CDS index, and web must deal with the survival vthe istime the of value at t of of a$1 paid offpremium at the time of a r 1 value 1 $1 paid where 0t where is the at t of off at default representative firm, provided 3v0t num´ 3 CDX (M ) × S (T ). Finally, the value of protection leg minus leg isof default contingent probability and the defaultable annuity market e raire to account for default risk. ctation under the num´ e raire probability of interest. the value t value of an annuity of $1 off paid at default occurs before theofpayoff: index where t and under the numéraire probability interest. bexpiry; nvaluevi 1t where v0t isatthe at t of 1$1 paid at the time of default where v0t is the atis t of $1 paid off at the time of default of a representative firm, ider a variance swap starting at t with the following We only present the percentage variance contract formulation here. The risk we are dealing with 1 is the valueprovi at t occursv1tbefore the index and where v1t paid isLGD the value at texpiry; of an annuity of $1 at default occurs before the index expiry;default and where inally, an index of interest rate swap volatility is periodic DSX = LGD · v − CDX (M ) · v , an index of rate swap volatility is , · · · , T , until either a default of the representative firm or the expiry of the index (whichever DSX = · v − CDX (M ) · v , t 0t t 1t isFinally, that of aT CDS index, forbM which ainterest buyer pays premium (the CDS index spread) and the 1 t and where 0t t isb the ofvalue at 1tt of an annuity of $1 paid default occurs before the Tindex expiry; v1tdefault ! " b 1 , · · · , T , until either a the representative firm or the ex seller insuresbp losses from defaults by any T of the index’s constituents during the term of the contract. 1 bM = LGD · · ·PVBP , (T until a default of the representative firm expiry (whichever DSX · v1tor −the ·ofv1texpiry ,index theCDX value atof)the t the an annuity of index $1 (whiche value of, TabM defaultable Voccurs T first)–the )− )1 , × ,· ·· ··either ·and πn (t, T )If ≡ tthe representative 0t is t (M n (t, T n )index TT n (t, a constituent defaults, the P defaulted obligor is $ removed from the the continues ,,TTannuity. , until either occurs a default of firm or of the 11,index, bM b first)–the ofeither atime defaultable annuity. 1a CDS where occurs first)–the value defaultable annuity. v of0tisa is the value at off atatthe default to be traded with abp notional amount. index are European-style, to paid TT$1 ,... ,TabM ,value until aofdefault of ofthe representative where the value att of t at of$1 off the time of default ofa arepresentative representativefirm, firm,prov pr 2 ×Options 1paid Anprorated CDS payer ison an option abuy CDS index at aspaid protection buyer with strike Poccurs IRS-VI (t,protection T )index ≡ at100 T ) to0tventer n (t, (payers) or sell (receivers) the strike spread upon option expiry.first)–the value of a defaultable annuity. T − t A CDS index payer is an option to enter a CDS index at T aspro a ere Pn (t, T ) is the fairWevalue the contract such that CDS index payer isis an option to enter CDS index at Ttime buyer with strike is the value trepresentative of anan annuity of of $1 pa default before index and vas where the valuethe at t index of $1aexpiry; paid off atwhere the ofprotection default ofat aat firm, is the value t of annuity $1 default before the expiry; and where 1tv a 0t, Toccurs ) with length assume of credit events may occur over aA sequence of regular intervals (Tvoccurs 0

Rt (T1 ,··· ,Tn )

i−1 buyer i 1t arising from spread K. Upon exercise, the protection would a front-end A CDS index payer also isspread anreceive option to enter aprotection CDS index at T buyer as a protection buyer with st K. Upon exercise, the protection receive a$1(wh fron of a defaultable annuity. ! " spread K. Upon exercise, the protection buyer would also receive a front-end protection from , · · · , T , until either a default of the representative or the ofalso the index T the value atexpiry twould ofarising an annuity of (which pa default occurs before the index expiry; and where v TbM , until either a default of the representative firm or the expiry of the index 1Tmatures. 1t is firm 1 , · · ,bM losses occurring the option Accordingly, consider a loss-adjusted forward position at A before bp e Pn (t, T ) is as in Eq. (23). spread K. Upon exercise, the protection buyer would also receive a front-end protection arising where Pnn (t,T (t, T ) = Et Vn (t, T) , T asindex occurring before the option matures. Accordingly, consider a(whic lossfr 11 occurring matures. Accordingly, consider a loss-adjusted forward at occurs value oflosses aa(20) defaultable annuity. , which · · · ,first)–the Tthe ,option until either offollowing: the representative firm or the expiryposition of the TT1before first)–the value of adefault defaultable annuity. bM t in a CDS indexlosses that starts atlosses ,occurs can be shown to equal the occurring before theta option matures. Accordingly, consider a loss-adjusted position protection buyerto with strike spread K. Upon exercise, theforward proaan CDS index that starts at Tindex , which shown to equal the folls t in a CDS index that starts at value Tpayer , payer which can beoption shown equal the following: A ACDS an option toannuity. a aCDS index atatT can buyer with occurs first)–the ofisin ais defaultable CDSindex index toenter enter CDS Tasasabeaprotection protection buyer with A denotes conditional Credit expectation under the annuity probability Q . Moreover, by Eq. (19) t in a CDS index that starts at T , which can be shown to equal the following: tection buyer would also receive a front-end protection arising " # edit A 1 spread K.K.Upon exercise, protection also receive protection A CDS index payer isthe an option to buyer enter awould CDS index at1 T aasfront-end protection buyerarising with spread Upon exercise, protection would also a afront-end protection arisins ) losses − CDX , before DSXL " τ (M the t (M ) buyer " occurring #receive t,T (τ ) ≡ N (τ ) v1τ CDX1 the matures. Accordingly, lemma, L option b DSXL (τ ) ≡ from CDX (τ ) ≡ N (τ ) v (M ) − CDX DSX " # 1τ τ t (M CDX N (τ ) v (M ) − CDX (M ) , t,T 1 losses occurring before the option matures. Accordingly, consider a loss-adjusted forward positi spread K. Upon exercise, the protection buyer would also receive a front-end protection arising 1τ τ t t,T losses occurring before theL option a loss-adjusted forward posi the nature of default credit risk new b it volatility can bealthough priced through credit swaptions, although the of nature ofDSX risk matures. Accordingly, consider bcredit ! calls for "a number t,T (τ ) ≡ N (τ ) v1τ CDXτ (M ) − CDXt (M ) , # $ bp 2 2 into account. A bwhich tlosses awe starts atat T ,that which can bebe shown toconsider equal following: the option matures. Accordingly, athe loss-adjusted to− take example, need cont) in aoccurring CDS index that starts T ,T can shown to equal following: forward positi the value of CDX (M ), set such a forward position at τ inthe the that starts at ,features · · · ,features Tn )where Rto (T ·(M , Tn)account. ) is = defined EtForVFor (t, Tin =CDS Pwe (t,index T )to. before (21) EA for a number of1 new take example, need to consider credit τthat n 1 , · ·τinto n as t RT (T t CDX (M ) is defined as the value of CDX setthe such that a where CDX τ τ (M Lτ (M ) is defined as the value of CDX (M ), set such that a forward position at τ), in where CDX sider credit variance swaps on loss-adjusted forward position τ index that the starts at T , which can be shown to equal the following: (τtand )in=awe 0,CDS is worthless, DSX nce swaps on loss-adjustedindex forward position in viz a CDS index, must(M deal with survival τ,T where # # position at τ in asLthe value1 of CDX (M"L), (τ set) = such that a forward CDX τ L ) is defined τ" 1 0, index is worthless, viz DSX L e second equality and follows Eq. (20). annuity (τ )default = 0, DSX index is worthless, DSXτ,T for τ,T (τ (τ )≡ (τ (τ ) v)1τv1τCDX ) −) − CDX ) ), , DSX ) ≡NN CDX CDX τ (M t (M t,Tt,T ngent probability theby defaultable market num´ eraire viz toworthless, account risk. τ (M t (M " # (τ )F= 0, b1 b index is viztoDSXL τ,T L probability and the defaultable annuity market numéraire ar to our derivation of Eq. (16), we now take the expectation under the annuity probability 1 v v CDX (M ) − CDX (M ) , (τ ) ≡ N (τ ) v DSX 0τ 1τ τ t We only present the percentage variance contract formulation here. The risk we are dealing with τ t,T CDXτ (M ) = LGD + . vb (M v0τ set vτF 2 account for default risk. remainder, obtaining: vτF (M ), 1set !0τ of CDX ! (M), , Tn ) with Taylor’s expansion b where v1τdefined N)as(τ ) v (M )1is defined the value such that a forward position atat where F 1τ T (T1 , ·a· ·buyer CDX (M ) = LGD + .τ in CDX )CDX is as the value of CDX (M ), set such that a forward position τ τ index τ at of a CDS index, of forRwhich pays periodic premium (the CDX CDS spread) and the τ τ (M τ (M = LGD + . 1 v0τ b vτ v1τ viz N (τ ) v1τ - τsuchL that b isviz N (τ ) v(M a=forward ! set in the index CDX ) =position LGD + . isaworthless, 1τ of 1τat), L(τthe τ v(M ) 0, index is worthless, DSX (M ) defined as value CDX such that forward position at τ i where CDX $ insures losses from defaults by any of the index’s constituents during the term of the contract. ) = 0, index is worthless, viz DSX τ τ τ,TLτ,T b (τIndeed, (τ ) v1τ CDXt (Mv1τ ) is aNmartingale RT2 (T1 , · · · , Tn ) − Rt2 (T1 , · · · Note , Tn ) that N (τ ) v1τ is the natural num´eraire in -this market. L !,T(!)=0, constituent defaults, fromN the and the index continues (τ )that = index worthless, viz'DSX natural num´ eraire ainmartingale this market. Indee Note N (τmarket. ) v1τ is the %& the defaulted obligor is removed τ,T natural num´ eraire in0,this Indeed, CDX (M Note&that (τ index, ) vis 1τ is the tF , )defined through Radon-Nikodym derivative, as ) is CDX under probability” Qsc(τ Rt (T1 ,··· ,Tn ) the “survival contingent ∞ 1 1thenum´ v0τvmarket. vτFvIndeed, the natural e raire in this ) is a marting Note that N v1τ isunder 2 with a prorated t (Mthrough p European-style, traded notional amount. Options on a CDS index are to buy 0τ r τ defined the R “survival probability” ) contingent = LGD ++ F .Q.sc , derivative, Swpn Tunder ) dK losses +the “survival Swpn (K,any Tprobability” ) dK defined through the Radon-Nikodym as contingent spread) follows: and the seller insures from defaults of , (22)Qthe CDX )= LGD τ (M sc ,CDX n,t (K, n,tby τ (M $ (T1sell , · · ·(receivers) , Tn ) BPt or b1 b v1τ NN (τvτ(τ ) the v)1τv1τ % T contingent probability” through Radon-Nikodym derivative “survival Qsc , defined v1τ 0 protection at the strike spread upon Rt (T1under ,··· ,Tn ) the 0τ ers) option expiry. $ follows: N (T ) v dQ + $ . %T sc $ 1T follows: − τ r (u) $du τ (M ) = LGD %b TCDX N (T ) v1T $ Nlength follows: sc $$(τ ) v1τ − ,the Ti )sc with We assume credit events may occur overthea defaulted sequence of regular intervals (Ti−1 $,N(!) $ r = efrom N (T ) natural v1T v1τ dQ dQ (u) du N %the − r (u) du constituent defaults, obligor is removed dQ (τ ) v p Note that v is numéraire in this market. Inτ r 1! r $ T 1τ = e , τ $ Note that N (τ ) v is the natural num´ e raire in this market. Indeed, CDX ) is a marti F = the e dQ is natural num´ eraire in,du this market. CDX (M Note that swaptions N (τ )1τv$1τ at t (M NdQ (T )$ vr1T Indeed, t, wpnn,t (K, T ) and Swpnn,t (K, T ) denote the prices of receiver and tN T payer sc $ − r (u) (τ) )isv1τa mar r i = 1, · · · , bM , where is the of years the index runs, T0 is the time τ ) v1τ dQ ofFrthe index t $(M)= (τ eaN , FT “survival index,Mand the number index continues to be traded with a prorated is martingale under the contingent T the natural r denotes through the Radon-Nikodym thethe “survival probability” Qsc is num´ raire in this N market. Indeed, CDXt (M ) is derivativ a deriva marti Note that N, (τ ) contingent v1τcontingent dQ (τ ) v1τonly. , defined through the Radon-Nikodym T and Fstrike K, and at under T .under “survival probability” Q,short-term ng a swap with tenor Tn −where scdefined theexpiring information set at time T which includes the path ofethe rate FrT T r denotes - the pat sc r denotes the information set at where F the information set at time T , which includes p where F time T , which includes the path of the short-term rate only. follows: , defined through the Radon-Nikodym derivativ under the “survival contingent probability” Q T $ $ τ ∈ [t,sc follows: Treceiver with rstrike T ;the M ) path ≡ of the short-term rate o The prices of a(receivers) payer andprotection K spread expiring tive, at T ,asare, forat any T τ (K, T%],T SW to buy (payers) or sell at Fthe strike where set time T− ,%which includes follows: $ 11 N (T(T ) vT dQ T denotes the information p N )1Tv],1TSW dQ rfor (u) dudu The prices ofsc a$$$scpayer and with strike at T , are, for any −τ receiver r (u) The prices of a payer and receiver with strike K expiring at T , are, any τ ∈ [t, $ follows: = e , K, expiring τ τ (K, T ; M ) ≡p = expiring e% T upon option expiry. $$ r$ K The prices of a payer and receiver with strike at T , are, for any τ ∈ [t, T ], SWτ (K, T ; M dQdQ (τ ) v N (T ) v dQ N (τ ) v 1τ r sc $FT F 1T1τ − r (u) du , $ r T= e τ dQ N (τ ) v 1τ r denotes FT time T , which includes the r where12 ...F,bM, the information setset at path of the short-term rate lar intervals (Ti-1,Ti) with length 1/b, forwhere i=1, Mthe where information at time T , which includes the path ra TFT denotes 12of the short-term 12 10 p p r time rset at is the number of years the index runs, The T0The isprices the of the (K, T of a payer and receiver with strike K expiring at T , are, for any τ ∈ [t, T ], SW where F denotes the information time T , which includes the path of the short-term rate F Twith denotes theK12 information at time T, τwhich prices of a payer andwhere receiver strike expiring at Tset , are, for any ∈ [t, Tin], SW τ τ (K,;TM T thewith pathstrike of the short-term only. a pτ (K, T ; M The prices of a payer andcludes receiver K expiring at Trate , are, for The any τprices ∈ [t, Tof], SW 1 b,

for i = 1, · · · , bM , where M is the number of years the index runs, T0 is the time of the index

1212 12

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payer and receiver with strike K expiring at T, are, for any indexes viable for serving as the underlying of tradable prodp + r ucts"such as volatility futures and options. M)N(!) v1! . E!sc ! (K,T T (M)-K)! ! ! !%[t,T +" + r sc sc ! " ! " ." . Er!scSW v1τ · Eτ (CDX )− K) v1!and ≡ N+].(τ ) v1τ sc · Eτ! (K − CDXT (M )) T (M !sc " + τ -(K, T ; MT)(M)) (K,T M)N(!) [K ! T (M ) − K)+ and" SWr (K, T ; M ) ≡ N (τ ) v1τ · E sc (K − !CDXT (M ))+ + " (CDX .

τ + SWττ (K, rT ; M ) ≡ N (τ ) v1τ · E and − CDXT (M )) . + K) sc (CDX T (M τ ·(K eτ·(CDX assume that)T − and SWτ (K, T ; M ) ≡ N (τ ) v1τ E (M ) − K) Esc τ (K − CDXT (M )) . # $ $ # sume that dCDXτ (M ) #= − #Esc ej(τ ;M ) $− 1 η $(τ ) dτ dCDXCDX $ $ τ (M ) (M ) # sc# τj(τ ;M ) $ dτ $ $ # ) − 1 η (τ )# dCDXdCDX Eτ # ej(τ ;M τ (M ) τ = − = −) =Esc −;M 1 ) η (τ ) dτ CDXτ (Mτ) (M τ− eEsc ej(τ sc − 1 ηj(τ ;M ) ) −$ dτ1 dJ sc (τ ) , τ ) · dW (τ )#+ e (τ CDXτCDX (M ) (M ) + σ (τ ; M

τthat me that

-

τ sc ;M ) income asset classes. + σ (τ ; M )" · dW sc (τ ) + #ej(τ # )− 1 $dJ! scsc(τ $ ), ! " j(τ ;M +) · dW r(τ ) sc sc (CDX + σ (τ ; M + e 1· E)scτ − dJ(K ) sc , T(τ j(τ sc sc ))+ . and SW −(τ CDX N (τ ) v · E (M ) − K) (K, T ; M ) ≡ N (τ ) v− (M 1τwhere 1τ;M T (! + σ (τ ; M ) · dW sc sc τ τ a methodological research perspective, the logical next (τ ) + e 1 dJ ) , W (!) W (τ ) is a multidimensional Brownian motion and J (τ ) is a Cox process (with From intensity We assume that sc jump . . step in supporting the creation of a market for standardized &( ) and size j( )) under the τnd ) is a multidimensional Brownian motion and J (τ ) is a Cox process (with intensity jump size j (·)) under the survival contingent probability. # $ sc$ # dCDX Brownian motion (τ ) is a Cox process (with intensity τ (M ) scis(τa) multidimensional sc sc j(τ ;M ) and J survival contingent probability. aprice multidimensional and (τ ) is = − Eoriginated − 1atηt, (τ and ) dτJpaying e motion mp sizeis j (·)) under the survivalBrownian contingent probability. e wish to a credit variance swap off aatCox T , asprocess follows: (with intensity τ CDXτ (M ) # $ sizeprice jsize (·)) under the survival contingent probability. t, follows: and options on these indexes in a way that is consistent with the unto a jcredit variance originated at t, and off jump (·)) under theswap survival contingent probability. sc j(τpaying ;M ) sc at T , as + σ (τ ; M ) · dW (τ ) + e − 1 dJ (τ ) ,

) − Pvar,M (t, )) × ) v1T , off at T , as follows: oish price a credit variance swap originated atT t, and paying paying offvariance at(VT, as Tfollows: M (t, to price a credit swap originated atNt,(T and paying off at T , as derlying follows: yield or credit curves, the indexes themselves and the where W sc (τ ) is a(V multidimensional Brownian motion and J sc (τ ) is a Cox process (with intensity term M (t, T ) − Pvar,M (t, T )) × N (T ) v1T ,

M

structure of volatility in order to facilitate risk managetrading strategies by end users.

η (·)(t,and sizefair underof thethe contingent Pvar,M T )jump is the value contract, we(T have the percentage variance as, formulation of ment and (Vj (·))(t, T) − Psurvival (t, T ))and × probability. N ) v defined ,

M (V (t, T var,M 1T )− Pvar,M (t, Tat))t, × (T ) voff1Tat, T , as follows: M variance We wish to price a credit swap originated andNpaying

% Tcontract, and we %have (t, T ) is the fair value of the defined the percentage variance as, T

) ≡(t,T) ∥σ (τ MP)∥ dτ(t,we + M, ) dJthe (τpercentage ) .we have variance as, Vvalue where fair value ofhave contract, and (t,the T );− T )) ×the Nj (T(τ ) v;1T (Vcontract, var,M M (t, T t, T ) is the fair and defined Mis var,M %of Tthe of ar,M (t, T ) is the fair value T t we have defined the percentage variance as, t the contract,% and FOOTNOTES 2 2 sc ≡% (τ ;ofMthe )∥ contract, dτ + %andT jwe (τ ; Mdefined ) dJ the (τpercentage ). M (t, where PVvar,M (t,T T) ) is theTfair∥σ value variance as, %have MO (2013d, Chapter 5, Appendix D), 2we showt that T t % T sc %M ≡ T ) ≡∥σ (τ ;∥σ )∥; M dτ)∥+2 dτ +%j 2T (τ ;jM VM (t,VT )(t, 2 ) dJ (τ ) sc T (τ (τ ; M ) dJ . (τ ) . 2

M

&

2

2

2

sc

'

sc

(t, T )(M ≡) we ∥σr show (τ ; M )∥that dτt+ j (τ ; M ) dJ (τ ) . p %VtMCDX 2013d, Chapter 5, 2Appendix t % ∞ tt D), SW SWt (K, T ; M ) t t (K, T ; M ) t Pvar,M (t, T ) = dK + dK showed the remarkable property that the fair value of a 2 2 & ' variance swap remains the same in this case. K K v 13d, Chapter 5, Appendix D), we show that 1t 0 CDX (M ) % CDX % ∞ In MO (2013d, Chapter 5, Appendix D),we we show show that t r D), O (2013d, Chapter 5, Appendix that t (M ) 2 SWpt * (K, T ; M +) (%& TSW ) t (K, T ; M ) ' dK + % ∞1 2 dK ' ar,M (t, T ) = &% p ) 2) r sc 2 % CDXtj(τ ;M sc ) '(24) SW− T2; M v pK − 2E er (MK 1 −T ;jM(τ) ;dK M%+ )CDX − ; Mt )(K, dJ (τ ) . CDX ) ∞ t (M t (K, 0& )(τSW %tt(M %j ∞ pTdK 2P1t T ; M ) SW (K, ; M ) ; M )2. Merener var,M (t, T ) =CDXSW t (M t) (K, r (2012) considers the replication of variance 2 2 2 t t v1t K TdK K SWt (K, 2 SW (K, ; M ) T (% ) * + 0 CDX (M ) t t (t, T ) = + dK M T 2 dK1CDX +2 (M ) * dK Pvar,M (t, T (% +2 K K v1t)−=2E j(τ ;M sc sc T) ) 2 2 0 t ; M) v1t e is −ej(τ1;M −K j1 (τ dJ (τ ) K . (24) ) j (τ ; M ) − CDX t dingly, a credit volatility −index 2Esc − 1 − j (τ ; M ) 2− j 2 (τ ; M )t (M dJ)sc (τ ) . + (24) t (% Tt0 (% ) * 2 )t * + approximation, and does not rescale for the relevant no-

1 2 1 T ;M ) j(τ sc , − 2Easc − 1 )−−j 1(τ− ;M )− j (τ ; M j(τ ;M sc e 2 ) dJ (τ ) sc. t− 2E e j (τ ; M Accordingly, credit volatility index is 1 2) − j (τ ; M ) dJ (τ ) a credit volatility index is tt C-VI 2 T) ) ≡ 100 × index isPvar,M (t, M (t, T volatility Accordingly, atcredit ,T − t 1 , credit volatility indexindex is C-VIisM (t, T ) ≡ 100 ×1 T − t Pvar,M (t, T ) gly, a credit volatility Pvar,M (t, T ) (t, T ) ≡ 100 × M (24). Pvar,M (t, T ) is as C-VI in Eq.

market . tion of(24) (24)numéraire in the manner we suggest in this

section, thereby providing neither the index formulae and pricing in this section nor the hedging details described in , T −,t where Pvar,M (t, T )RHS is as in ers variance swaps priced and hedged based on parametric he first term on the of Eq. Eq.(24). (24) is model-free, once we estimate the CDX default intensity 1 1 we (t, The firstwhere term on M the(t, RHS model-free,Ponce estimate default intensity C-VI T )of(t, ≡Eq. T )(t,the var,M assumptions. (t,T var,Mterm second isT100 small for all intents and purposes, and should not materially porated by v . The Pvar,M C-VI )(24) ≡×is100 T )CDX T× − t M (t, T ) is as1tin Eq. (24). M incorporated by v1t . The second term is small for all intents T −and t purposes, and should not materially

the value of the index approximated by by only retaing thefirst firstterm. term. term on the RHS is model-free, once we the CDX default intensity affect the value of of theEq. index(24) approximated only retaing the estimate

. The

1t

3. Interestingly, our model-free expression for the variance ment of the forward swap rate, as it turns out by compar-

t,by T )(t, asThe Eq.in(24). second term is small for all intents and purposes, and should not materially dar,M vis 1t .T ) in is as Eq. (24). second term is small for all intents and purposes, the and should erm on the RHS of Eq. (24) model-free, once we default intensity lue of the index approximated by only retaing theonce firstestimate term. rst term on the RHS of Eq. is (24) is model-free, we estimateCDX the CDX default intensity

not materially affect the value of the index approximated by

. The term term is small for allfor intents and purposes, and should not materially y v1tby The second is small all intents and purposes, and should not materially ted v1t . second evalue of theofindex approximated by only retaing the first term. 13 the index approximated by only retaing the first term. 13

Closing Thoughts The viability of volatility indexing using the methodologies 13 13

that complements the asset pricing foundations laid down describe the behavior of variance risk-premiums under both interpretations of n (t,T

13

options used in the various index formulas render the resulting

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REFERENCES Journal of Financial Economics

able from www.garp.org/risk-news-and-resources/2013/ december/

Interest Rate Finance Institute. Journal of Finance Energy & Power Risk Management 4,

Quantitative Finance 12, 249-261. Journal of Portfolio Management 20, 7480. -

Quantitative Finance 1, 19-37.

script.

-

Institute. Journal of Financial Economics

Mele, Antonio, 2013. Lectures on 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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