# Variance Swap. by Fabrice Douglas Rouah

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1 Variance wap by Fabrice Douglas Rouah In his Noe we presen a deailed derivaion of he fair value of variance ha is used in pricing a variance swap. We describe he approach described by Demeer e al. [] and ohers. We also show how a simpler version can be derived, using he forward price as he hreshold in he payo decomposiion ha is used in he derivaion. he variance swap has a payo equal o N var R K var () where N var is he noional, R is he realized annual variance of he sock over he life of he swap, and K var E[ R ] is he delivery (srike) variance. he objecive is o nd he value of K var : ock Price DE he variance swap sars by assuming a sock price evoluion similar o Black- choles, bu wih ime-varying volailiy parameer d d + dw : Consider f() ln and apply Iō s Lemma d ln d + dw so ha d d ln : () he Variance In equaion () ake he average variance from o V Z " Z d Z # d d ln " Z # d ln : (3)

2 he variance swap rae K var is he fair value of he variance; ha is, i is he expeced value of he average variance under he risk neural measure. Hence " Z # K var E[V ] E d (4) # E " Z d ln r E ln : he erm d represens he rae of reurn of he underlying, so under he risk neural measure he average expeced reurn over [; ] is he annual risk free rae r imes he ime period, namely r h. Mos i of he res of his noe will be devoed o nding an expression for E ln : 3 Log Conrac he log conrac has he payo funcion f( ) ln + : (5) Noe ha f ( ) ( ) and f ( ). 4 Payo Funcion Decomposiion Any payo funcion f( ) as a funcion of he underlying erminal price > can be decomposed as follows f( ) f( ) + f ( )( ) + (6) Z f (K)(K ) + dk + Z f (K)( K) + dk where > is an arbirary hreshold. ee he Noe on for a derivaion of equaion (6) using hree di eren approaches. Apply equaion (6) o he log conrac (5) o ge ln + ln + + ( ) + Z K (K ) + dk + Z K ( K) + dk:

3 Cancel from boh sides and re-arrange he erms o obain ln Z + K ( Z K (K ) + dk + (7) K) + dk: his is equaion (8) of Demeer e al. []. ake expecaions on boh sides of equaion (7), bringing he expecaions inside he inegrals where needed E ln E[ ] + (8) Z K E (K ) + dk + er + e r Z P (K)dK + er K Z Z K E ( K C(K)dK K) + dk where P (K) e r E [(K ) + ] is he pu price, C(K) e r E [( K) + ] is he call price, and where E[ ] e r F is he ime- forward price of he underlying a ime zero when he underlying price is. Now wrie ln ln + ln which implies ha E ln ln E ln : (9) ubsiue equaion (8) ino (9) o obain E ln ln e r + () e r Z 5 Fair Value of Variance P (K)dK + er K Z K C(K)dK: Recall equaion (4) for he fair value K var r E ln : () 3

4 h i ubsiue equaion () for E ln o obain he fair value of variance a incepion, namely, a ime. K var r e r ln + () e r Z P (K)dK + er K his is equaion (9) of Demeer e al. []. Z 6 Fair Value Using Forward Price K C(K)dK omeimes K var is wrien in a simpli ed form. o see his, le he hreshold in equaion () be de ned as he forward price, e r F. Afer some minor algebra, we arrive a ( Z K var F Z ) er K P (K)dK + F K C(K)dK : (3) 7 Mark-o-Marke Value of a Variance wap In his secion we use he noaion of Jacquier and laoui [6]. A incepion of he variance swap, he swap srike is se o he expeced value of he fuure variance, so he swap has value zero. Going forward, however, he value of he swap can become non-zero. o see his, rs denoe he denoe he average expeced variance over he ime inerval (; ) as " Z # Kvar ; E udu where E [] denoes he expecaion a ime. A incepion ( ) we wrie " Z # E udu K ; var which is K var ha appears in equaion (4), (), (), and (3), he value a incepion of he variance swap srike. he value a ime of he variance swap srike, denoed, is he ime- expeced value " Z # e r( ) E udu Kvar ; : (4) ) 4

5 A incepion his expeced value is zero, bu a ime i is no necessarily so. We wrie equaion (4) by breaking up he inegral, which produces " e r( ) E Z udu + Z # udu Kvar ; (5) ( " e r( ) ; + Z # ) E udu Kvar ; e r( ) ; + K; var Kvar ; e r( ) ; Kvar ; + Kvar ; K ; var R where ; udu is he realized variance a ime, which is known. he las equaion in (5) for is one ha is ofen encounered, such as ha which appears in ecion 8.6 of Flavell [3], for example. I indicaes ha he ime mark-o-marke value of he variance swap is a weighed average of wo componens. he erm ; Kvar ;, which represens he "accrued value" of he variance swap. Indeed, his is he realized variance up o ime minus he conraced srike.. he erm K ; var K ; var, which represens he di erence in fair srikes calculaed a ime, and calculaed a ime. Hence, a ime, o obain, we need o calculae R udu, which involves only variance ha has already been realized. We also need o calculae Kvar ;. In he same way ha equaions () or (3) are used wih opions of mauriy o obain Kvar ;, hose same equaions can be used wih opions of mauriy ( ) o obain K ; var : 8 Consan Vega of a Variance wap Exhibi of Demeer e al. [] shows ha a porfolio of opions weighed inversely by he square of heir srikes has a vega which becomes independen of he spo price as he number of opions increases. his can be demonsraed by seing up a porfolio of weighed opions C de ned as Z w(k)c (; K; ) dk (6) where w(k) is he weigh associaed wih he opions and C(; K; ) are heir Black-choles prices. his porfolio is similar o ha appearing in equaion (3). However, since we are concerned wih vanna, which is idenical for calls and pus, we do no have o spli up he porfolio ino calls and pus eiher will 5

6 do. his implies ha we can de ne C(; K; ) o be eiher calls or pus, and we don need o spli up he inegral in equaion (6) ino wo inegrals. o explain heir Exhibi, Demeer e al. [] demonsrae ha he vanna of he porfolio, is zero only when he weighs are inversely proporional o K. 8. Vega of he Porfolio he he porfolio is idenical for vanilla calls and pus so he vega of Z K; ) dk: he Black-choles vega of an individual opion p exp d where d p ln(x) + and x K. Hence we can wrie equaion (7) as p V p Z Change he variable of inegraion o x. and we can wrie p Z V p w(k) exp d dk: Hence dx dk so ha dk dx w(x) exp d dx: (8) In his las equaion, d depends on x only. Hence when we di ereniae V wih respec o we only need o di ereniae he erm w(x). his derivaive is, by he 8. Vanna of he Porfolio w(x) w(x) x: ubsiuing equaion (9) ino (8) and di ereniaing wih respec o, he sensiiviy of he porfolio vega p p w(x) + d dx: he erm inside he square brackes can be wrien w(x) w(k) : 6

7 @ implies ha w + or ha w w K. he soluion o his di erenial equaion is w(k) / K : Hence, when he weighs are chosen o be inversely proporional o K, he porfolio vega is insensiive o he spo price so ha is vanna is zero. his is illusraed in he following gure, which reproduces par of Exhibi of Demeer e al. []. A porfolio of calls wih srikes ranging from \$6 o \$4 in incremens of \$ is formed, and he Black-choles vega of he porfolio is calculaed by weighing each call equally (doed line) and by K (solid line). he ineres rae is se o zero, he spo price is se o \$, he mauriy is 6 monhs and he annual volailiy is %. 4 Weighed by inverse srike Equally weighed Porfolio Vega ock Price he vega of he equally-weighed porfolio (doed line) is clearly no consan bu increases wih he sock price. he vega of he srike-weighed porfolio, on he oher hand, is a in he \$6 o \$4 region, which indicaes ha is vanna is zero here. 7

8 9 Volailiy wap his is a swap on volailiy insead of on variance, so he payo is N vol ( R K vol ) () where N vol is he noional, R is he realized annual volailiy, and K vol is he srike volailiy. he values of N vol and K vol can be obained by wriing equaion () as N vol ( R K vol ) N vol R Kvol N vol R + K vol K vol R Kvol : () 9. Naive Esimae of rike Volailiy Comparing he las erm in equaion () wih equaion (), we see ha N vol K vol N var and K var K vol from which we obain he naive esimaes K vol p K var and N vol N var K vol. 9. Vega Noional and Convexiy he payo of a volailiy swap is linear in realized volailiy, bu he payo of a variance swap is convex in realized volailiy. he noional on volailiy, N vol is usually called vega noional. his is because N vol represens he change in he payo of he swap wih a poin change in volailiy. his is bes illusraed wih an example. uppose ha he variance noional is N var \$;, and ha he fair esimae of volaily is 5, so ha he variance srike is K var 5. he srike on he volailiy is K vol 5, and he vega noional is N vol \$; 5 \$5;. In he following gure we plo he payo from he volailiy swap and from he variance swap, as R varies from o 5: 8

9 5 Variance Payoff NoionalVariance x (Realized Vol^ K^) Volailiy Payoff NoionalVega x (Realized Vol K) 5 Payoff in \$M Realized Volailiy in Percen he payo from he volailiy is Volailiy Payo N vol ( R K vol ) \$5; ( R 5) ; which is linear in R. When he realized volailiy increases by a single poin, he payo increases by exacly N vol \$5;. his is represened by he dashed line. he payo from he variance is Variance Payo N var R K var \$; R 5 ; which is convex in R and is represened by he solid line. uppose ha he realized variance experiences a wo-poin increase from R 38 o R 4. hen he volailiy payo increases from \$65; o \$75; an increase of \$; which is exacly N vol. he variance payo increases from \$89; o \$975; an increase of \$56;. 9

10 9.3 Convexiy Adjusmen he convexiy bias described in [] is he di erence beween he las and rs erm in equaion (). Wihou loss of generaliy we can se N vol. Hence Convexiy Bias R K vol ( R K vol ) K vol ( R K vol ) : K vol We approximae he expeced value of volailiy by he square roo of he expeced value of variance. Hence we use he approximaion E [ R ] p E [ R ]. We can nd a beer approximaion o he volailiy swap by considering a second order aylor series expansion of p x p p x x + p (x x ) + x 8 p (x x x 3 ) : () e x R v and x E R v in equaion () o obain p v p v + p v (v v) + 8 p v 3 (v v) : (3) ake expecaions and he middle erm on he righ hand side of equaion (3) drops ou o become E p v p v + 8 p v 3 E (v v) : In he original noaion q E [ R ] E [ R ] + V ar R 8E [ : R ]3 Hence, he loss of accuracy brough on by approximaing E [ R ] wih p E [ R ] can be miigaed by adding he adjusmen erm V ar[ R] 8E[ R] 3 : Oher Issues. Implemening he Variance wap Formula o come. Requires an approximae due o he fac ha marke prices of pus and calls are no available on a coninuum of srikes, bu are insead available a discree srikes, ofen in incremens of \$5 or \$.5.. Variance wap Greeks o come. form. Many of he Greeks for variance swaps are available in analyical

11 .3 Variance wap as Model-Free Implied Volailiy o come..4 he VIX o come. he VIX index is a 3-day variance swap on he &P 5 Index, wih convexiy adjusmen. Variance wap in Heson s Model In he Heson (993) model he sock price and sock price volailiy v each follow heir own di usion, and hese di usions are driven by correlaed Brownian moion. Hence d d + p v dz () dv ( v )d + p v dz () h i wih E dz () dz() d. he volailiy v follows a CIR process, and i is sraighforward o show ha he expeced value of v, given v s (s < ) is E [v j v s ] v s e ( s) ( + e s) + (v s ) e ( s) : ee, for example, Brigo and Mercurio []. As explained by Gaheral [4], a variance swap requires an esimae of he fuure variance over he (; ) ime period, namely of he oal (inegraed) variance w R v d. A fair esimae of w is is condiional expecaion E [w j v ]. his is given by " Z # E [w j v ] E v d v Z Z E [v j v ] d + (v ) e d + e (v ) : ince v represens he oal variance over (; ), i mus be scaled by in order o represen a fair esimae of annual variance (assuming ha is expressed in years.) Hence he srike variance for a variance swap is given by E [w j v ] e his is he expression on page 38 of Gaheral [4]. (v ) + :

12 References [] Brigo, D., and F. Mercurio (6). Ineres Rae Models - heory and Pracice: Wih mile, In aion, and Credi. econd Ediion. New York, NY: pringer. [] Demeer, K., Derman, E., Kamal, M., and J. Zou (999). "A Guide o Volailiy and Variance waps," Journal of Derivaives, ummer 999; 6, 4, pp 9-3. [3] Flavell, R. (). waps and Oher Derivaives. econd Ediion. New York, NY: John Wiley & ons. [4] Gaheral, J. (6) he Volailiy urface: A Praciioner s Guide. New York, NY: John Wiley & ons. [5] Heson,.L. (993). "A Closed-Form oluion for Opions wih ochasic Volailiy wih Applicaions o Bond and Currency Opions." Review of Financial udies, Vol. 6, pp [6] Jacquier, A., and. laoui (7). "Variance Dispersion and Correlaion waps," Working Paper, Universiy of London and AXA Invesmen Managemen.

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