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 imevarying 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 rearrange 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 MarkoMarke 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 nonzero. 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 markomarke 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 Blackcholes 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 Blackcholes 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 Blackcholes 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 equallyweighed porfolio (doed line) is clearly no consan bu increases wih he sock price. he vega of he srikeweighed 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 wopoin 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 ModelFree Implied Volailiy o come..4 he VIX o come. he VIX index is a 3day 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 93. [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 ClosedForm 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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