ARIANCE DISPERSION AND CORRELAION SWAPS ANOINE JACQUIER AND SAAD SLAOUI Abstract In the recent years, banks have sold structured products such as worst-of optons, Everest and Hmalayas, resultng n a short correlaton exposure hey have hence become nterested n offsettng part of ths exposure, namely buyng back correlaton wo ways have been proposed for such a strategy : ether pure correlaton swaps or dsperson trades, takng poston n an ndex opton and the opposte poston n the components optons hese dsperson trades have been set up usng calls, puts, straddles, varance swaps as well as thrd generaton volatlty products When consderng a dsperson trade usng varance swaps, one mmedately sees that t gves a correlaton exposure Emprcal analyss have showed that ths mpled correlaton was not equal to the strke of a correlaton swap wth the same maturty he purpose of ths paper s to theoretcally explan such a spread In fact, we prove that the P&L of a dsperson trade s equal to the sum of the spread between mpled and realsed correlaton - multpled by an average varance of the components - and a volatlty part Furthermore, ths volatlty part s of second order, and, more precsely, s of volga order hus the observed correlaton spread can be totally explaned by the volga of the dsperson trade hs result s to be revewed when consderng dfferent weghtng schemes for the dsperson trade 1 Introducton For some years now, volatlty has become a traded asset, wth great lqudty, both on equty and ndex markets Its growth has been such that some optons on t have been created and traded n huge quanttes Indeed, varance swaps are very lqud nowadays for many stocks, and optons on varance and on volatlty have been the subject of several research, both from academcs and practtoners Furthermore, these products have gven brth to postons on correlaton, whch had to be hedged Hence, products such as correlaton swaps have been proposed to answer these needs Our purpose here s to compare the far correlaton prced n a correlaton swap and the mpled correlaton of a dsperson trade Indeed, a dsperson trade can be bult upon varance or gamma swaps, hence creatng an almost pure exposton to correlaton, ndependent of the level of the stock Furthermore, we wll explan the observed spread between these two correlatons n terms of the second-order dervatves of such dsperson trades We ndeed beleve that the moves n volatlty, both of the ndex and of ts components, have a real mpact on the mpled correlaton Consttuents of dsperson trades In ths secton, we brefly revew the dfferent fnancal products whch wll serve as a bass n the constructon of dsperson trades We consder a probablty space Ω, F, P and a stock prce process S := S t t defned on t such that ds t = σ t dw t for any t, wth S >, and where W t t denotes a standard Brownan moton We assume that the volatlty process σ t t s smooth enough and not explodng so that a unque soluton S exsts We wll consder European optons wrtten on S, maturty and rsk-free nterest rate r N wll denote the cumulatve standard Gaussan probablty 1
ANOINE JACQUIER AND SAAD SLAOUI functon and φ ts densty For a European opton, we wll denote Ψ := Sσ the anna, Υ σ := σ the vega, Υ v := σ the vega wth respect to the varance, Ω := r the dervatve of wth respect tothe nterest rate, and Λ σ := σσ or Λ v := σ σ We recall the Black-Scholes prce of a Call opton C S, K, at tme t, on S wth strke K > and maturty > assumng σ t s constant for all t : wth C S, K, = S t N d 1 Ke rτ N d, d 1 := log S t/k + r + σ / τ σ, d := d 1 σ τ, and τ := t τ We wll also consder straddle optons, whch consst of a long call and a long put optons wth the same characterstcs strke, stock, maturty A varance swap wth a notonal of N s a swap on the realsed varance of a stock prce, so that ts payoff s worth = N 1 σt dt K, where K s a fxed amount specfed n the contract Referrng to [3], the far prce K, swap s equal to S e r S K, = r S 1 + log S S + er where P S, K, represents a European put opton and S of the varance + K P S, K, + S K C S, K,, > a lqudty threshold Hence, the varance swap s fully replcable by an nfnte number of European calls and puts Moreover, f we take S := S e r, such that the lqudty threshold s equal to the forward value of the stock prce, the above formula smplfes to K, = er S + K P S, K, + S K C S, K, A varance swap s nterestng n terms of both tradng and rsk management as : - t provdes a one-drecton poston on the volatlty / varance - t allows one to speculate on the dfference between the realsed and the mpled volatlty Hence, f one expects a rse n volatlty, then he should go long a varance swap, and vce-versa - As the correlaton between the stock prce and ts volatlty has proven to be negatve, varance swaps are also a means to hedge equty postons From a mark-to-market pont of vew, the value at tme t τ = t of the varance swap strke wth maturty s Π t = e rτ 1 E t = e rτ 1 t t = e rτ t 1 t t σu du K, σu du K, + 1 τ τ E t t σudu K, + τ E t σu du t 1 τ t σ udu K,
But K t, = E t τ 1 t σ u du, and so ARIANCE DISPERSION AND CORRELAION SWAPS 3 Π t = e rτ 1 t σu du K, + τ Kt, Hence, we just need to calculate the new strke of the varance swap wth the remanng maturty τ = t A gamma swap looks lke a varance swap, but weghs the daly square returns by the prce level Formally speakng, ts payoff s = N 1 σ t S t dt K, Γ S where smlarly K, Γ represents the far strke of the gamma swap as defned n the contract he replcaton for ths opton s based on both the Itô formula and the Carr-Madan formula see [] Let us consder the functon f : R + R defned by f F t := e rt F t log F t /F F t + F where F := F t t represents the forward prce process of S We have df t = σ t F t dt, and, by Itô s formula, we have f F = = r t fdt + F fdf t + FF f σt Ft dt e rt F t log F t /F F t + F dt +, e rt log F t /F df t + 1 Hence, the floatng leg of the gamma swap can be wrtten as 1 σt S t dt = f F r e rt F t log F t /F F t + F dt S S e rt σ t F tdt e rt log F t /F df t where we used the fact that S t = F t e rt Now, Carr and Madan [] proved that for any contnuous functon φ of the forward F, we can wrte φf = φκ+φ κ κ F κ + κ F + + φ KF K + + κ φ KK F +, where κ > represents a threshold for example a lqudty threshold n the case of a varance or a gamma swap We now consder the functon φ : R + R defned by φx := e r fx, where f s defned above akng κ = F, the at-the-money forward spot prce, we obtan φf = F K K F + + F K F K + Pluggng ths equaton nto the above floatng leg of the gamma swap, we eventually have 1 σt S t dt = F e r 1 + S S K K F 1 + + F K F K + S r e rt F t log F t /F F t + F dt + e rt log Ft F df t Hence, a long poston n a gamma swap conssts of a long contnuum of calls and puts weghted by the nverse of the strke, rollng a futures poston logf t /F and holdng a zero-coupon bond, worth r F t log F t /F F t + F at tme t,
4 ANOINE JACQUIER AND SAAD SLAOUI At tme t =, the far value of the gamma swap s hence: 1 Γ = E K, σt S t S dt = er S F We can also calculate the prce of the gamma swap at tme t = τ, 1 t σu S u du K, Γ = 1 S t + K P S, K, + F K C S, K, σu S u du + 1 S t σu S u du K, Γ S he frst term of the rght sde of the equaton s past, and the two other terms are strkes, hence 1 E t σu S u du K, Γ S = 1 t σu S u du + τ S Kt, Γ K, Γ Both varance and gamma swaps provde exposure to volatlty However, one of the man dfference, from a management pont of vew, s that varance swaps offer a constant cash gamma, whereas gamma swaps provde a constant share gamma, and hence does not requre a dynamc reallocaton Furthermore, as gamma swaps s weghted by the performance of the underlyng stock, t takes nto account jumps n t, hence traders do not need to cap t, as t s the case for varance swaps through the use of condtonal varance swaps, pp varance swaps, corrdor varance swaps 3 Correlaton tradng 31 Impled correlaton Consder an ndex e a basket wth n stocks σ represents the volatlty of the th stock, w ts weght wthn the ndex, and ρ the correlaton between stocks and j If we replcate the ndex, we create a basket wth the followng volatlty σi := w σ + w w j σ σ j ρ j,j We can then defne the mpled correlaton n ths portfolo, namely an average level of correlaton, as ρ mp := 1 w w j σ σ j σi,j w σ, where σ I represents the volatlty of the ndex We can rewrte ths formula as ρ mp = σ σ j 1 n j> j> ρ j w w j σ σ j In [1], Bossu assumed that, under some reasonable condtons, the term n w σ s close to zero and hence, a good proxy for the mpled correlaton s n ρ mp = w σ σi
ARIANCE DISPERSION AND CORRELAION SWAPS 5 3 Correlaton Swaps A correlaton swap s an nstrument smlar to a varance swap, and pays at maturty the notonal multpled by the dfference between the realsed correlaton and a strke Mathematcally speakng, the payoff of such an opton s Π = K + w w j 1 <j n 1 1 <j n w w j ρ j Smlar to the mpled correlaton above, the realsed correlaton ρ above can be approxmated as n n 1 ρ = w σ σi w σ σi, where σ I, σ 1,, σ n account for realsed volatltes We refer to Bossu [1] and the correspondng presentaton for the detals of ths approxmaton whch s, n fact, a lower bound, thanks to Jensen s nequalty Hence, the realsed correlaton can be seen as the rato between two traded products, through varance swaps, or varance dsperson trades Based on ths proxy, Bossu [1] proved that the correlaton swap can be dynamcally quas-replcated by a varance dsperson trade, and that the P&L of a varance dsperson tradng s worth n w σ 1 ρ, where ρ represents the realsed correlaton hough ths result s really nce, several ssues need to be ponted out: frst, lqudty s not enough on all markets for varance swaps, nether for every ndex and ts components; then ths model does not specfy the form of the volatlty Indeed, t does not take nto account the possble random moves n the volatlty, namely a volatlty of volatlty parameter 4 Dsperson radng 41 P&L of a delta-hedged portfolo, wth constant volatlty We here consder an opton t, valued at tme t wrtten on the asset S he hedged portfolo conssts of beng short the opton and long δ t of the stock prce, resultng n a certan amount of cash Namely, the P&L varaton of the portfolo Π at tme t s worth Π t = t δ S t + δs t t r t he frst part corresponds to the prce varaton of the opton, the second one to the stock prce movements, of whch we hold δ unts, and the thrd part s the rsk-free return of the amount of cash to make the portfolo have zero value Now, aylor expandng the opton prce gves t = δ t S + 1 Γ t S + θ t t, where θ t := t t and Γ t := SS t, hence, the P&L varaton reads Π t = δ t S + 1 Γ t S + θ t t δ t S t + δ t S t t r t Moreover, as the opton prce follows the Black & Scholes PDE θ t + rs t δ t + 1 σ S t Γ t = r t, we obtan the fnal P&L for the portfolo on t, t + dt : P&L t,t+dt = 1 Γ ts t dst S t σ t dt
6 ANOINE JACQUIER AND SAAD SLAOUI 4 P&L of a delta-hedged portfolo, wth tme-runnng volatlty We here consder the P&L of a trader who holds an opton and delta-hedges t wth the underlyng stock As we do wsh to analyse the volatlty rsk, we stay n ths ncomplete market, as opposed to tradtonal stochastc volatlty opton prcng framework he dynamcs for the stock prce s now ds t /S t = µdt + σ t dw t, wth µ R For the volatlty process, we assume the followng dynamcs: dσ t = µ σ,t dt + ξσ t dw σ t, wth σ, ξ >, µ σ R and d < W, W σ > t = ρdt As before, the P&L of the trader on the perod t, t + dt s Π t = t δ t S t + δ t S t t r t We now use a aylor expanson of wth respect to the tme, the stock and the volatlty t = θ t dt + S t S t + σ t σ t + 1 SS t S t + σσ t σ t + Sσ t S t σ t Now, n the P&L formula, we can replace the r t dt term by ts value gven n the Black-Scholes PDE, calculated wth the mpled volatlty Indeed, ths s the very volatlty that had to be nput to determne the amount of cash to lock the poston Hence, we obtan P&L t =θ t dt + S t ds t + σ t dσ t + 1 SS t ds t + σσ t dσ + Sσ t ds t dσ t δ t ds t + rδ t S t dt θ t + 1 41 σt St SS t + rs t S t dt, where σ t represents the mpled volatlty of the opton We can rewrte 41 as P&L t = 1 dst ΓS t σ t dt + σ t dσ + 1 σσ t dσ t + SS t S t σ t dw t dσ t S t In tradng terms, ths can be expressed as 4 P&L t = 1 dst ΓS t σ t dt + ega dσ + 1 olga ξ σt dt + anna σ ts t ρξdt, S t where ega = σ, olga = σσ, anna = Sσ, ρ represents the correlaton between the stock prce and the volatlty and ξ s volatlty of volatlty as defned above 43 Delta-hedged dsperson trades wth dσ = We consder the dsperson trade as beng short the ndex opton and long the stock optons We also consder t delta-hedged he P&L of a delta-hedged opton Π n the Black-Scholes framework s ds P&L = θ t 1 Sσ t dt he term n := ds t / S t σ t dt represents the standardsed move of the underlyng on the consdered perod Let us now consder an ndex I composed by n stocks,,n For convenence, we drop the tme ndex on the stock prce processes We frst develop the P&L of a long poston n the ndex, n terms of ts consttuents, then decompose t nto dosyncratc rsk and systematc rsk Let us denote n := d / σ dt the standardsed move of the th stock, n I := ds I / S I σ I dt the standardsed move of the ndex, p the number of shares n the ndex, w the weght of share n the ndex, σ the volatlty of stock, σ I the volatlty of the ndex, ρ j the correlaton between stocks and j, θ the theta
ARIANCE DISPERSION AND CORRELAION SWAPS 7 of the opton wrtten on stock, θ I the theta of the opton wrtten on the ndex he P&L of the ndex can hence be wrtten as : P&L t = θ I n I 1 n = θ I σ w n 1 σ I = θ I σ w n + w w j σ σ j n n j 1 σ I σ I where we used the fact that and n I := di Iσ I dt = = θ I n w σ σ I n 1 + θ I σ I dt 1 n p d n j=1 p js j = σ 1 I σ I = σ σ j σi w w j n n j ρ j, p S d n j=1 p σ = js j σ dt w σ n + w w j σ σ j ρ j w σ σ I n, Hence, the dsperson trade, namely shortng the Index opton and beng long the optons on the stocks has the followng P&L P&L = P&L P&L I = θ n 1 + θ I n I 1 he short and long poston n the optons wll be reflected n the sgn of the θ 1,,θ n, θ I More precsely, a long poston wll mean a postve θ whereas a short poston wll have a negatve θ 44 Weghtng schemes for dsperson tradng Here above, when consderng the weghts of the stocks n the ndex, we dd not specfy what they precsely were In fact, when buldng a dsperson trade, one faces two problems : frst, whch stocks to pck? hen, how to weght them? As there may lack lqudty on some stocks, the trader wll not take nto account all the components of the ndex hus, he would rather select those that show great characterstcs and lqudty From hs pont of vew, he can buld several weghtng strateges: ega-hedgng weghtng: the trader wll buld hs dsperson such that the vega of the ndex equals the sum of the vegas of the consttuents Hence, ths wll mmune hm aganst short moves n the volatlty Gamma-hedgng weghtng: he gamma of the ndex s worth the sum of the gammas of the components As the portfolo s already delta-hedged, ths weghtng scheme protects the trader aganst any move n the stocks, but leaves hm wth a vega poston heta-hedgng weghtng: ths strategy s rather dfferent from the prevous two, as t wll result n both a short vega as well as a short gamma poston 5 Correlaton Swaps vs Dsperson rades We here focus of the core topc of our paper, namely the dfference between the strke of a correlaton swap and the mpled correlaton obtaned through dsperson tradng Emprcal proofs do observe a spread of approxmately ten bass ponts between the strke of a correlaton swap and the dsperson
8 ANOINE JACQUIER AND SAAD SLAOUI mpled correlaton We frst wrte down the relaton between a dsperson trade through varance swaps and a correlaton swap; thanks to ths relaton, we analyse the nfluence of the dynamcs of the volatlty on ths very spread In the whole secton, we wll consder that the nomnal of the ndex varance swap s equal to 1 51 Analytcal formula for the spread We keep the prevous notatons, namely an ndex I, wth mpled volatlty σ I, realsed volatlty ˆσ I composed of n stocks wth characterstcs σ, ˆσ, w,n he mpled and the realsed correlaton are obtaned as prevously, σi = w w j ρσ σ j, and ˆσ I = w σ + w ˆσ + If we subtract these two equaltes, we obtan ˆσ I σi = ˆσ I σ + w w j ˆσ ˆσ j ˆρ σ σ j ρ and hence n w = = w w w ˆσ I σ + ˆσ I σ + w w j ˆρˆσ ˆσ j w w j σ σ j ˆρ ρ + ˆσ ˆσ j σ σ j ˆρ w w j σ σ j ˆρ ρ + ˆρ ˆσ ˆσ j σ σ j, ˆσ I σ ˆσ I σi w w j σ σ j ˆρ ρ = ˆρ ˆσ ˆσ j σ σ j From a fnancal pont of vew, the above formula evaluates the P&L of a poston consstng of beng short a dsperson trade through varance swaps short the ndex varance swap and long the components varance swaps and long a correlaton swap he rght-hand sde, the P&L, s the spread we are consderng 5 Gamma P&L of the dsperson trade We here only consder the gamma part of the P&L of the varance swap of the ndex We have as n secton 43 P&L Γ I = 1 Γ IIt di t /I t σidt = 1 n Γ IIt d w w σ + w w j σ σ j ρ j dt S = 1 Γ IIt w ds + d ds j w w j w S S σ w w j σ σ j ρ j j = 1 ds Γ IIt w σ dt + ds ds j w w j σ σ j ρ j dt S S j σ σ j Let us make a pause to analyse ths formula As before, n the context of dsperson, we assume that the correlatons ρ j between the components are all equal to an average one ρ Furthermore, as ths correlaton s the one that makes the mpled varance of the ndex and the mpled varance of the weghted sum of
ARIANCE DISPERSION AND CORRELAION SWAPS 9 the components equal, t exactly represents the mpled correlaton hen d ds j / S j accounts for the nstantaneous realsed covarance between the two stocks and S j, and hence d ds j / S j σ σ j s precsely the nstantaneous realsed correlaton between the two stocks Agan, we assume t s the same for all pars of stocks, and we note t ˆρ hen, we can replace the weghts w = p /I We therefore obtan P&L Γ I = 1 Γ I p ds σ dt + I t w w j σ σ j ˆρ ρdt Suppose we consder a poston n a dsperson trade wth varance swaps, where α represents the proporton of varance swaps for the th stock, the gamma P&L s then worth α P&L Γ P&L Γ I = 1 ds α σ dt Γ p 1 Γ I + Γ I p p j σ σ j S j ρ ˆρdt he P&L of the dsperson trade s hence equal to the sum of a spread between the mpled and the realzed correlaton over a perod of tme t, t + dt pure correlaton exposure and a volatlty exposure Now, we recall that the gamma of a varance swap for a maturty s Γ = / S Hence, we can rewrte the gamma P&L for the ndex varance swap as P&L Γ I = 1 S p It ds σ dt + 1 It p p j σ σ j S j ˆρ ρdt, whch mples α P&L Γ P&LΓ I = 1 ds σ S dt α p S It + 1 w w j σ σ j ρ ˆρdt, and hence α P&L Γ P&L Γ I = 1 ds σ dt α w + 1 w w j σ σ j ρ ˆρdt he sum that multples the correlaton spread does not depend on the correlaton, but only on the components of the ndex Hence, we can defne β := 1 w w j σ σ j and eventually wrte 51 P&L Γ Dsp = 1 ds α σ dt w + β ρ ˆρdt If we take α = w, then we see that the gamma P&L of the dsperson trade s exactly the spread between mpled and realzed correlaton, multpled by a factor β whch corresponds to a weghted average varance of the components of the ndex 5 P&L Γ Dsp = β ρ ˆρdt In fact, as we wll analyse t later, ths weghtng scheme s not used However, ths approxmaton consderng that the gamma P&L s pure correlaton exposure s qute far, and we wll measure the nduced error further n ths paper
1 ANOINE JACQUIER AND SAAD SLAOUI 53 otal P&L of the dsperson trade In the prevous subsecton, we proved that the gamma P&L of a dsperson trade s exactly a correlaton P&L Hence, the observed dfference between the mpled correlaton of a dsperson trade and the strke of the correlaton swap wth the same characterstcs about ten bass ponts s precsely due to the volatlty terms, namely the combned effects of the vega, the volga vomma and vanna Usng 4 and 5, we can now wrte P&L Dsp = P&L Γ Dsp + P&Lol Dsp, Where the P&L Γ contans the correlaton exposure, and the P&L ol contans all the vegas, volgas and vannas More precsely, P&L ol Dsp = n α ega dσ + 1 olga ξ σ dt + anna σ ρ ξ dt ega I dσ I + 1 olga ξiσ Idt + anna I σ I Iρ I ξ I dt When replacng the greeks by ther values for a varance swap, we obtan the vanna beng null: P&L Dsp = P&L Γ Dsp + τ n α σ mp dσ σ mp dσ dt + τ n α ξ σ,t ξ IσI dt 54 P&L wth dfferent weghtng schemes In the followng weghtng schemes strategy, we wll consder that the gamma P&L of the dsperson trade s pure correlaton exposure, hence respects 5 Concernng the notatons, α s stll the proporton of varance swaps of the th stock α = N /N I, and we consder N I = 1, w the weght of stock n the ndex and N represents the notonal of the th varance swap We also do not wrte the negatve sgns for the greeks ; therefore, when wrtng the greek of a product, one has to bear n mnd that ts sgn depend on the poston the trader has on ths very product ega flat Strategy In ths strategy, the vega notonal of the ndex varance swap s equal to the sum of the vega notonals of the components : N Υ σ, = N I Υ σ,i w, and α = σ I w /σ he vegas of the P&L dsappear and we are left wth P&L Dsp = P&L Γ Dsp + τ n α ξ σ,t ξ IσI dt, wth the above mentoned approxmaton, we therefore have P&L Dsp = β ρ ˆρdt + τ n σ I w ξ σ σ,t ξi σ I dt Now, the error due to the approxmaton n the gamma P&L s worth ds α σ dt w We now focus on the α w part Here we have α w = w σ I w σ = w σi 1, w σ whch s ndeed very close to From a very theoretcal pont of vew, ths formula tells us that the observed dfference between the strke of a correlaton swap and the mpled correlaton va varance
ARIANCE DISPERSION AND CORRELAION SWAPS 11 swaps dsperson trades can be smply explaned by the volga of the dsperson trade, hence, by the volatlty of volatlty terms ega weghted flat strategy n 1 In ths strategy, we have : N Υ σ, = j=1 w n 1 jσ j NI Υ σ,i w σ I, and α = j=1 w jσ j w σ I σ I /σ We are left wth P&L Dsp = P&L Γ Dsp + τ n α ξ σ,t ξi σ I dt + τ n α ξ σ,t ξi σ I dt, wth the above mentoned approxmaton, we therefore have : P&L Dsp = β ρ ˆρdt + τ σ I w σ I w j σ j σ + τ σ I w σ I w j σ j σ j=1 j=1 1 j=1 ξ σ,t 1 ξi σ I dt he error due to the approxmaton n the gamma P&L s then worth 1 α w = w σ I σ I w j σ j w σ = w σ I σ I w j σ j w σ ξ σ,t ξiσ I dt j=1 1 1, whch s ndeed very close to From a very theoretcal pont of vew, ths formula tells us that the observed dfference between the strke of a correlaton swap and the mpled correlaton va varance swaps dsperson trades can be smply explaned by the volga of the dsperson trade, hence, by the volatlty of volatlty terms heta/gamma flat Strategy Suppose we want to get rd of the gamma P&L of the dsperson Recallng that t s worth P&L Γ Dsp = 1 ds α Γ S σ dt 1 dit Γ IIt σidt We thus need to set α = Γ I I t Γ S dit and replacng the Γ and Γ I by ther values, we get I t σ I dt, σ dt ds I t α = di t /I t σi ds dt / σ dt On a very short perod, we almost have ds/s =, and hence ths s also a theta flat strategy wth no nterest rate Actually the dfference between the gamma and the theta flat strateges s the dfference between the rsk-free rate and the return of the stocks over the perod we consder Furthermore, the dsperson trade s fully exposed to moves n volatlty, namely through the vegas and the vannas of the varance swaps
1 ANOINE JACQUIER AND SAAD SLAOUI 6 Concluson We have here dealt wth dsperson tradng, and we showed the P&L of such a strategy, consderng both varance swaps and gamma swaps he frst one are partcularly appealng because of ther greeks, whch enable us to have a clear vson of our exposure he man result of our paper s that we proved that the observed spread between mpled correlaton through varance swaps dsperson trades and far values of correlaton swaps s totally explaned by a vol of vol parameter We also developed results for gamma swaps dsperson trades and dfferent weghtng schemes, one of them, the vega flat weghtng strategy, beng an arbtrage bound hs also gves us a way of estmatng the volatlty of volatlty parameter, based on the observed prces of varance and correlaton swaps hs work could be analyzed deeper when consderng thrd-generaton exotc product such as corrdor varance swaps, up varance swaps, hey ndeed allow nvestor to bet on future realsed varance at a lower cost Smlar results should be found, but wth less elegant formulas, as the stock prce - just lke for gamma swaps - wll have to be taken nto account References [1] Bossu, S, 6, A new approach for modellng and prcng correlaton swaps n equty dervatves, Global Dervatves radng & Rsk Management, May 6 [] Carr, P, Madan, D, 5, FAQ n opton prcng theory, Journal of Dervatves, forthcomng [3] Demeterf, B, Derman, E, Kamal, Zou, 1999, More than you ever wanted to know about arance Swaps, Goldman Sachs, Quanttatve Strateges Research Notes Appendx A Greeks of a varance swap Let > represent the maturty of the varance swap and t the valuaton date We denote t := τ, and the followng greeks are straghtforward, = 1 S 1 St 1, Γ = S t /, Θ = σ /, Υ σ = τ/, S Γ = 4S 3 t /, S σ =, τ Γ =, τ σ = 1, Υ σ = στ/ Appendx B Greeks of a gamma swap Let > represent the maturty of the varance swap and t the valuaton date We denote t := τ, and the followng greeks are as follows, Υ σ = 1 τσe r+τ S t /S, Ψ = 1 τσe r+τ /S, Λ σ = 1 τe r+τ S t /S, Γ = S S t er+τ We prove below the value for the vega Υ σ and the gamma Γ of the gamma swap we know that the value of a gamma swap at ncepton s worth 1 E σt S t dt = er F + S S K P S, K + F K C S, K
ARIANCE DISPERSION AND CORRELAION SWAPS 13 Its vega at ncepton s then Υ Γ σ = er S = er S = er S S O K K σ = er S S t K π exp K π exp 1 K S φd1 log S /K + r + σ / σ 1 log S σ log K + r + σ / where we used the fact that the vega of a call opton s equal to the vega of a put opton Let us do the followng change of varable x = log S log K + r + σ / σ We then have Υ Γ σ = σer S S Moreover, at tme t = τ, we have 1 E t σ S u u du S dx π exp x / = σe r = 1 t and hence, the vega of the gama swap at tme t s worth Υ Γ σ t = σe rτ τ Concernng the gamma of a gamma swap, we have Γ Γ = er S K SSO K = er S σu S u du + τ S t K t, Γ S S, S t S K φd 1 S σ er = S where, for ease of notaton, O K represents ether a put or a call of strke K We here used the same change of varable as for the vega of the gamma swap, Appendx C P&L of a gamma swap We here consder the P&L of a gamma swap he calculatons below are almost dentcal to those of Part 53 α P&L Γ P&L Γ I = 1 ds α σ dt Γ p 1 S Γ I + Γ IIt w w j σ σ j ρ ˆρdt Snce the gamma of a gamma swap for a maturty, at tme t s Γ = exprτ / S S t, we have P&L Γ Dsp = 1 ds α σ dt Γ S w Γ I I 1 t + Γ IIt w w j σ σ j ρ ˆρdt = n ds erτ σ dt α w S I t I + 1 erτ I t w w j σ σ j ρ ˆρdt I he sum that multples the correlaton spread does not depend on the correlaton, but only on the components of the ndex Hence, we can defne β Γ := 1 e rτ I 1 I t w w j σ σ j, and eventually P&L Γ Dsp = n ds erτ σ dt α w I t + β Γ ρ ˆρdt S I
14 ANOINE JACQUIER AND SAAD SLAOUI We can now wrte the total P&L of the gamma swap, as well as the one for the dsperson trade va gamma swaps Hence P&L Dsp = P&L Γ Dsp+e rτ n P&L I = P&L Γ I + ega I dσ I + 1 olga IξIσ Idt + anna I σ I Iρ I ξ I dt = P&L I = P&L Γ I + σe rτ I t dσ I + 1 I ξ Iσ I dt + ρ I ξ I dt α σ dσ + 1 I t S ξ σ dt + ρ ξ dt σ I dσ I + 1 I ξ Iσ I dt + ρ I ξ I dt Appendx D Arbtrage opportunty condton and vega weghted flat strategy for varance swap dsperson We here analyse the vega weghted flat strategy n terms of arbtrage opportuntes We only consder a dsperson trade bult upon varance swaps A portfolo α = α 1,, α n represents an arbtrage opportunty f and only f for any ˆσ = ˆσ I, ˆσ 1,, ˆσ n, we have σ I ˆσ I + n Rearrangng the terms, we have n ˆσ ˆσ I + σi In partcular, f ˆσ =, then σ I n weghted flat strategy, we have α = w σi / ˆσ σ σ α σ, e, n α σ /σ I 1 If we consder the vega n σ j=1 α jσ j, and wth ths weghtng schemes, we see that n α σ /σ I = 1, and the vega weghted flat strategy represents the boundary condton for arbtrage opportunty Department of Mathematcs, Imperal College London and Zelade Systems, Pars E-mal address: ajacque@mperalacuk