A Consistent Pricing Model for Index Options and Volatility Derivatives

Transcription

1 WOKING PAPE F-29-5 ama Cont & Thomas Kokholm A Consstent Prcng Model for Index Optons and Volatlty Dervatves Fnance esearch Group

2 A Consstent Prcng Model for Index Optons and Volatlty Dervatves ama Cont Center for Fnancal Engneerng Columba Unversty, New York Thomas Kokholm Aarhus School of Busness Aarhus Unversty September 7, 29 Abstract We propose and study a flexble modelng framework for the jont dynamcs of an ndex and a set of forward varance swap rates wrtten on ths ndex, allowng optons on forward varance swaps and optons on the underlyng ndex to be prced consstently. Our model reproduces varous emprcally observed propertes of varance swap dynamcs and allows for jumps n volatlty and returns. An affne specfcaton usng Lévy processes as buldng blocks leads to analytcally tractable prcng formulas for optons on varance swaps as well as effcent numercal methods for prcng of European optons on the underlyng asset. The model has the convenent feature of decouplng the vanlla skews from spot/volatlty correlatons and allowng for dfferent condtonal correlatons n large and small spot/volatlty moves. We show that our model can smultaneously ft prces of European optons on S&P 5 across strkes and maturtes as well as optons on the VIX volatlty ndex. The calbraton of the model s done n two steps, frst by matchng VIX opton prces and then by matchng prces of optons on the underlyng. Presented at the 9th Annual Dervatves Securtes and sk Management Conference 29 and the Nordc Fnance Network Workshop 29. A frst draft of ths paper was completed whle Thomas Kokholm held a vstng scholar poston at the Graduate School of Busness GSB), Columba Unversty, and he wshes to thank Bjørn Jørgensen and the accountng dvson at GSB for placng ther facltes at hs dsposal. Moreover, thanks to Bjarne Astrup, Peter Løchte Jørgensen and Elsa Ncolato for comments.

3 Contents Introducton. Contrbuton Outlne Varance Swaps and Forward Varances 5 2. Varance Swaps Forward Varances Optons on Forward Varance Swaps A Model for the Jont Dynamcs of Varance Swaps and the Underlyng Index 7 3. Varance Swap Dynamcs Dynamcs of the Underlyng Asset Prcng of Vanlla Optons Examples Normally Dstrbuted Jumps Exponentally Dstrbuted Jumps Impact of Jumps on the Valuaton of Varance Swaps The VIX Index 6 4. Descrpton VIX Futures Implementaton 9 5. Data Calbraton Concluson 2 A Characterstc Functons 25 A. Normally Dstrbuted Jumps A.2 Exponentally Dstrbuted Jumps B Proof of proposton 2 27 C Tables 28

4 Introducton Volatlty ndces such as the VIX ndex and dervatves wrtten on such ndces have ganed popularty n markets as tools for hedgng volatlty rsk and as market-based ndcators of volatlty. Varance swap contracts [9] are ncreasngly used by market operators to take a pure exposure to volatlty or hedge the volatlty exposure of optons portfolos. The exstence of a lqud market for volatlty dervatves such as VIX optons, VIX futures and a well developed over-the-counter market for optons on varance swaps, and the use of varance swaps and volatlty ndex futures as hedgng nstruments for other dervatves have led to the need for a prcng framework n whch volatlty dervatves and dervatves on the underlyng asset can be prced n a consstent manner. In order to yeld dervatve prces n lne wth ther hedgng costs, such models should be based on a realstc representaton of the jont dynamcs of the underlyng asset and varance swaps wrtten on ths, whle also be able to match the observed prces of the lqud dervatves futures, calls, puts and varance swaps used as hedgng nstruments. In prncple, any contnuous-tme model wth stochastc volatlty and/or jumps mples some jont dynamcs for varance swaps and the underlyng asset prce. Broade and Jan [8] studed the valuaton of volatlty dervatves n the Heston model; Carr et al. [] study the prcng of volatlty dervatves n models based on Lévy processes. However, n many commonly used models the dynamcs mpled for varance swaps s unrealstc [, 5]: for example, exponental Lévy models mply a constant varance swap term structure whle one-factor stochastc volatlty models predct perfect correlaton among movements n varance swaps at all maturtes. Also, as ponted out by Bergom [5, 7], the jont dynamcs of forward volatltes and the underlyng asset s nether explct nor tractable n most commonly used models, whch makes parameter selecton/calbraton dffcult and does not enable the user to choose a parameter to take a vew on forward volatlty. As a result, classcal models such as the Heston model or tme changed) exponental-lévy models are unable to match emprcal propertes of varance swaps and VIX optons [5, 7, 6,, 2]. Some of these ssues can be tackled usng mult-factor stochastc volatlty models [, 2] but these models reman ncapable of reproducng fner features of the data such as the magntude of the VIX opton skew or the dfferent condtonal correlatons n large and small spot/volatlty moves see Table ). A new modelng approach, recently proposed by Bergom [5, 7, 6] see also related work of Bühler [] and Gatheral [2]) s one n whch, nstead of modelng nstantaneous volatlty, one models drectly the forward) varance swaps for a dscrete tenor of maturtes. Ths approach, whch can be seen as the analog of the LIBO market model for volatlty modelng, turns out to be qute flexble and allows to deal wth the ssues rased above

5 whle retanng some tractablty. These models are based on dffuson dynamcs where market varables are drven by a multdmensonal Brownan moton. ecent prce hstory across most asset classes has ponted to the mportance of dscontnutes n the evoluton of prces; ths anecdotal evdence s supplemented by an ncreasng body of statstcal evdence for jumps n prce dynamcs [, 2, 3, 7]. Volatlty ndces such as the VIX have exhbted, especally durng the recent crss, large fluctuatons whch strongly pont to the exstence of jumps, or spkes, n volatlty. Fgure, whch depcts the daly closng levels of S&P 5 and the VIX volatlty ndex from September 22nd, 23 to February 27th, 29, reveals to the smultanety of large drops n the S&P 5 wth spkes n volatlty, whch corresponds to the well-known leverage effect. Comparng the relatve changes n the two seres Fgure 2) reveals that, whle there s already a negatve correlaton n small changes n the seres, large changes jumps exhbt an even stronger negatve correlaton, close to - see Table ). These observatons are confrmed by a recent study of Tauchen and Todorov [27], who fnd sgnfcant statstcal evdence of smultaneous jumps of opposte sgn n the VIX and the underlyng ndex. These emprcal facts need to be accounted for n a realstc model for varance swap dynamcs. They are also mportant from a prcng perspectve: Broade and Jan [9] fnd that, for a wde range of models and parameter specfcatons, the effect of dscrete samplng on the valuaton of volatlty dervatves s typcally small whle the effect of jumps can be sgnfcant. Jumps n volatlty are also mportant n order to produce the postve skew of mpled volatltes of VIX optons [6, 2].. Contrbuton Followng the approach proposed by Bergom [5, 7, 6], the present work proposes an arbtrage-free modelng framework for the jont dynamcs of forward varance swap rates along wth the underlyng ndex, whch. captures the nformaton n ndex opton prces by matchng the ndex mpled volatlty smles. 2. s capable of reproducng any observed term structure of varance swap rates. 3. captures the nformaton n optons on VIX futures by matchng ther prces/ mpled volatlty smles. 4. mples a realstc jont dynamcs of spot and forward mpled volatltes, allowng n partcular for jumps n volatlty and returns. 5. allows for the spot/volatlty correlaton and the mpled volatlty skews of vanlla optons) to be parametrzed ndependently. 2

6 6. s able to handle the term structure of vanlla skew separately from the term structure of volatlty of volatlty. 7. s tractable and enables effcent prcng of vanlla optons, whch s a key pont for calbraton and mplementaton of the model. We address these dfferent ssues, whle retanng tractablty, by ntroducng a common jump factor whch affects varance swaps and the underlyng ndex wth opposte sgns. Descrbng these jumps n terms of a Posson random measure leads to an analytcally tractable framework, where VIX optons and calls/puts on the underlyng ndex can be smultaneously prced usng Fourer-based methods. Our framework s n fact a class of models and allows for varous specfcatons of the jump sze dstrbuton; we gve two worked-out examples and detal ther mplementaton. The dfference between our modelng framework and the Bergom model [7] s manly the ablty to meet ponts 5), 6) and 7) above. Also, thanks to a sem-analytc representaton of call opton prces, our model also satsfes the tractablty property ), allowng effcent calbraton to the whole mpled volatlty surface. Ths s an advantage over the Bergom model where only the short-term mpled volatlty smle can be matched [5, 7]. S&P Sep-3 25-May-4 27-Jan-5 29-Sep-5 5-Jun-6 7-Feb-7 -Oct-7 3-Jun-8 7-Feb-9 Date VIX Sep-3 25-May-4 27-Jan-5 29-Sep-5 5-Jun-6 7-Feb-7 -Oct-7 3-Jun-8 7-Feb-9 Date Fgure : Tme seres of the VIX ndex bottom) depcted together wth the S&P 5 top) coverng the perod from September 22nd, 23 to February 27th, Outlne Secton 2 descrbes varance swap contracts, forward varance swap rates and optons on varance swaps. The model s descrbed n Secton 3 and two 3

7 Fgure 2: Daly relatve changes n the VIX vertcal axs) vs daly relatve changes n the S&P 5 ndex horzontal axs), September 22nd, 23 to February 27th, 29. Table : Condtonal correlatons between the daly returns on S&P 5 and the VIX from September 22nd, 23 to February 27th, 29. Abs. eturn Uncondtonal <.5% >.5% < % > % < 5% > 5% Correlaton Observatons

8 model specfcatons are presented. Secton 4 dscusses the VIX ndex and the connecton between forward varance swap rates and VIX ndex futures. Secton 5 mplements the two specfcatons of the model and examnes ther performance n jontly matchng VIX optons and optons on the S&P 5. Secton 6 concludes. 2 Varance Swaps and Forward Varances Consder an underlyng asset whose prce S s modeled as a) stochastc process S t ) t on a fltered probablty space Ω, F, {F t } t, P, where {F t } t represents the hstory of the market. We assume the market s arbtrage-free and prces of traded nstruments are represented as condtonal expectatons wth respect to an equvalent prcng measure Q. We shall neglect n the sequel correctons due to stochastc nterest rates. 2. Varance Swaps The annualzed realzed varance of a prce) process S over a tme grd t = t <... < t k = T s gven by V t,t = M k log S ) 2 t, ) k S t = where M s the number of tradng days per year. A varance swap VS) wth maturty T ntated at t < T pays the dfference between the annualzed realzed varance of the log-returns V t,t, determned such that the contract has zero value at the tme of ntaton t. For any semmartngale S, as sup =,...,k t t the realzed varance converges to the quadratc varaton of the log prce: less a strke called the varance swap rate V T t k = log S ) 2 t Q [log S]T [log S] S t. t Approxmatng the realzed varance by the quadratc varaton of the log returns s justfed when the samplng frequency s daly, as s the case n most varance swap contracts, to warrant replacng the realzed varance by ts contnuous counterpart [9]. The fxed leg pad n a varance swap s then approxmated by V T t = T t E [log S] T [log S] t F t), 2) and we refer to ths quantty as the spot) varance swap VS) rate prevalng at date t for the maturty T. An example of a varance swap term structure s gven n Fgure 3. 5

9 aug-8 sep-8 okt-8 nov-8 dec-8 jan-9 feb-9 mar-9 apr-9 maj-9 jun-9 Fgure 3: The term structure of month forward varance swaps for the S&P 5 on August 2th, Forward Varances The forward varance, quoted at date t for the perod [T,T 2 ] s the strke that sets the value of a forward varance swap runnng from T to T 2 to zero at tme t; t s gven by V T,T 2 t = E ) [log S] T 2 T T2 [log S] T F t 3) = T 2 t)v T 2 t T t)v T t, 4) T 2 T where t < T < T 2. The last equalty follows easly by substtuton of 2). In partcular, as noted n [5, 6] forward varances have the martngale property under the prcng measure. Choosng t < s < T < T 2 t follows by substtuton of 3) and the use of the law of terated expectatons that E V T,T 2 s F t ) = V T,T 2 t 5) whch shows that forward varances are martngales under the prcng measure Q. Assume that a set of settlement dates s gven T < T <... < T n known as the tenor structure and that the nterval between two tenor dates s fxed, τ = T + T = τ equal to 3 days f we use the tenor structure of the VIX futures). We defne the forward varance over the tme nterval [T,T + ] as Vt = V T,T + t. 6) 6

10 2.3 Optons on Forward Varance Swaps A call opton wth strke K and maturty T on a forward varance swap for the perod [T,T 2 ] gves the holder the opton to enter at date T nto a varance swap runnng from T to T 2 wth some predetermned strke K. Hence, the value at tme T s πt,t,t 2,K) = e ) T + 2 T r sds E VT,T 2 K F T ) whch gves us the tme t value =e T 2 T r sds V T,T 2 T K) +, 7) πt,t,t 2,K) = e T 2 ) ) + t r sds E V T,T 2 T K Ft. 8) From ths expresson t s clear that havng tractable dynamcs for the forward varance swap rates enable the prcng of optons on varance swaps. 3 A Model for the Jont Dynamcs of Varance Swaps and the Underlyng Index Our goal s to construct a model whch allows for an arbtrary ntal varance swap term structure. allows to specfy drectly the dynamcs of the observable forward varance swaps V t on a dscrete tenor of maturtes T, =..n). allows for flexble modelng of the varance swap curve: for example, t should be able to accommodate the fact that varance swaps at the long end of the maturty curve have a lower varablty than those at the short end. allows for jumps n volatlty [27] and n the underlyng asset [, 2, 3, 7]. allows for a flexble specfcaton of the spot prce dynamcs. s analytcally tractable.e. leads to effcent numercal methods for prcng/calbraton of calls/puts both on the underlyng and volatlty dervatves such as optons on forward varance swaps and optons on forward volatlty. In order to acheve these goals, we frst specfy the dynamcs of a dscrete tenor of) forward varance swaps Secton 3.) usng an affne specfcaton 7

11 whch allows Fourer-based prcng of European-type volatlty dervatves. Once the dynamcs of forward varance swaps has been fxed, we specfy a jump-dffuson dynamcs for the underlyng asset whch s compatble wth the varance swap dynamcs Secton 3.2). Presence of a jump component as well as a dffuson component n the underlyng asset allows us to satsfy ths compatblty condton whle smultaneously matchng values/mpled volatltes of optons on the underlyng asset Secton 3.3). 3. Varance Swap Dynamcs Gven that the forward varance swap rate s a postve) martngale under the prcng measure, we model t as V t = V e X t = V exp { t µ sds + t ωe k T s) dz s + t } e k 2T s) xj dx ds), where J dxdt) s a Posson random measure wth compensator ν dx) dt and Z a Wener process ndependent of J. The martngale condton mposes { } ) µ t = 2 ω2 e 2k T t) exp e k 2T t) x ν dx). ) For t > T we let V t = V T. Ths specfcaton allows for jumps and the exponental functons nsde the ntegrals allow to control the term structure of volatlty of volatlty. For example, f k s large and k 2 small, the dffuson Z results mostly n fluctuatons at the short end of the curve, whle the jumps mpact the entre varance swap curve. Correlated Brownan factors can be added f more flexblty n the varance swap curve dynamcs s desred, e.g. correlaton between movements n the short and the long end of the curve. Lkewse, functonal forms other than exponentals can be used n 9), although the exponental specfcaton s qute flexble as we wll observe n the examples. Expressons such as 8) can be evaluated usng Fourer-based methods [3, 24, 8] gven the characterstc functon of X T, whch n ths case has a smple form: ] { E [e ux T = exp T 2 u2 T + exp T ω 2 e 2k T s) ds + u { } ue k 2T s) x µ s ds 9) ) } ν dx) dt. ) We hope that the reader wll not confuse the complex number wth the ndex n the forward varance swap rate V t. 8

12 3.2 Dynamcs of the Underlyng Asset Once the dynamcs of forward varance swaps V t for a dscrete set of maturtes T, =..n has been specfed, ths mposes some constrants on the rsk neutral) dynamcs of the underlyng asset S t ) t. S should be such that. the dscounted spot prce Ŝ t = exp t r s q s ) ds)s t s a postve) martngale, where q t s the dvdend yeld. 2. the dynamcs of the spot prce s compatble wth the specfcaton of Vt, whch puts the followng constrant on the quadratc varaton process [ln S] of the log-prce: E[ [ln S] T+ [ln S] T F t ] = V t 2) To these constrants we add a thrd requrement, namely that: 3. the model values of calls/puts on S match the observed prces across strkes and maturtes. Typcally we need at least two dstnct parameters/degrees of freedom n the dynamcs of the underlyng asset n order to accommodate ponts 2) and 3) above. The Bergom models [5] and [7] propose to acheve ths by ntroducng a random local volatlty functon whch s reset at each tenor date and chosen such at tme T to match the observed value of VT. Ths procedure guarantees coherence between the varance swaps and the underlyng asset dynamcs but leads to a loss of tractablty: even vanlla call optons need to be prced by Monte Carlo smulaton for maturtes T > T. We adopt here a dfferent approach whch allows a greater tractablty whle smultaneously allowng for jumps n the volatlty and the prce. In fact, as we wll see, ntroducng jumps s the key to tractablty. The underlyng asset s drven by a Brownan moton W, correlated wth the dffuson component Z drvng the varance swaps: < W,Z > t = ρ t. a jump component, whch s drven by the same Posson random measure J whch drves jumps n the varance. However, we allow dfferent jump ampltudes n the underlyng and the forward varance. Presence of a jump component as well as a dffuson component n the underlyng asset allows us to satsfy ths compatblty condton whle smultaneously matchng values/mpled volatltes of optons on the underlyng asset Secton 3.3). For each nterval [T,T + ) we ntroduce a stochastc dffuson coeffcent σ and a functon u x,v ) whch expresses the sze of the jump n the 9

13 underlyng asset n terms of the jump sze x n forward varance level V and V tself. We wll gve n secton 3.4 some smple and flexble specfcatons for ths functon u but most developments below hold for arbtrary choce of u. We wll present here the detaled computatons n the typcally useful) case where T = T m, for m =,...,n, but the analyss can be easly generalzed to any maturty. The dynamcs of the underlyng asset s then specfed as S Tm = S exp { Tm m ) r s q s )ds + µ T + T ) + σ WT+ W T = m T+ + = T u x,v T ) J dx ds) } 3) where µ = 2 σ2 ) ) e u x,vt ν dx) and the σ s are stochastc and revealed at tme T by relaton 6) below to match the known varance swap value VT at ths tme pont. The drft term µ s also stochastc and F T -measurable. The random jump measure J n the ndex dynamcs s the same as that n the VS dynamcs, so the ndex and the varance swaps jump smultaneously. u s a determnstc functon of x and VT chosen to match the observed volatlty smles mpled by prces of optons on the spot. W s correlated wth Z wth a correlaton ρ. As far as prcng of vanlla nstruments s concerned, the model can be vewed smply as a flexble and analytcally tractable) parameterzaton of the jont dstrbuton of forward varance swap rates and the underlyng asset on a set of tenor dates. At ths level the only assumpton we are makng s that ths jont dstrbuton s nfntely dvsble [26] and our model s just a flexble parameterzaton of the Lévy trplet of the dstrbuton. Ths makes t partcularly easy to prce any payoff whch nvolves these varables only at tenor dates. The assumpton of a common jump factor whch affects the varance swaps and the underlyng n opposte drectons s not only analytcally convenent but qute realstc from an emprcal perspectve. As revealed n Table, whle the uncondtonal correlaton of daly returns of the VIX ndex and the S&P 5 from September 22nd, 23 to February 27th, 29 s 74%, the condtonal correlaton between the two sub-seres for daly moves n the S&P 5 less than.5% drops to.45 whle the condtonal correlaton of moves greater than 5%, whch can be nterpreted as jumps, s 93%, close to -%. Ths observaton s also n agreement wth the fndngs reported by Tauchen and Todorov [27] usng non-parametrc methods. Ths feature, whch has no equvalent n dffuson-based stochastc volatlty models such as [7, 2], s a generc property of our framework.,

14 Gven the dynamcs 3) of the underlyng asset, the quadratc varaton over the tme nterval [T,T m ] s gven by m T+ [log S] Tm [log S] T = σ 2 ) T + T ) + u x,v 2 T J ds dx). = T 4) By takng expectaton on 4) the varance swap rate n 2) equals m T V Tm + T T = E [ [ ]) σ 2 ] ) F T + E u x,v 2 T = m T T ν dx) FT, and the forward varance at tme t for the nterval [T,T + ] equals Vt = E [ [ ] σ 2 F ] ) t + E u x,v 2 T ν dx) Ft. 5) Equaton 5) has to hold at all tmes, but snce Vt s a martngale we just need to ensure that at tme T VT = σ 2 ) + u x,v 2 T ν dx). 6) Havng fxed ν dx) t s seen that any observed forward varance structure can be matched by an approprate choce of the σ s whch leaves the parameters n u free to calbrate to opton prces. 3.3 Prcng of Vanlla Optons Let us now explan the procedure for prcng European calls and puts on S n ths framework. The am s to compute n an effcent manner the value of call optons of varous strkes and maturtes at, some ntal date t = : C,S,K,T m ) = e Tm r sds E[S Tm K) + F ]. 7) Denote by F Z,J) t the fltraton generated by the Wener process Z and the Posson random measure J. By frst condtonng on the factors drvng the varance swap curve and usng the terated expectaton property C,S,K,T m ) = e Tm r sds E[E[S Tm K) + F Z,J) T m ] F ] 8) we obtan a mxng formula for valung call optons: Proposton The value C,S,K,T m ) of a European call opton wth maturty T m and strke K s gven by C,S,K,T m ) = E[C BS U Tm,K,T m ;σ )], 9)

15 where C BS S,K,T;σ) denotes the Black-Scholes formula for a call opton wth strke K and maturty T: σ 2 = m T m = σ 2 ρ 2 ) T + T ), 2) and U Tm s a F Z,J) T m -measurable random varable gven by U Tm = S exp { m = ) ) ) 2 σ2 ρ2 + e u x,vt ν dx) T + T ) ρ Z T+ Z T ) σ + T+ T } u x,vt )Jdx ds) and the expectaton n 9) s taken wth respect to the law of Z,J). 2) Proof. Condtonal on F Z,J) T m, the ncrements of Z and the paths of the varance swap rates, and n partcular VT, thus the correspondng σ are known for =,,...,m from equaton 6). Moreover d ) ) W T+ W T = ρ ZT+ Z T + ρ 2 ) ŴT+ ŴT, where the Wener processes Z and Ŵ are ndependent. The process S t can therefore be decomposed as { Tm S Tm = U Tm exp r s q s )ds + m = +σ ρ 2 2 σ2 ρ 2 ) T + T ) ŴT+ ŴT ))}, where σ 2 = V T u x,v T ) 2 ν dx) and U Tm = S exp { m = ) ) ) 2 σ2 ρ 2 + e u x,vt ν dx) T + T ) ρ Z T+ Z T ) σ + s F Z,J) T m -measurable. In dstrbuton we have T+ T } u x,vt )Jdx ds) S Tm d Tm = UTm e r s q s)ds+µ T m+σ Ŵ Tm, 2

16 where σ s gven n 2) and µ = m T m 2 σ2 = ρ 2 ) T + T ), 22) where we notce that µ = 2 σ ) 2. Hence, gven F Z,J) T m, the nner condtonal expectaton n 8) reduces to the evaluaton of the Black-Scholes formula E[S Tm K) + F Z,J) T m ] = C BS U Tm,K,T m ;σ ), where σ depends on all the u s through 6) and 2). Ths result has nterestng consequences for prcng and calbraton of vanlla contracts. Note that the outer expectaton can be computed by Monte Carlo smulaton of the Z and J: wth N smulated sample paths for Z and J we obtan the followng approxmaton Ĉ N = N N k= ) C BS U k) N T m,k,t m ;σ k) C,S,K,T m ). 23) Snce the averagng s done over the varance swap factors Z and J, ths s a determnstc functon of the parameters n the u s. Ths wll prove useful when calbratng the model usng opton data, snce we do not have to run the N Monte Carlo smulatons for each calbraton tral. Equaton 23) thus allows to calbrate the model to the entre mpled volatlty surface n an effcent manner, n contrast to the Bergom model where t s only possble to calbrate to at-the-money slopes of the mpled volatltes ATM skews). 3.4 Examples Dfferent classes of models n the above framework can be obtaned by varous parameterzatons of the Lévy measure ν dx) descrbng the jumps and for the functons u appearng n 3). In ths paper we mplement two model parameterzatons; normal jumps and double exponental jumps n the varance swaps and the ndex Normally Dstrbuted Jumps In the frst example, we specfy the Lévy measure as ν dx) = λ δ x m) 2 2π e 2δ 2 dx, 24) where m and δ 2 s the mean and varance n a normal dstrbuton and λ the ntensty of the jumps. Wth ν dx) specfed the characterstc functon 3

17 n ) can be calculated see 4) n Appendx A) and prces of optons on the varance swaps can be computed by Fourer transform methods. We now consder the followng specfcaton for u, whch relates the jumps n the varance swap to the jumps n the underlyng asset u x,v T ) = V T V ) 2 b x, 25) but any functonal form can be used as long as 6) leads to postve values for σ. Ths gves us the σ s at tme T σ 2 = V T V T V = V T λ V T V b x) 2 ν dx) b 2 m 2 + b 2 δ 2). In order to acheve non-negatve values for σ 2 we requre Exponentally Dstrbuted Jumps λ b 2 m2 + b 2 δ2) V. 26) To ntroduce an asymmetry n the tal of the jump sze dstrbuton we can use a double exponental dstrbuton [23]: ) ν dx) = λ pα + e α +x x + p)α e α x x< dx, 27) where p denote the probablty of a postve jump, /α + and /α the expected postve and negatve jump szes, respectvely. In ths specfcaton the characterstc functon for the forward varance swap s gven by 43). As above, we specfy the u functon as n 25), whch gves rse to the followng constrant on the σ s σ 2 = V T λ V T V 2pb 2 α p)b2 α 2 To ensure postve σ s we constran the calbraton by 2pb 2 λ α ) p)b2 + α 2 V. 28) The parameters b n the above specfcatons can take any value as long as 26) respectvely 28) are satsfed. We wll see later, how the calbraton of b entals that jumps n the underlyng have opposte drecton to the ). 4

18 jumps n the VIX futures. Together wth negatve correlaton between Z and W, ths feature enables the model to generate postve skews for the mpled volatlty of VIX optons and negatve skews for the mpled volatlty of calls and puts on the underlyng ndex, as observed n emprcal data. Ths property s acheved wthout the need to add extra volatlty factors, as proposed recently n e.g. [, 2, 2] where mult-factor mean-revertng dynamcs on volatlty and volatlty of volatlty are mposed to accommodate ths behavor, resultng n a loss of tractablty. 3.5 Impact of Jumps on the Valuaton of Varance Swaps As noted above, recent prce hstory across most asset classes has ponted to the mportance of dscontnutes n the evoluton of prces; ths s supplemented by an ncreasng body of statstcal evdence for jumps n prce dynamcs [, 2, 3, 7]. Before movng on we would lke to dgress on the effect of jumps n the underlyng ndex dynamcs when prcng varance swaps. As noted by Neuberger [25], f the underlyng asset follows contnuous dynamcs, ts quadratc varaton and hence also the payoff from a varance swap can be replcated by contnuous rebalancng of a poston n the underlyng asset and a statc poston n a log contract, whch n turn can be replcated by a statc portfolo of calls and puts [2, 9]: T t [log S] T [log S] t ) = 2 T T t t and then under dscrete samplng ds t log S ) T S t S t M k k = log S ) 2 t = 2 M S t k k = S log S ). S S The left hand sde s the realzed varance pad and the rght hand sde s the payoff from the hedgng strategy. Fgure 4 depcts the day to day proftloss from holdng a long poston n the replcatng portfolo and a short poston n the three month varance swap. Varance Pont VP) s defned as realzed varance multpled by,. As can be seen, the replcatng portfolo performs relatvely well most of the days, but days n whch the ndex moved consderably show an error of up to -.5 VP. Table 2 shows the error from holdng the portfolo relatve to the realzed varance over the three month perod. The maxmum error from followng ths strategy arses n the June 28 - September 28 varance swap and results n an error of.243%. These four real examples ndcate that the error from assumng contnuous dynamcs have been hstorcally small. Sometmes ths s taken as an argument that jumps are not sgnfcant when varance swaps are prced. Ths can very well be a false concluson, 5

19 whch can be realzed from the followng. When assumng contnuous dynamcs, the error that arses when valung varance swap rates s gven [4] by 2 [ T T t E t ] e y y )νdy)ds y2 F t 2 where y s the jump sze n the ndex. That ths expectaton s small under the physcal measure does not ental that the mean of the error under the prcng measure should be zero. Ths common fallacy s akn to sayng that snce crashes occur nfrequently deep out-of-the-money puts should be prced at zero: recent experence shows that such mstaken assertons can be costly. In fact addng jumps wth negatve mean value to the log-prce dynamcs should ncrease the value of the varance swap rate, snce beng long a varance swap can be seen as an nsurance aganst downward movements n the underlyng asset and therefore have a rsk premum attached to t [5]. 4 The VIX Index As mentoned, the market for optons on the VIX ndex s well developed, and unless data on optons on varance swaps s avalable, nformaton stemmng from ths market should be used when the dynamcs of the varance swaps are specfed. For completeness we descrbe the VIX ndex n the frst subsecton, but the reader s referred to [6] and [4] for a detaled dscusson. In the second subsecton we derve our choce for the VIX futures dynamcs gven the dynamcs of the forward varance swaps. 4. Descrpton The Chcago Board Optons Exchange CBOE) ntroduced the volatlty ndex, VIX, n 993, and t has snce become the ndustry benchmark for market volatlty. The VIX ndex provdes nvestors wth a quote on the expected market volatlty over the next 3 calendar days. In September 23 the VIX was revsed such that the ndex s now calculated on the bass of put and call optons on S&P 5 nstead of the S&P ndex. Furthermore, the ndex was changed to calculate the expected volatlty on the bass of optons n a wde range of strke prces, where the orgnal VIX ndex was based on at-the-money strkes only. The general formula used n the VIX calculaton at tme t s V St T = 2 T t K K 2 e rt t T t) QK,T;t) ) 2 Ft, 29) T t K where T s tme of expraton, K the strke prce of the th out-of-the money, 6

20 opton, K the resoluton of the strke grd: K = K + K 2 For the lowest strke, K s the dfference between the lowest and second lowest strkes. Ths also apples to the hghest strke. rt T s the rsk-free nterest rate to expraton, QK,T;t) the mdpont between the bd and ask prces on the opton wth strke K and maturty T, ether a call f K > F t or a put f K < F t, where F t s the forward S&P 5 ndex prce. K s the frst strke below the forward ndex level F t. Normally, no optons wll expre n exactly 3 days. Therefore, CBOE nterpolates between V S T t and V S T 2 t ) 365 V IX t = T t)v S T N T2 3 t + T 2 t) V S T 3 N 2 T t, 3 N T2 N T N T2 N T 3) where N T and N T2 denote the actual number of days to expraton for the two optons maturtes, N T < 3 < N T2. The VIX s often presented as an ndcator of the expected volatlty over the next 3 days. It s not mmedately clear from ether 29) or 3) that ths s n fact the case. For notatonal smplcty assume that the jumps exhbt fnte varaton. Then t can be shown that the realzed varance of returns can be decomposed nto three components [log S] T [log S] t = 2 T t + Ft F t T t K 2 K S T) + dk K 2 S T K) + dk + e y y y2 2. T t ) df s F s F t ), 3) ) J dy ds) where J s a Posson random measure wth Lévy measure ν dx), drvng the jumps n the underlyng asset. Notce how the realzed varance can be replcated by tradng n optons and futures up to a dscontnuous jump component. Takng expectatons wth respect to the prcng measure n 3) we obtan Vt T = 2 T t e T t r sds 2 T t E [ T t QK,T;t) K 2 dk ] e y y )ν y2 dy)ds F t. 32) 2 Equaton 29) s a dscretzaton of the frst term n 32). The extra term 2 Ft T t K ) n 29) s a contrbuton due to the dscretzaton around Ft. 7

21 The square of the VIX ndex s thus a model free estmate of the expected volatlty over the next 3 days gven contnuous dynamcs of the underlyng but under general prce processes the error term s gven by the last term n 32): V IX 2 t = V t+3days t + 2 T t E 4.2 VIX Futures [ T t ] e y y )ν y2 dy) ds F t. 2 33) Let V IXt denote the VIX futures prce for the nterval [T,T + ] seen at tme t. For t = T we have ) ) ) V IX 2 T = V T + 2 e u x,vt ) u x,v 2 T u x,v T )ν dx). 2 We also have the martngale property for t < T V IX t = E [ V IX T F t ] By Jensen s nequalty for convex functons V IX t ) 2 E [ V IX T ) 2 Ft ] = V t + 2E [ e u x,v T ) ) u u x,v x,v T 2 T ) 2 ν dx) F t ] 34) snce V t s a martngale. Equaton 34) shows that there s a convexty correcton connectng VIX futures to forward varance swap rates. To obtan a tractable framework, we frst characterze the dynamcs of V t then propose an approxmaton whch has a closed-form characterstc functon. Proposton 2 The process Vt has the multplcatve decomposton Vt = V M ta t where A t s a fnte varaton process and Mt s an exponental martngale gven by { t Mt = exp ηs ds + t t } ωe k T s) dz s e k 2T s) xj dx ds), 35) where ηt = 8 ω2 e 2k T t) { } ) exp 2 e k 2T t) x ν dx). 36), 8

22 The proof of ths result s gven n Appendx B. Usng ths result we approxmate V IX t wth V IX t V IX M t = V IX expy t ). 37) where Yt = ln M t. Ths approxmaton leaves the ntal level of the V IX and the volatlty of V IX untouched and thus s relevant for prcng VIX dervatves. As n the case of the forward varance swaps, once the Lévy measure ν dx) s specfed, the characterstc functon for the exponent n 37) can be easly computed, so optons on the VIX ndex can be prced by Fourer transform methods. 2 5 Implementaton In ths secton the model specfcatons descrbed n Secton 3.4 are mplemented on data on the VIX and the S&P 5 mpled volatlty smles. 5. Data We assess the performance of the model usng prces from August 2th, 28 on a range of VIX put and call optons for fve maturtes, VIX futures for the same maturtes, the dvdend yeld on S&P 5, call and put optons on S&P 5 for sx maturtes and for the varous maturtes we also have the correspondng prces of the futures on S&P 5, from whch we also derve a dscount curve. We also observe 3 month forward varance swap rates for varous maturtes. The varance swap rates have been converted to forward month varance swap rates by smple lnear extrapolaton and these are depcted n Fgure 3. Optons for whch the bd prce s zero were removed. 5.2 Calbraton The calbraton of the model conssts of three steps: A.. Determne the parameters controllng the varance swap dynamcs by calbraton to VIX optons usng the characterstc functon n 42) or 44) and Fourer transform methods. 2. Use the parameters from the frst step to smulate N paths of the varance swaps and store the ncrements of Z, the jump tmes and jump szes along wth the V T s. 3. Calbrate to optons on the ndex recursvely by use of 23). 2 The characterstc functons for the specfcatons n Secton 3.4 are gven n Appendx 9

23 In the frst step we compute model prces by the fast Fourer transform technque descrbed n [3] usng the revsed method n [8]. The calbraton s performed by mnmzng the sum of squared errors weghted by the nverse bd-ask spread across all maturtes and strkes on out-of-the-money optons on the VIX futures SE = optons Q Ask Q Bd Q Market,Md Q Model ) 2, 38) usng a gradent-based mnmzaton algorthm. The resultng parameters for the two specfcatons are shown n Table 3 and t also ncludes the resultng average relatve percentage bd-ask error from the calbraton, Error = # {optons} optons max { Q Model Q Ask ) +,Q Bd Q Model ) +} Q Market,Md. 39) We see how both model specfcatons are able to acheve very low calbraton error. Note the hgh value of k compared to k 2, whch mples n the rsk-neutral dynamcs) that the dffuson component Z s manly causng fluctuatons at the short end of the varance swap curve, whle the jumps mpact the entre curve. Fgures 5 and 6 depct the Black) mpled volatlty for the bd, ask, md and model prces of VIX optons as a functon of moneyness. We observe that model prces ft very well wthn the bd-ask spread for almost all the observatons n both specfcatons. Ths performance should be compared wth flat model mpled volatltes n the Bergom model [5] and downward slopng volatltes n the Heston model [22]. The Bergom model [7] s able to generate postve skews on volatlty of volatlty va the use of a Markov functonal mappng for the forward varances, but t should be noted that the model can only prce vanlla optons on the underlyng ndex by Monte Carlo smulaton. Step 2 s done by dscretzaton of 9). In ths example we have used 2 5 smulated sample paths. Smulaton of the jumps s straghtforward, snce they arrve at constant ntensty and the jump szes are computed by draws from a normal/exponental dstrbuton tmes the scalng e k 2T τ), where τ s the jump tme. For detals on how to smulate the ncrements n the varance swaps due to the dffuson the reader s referred to [5]. In step 3 we have fxed the correlaton parameter between the contnuous components to ρ = Then, the last step s mplemented step-wse by matchng the model prces 23) to market prces on the shortest maturty 3 Instead one could also specfy a correlaton ρ on each nterval and nclude that parameter n the calbraton for each nterval. We tred ths, but the performances of the models dd not mprove wth ths added flexblty. 2

24 out-of-the-money optons. Agan, ths s acheved by mnmzng the SE 38). Ths yelds b. Then b are found n the same way by calbratng to the next maturng optons by the use of b for the frst nterval n 2). The remanng b s are estmated n a smlar manner. The estmated parameters for the two dfferent specfcatons of the jump sze dstrbutons are shown n Table 4 along wth the prcng error n 39) for each maturty. The mean and standard devatons of the jumps before ) scalng wth VT /V 2 are also ncluded. The values of b for =,...,3 relate to the monthly dstrbutons of the ndex up to the maturty of the optons exprng on December 9th, 28. For the last two opton maturtes the nterval between the expratons are three months and hence b for = 4, 5 relate to the three month dstrbutons of the ndex. Double exponental jumps perform slghtly better on the frst nterval, but there s not a strong dfference between the two specfcatons, showng that the model performance s not very senstve to the choce of the jump sze dstrbuton. Fgures 7 and 8 show the result of the calbraton to the S&P 5 optons. The ft of the two specfcatons are practcally ndstngushable and overall the calbratons perform well for both wth somewhat less success on the shortest maturty. We conclude ths secton wth an examnaton of the sze of the jump error term when valung the forward varance swap rates V under the assumpton of contnuous dynamcs gven the models mplemented n ths paper are true. ecall that the error s gven by [ ) ) ε = 2E e u x,vt ) u x,v 2 ) ] T u x,v T ν dx) F. 2 4) In Table 5 the absolute errors mpled by the calbraton of the two model specfcatons are computed for each nterval. Moreover, the errors relatve to the ntal forward varance swap value are reported. For the frst nterval t s 2.2%/2.3% and then t roughly ncreases as ncreases. The reported errors for the last two ntervals are taken as the average of the three monthly errors relatve to the average of the three monthly forward varances coverng the same ntervals. 6 Concluson We have presented a model for the jont dynamcs of a set of forward varance swap rates along wth the underlyng ndex. Usng Lévy processes as buldng blocks, ths model leads to a tractable prcng framework for varance swaps, VIX futures and vanlla call/put optons, whch makes calbraton of the model to such nstruments feasble. Ths tractablty feature dstngushes our model from prevous attempts [5, 4, 2] whch only allow 2

25 for full Monte Carlo prcng of vanlla optons. Our model reproduces salent emprcal features of varance swap dynamcs, n partcular the strong negatve correlaton of large ndex moves wth large moves n the VIX and the postve skew observed n mpled volatltes of VIX optons, by ntroducng a common jump component n the varance swaps and the underlyng asset. Usng two dfferent specfcatons for the jump sze dstrbuton Lévy measure) we have llustrated the feasblty of the numercal mplementaton, as well as the capacty of the model to match a complete set of market prces of vanlla optons and optons on the VIX. Our model can be used to prce and hedge varous payoffs senstve to forward volatlty, such as clquet or forward start optons, as well as volatlty dervatves, n a manner consstent wth market prces of smpler nstruments such as calls, puts or varance swaps whch are typcally used for hedgng them. eferences [] Y. At-Sahala and J. Jacod, Estmatng the Degree of Actvty of Jumps n Hgh Frequency Fnancal Data, Annals of Statstcs, forthcomng, 28). [2] T. Andersen, L. Benzon, and J. Lund, An Emprcal Investgaton of Contnuous-Tme Equty eturn Models, Journal of Fnance, 57 22), pp [3] O. Barndorff-Nelsen and N. Shephard, Varaton, Jumps, Market Frctons and Hgh-Frequency Data n Fnancal Econometrcs, n Advances n Economcs and Econometrcs: Theory and Applcatons, Nnth World Congress,. Blundell, W. K. Newey, and T. Persson, eds., Econometrc Socety Monographs, Cambrdge Unv Press, 27, pp [4] L. Bergom, Smle Dynamcs, sk, 7 24), pp [5], Smle Dynamcs II, sk, 8 25), pp [6], Dynamc propertes of smle models, n Fronters n Quanttatve Fnance: Volatlty and Credt sk Modelng,. Cont, ed., Wley, 28, ch. 3. [7], Smle Dynamcs III, sk, 2 28), pp [8] M. Broade and A. Jan, Prcng and Hedgng Volatlty Dervatves, The Journal of Dervatves, 5 28), pp

26 [9], The Effect of Jumps and Dscrete Samplng on Volatlty and Varance Swaps, Internatonal Journal of Theoretcal and Appled Fnance, 28), pp [] H. Buehler, Consstent Varance Curve Models, Fnance and Stochastcs, 26), pp [] P. Carr, H. Geman, D. Madan, and M. Yor, Prcng Optons on ealzed Varance, Fnance and Stochastcs, 9 25), pp [2] P. Carr and D. Madan, Towards a Theory of Volatlty Tradng, n Volatlty: New Estmaton Technques for Prcng Dervatves,. Jarrow, ed., sk Publcatons, 998. [3] P. Carr and D. Madan, Opton Valuaton usng the Fast Fourer Transform, Journal of Computatonal Fnance, 2 999), pp [4] P. Carr and L. Wu, A Tale of Two Indces, Journal of Dervatves, 3 26), pp [5], Varance sk Premums, evew of Fnancal Studes, 22 29), pp [6] CBOE, VIX: CBOE Volatlty Index, 23). [7]. Cont and C. Mancn, Nonparametrc Tests for Analyzng the Fne Structure of Prce Fluctuatons, Fnancal Engneerng eport 27-3, Columba Unversty, 27. [8]. Cont and P. Tankov, Fnancal Modellng wth Jump Processes, Chapman & Hall/CC, 24. [9] K. Demeterf, E. Derman, M. Kamal, and J. Zou, A Gude to Volatlty and Varance Swaps, Journal of Dervatves, 6 999), pp [2] D. Duffe, J. Pan, and K. Sngleton, Transform Analyss and Asset Prcng for Affne Jump-dffusons, Econometrca, 68 2), pp [2] J. Gatheral, Consstent Modelng of SPX and VIX Optons, n Bacheler Congress, July 28. [22] S. Heston, A Closed-Form Soluton for Optons wth Stochastc Volatlty wth Applcatons to Bond and Currency Optons, evew of Fnancal Studes, 6 993), pp

27 [23] S. Kou, A Jump-Dffuson Model for Opton Prcng, Management Scence, 48 22), pp. 86. [24] A. Lews, A Smple Opton Formula For General Jump-Dffuson And Other Exponental Lévy Processes, tech. report, OptonCty.net, 2. [25] A. Neuberger, The Log Contract: A New Instrument to Hedge Volatlty, Journal of Portfolo Management, 2 994), pp [26] K. Sato, Lévy Processes and Infntely Dvsble Dstrbutons, Cambrdge Unversty Press, 999. [27] V. Todorov and G. Tauchen, Volatlty Jumps, workng paper, Duke Unversty,

28 A Characterstc Functons A. Normally Dstrbuted Jumps Varance Swaps We have from [26] that the characterstc functon of XT s gven by ] { E [e ux T = exp T T 2 u2 ω 2 e 2k T s) ds + u µ sds Notce now that and T + T exp { } ue k 2T s) x e 2k T s) ds = e 2k T 2k E [ e Y ] = e m+ δ2 2 ) } ν dx) dt. for Y N m,δ 2). Insertng ths nto the expresson for the characterstc functon we arrve at ] { E [e ux T = exp 2 ω2 u e 2k T 2k 2 ω2 u 2 e 2kT 2k T uλ exp {e k 2T s) m + 2 } ) e 2k 2T s) δ 2 ds VIX Futures T +λ exp {ue k 2T s) m 2 } ) } u2 e 2k 2T s) δ 2 ds. 4) ecall from 37) that V IXt V IX M t where Mt = expyt ) s a postve exponental martngale. The characterstc functon of Y T = T η sds + 2 T ωe k T s) dz s + T can be found n the same way as for X. It s gven by ] { E [e uy T = exp 8 ω2 u e 2k T 2k 8 ω2 u 2 e 2kT 2k T { uλ exp 2 e k 2T s) m + 8 e 2k 2T s) δ 2 T +λ exp 2 e k 2T s) xj dx ds) { 2 ue k 2T s) m 8 u2 e 2k 2T s) δ 2 } ) ds } ) } ds. 42) 25

29 A.2 Exponentally Dstrbuted Jumps Varance Swaps The characterstc functon n ) takes the form ] { E [e ux T = exp 2 ω2 u e 2k T 2k VIX Futures T uλ T +λ 2 ω2 u 2 e 2kT 2k pα + α + e k 2T s) + p)α α + e k 2T s) ) ds pα + α + ue k 2T s) + p) α α + ue k 2T s) ) } ds. 43) ecall from 37) that V IX t V IX M t where M t = expy t ) s a postve exponental martngale. In ths specfcaton the characterstc functon of Y T s gven by ] { E [e uy T = exp 8 ω2 u e 2k T T uλ T +λ 2k 8 ω2 u 2 e 2kT 2k pα + α + + p) α 2 e k 2T s) α + 2 e k 2T s) pα + α + + p)α 2 ue k 2T s) α + 2 ue k 2T s) ) ds ) ds }. 44) 26

30 B Proof of proposton 2 We can express V t n 9) as V t = V + t + t t V s ωe k T s) dz s Applyng the Itô formula to Vt we obtan Vt = V + 2 = + + t t V t V t t t V s ) V 2 s { } ) Vs exp e k 2T s) x J ds dx) { } ) Vs exp e k 2T s) x ν dx) ds. V s ) 2 ω 2 e 2k T s) ) V s ) ds { } ) exp e k 2T s) ) x V 2 2 s ν dx) ds s ωe k T s) 2 exp t V s V s V s ) 2 dz s { 2 e k 2T s) x Vs ωe k T s) dz s 2 exp Ths can be wrtten as { t Vt = V exp where } ) J ds dx) { } ) exp e k 2T s) x 8 ω2 e 2k T s) { } ) 2 e k 2T s) x J ds dx). + t µ t = 4 ω2 e 2k T t) 2 t ) ν dx)ds µ s ds + ωe k T s) dz s 45) 2 } 2 e k 2T s) xj dx ds), { } ) exp e k 2T t) x ν dx). The stochastc ntegrals n 45) beng processes wth ndependent ncrements, a straghtforward applcaton of the exponental formula for Posson random measures [8, Prop. 3.6.] yelds the multplcatve decomposton n Proposton 2 where M s gven by 35)-36). 27

31 C Tables Table 2: Error from followng the replcatng strategy relatve to the realzed varance over the three month wndows. Sep-Dec 27 Dec 27-Mar 28 Mar-Jun 28 Jun-Sep 28 Error.37%.25%.6%.243% Table 3: Calbrated parameters for the two models from the VIX volatlty smles on August 2th, 28. The top panel corresponds to the normally dstrbuted jumps and the bottom to the double exponentally dstrbuted jumps. Normal jumps λ ω k k 2 m δ Error %) Double exponental jumps λ ω k k 2 p α + α Error %)

32 Table 4: Model parameters calbrated from the S&P 5 volatlty smles on August 2th, 28 wth the correlaton between the two Brownan components set to Gaussan jumps b b m b δ Error %) Double exponental jumps b ) b p α + b p) α ) b 2 p + b2 α 2 p) 2 + α Error %) Table 5: The errors arsng from approxmatng the ndex dynamcs wth a contnuous process gven the model specfcatons descrbed n ths paper are true Gaussan jumps ε %) ε V Double exponental jumps ε %) ε V 29

33 .5.5 VP Sep 27 - Dec 27 Dec 27 - Mar 28 Mar 28 - Jun 28 Jun 28 - Sep 28-2 Fgure 4: The proft-loss from day to day n varance ponts VP) from holdng a long poston of the hedge portfolo and a short varance swap for four dfferent three month varance swaps. The x-axs s tradng days durng the three month perod. 3

34 Expry: 798 Expry: Expry: 98 Expry: Expry: 29 Md Model Bd Ask Fgure 5: VIX mpled volatlty smles on August 2th 28 for the model wth normally dstrbuted jump szes plotted aganst moneyness m = K/V IX t on the x axs. 3

35 Expry: 798 Expry: Expry: 98 Expry: Expry: 29 Md Model Bd Ask Fgure 6: VIX mpled volatlty smles on August 2th 28 for the model wth double exponentally dstrbuted jump szes plotted aganst moneyness m = K/V IX t on the x axs. 32

36 Expry: 998 Md Model Bd Ask Expry: Expry: 28.5 Expry: Expry: Expry: Fgure 7: S&P 5 mpled volatlty smles on August 2th 28 for the model wth normally dstrbuted jump szes plotted aganst moneyness m = K/S t on the x axs. 33

37 .5.4 Expry: 998 Md Model Bd Ask.5.4 Expry: Expry: 28.5 Expry: Expry: Expry: Fgure 8: S&P 5 mpled volatlty smles on August 2th 28 for the model wth double exponentally dstrbuted jump szes plotted aganst moneyness m = K/S t on the x axs. 34

38 Workng Papers from Fnance esearch Group F-29-5 F-29-4 F-29-3 F-29-2 F-29- F-28-7 F-28-6 F-28-5 F-28-4 F-28-3 F-28-2 F-28- F-27-3 F-27-2 F-26-9 F-26-8 F-26-7 ama Cont & Thomas Kokholm: A Consstent Prcng Model for Index Optons and Volatlty Dervatves. Stefan Hrth & Marlese Uhrg-Homburg: Investment Tmng, Lqudty, and Agency Costs of Debt. Lasse Bork: Estmatng US Monetary Polcy Shocks Usng a Factor- Augmented Vector Autoregresson: An EM Algorthm Approach. Leondas Tsaras: The Forecast Performance of Competng Impled Volatlty Measures: The Case of Indvdual Stocks. Thomas Kokholm & Elsa Ncolato: Sato Processes n Default Modelng. Esben Høg, Per Frederksen & Danel Schemert: On the Generalzed Brownan Moton and ts Applcatons n Fnance. Esben Høg: Volatlty and realzed quadratc varaton of dfferenced returns. A wavelet method approach. Peter Løchte Jørgensen & Domenco De Govann: Tme Charters wth Purchase Optons n Shppng: Valuaton and sk Management. Stg V. Møller: Habt persstence: Explanng cross-sectonal varaton n returns and tme-varyng expected returns. Thomas Poulsen: Prvate benefts n corporate control transactons. Thomas Poulsen: Investment decsons wth benefts of control. Thomas Kokholm: Prcng of Traffc Lght Optons and other Correlaton Dervatves. Domenco De Govann: Lapse ate Modelng: A atonal Expectaton Approach. Andrea Consglo & Domenco De Govann: Prcng the Opton to Surrender n Incomplete Markets. Peter Løchte Jørgensen: Lognormal Approxmaton of Complex Pathdependent Penson Scheme Payoffs. Peter Løchte Jørgensen: Traffc Lght Optons. Davd C. Porter, Carsten Tanggaard, Danel G. Weaver & We Yu: Dspersed Tradng and the Preventon of Market Falure: The Case of the Copenhagen Stock Exhange.

39 F-26-6 F-26-5 F-26-4 F-26-3 F-26-2 F-26- F-25-5 F-25-4 F-25-3 F-25-2 F-25- F-24- Amber Anand, Carsten Tanggaard & Danel G. Weaver: Payng for Market Qualty. Anne-Sofe eng asmussen: How well do fnancal and macroeconomc varables predct stock returns: Tme-seres and cross-sectonal evdence. Anne-Sofe eng asmussen: Improvng the asset prcng ablty of the Consumpton-Captal Asset Prcng Model. Jan Bartholdy, Denns Olson & Paula Peare: Conductng event studes on a small stock exchange. Jan Bartholdy & Cesáro Mateus: Debt and Taxes: Evdence from bankfnanced unlsted frms. Esben P. Høg & Per H. Frederksen: The Fractonal Ornsten-Uhlenbeck Process: Term Structure Theory and Applcaton. Charlotte Chrstansen & Angelo analdo: ealzed bond-stock correlaton: macroeconomc announcement effects. Søren Wllemann: GSE fundng advantages and mortgagor benefts: Answers from asset prcng. Charlotte Chrstansen: Level-ACH short rate models wth regme swtchng: Bvarate modelng of US and European short rates. Charlotte Chrstansen, Juanna Schröter Joensen and Jesper angvd: Do more economsts hold stocks? Mchael Chrstensen: Dansh mutual fund performance - selectvty, market tmng and persstence. Charlotte Chrstansen: Decomposng European bond and equty volatlty.

40 ISBN Department of Busness Studes Aarhus School of Busness Aarhus Unversty Fuglesangs Allé 4 DK-82 Aarhus V - Denmark Tel Fax