Optimal Portfolios when Volatility can Jump

Transcription

1 Opimal Porfolios when Volailiy can Jump Nicole Branger Chrisian Schlag Eva Schneider Finance Deparmen, Goehe Universiy, Meronsr. 7/Uni-Pf 77, D Frankfur am Main, Germany. Fax: +49-(0) Phone: +49-(0) Earlier versions of he paper were presened a Vanderbil Universiy, Universiä Göingen, he European Summer Symposium in Financial Markes in Gerzensee, 2005, and a Annual Meeings of he German Finance Associaion in Augsburg, We hank he paricipans for helpful commens. Special hanks go o Holger Kraf.

2 Opimal Porfolios when Volailiy can Jump Absrac We consider an asse allocaion problem in a coninuous-ime model wih sochasic volailiy and (possibly correlaed) jumps in boh, he asse price and is volailiy. Firs, we derive he opimal porfolio for an invesor wih consan relaive risk aversion. One main finding is ha he demand for jump risk now also includes a hedging componen, which is no presen in models wihou jumps in volailiy. Second, we show in a parial equilibrium framework ha he inroducion of nonlinear derivaive conracs can have a subsanial economic value. Third, we analyze he disribuion of erminal wealh for an invesor who uses he wrong model when making porfolio choices, eiher by ignoring volailiy jumps or by falsely including such jumps alhough hey are no presen in he rue model. In boh cases he erminal wealh disribuion exhibis faer ails han under he correcly specified model, as well as significan defaul risk. Volailiy jumps are hus an imporan risk facor in porfolio planning. JEL: G2, G3 Keywords: Dynamic asse allocaion, jump risk, volailiy jumps, sochasic volailiy, model mis-specificaion

3 Inroducion and Moivaion The key risk facors considered in opion pricing models, besides he diffusive price risk of he underlying asse, are sochasic volailiy, jumps in he asse price, and also jumps in volailiy. Models ha include some or all of hese facors were developed by Meron (976), Heson (993), Baes (996), Bakshi, Cao, and Chen (997), and Duffie, Pan, and Singleon (2000). The imporance of jumps in volailiy has become apparen in recen sudies, which ry o explain he ime series properies of sock and opion prices simulaneously, like Eraker, Johannes, and Polson (2003), or Broadie, Chernov, and Johannes (2005). In his paper, we analyze he imporance and impac of jumps in volailiy for porfolio planning. Firs of all, we invesigae he impac of jumps in volailiy on he invesor s opimal porfolio and on is srucue. Second, we assess he uiliy gain generaed by he availabiliy of derivaives in his economy. Third, we analyze he disribuion of erminal wealh for an invesor who uses he wrong model, namely eiher one ha does no conain volailiy jumps alhough he rue model does, or one conaining such jumps alhough hey are no par of he rue model. Our resuls show ha jumps in volailiy are an imporan risk facor when i comes o porfolio planning. Their inclusion or omission changes he srucure of he opimal porfolio. Furhermore, he use of a wrong model ha eiher ignores volailiy jumps or wrongly includes hem resuls in economically significan uiliy losses. There are imporan fundamenal differences beween sochasic volailiy and sochasic jumps, as shown in he area of coningen claim pricing by Das and Sundaram (999) and Carr and Wu (2002), and concerning opion hedges by Branger and Schlag (2004). In an asse allocaion conex he main papers analyzing he impac of jumps are Liu, Longsaff, and Pan (2003), Liu and Pan (2003) and Dieckmann and Gallmeyer (2004). Whereas Dieckmann and Gallmeyer (2004) consider he allocaion of diffusive and jump risks beween heerogeneous agens in a pure exchange economy, Liu, Longsaff, and Pan (2003) and Liu and Pan (2003) are operaing in a parial equilibrium framework. Our analysis ies up some loose ends in he lieraure on asse allocaion in coninuousime models. We consider he porfolio planning problem in a very general seup wih sochasic volailiy, jumps in he sock price, and, in paricular, jumps in volailiy. Thereby, we exend he comparison of diffusion risk and jump risk in Liu, Longsaff, and Pan (2003) o he more realisic case when derivaives are acually available o he invesor. By considering a model ha includes jumps in volailiy, we also exend he framework in Liu and Pan (2003) who sudy he benefis from rading derivaives in a model wihou jumps in volailiy. Our framework represens a significan generalizaion of boh of hese papers. We solve he model in closed form for he case of imperfecly correlaed jumps in he sock price and in volailiy. Our model can capure jumps in he sock price only, jumps in volailiy only, and simulaneous jumps in boh processes. We can hus analyze srucural differences beween hese kinds of jumps. For he numerical analysis, we resric he model

4 o he more simple case where jump sizes are deerminisic and where boh he sock and he volailiy jump simulaneously. This allows us o focus on he key aspecs of our model and i also allows for an easy comparison wih he papers by Liu, Longsaff, and Pan (2003) and by Liu and Pan (2003). Firs, we derive he opimal porfolio of an invesor wih consan relaive risk aversion and analyze is srucure. To concenrae on he impac of jumps, we assume ha he marke is complee, i.e. enough derivaives are raded. In he spiri of Meron (97) we separae he overall demand for a risk facor ino a speculaive componen, which represens he invesor s desire o earn he associaed risk premium, and he hedging componen, which serves o proec he invesor agains unfavorable changes in he invesmen opporuniy se. Our main finding is ha, wih jumps in volailiy, he opimal demand for jump risk now also conains a hedging componen, which is no presen in he Liu and Pan (2003) economy wihou jumps in volailiy. Inuiively, he hedging demand can be explained by he desire of he invesor o hedge agains fuure unfavorable changes in volailiy. When jumps have an impac on volailiy, par of his hedging can be achieved by rading jump risk, while in he case wihou jumps in volailiy, all he hedging has o be done by rading diffusion risk. The omission of jump risk in volailiy will hus lead o an overesimaion of he hedging demand for diffusion risk. Second, we deermine he economic value of rading in derivaives. Derivaives are a vehicle o achieve he opimal exposure o he fundamenal risk facors in an economy, i.e. o diffusions and jump componens. The inroducion of derivaives hus always increases he invesor s uiliy in a parial equilibrium model, where we assume he marke prices of exising asses o remain unchanged. We measure he economic gain due o derivaives by he annualized percenage increase in cerainy equivalen wealh, which can be hough of as a kind of an addiional ineres rae. Our resuls show ha his gain is economically significan. Third, we analyze he impac of model mis-specificaion on he porfolio planning problem. Given ha he rue model is no known and has o be esimaed, we assume ha he invesor eiher wrongly uses a model wihou jumps in volailiy which is calibraed o marke daa, or ha he includes volailiy jumps in he asse allocaion alhough hey are no presen in he rue daa generaing process. In boh siuaions wih model misspecificaion, we show ha he disribuion of erminal wealh exhibis more mass in boh, he lef and he righ ail. In paricular, he risk of obaining a very low erminal wealh increases significanly. Our resuls show ha neiher he use of a oo parsimonious nor of a oo sophisicaed model provides a simple robus heg wih respec o model risk. Our paper is mainly relaed o Liu, Longsaff, and Pan (2003) and Liu and Pan (2003). In a sochasic volailiy model wih deerminisic jumps in he sock price and no jumps in volailiy, Liu and Pan (2003) derive closed-form soluions for he opimal porfolio composiion in he case of a CRRA uiliy funcion. They show ha derivaives can be used o achieve he desired exposure o each risk facor. The reason is ha such nonlinear conracs can be employed o disenangle jump risk and diffusion risk which, in he case of he sock, are only available as one package. Furhermore, derivaives allow o rade volailiy risk. Liu, Longsaff, and Pan (2003) propose a more general framework 2

5 for he dynamics of he sae variables by including volailiy jumps, bu resric he se of raded asses o he sock and he money marke accoun. They find ha compared o he case of no sock price jumps, he invesor reduces he posiion in he risky asse even when jumps are upward. In he mos general case wih jumps in boh price and volailiy he posiion in he risky asse can increase or decrease compared o he no jump case, since jumps in volailiy enable a leas a parial hedging of jumps in he sock price. Anoher paper relaed o ours is he sudy by Daglish (2002). He considers he same fundamenal model as Liu, Longsaff, and Pan (2003), bu allows for sochasic jumps insead of assuming deerminisic jump sizes. Wih he sock and he money marke accoun only, he marke is incomplee. There are no closed-form soluions and he analysis has o be done numerically. For he case of log-normally disribued jumps, his resuls confirm hose of Liu, Longsaff, and Pan (2003). Since he marke is incomplee, i is hus no longer possible o work in erms of risk facors, bu one has o consider demands for exogenously given specific asses insead. In conras, we reain a complee marke by assuming ha a sufficienly large number of derivaives is raded o span even sochasic jumps which are drawn from a discree disribuion. This allows us o focus on he srucural impac of jumps. Wu (2003) invesigaes a jump-diffusion model, in which he diffusion risk premium follows an Ornsein-Uhlenbeck process. Wih he sock and he money marke accoun as raded asses, he finds ha here is ineracion beween reurn predicabiliy and he impac of jump risk. He also provides a deailed analysis of he impac of jumps as he source of non-normaliy on porfolio decisions in one-period and muli-period models. Ilhan, Jonsson, and Sircar (2004) compue asympoic approximaions o he opimal derivaive holdings based on an indifference argumen in a world wih purely diffusive sochasic volailiy. Finally, Das and Uppal (2004) analyze he impac of sysemic jump risk on he opimal porfolio and on he uiliy of an invesor wih CRRA, as well as he uiliy loss when his risk is ignored. In heir paper, sysemic jump risk is modeled as a common jump across all asse prices bu does no comprehend jumps in volailiy. The remainder of he paper is srucured as follows. In Secion 2 we presen he model. Secion 3 conains he soluion o he porfolio planning problem and is economic inerpreaion. Secion 4 provides a numerical example for he impac of jumps in volailiy. The economic value of derivaives in he conex of our model is discussed in Secion 5 and model mis-specificaion is analyzed in Secion 6. Secion 7 concludes. 3

6 2 Model Seup The dynamics of he sock price S and he insananeous variance V under he rue measure P are given by he following sysem of sochasic differenial equaions: ) ds = µ S d + ( V S db () + S x (j) dn (j,k) E P [X]λ P V d () j,k ( dv = κ P ( v P V )d + σ V V ρdb () + ) ρ 2 db (2) ( ) + y (k) dn (j,k) E P [Y ]λ P V d. (2) j,k The asse price and variance are driven by he independen Brownian moions B () and B (2) and by M J K independen Poisson processes N (j,k), each wih (sochasic) inensiy λ P V p jk. The physical probabiliy ha a jump occurs over he nex inerval of lengh d a all is equal o λ P V d, and given ha a jump has occurred, he random jump sizes (X,Y ) have realizaions (x (j),y (k) ) wih probabiliies p jk. In general, he jump sizes for boh, he asse price and he variance, are sochasic. We assume ha hese jump sizes are discree random variables wih possible realizaions x (j), j =,...,J for he sock and y (k), k =,...,K for he variance. The variance jumps have o be resriced o y (k) 0 for k =,...,K, in order o avoid negaive values for V. v P is he long-run mean of he variance. This seup allows us o model hree differen kinds of jumps: jumps in he sock price only, jumps in he variance only, and simulaneous jumps in boh processes. We se x () = 0 and y () = 0. Table summarizes he srucure. Jumps in he sock price only can be described by pairs (x (j),y () ) = (x (j), 0) for j 2. These jumps have an individual inensiy under he P-measure equal o λ P V p j, so ha he inensiy for a pure sock price jump is given by λ P V J j=2 p j. Analogously, pure variance jumps are represened by pairs (x (),y (k) ) = (0,y (k) ) for k 2. The P-inensiy for such a jump is given by λ P V K k=2 p k. The res of he probabiliy mass is disribued over all possible realizaions of simulaneous jumps (x (j),y (k) ), j,k 2 in he sock price and in V. The correlaion srucure of price and variance jumps can be generaed by an appropriae specificaion of he join probabiliies. Noe ha he even (x (),y () ) = (0, 0) is no considered in he jump size disribuion (i.e. is assigned a zero probabiliy), since i obviously represens he case of no jump a all. Even if we do no resric he join disribuion of X and Y besides y (k) 0 for all k =,...,K, he mos naural and empirically well-suppored srucure would be one wih a negaive jump size for prices (a leas on average) and upward jumps in volailiy. A ypical jump even would hus decrease prices and simulaneously increase volailiy, which can be regarded as an increase in uncerainy afer a marke crash. The ineres rae r is consan. The marke prices of risk in our model are no unique, bu have o be given exogenously. Following Liu and Pan (2003), we specify he pricing 4

7 kernel ξ via he sochasic differenial equaion ( dξ = ξ rd + η B V db () + η B2 ) V db (2) ) ( λ Q + ξ { j,k ( λ Q q jk λ P p jk dn (j,k) ) } λ P λ P V d. The marke price of risk η B V represens he compensaion per uni of V db (), while η B2 V is he expeced reward for bearing one uni of V db (2). The compensaion for an exposure of +α o a jump of size (x (j),y (k) ) (i.e. for an increase in wealh of α 00% if such a jumps occurs) is given by α [p jk λ P q jk λ Q] V. From hese specificaions we obain he following dynamics under he risk-neural measure Q: ) ds = rs d + ( () V S d B + S j,k ( dv = κ Q ( v Q V )d + σ V V ρd + ( j,k y (k) dn (j,k) B () E Q [Y ]λ Q V d x (j) dn (j,k) + ρ 2 d ), E Q [X]λ Q V d ) (2) B where he inensiy of he Poisson process dn (j,k) is now λ Q V q jk. In he general case, boh, he jump inensiy and he jump size disribuion, are differen under P and Q. The parameers of he volailiy process under he measures P and Q are relaed via ( κ Q ( v Q V ) E Q [Y ]λ Q V = κ P ( v P V ) E P [Y ]λ P V σ V ρη B + ) ρ 2 η B2 V, so ha κ Q = κ P + σ V ( ρη B + ρ 2 η B2 ) + ( E P [Y ]λ P E Q [Y ]λ Q) (3) κ Q v Q = κ P v P. (4) The expeced excess reurn on equiy is given by µ r = ( η B + E P [X]λ P E Q [X]λ Q) V. I combines he compensaion for diffusion risk and he compensaion for jump risk and is proporional o he local variance V. In he seup of Liu, Longsaff, and Pan (2003), where only he sock and he money marke accoun are raded, he relaive size of hese wo risk premia does no maer. However, in a model where boh risk facors can be raded separaely, he decomposiion of he equiy risk premium ino he compensaions for hese wo risk facors becomes imporan. 5

8 A very aracive feaure of he analysis in Liu and Pan (2003) is ha he marke is complee, so ha insead of pre-specifying radable asses and focusing on he invesor s demand for his paricular se of asses, one can work wih he more general concep of demands for he various risk facors driving he economy. In he seup considered here, he marke is complee when he number of non-redundan derivaives is equal o he number of possible sock-volailiy jump realizaions ha occur wih posiive probabiliy plus one o hedge he sochasic volailiy facor. In he general case we hus need M addiional raded insrumens besides he sock and he money marke accoun. To achieve marke compleeness wih a finie number of derivaives, we have o assume ha he jump size disribuion is discree wih a finie number of possible values bu we canno assume coninuous jump size disribuions. However, our seup is rich enough o sudy he case of sochasic jumps and o analyze possible differences o he case of deerminisic jumps only, and addiionally, beween differen ypes of jumps. Furhermore, we assume ha enough coningen claims are raded. While his may no be he case for all underlyings, we do hink his assumpion o be jusified when we focus on sock indices e.g.. Here we usually observe a large number of opions wih sufficien rading volume so ha an invesor could use hem for his asse allocaion. We now consider he coningen claims ha are raded in our economy. Le O (i) = g (i) (S,V ) (i =, 2,...,M) denoe he price of he i-h derivaive as a funcion of he sae variables. The dynamics of he price follow from Io and he fundamenal parial differenial equaion, and we obain do (i) = ro (i) d + ( ) ( g s (i) S + σ V ρg v (i) η B V d + ) V db () ( + σ V ρ2 g v (i) η B2 V d + ) V db (2) ( + (j,k) g (i) dn (j,k) E [ ) P g (i)] λ P V d j,k + ( E P [ g (i)] λ P E Q [ g (i)] λ Q) V d where he exposures o sock price diffusion risk, volailiy diffusion risk, and jump risk are given by g s (i) = g(i) (s,v) s (S,V ) g v (i) = g(i) (s,v) v (j,k) g (i) (S,V ) = g (i) (( + x (j) )S,V + y (k) ) g (i) (S,V ). For sandard European call and pu opions hese expressions can be calculaed using he opion pricing model in Duffie, Pan, and Singleon (2000). In conras o Liu and Pan (2003), closed form soluions for he Fourier ransforms needed o price opions are no 6

9 longer available. However, he numerical evaluaion of he respecive differenial equaions is raher sraighforward. 3 Porfolio Planning Problem The objecive of he invesor is o maximize he expeced uiliy of erminal wealh, i.e. here is no inermediae consumpion. The assumed uiliy funcion is of he CRRA ype wih risk aversion parameer γ. The radable asses in our economy are he money marke accoun, which earns ineres a he consan rae r, he sock, and a sufficien number of derivaive asses wrien on he sock, so ha he marke is complee. Kraf (2003) and Korn and Kraf (2004) focus on he echnical aspecs of such coninuous-ime asse allocaion problems in he conex of sochasic opporuniy ses. The invesor s opimizaion problem and is soluion are srucurally similar o Liu and Pan (2003). Compared o heir paper we presen a significan generalizaion of he jump componen in he dynamics of he sae variables. Firs, we include jumps in volailiy in addiion o price jumps, and, second, we allow for non-deerminisic jump sizes. As a resul, our model represens an economy where jumps may occur in he sock price only, in he variance only, or in boh processes simulaneously. The srucure of he jump size disribuion shown in Table makes i possible o generae differen correlaion srucures for he jump sizes. Le φ and ψ (i), i =, 2...,M, represen he fracions of wealh invesed in he sock and in he M derivaive asses, respecively. In he case ha φ + M i= ψ(i) he remaining wealh is invesed in he money marke accoun. The sochasic differenial equaion for wealh is hen given by ) } M dw = W {( φ ψ (i) ds M rd + φ + ψ (i) do (i). S O (i) i= For he following analysis, i is useful o work wih exposures o he fundamenal risk facors B (), B (2), and o he M differen jump evens insead of porfolio weighs. Rewriing he dynamics of wealh in erms of hese exposures, one obains ( dw = rw d + θ B W η B V d + ) ( V db () + θ B2 W η B2 V d + ) V db (2) ) + W ( j,k θ N(j,k) dn (j,k) j,k q jk θ N(j,k) λ Q V d i=. (5) The fracion of wealh θ B invesed in risk facor V db () in he raded asses via θ B = φ + M i= ψ (i) 7 ( g (i) s S O (i) is relaed o he invesmen ) + σ V ρ g(i) v. (6) O (i)

10 Analogously, we find θ B2 = σ V ρ 2 M i= ψ (i) g (i) v O (i) (7) wih θ B2 represening he fracion of wealh invesed in V db (2), and M θ N(j,k) = φ x (j) + ψ (i) (j,k) g (i) O (i) where θ N(j,k) i= sands for he fracion of wealh invesed in he risk facor dn (j,k) wih jump gives he relaive jump in wealh if here is a jump of size x (j) sizes (x (j),y (k) ). So θ N(j,k) in he sock price and of size y (k) in volailiy. For example, θ N(j,k) invesor s wealh will decrease by θ N(j,k) (8) < 0 means ha he 00 percen when a jump of ype (j,k) occurs. This noaion differs from he one used in Liu and Pan (2003) in ha he sign of θ N immediaely indicaes he direcion of he wealh change in case of a jump. When he marke is complee, any exposure (θ B,θ B2,θ N(j,k) ) can be obained by suiable posiions in he sock, he money marke accoun, and he coningen claims. The posiions in he raded asses follow from solving he sysem of equaions (6), (7) and (8) for φ and ψ (i) (i =, 2,...,M). The invesor s opimizaion problem is given by [ ] { max γ W γ T θ B,θ B2 } E,θ N(j,k) (j,k):p jk >0,0 T subjec o he wealh dynamics in (5). The associaed indirec uiliy funcion J(, w, v) is hen obained as [ ] J(,w,v) = { max γ W γ T W = w,v = v, θ B s,θs B2,θ N(j,k) s (j,k):p jk >0, s T } E again subjec o (5). From his we can immediaely derive he Hamilon-Jacobi-Bellman (HJB) equaion: ( { max J + wj W r + θ B η B v + θ B2 η B2 v ) q jk θ N(j,k) λ Q v {θ B,θ B2,θ N(j,k) (j,k):p jk >0 } j,k + 2 w2 J WW v [ (θ B ) 2 + (θ B2 ) 2] + λ P v [ ] p jk J(,w( + θ N(j,k) ),v + y (k) ) J(,w,v) j,k + [ κ P ( v P v) E P [Y ]λ P v ] J V + 2 σ2 V vj V V ( + σ V vwj WV ρθ B + ) } ρ 2 θ B2 = 0, (9) 8

11 where subscrips of J denoe parial derivaives. To find he opimal porfolio composiion, one firs needs o know he indirec uiliy funcion J. The usual guess for his funcion, as in Liu and Pan (2003) and numerous oher papers, is J(,w,v) = w γ exp {γh(τ) + γh(τ)v}, (0) γ where τ = T. Afer compuing he necessary parial derivaives one can deduce he opimal exposures θ B, θ B2, and θ N(j,k) o he fundamenal risk facors V db (), V db (2), and jumps of size x (j) in he sock price and y (k) in volailiy from he sandard firs-order condiions. They are given in he following lemma. Lemma (Opimal exposures o fundamenal risk facors) The opimal exposures o he fundamenal risk facors are given by θ B θ B2 θ N(j,k) = = ηb γ + ρσ V H(τ) () = ηb2 γ + ρ 2 σ V H(τ) (2) [ (pjk ) λ P /γ ( ) pjk λ ] P /γ [ ] + e H(τ)y(k) (3) q jk λ Q q jk λ Q wih θ N(j,k) for all j,k. Noe he addiional resricion for θ N(j,k), which assures ha wealh canno become negaive. The opimal porfolio weighs for he risky asses follow immediaely from hese opimal weighs via Equaions (6), (7) and (8). Finally, he Expressions (), (2), and (3) are plugged back ino he HJB equaion (9). Collecing erms wih and wihou v one obains he following sysem of ordinary differenial equaions for he funcions h and H: h (τ) = κ P v P H(τ) + γ r (4) γ H (τ) = a + bh(τ) + c H 2 (τ) + λ [ (pjk ) Q λ P /γ q jk exp{y H(τ)}] (k) (5) q jk λ Q j,k wih he boundary condiions h(0) = H(0) = 0 and a = γ 2γ 2 [ (η B ) 2 + (η B2 ) 2] + γ λ Q γ γ λp (ρη B + ) ρ 2 η B2 b = ( κ P + E P [Y ]λ P) + γ γ σ V c = 2 σ2 V. 9

12 In Liu and Pan (2003), Y is idenically equal o zero, and he parial differenial equaion (5) for H is a Ricai equaion wih consan coefficiens. In his case, a closed form soluion is available for boh H and h. This is no longer rue in he general case where he volailiy jump size is differen from zero or even sochasic. Now, he sysem (4, 5) has o be solved numerically, e.g. via he Runge-Kua mehod. Neverheless, our funcion H shares an imporan propery wih he corresponding funcion in Liu and Pan (2003), namely ha H(τ) 0 for γ <, H(τ) 0 for γ >, and H(τ) = 0 for he log-invesor wih γ =. A proof is given in Appendix A. This propery of H implies ha he indirec uiliy of he invesor is increasing in V. To ge he inuiion, noe ha we have made he by now sandard assumpion ha he size of he marke price of risk is increasing in V for all risk facors. The higher V, he larger he compensaion earned by he invesor per uni of risk, and he larger herefore his uiliy. In he economy analyzed by Liu, Longsaff, and Pan (2003) only he sock and he money marke accoun are raded, which is in our case equivalen o imposing he addiional resricions θ B = φ, θ N(j,k) = φ x (j) and θ B2 = 0 when solving for he opimal exposure. Wihou derivaives, he exposure o he second diffusion risk wih impac on volailiy only is equal o zero, and he relaion beween he exposure o he firs diffusion risk and o jump risk is fixed a he relaion of hese risk facors for he sock. The inroducion of derivaives complees he marke and allows he invesor o rade he risk facors separaely. In Secion 5, we will analyze he economic value of his possibiliy. The opimal demand in (), (2), and (3) has wo basic sources which were already discussed by Meron (97). Firs, he invesor wans o earn he risk premium for he respecive facor, which represens he myopic or speculaive demand, given by he firs summand. Second, here is also a desire o hedge agains unfavorable changes in he sae variables deermining he invesmen opporuniy se, i.e. in our framework agains changes in V. This is he invesor s hedging demand, given by he second summand. The mos imporan difference beween a model ha allows for jumps in volailiy and he seup sudied by Liu and Pan (2003) where jumps affec he sock price only is ha he opimal demand for jump risk now conains a hedging componen which is no presen in an economy wihou variance jumps. This hedging demand is given by he second summand in Equaion (3). For y (k) > 0, here will hus be demand for jump risk (θ N(j,k) 0) even in he case when λ P p jk = λ Q q jk, i.e. when he jump risk premium is zero. To ge he inuiion, noe ha he hedging demand arises because he invesor wans o hedge agains unfavorable changes in volailiy. Expressing his in erms of exposures o risk facors, here will hus be a hedging demand in all facors ha have an impac on volailiy. For a jump wih y (k) 0, his generaes a hedging demand in his jump risk facor. The hedging demand of he invesor arises from he impac of V on he compensaion per uni of risk, as, e.g. discussed in Munk (2004) or Munk and Sørensen (2004). For small V, he expeced reurn earned by he invesor on his myopic exposure is low, and he risk of a large negaive reurn and hus a low erminal wealh is comparably high. This induces 0

13 he invesor o shif wealh from saes wih high V o saes wih low V. On he oher hand, for high V, he expeced reurn is high. This induces he invesor o shif wealh o saes wih a high V o grasp hese good invesmen opporuniies. Taken ogeher, his ulimae hedging demand depends on he rade-off beween hese wo opposie effecs. As described in Liu (200), for γ >, he uiliy funcion of an invesor is unbounded from below, bu bounded from above. He cares more abou saes wih low V since his implies a higher probabiliy of losses due o a lower expeced reurn. Thus, his hedge is o ake a shor posiion in V. In line wih his inuiion, he marginal indirec uiliy J W of he invesor is decreasing in V, which formally follows from he fac ha H is non-posiive. A low volailiy hus corresponds o a high indirec uiliy, implying a bad sae. For γ <, on he oher hand, he uiliy funcion of he invesor is bounded from below and unbounded from above, and he speculaes by aking a long posiion in volailiy. Finally, for γ =, boh effecs exacly offse each oher, and he invesor is neural wih respec o changes in he invesmen opporuniy se. In conras o Liu and Pan (2003) his hedging demand can now be me by a posiion in all risk facors and no only in he wo diffusive risks. Consider he case γ >. H is non-posiive and he invesor wans o hedge by aking a shor posiion in variance. The hedging demand for he firs diffusion db () will be posiive if ρ < 0, and he hedging demand for he second diffusion db (2) will be negaive. Given ha any jump in volailiy is an upward jump, he hedging demand for all jumps ha also affec volailiy will be negaive, while here will be no hedging demand for jumps ha occur only in he sock price. The analysis for γ < proceeds along he same lines, showing ha he invesor now hedges by aking a long posiion in variance, which resuls in a posiive hedging demand for jump risk. To explain he srucure of he jump demand in our model in more deail, consider wo pairs of jump size realizaions (x (a),y (a) ) and (x (b),y (a) ) wih p aa = p ba = 0.5. To focus on price jumps he volailiy jump size is he same for boh pairs. Assume γ >, x (b) < x (a) < 0, y (a) > 0 and λ P = λ Q, i.e. here is no jump inensiy premium. If p aa /q aa = p ba /q ba =, he jump size premium is also zero. There is no myopic demand here, and he negaive hedging demand is equal for boh jump realizaions, i.e. = θ N(b,a) < 0. For he hedging demand, i is hus only he size of y () ha maers. θ N(a,a) Noe ha here is no insurance agains price jumps, bu he invesor is willing o give up wealh in siuaions when volailiy has jumped upwards. Now le p aa /q aa > > p ba /q ba. This means ha he jump size premium is relaively higher for large negaive jumps, so ha insurance agains hese jumps is relaively more expensive. This corresponds o he sylized facs repored in he lieraure, namely ha ou-of-he-money pu opions seem very expensive. In his scenario we obain a posiive myopic demand for jump ype (a,a) and a negaive myopic demand for jump ype (b,a), where he invesor earns a risk premium in boh cases. The hedging demand is sill negaive for boh jump realizaions. In absolue erms i is larger han in Case I for jump (a,a) and smaller for jump (b,a). In oal here is definiely no insurance agains large jumps, since θ N(b,a) < 0, bu he sign of θ N(a,a) is no deermined.

14 When p aa /q aa < < p ba /q ba, he siuaion is exacly reversed. The oal posiion of he invesor will no conain proecion agains small jumps, bu here may be insurance agains large negaive jumps, depending on he rade-off beween risk premium and addiional uiliy derived from hedging. 4 Numerical Example: Deerminisic Jumps For he numerical example, we focus on he framework used for he examples in Liu, Longsaff, and Pan (2003), where he jump sizes for boh he asse price and he variance are deerminisic, i.e. we se X µ X and Y µ Y. This also implies ha he jump size disribuion degeneraes ino a single poin, which implies ha he pricing of jump risk depends on he difference beween λ P and λ Q only, while he jump size disribuions are degenerae. Assuming he usual case of a negaive jump size for prices (µ X < 0) and upward jumps in volailiy (µ Y > 0), a jump decreases prices and simulaneously increases volailiy, which can be regarded as an increase in uncerainy afer a marke crash. Due o he simpler srucure of he model compared o he general case, now only wo non-redundan derivaives are needed o complee he marke. Liu and Pan (2003) show ha he marke is complee if he claims saisfy he resricion D 0, where D = ( g () µ X O () g() s S O () ) g v (2) O (2) ( g (2) µ X O (2) g(2) s S O (2) ) g v () O () and g (i) = g (i) (( + µ X )S,V + µ Y ) g (i) (S,V ). This condiion says ha he deerminan of he local sensiiviy marix of he wo derivaive conracs mus no be zero, implying ha he wo conracs are linearly independen. The ransformaion from risk exposures o asse posiions works as in Liu and Pan (2003). For he risk exposure o he deerminisic jumps, we now ake he simpler noaion θ N insead of θ N(j,k). Solving Equaions (6), (7) and (8) for he opimal posiions in he sock and in he derivaive asses gives: ( ) 2 φ = θ B ψ (i) g s (i) S + σ O (i) V ρ g(i) v (7) O (i) i= [ ψ () = g v (2) D O (2) [ ψ (2) = D ( θ B2 θ N σ V ρ 2 θ B ( ) ( ρ θ B2 ρ 2 σ V ρ 2 ) ( + θb2 g () µ X O () g() s S O () g() v O () θ N g (2) µ X O (2) θ B g(2) s S + θb2 O (2) )] (6) (8) )] ρ (9) ρ 2 Noe he ypo in Liu and Pan (2003), where he erm θ B2 ρ/ ρ 2 is subraced raher han added in boh he second and he hird equaion. One of he main iems of ineres in our paper is he impac of variance jumps on he srucure of he opimal demand funcions. To analyze his impac, we firs perform 2

15 a sensiiviy analysis wih respec o he deerminisic volailiy jump size µ Y. All oher parameers are unchanged, which implies in paricular ha he variance of variance and he mean variance will change. Therefore, we refer o his analysis as he uncalibraed case. The key new resul derived in Secion 3 is ha wih jumps in volailiy, he jump demand exhibis a hedging componen in addiion o he familiar speculaive par. Since he size and direcion of hedging demand primarily depends on he funcion H(τ), we firs ake a closer look a he impac of he variance jump size on his funcion. The comparison is based on he benchmark paramerizaion in Liu, Longsaff, and Pan (2003). In our noaion, he parameers are κ P = 5.3, v P = 0.52 = , 5.3 σ V = , ρ = 0.57, λ P =.8456, µ X = 0.25, and µ Y = Furhermore, we assume ha jump risk and diffusion risk each accoun for half of he expeced excess reurn on he sock, and we choose η B2 = 2.0, where he negaive sign can be jusified based on he discussion in Liu and Pan (2003). This yields he following values for he oher parameers in he model: κ Q = , v Q = , λ Q = and η B = Table 2 summarizes hese values as Paramerizaion I. Figure shows he funcion H as a funcion of he planning horizon τ for he benchmark case µ Y = as well as for µ Y = 0.0 and µ Y = 0.. The coefficien of risk aversion is equal o 3. For all hree values of µ Y, he funcion H is zero for τ = 0 and decreases sharply for shor planning horizons. I hen approaches an asympoic value for planning horizons of more han one year, where he asympoic value is deermined by he parameers. The hedging demand, which depends on he planning horizon only hrough H, is hus zero for τ = 0 and sabilizes a some value for increasing ime o mauriy. Inuiively, his can be explained by he exisence of some upper bound on he hedging demand of he invesor. Even if he invesor is risk averse, he is sill willing o ake some risk o earn he risk premia. The absolue value of H is increasing in µ Y, so ha he hedging demand is also increasing in µ Y. Inuiively, an increase in he variance jump size increases he variance of variance. The higher variance risk hen leads o a higher hedging demand agains his risk facor. This is no only rue for he hedging demand in jump risk, bu also for he hedging demand in diffusion risk, which can be explained by a higher overall concern of he invesor abou variance risk. Noe ha he analysis performed here is a sensiiviy analysis in which we assume ha only µ Y changes. In he nex sep, we invesigae he economic consequences of he presence of a jump componen in he volailiy process, i.e. he impac of variance jumps and heir size on opimal exposures o risk facors and opimal porfolio decisions. Again, we consider he case of jumps wih deerminisic sizes µ X and µ Y. As in he above analysis µ Y will be aken from he se {0, 0., }. However, now we will no simply leave all oher parameers unchanged when we compare economies wih volailiy jumps o one wihou. When varying µ Y one has o make sure ha he marke informaion he invesor could use o calibrae he model of his choice is correcly represened in he paramerizaion. This is wha we call he calibraed case. For example, he parameers plugged in he model chosen by he invesor mus resul in correc values for expeced sock reurns, opion prices, or risk premia. In more deail, he following parameers were resriced o 3

16 be idenical across models: he insananeous expeced excess reurn on he sock (given by ( η B + µ X ( λ P λ Q)) v P in all of he models), he insananeous variance of sock reurns (given by v P + µ 2 X λp v P in he wo models wih volailiy jumps and by v P in he one wihou), he insananeous variance of variance (given by σ 2 V vp +µ 2 Y λp v P in he models wih volailiy jumps and by σ 2 V vp in he one wihou), he average ime beween wo jumps (given by (λ P v P ) in all models), and he relaive jump size in he sock price (given by µ X in all models). Furhermore, we assume ha V 0 is he same in all models and equal o v P. To ulimaely calibrae he model, hree addiional opion prices are needed. We use wo European call opions wih a ime o mauriy of hree monhs and srike prices equal o 90% and 00% of he iniial sock price. The hird opion is a European call wih one monh o mauriy and a srike price equal o 90% of he iniial sock price. The prices of hese opions are compued based on he model developed by Duffie, Pan, and Singleon (2000). As above, he benchmark case is given by he model and he associaed parameers described in Liu, Longsaff, and Pan (2003). Table 2 shows he benchmark case, denoed by Paramerizaion I, as well as Paramerizaion II (µ Y = 0.), and Paramerizaion III (µ Y = 0). The scenario µ Y = 0. was included o represen he case, where he invesor correcly assumes ha volailiy can jump, bu uses a wrong jump size. Due o he fac ha jumps are rare, he momens of he jump size are subjec o severe esimaion risk, so ha such an error can occur easily. In general, wih a decreasing volailiy jump size µ Y, he calibraion yields a lower speed of mean reversion κ P and a higher level of he volailiy of volailiy σ V. This resul can be explained inuiively by noing ha, wih smaller jumps in volailiy, he insananeous variance of variance has o be generaed o a larger degree (or even compleely) by he diffusive par of he volailiy dynamics. This resuls in higher levels of σ V and lower levels of κ P. We assume ha he derivaives used by he invesor o form his porfolio are he wo 3-monh call opions described above. Our choice of hese opions is moivaed by wo consideraions. Firs, opions wih hese characerisics are usually highly liquid. Second, an ATM call is a sandard choice for a volailiy sensiive insrumen, while he opion wih he lower srike price has a large exposure o jump risk. We firs analyze he properies of he funcion H(τ) for differen values of µ Y. From Figure 2 we can see ha he impac of τ is quie similar o he case analyzed in Figure. Again, H(τ) is zero for τ = 0 and increases sharply in absolue erms for increasing τ, before i approaches an asympoic value. The approximaion o his asympoic value is much slower for µ Y = 0.0 han for he benchmark case of µ Y = This can be explained by he smaller speed of mean reversion of volailiy, due o which i akes longer for volailiy shocks o die ou. The impac of µ Y, however, is significanly differen depending on wheher we only vary µ Y as in Figure or recalibrae he model as in Figure 2. While in he firs case, H(τ) in increasing in absolue value in µ Y, i is now decreasing in µ Y. To ge he inuiion, noe ha we fix he variance of variance a some level. If we wrongly assume a model wihou jumps in volailiy, hen he whole variance of variance has o be explained by he diffusion componens, and he hedging demand of he invesor in he diffusion componen 4

17 increases, which implies ha H(τ) increases in absolue value for µ Y decreasing. In a second sep, we analyze he impac of µ Y on he decisions made by he invesor. The opimal exposures are given in (), (2) and (3). For he special case Y = µ Y = 0.0 and X = µ X, he expressions coincide wih he formulas given in Liu and Pan (2003), where he funcion H(τ) can be obained in closed form. The conversion of opimal exposures ino opimal asse posiions ψ (), ψ (2), and φ works as described above via Equaions (7), (8) and (9). Noe, however, ha he opimal exposures represen he basic resul, whereas he opimal asse posiions are he derived resul. These numbers naurally depend on he choice of asses, and here is no unique represenaion. So our resuls concerning he sock and he wo calls represen only one possible example for he porfolio composiion. Neverheless, in realiy he invesor has o buy (or sell) asses o achieve he desired exposure, so ha differences wih respec o his oupu of he model acually describe he differences in behavior one would see in he real world. Figure 3 shows he opimal exposures o he fundamenal risk facors for varying ime horizons and differen values of µ Y. For very shor horizons he opimal exposures almos exclusively reflec myopic demand, so all differences beween he paramerizaions can be aribued o he differen risk premium for V db (2). Noe ha he risk premia for he oher wo risk facors coincide for all calibraions by assumpion. When µ Y = 0.0, all he variance risk is aribued o he wo diffusion risk facors, which increases he hedging componen of he demand compared o he cases wih µ Y > 0. On he oher hand, here is only myopic demand in jump risk, so ha he planning horizon is irrelevan for he opimal exposure. The larger µ Y, he more imporan jump risk is for hedging volailiy risk, and he less imporan are he diffusions. Consequenly, he opimal exposure o jump risk depends more on he planning horizon, while he dependence of he opimal exposure o he wo diffusions on he planning horizon becomes less pronounced. Finally, wih increasing planning horizon he opimal exposures are more or less consan for all hree values of µ Y. The invesor ends o change his porfolio allocaions o a smaller degree when he invesmen horizon is sill long. The picure changes when we look a asse posiions which are given in Figure 4. Now, he differences beween he hree models are much more pronounced. The posiions in he asses vary significanly for τ = 0, despie he fac ha he opimal exposures o he risk facors db () and o jump risk are he same. This can be aribued o he fac ha he sensiiviies of he derivaives are calculaed in differen models and vary subsanially. The value of µ Y has a significan impac on he opimal porfolio. For µ Y = 0.0, e.g., he opimal posiion in he sock is negaive for longer planning horizons, while i is posiive in he oher wo cases wih µ Y > 0. Finally, for he opimal holdings of he raded asses, we observe ha for longer planning horizons he opimal posiions end owards an asympoic value which again depends on µ Y. 5

18 5 Economic Value of Derivaives In his secion we assess he economic value of derivaives. As saed in he inroducion, we augmen he choice se of he invesor relaive o he analysis in Liu, Longsaff, and Pan (2003), since now all he necessary marke compleing derivaives are assumed o be raded. A comparison o he case where only he sock and he money marke accoun are raded allows us o assess he economic value of rading derivaives, hus exending he analysis of Liu and Pan (2003) o he case of jumps in volailiy. Clearly, his represens a firs sep, since we perform he analysis in a parial equilibrium conex, where he prices of asses already raded are implicily assumed o remain unchanged. As discussed above derivaives allow he invesor o rade he risk facors separaely. Compared o a siuaion where he can only rade he sock and he money marke accoun, his uiliy will hus necessarily increase. To measure he economic value of derivaives, we use he porfolio improvemen measure R W as proposed by, among ohers, Liu and Pan (2003). I is defined as he annualized percenage difference in cerainy equivalen wealh ( ) ln W/Ŵ R W =, T where W (Ŵ) is he cerainy equivalen wealh for he case wih (wihou) derivaives. W and Ŵ are defined implicily via and J(0,W 0,V 0 ) = W γ γ Ĵ(0,W 0,V 0 ) = Ŵ γ γ wih J (Ĵ) represening he indirec uiliy funcion wih (wihou) derivaives. Since he invesor has consan relaive risk aversion, R W does no depend on his iniial wealh W 0. As in Secion 4 we analyze he special case wih simulaneous and deerminisic jumps in he sock and in volailiy. The indirec uiliy funcion Ĵ can be compued as in Liu, Longsaff, and Pan (2003), while for he compuaion of J we firs need o solve Equaions (4) and (5) numerically and hen plug he soluion o his sysem ino (0). The comparison beween he siuaions wih and wihou derivaives is based on Paramerizaion I from Table 2. Derivaives allow he invesor o achieve his opimal exposure o he individual risk facors. Wihou derivaives, his exposure o he second diffusion facor B (2) is zero, and he relaive exposure o diffusion risk B () and jump risk is fixed a he relaion of hese wo risk facors in he sock. Figure 5 illusraes he porfolio improvemen R W as a funcion of he planning horizon, he speed of mean reversion, he jump risk premium (capured by he relaion of he risk-neural and he rue jump inensiy), and he variance jump size. The upper lef 6

19 graph shows he porfolio improvemen measure R W for varying planning horizon τ. For τ = 0, here is only myopic demand, and he porfolio improvemen of 3% arises from he abiliy of he invesor o achieve his opimal demand. For increasing ime horizons, here is an addiional gain from achieving he opimal hedging demand, oo, and he porfolio improvemen sabilizes a nearly 5.5% for a horizon of more han 2 years. The upper righ graph shows he impac of he speed of mean reversion on he porfolio improvemen. The higher κ P, he less impac shocks in variance have, and he lower he variance of variance. Consequenly, hedging becomes less imporan for increasing κ P. For κ P beween one and wo, he opimal demand of he invesor which can be reached by rading derivaives is he mos differen from he risk package offered by he sock and he money marke accoun only. For oher values of κ P, he change in he absolue hedging demand lowers his difference beween opimal exposure and exposure aainable wihou rading derivaives, so ha he porfolio improvemen becomes smaller. For very high values of κ P, he hedging demand goes o zero, and he porfolio improvemen can be aribued o he possibiliy o achieve he opimal myopic exposure, only. The impac of he risk-neural jump inensiy λ Q is shown in he lower lef graph. The more λ Q differs from he (fixed) λ P, he larger he compensaion for a posiion in jump risk. The impac of a given difference beween he opimal exposure and he package offered by he sock on he porfolio improvemen hus increases. Furhermore, a change in λ Q changes he opimal myopic exposure o jump risk and he hedging exposure o all hree risk facors. Similar o he discussion for κ P, he porfolio improvemen ends o increase in he differences beween he resuling opimal exposure and he package offered by he sock. In our example, he minimal improvemen is realized for λ Q /λ P.6. Finally, he lower righ graph shows he porfolio improvemen for a varying variance jump size µ Y. The larger µ Y, he larger he variance of variance, and he larger he invesor s hedging demand. Trading in derivaives, which allows him o mee his hedging demand in paricular in jump risk, hus becomes more valuable, and he porfolio improvemen increases in µ Y. 6 Model Mis-Specificaion We now invesigae he consequences of model mis-specificaion in he conex of volailiy jumps. Noe ha his mis-specificaion can go wo ways. The invesor eiher uses a model ha is oo small, e.g. by ignoring volailiy jumps, or one ha is oo large, e.g. by including such jumps alhough hey are no a par of he rue model. We have seen in Secion 4 ha asse posiions and opimal exposures o risk facors usually change noiceably when differen models are used. However, he ulimae measure for he impac of model mis-specificaion is he loss in uiliy he invesor has o suffer when using incorrec dynamics for he sock price or for volailiy. Again he analysis considers he case of deerminisic jumps, and he differen models are calibraed o he same se of prices, risk premia, and momens for he sock price and is variance. 7

20 To analyze he impac of model mis-specificaion, we proceed in several seps. Firs, we calculae he (seemingly) opimal exposure o he risk facors, derived from he improper model. Then hese exposures are ransformed ino asse demands, using Equaions (7), (8) and (9), wih he sensiiviies sill based on he incorrec model. This gives he porfolio he invesor will acually buy a he marke. In he final sep, he asse posiions are convered back ino realized exposures, now based on he sensiiviies in he rue model, using Equaions (6), (7) and (8). Noe ha his las sep in he calculaion can only be done o analyze he impac of model mis-specificaion, bu no by he invesor who does no know he rue model. Firs, we consider he siuaion where he rue model is given by Equaions () and (2) and includes volailiy jumps, bu where he invesor bases his decision on a model wihou jumps in volailiy. The rue model is hus given by Paramerizaion I from Table 2, while he invesor ignores volailiy jumps, i.e. uses Paramerizaion III from Table 2. Figure 6 shows he realized exposures as a funcion of he planning horizon, where we have se he local variance equal o is long run mean v P. A comparison of his figure wih he (seemingly) opimal exposures in he lower graph of Figure 3 and he ruly opimal exposure in he upper graph shows ha he use of he wrong sensiiviies can have a significan impac. In paricular, in he correc model wih µ Y = , he opimal exposure o jump risk is increasing in he planning horizon (in absolue erms), while in he improper model wih µ Y = 0.0, he invesor considers a consan exposure o be opimal, and he ends up wih an exposure o jump risk ha is acually decreasing in he planning horizon (in absolue erms). Figure 7 shows he resuls for he opposie case when he rue model is wihou jumps in volailiy. The realized exposures o he risk facors are increasing in absolue value compared o he opimal case, so ha he invesor holds posiions wih a higher level of risk. Whereas he opimal exposure o jump risk in he rue model is now consan for all invesmen horizons, he realized exposure increases wih he invesmen horizon in absolue value. Knowledge of realized exposures under a mis-specified model is he necessary prerequisie for he deerminaion of he uiliy loss suffered by an invesor who bases his decision on an incorrec specificaion. The difference beween opimal and realized risk exposures will in general depend on he differences in he parameers and risk premia as well as on he differences of he sensiiviies of he raded asses under he rue and he assumed model. This implies ha he differences also depend on he curren level of volailiy. Due o his addiional dependence on V, he indirec uiliy for he mis-specified model canno be compued in closed-form as in Liu, Longsaff, and Pan (2003), who only consider rading in he sock and he money marke accoun, i.e. in linear claims whose sensiiviies canno be mis-esimaed. Insead, we have o deermine he disribuion of erminal wealh via a Mone Carlo simulaion. Furhermore, he lower bound on he jump risk exposure which is supposed o preven defaul can be imposed on he (seemingly) opimal exposure, bu no on he realized exposure. Thus, defaul becomes possible in a mis-specified model. In he realisic case of γ, he levels of uiliy of erminal wealh will go o when erminal wealh goes o zero. The indirec uiliy may hus ake on 8