Option Pricing with Lévy-Stable Processes Generated by Lévy-Stable Integrated Variance

Transcription

1 Opion Pricing wih Lévy-Sable Processes Generaed by Lévy-Sable Inegraed Variance Álvaro Carea 1 and Sam Howison 1 Birkbeck, Universiy of London Mahemaical Insiue, Universiy of Oxford, Oxford Man Insiue of Quaniaive Finance May 13, 9 Absrac We show how o calculae European-syle opion prices when he log-sock price process follows a Lévy-Sable process wih index parameer 1 α and skewness parameer 1 β 1. Key o our resul is o model inegraed variance σ sds as an increasing Lévy-Sable process wih coninuous pahs in T. Keywords: Lévy-Sable processes, sable Pareian hypohesis, sochasic volailiy, α-sable processes, opion pricing, ime-changed Brownian moion. We are very graeful for commens from Hu McCulloch, Mahias Winkel, Fred E. Benh, wo anonymous referees and seminar paricipans a he Universiy of Torono, Universiy of Oxford, Birkbeck College, Universiy of Chicago. Corresponding auhor: a.carea@bbk.ac.uk 1

2 1 Inroducion Up unil he early 199 s mos of he underlying sochasic processes used in he financial lieraure were based on Brownian moion, modelling in coninuous ime a large number of independen microscopic price changes, wih finie oal variance; and Poisson processes, modelling occasional large changes. These wo processes are he canonical models for coninuous sample pahs and hose wih a finie number of jumps, respecively. More generally, dropping he assumpion of finie variance, he sum of many iid evens always has, afer appropriae scaling and shifing, a limiing disribuion ermed a Lévy-Sable law; his is he generalised version of he Cenral Limi Theorem (GCLT) [ST94], and he Gaussian disribuion is one example. Based on his fundamenal resul, i is plausible o generalise he assumpion of Gaussian price incremens by modelling he formaion of prices in he marke by he sum of many sochasic evens wih a Lévy-Sable limiing disribuion. An imporan propery of Lévy-Sable disribuions is ha of sabiliy under addiion: when wo independen copies of a Lévy-Sable random variable are added hen, up o scaling and shif, he resuling random variable is again Lévy-Sable wih he same shape. This propery is very desirable in models used in finance and paricularly in porfolio analysis and risk managemen, see for example Fama [Fam71], Ziemba [Zie74] and he more recen work by Toka and Schwarz [TS], Orobelli e al [OHS] and Minik e al [MRS]. Only for Lévy-Sable disribued reurns do we have he propery ha linear combinaions of differen reurn series, for example porfolios, again have a Lévy-Sable disribuion [Fel66]. Based on he GCLT we have, in general erms, wo ways of modelling sock prices or sock reurns. If i is believed ha sock reurns are a leas approximaely governed by a Lévy-Sable disribuion he accumulaion of he random evens is addiive. On he oher hand, if i is believed ha he logarihm of sock prices are approximaely governed by a Lévy-Sable disribuion hen he accumulaion is muliplicaive. In he lieraure mos models have assumed ha log-prices, insead of reurns, follow a Lévy-Sable process. McCulloch [McC96] assumes ha asses are log Lévy-Sable and prices opions using a uiliy maximisaion argumen; more recenly Carr and

3 Wu [CW3] priced European opions when he log-sock price follows a maximally skewed Lévy-Sable process. Finally, based on Mandelbro and Taylor [Man97], Plaen, Hurs and Rachev [HPR99] provide a model o price European opions when reurns follow a (symmeric) Lévy-Sable process. In heir models he Brownian moion ha drives he sochasic shocks o he sock process is subordinaed o an inrinsic ime process ha represens operaional ime on which he marke operaes. Opion pricing can be done wihin he Black-Scholes framework and one can show ha he subordinaed Brownian moion is a symmeric Lévy-Sable moion. The moivaion of his paper is as follows. I is sandard o ake as a saring poin a model for he risk-neural evoluion of he asse price in he form ds S = r d + σ dw Q, where W Q is he underlying Brownian moion, r is he (consan) ineres rae and σ is he volailiy process; he case when σ is consan is he usual Black Scholes (BS) model. I is hen sandard o specify a sochasic process for σ, resuling in one of a number of sandard sochasic-volailiy models. When σ and W Q are independen for all T (as is ofen approximaely he case for FX markes), we have S T = S e r(t ) 1 R T σ s ds+r T σ s dw Q s, (1) and hen he value of a European vanilla opion wrien on he underlying sock price S is given by V (S, ) = E Q [ ( 1 V BS (S,, K, T σ sds) 1/, T )], () where he expeced value is wih respec o he random variable σ sds, he inegraed variance, under he risk-neural measure Q and V BS is he usual Black-Scholes value for a European opion. In general, he disribuion or characerisic funcion of he inegraed variance is no known, so evaluaing () is no sraighforward, alhough given he characerisic funcion of he inegraed variance we can use sandard ransform mehods o evaluae V (S, ) given by equaion (). 3

4 Nowihsanding hese difficulies, he inegraed variance is an imporan quaniy, represening a measure of he oal uncerainy in he evoluion of he asse price, and we use i as he saring poin for our model. We invesigae he properies of a wo-facor model in which he inegraed variance follows a Lévy-Sable process, while he shocks o he sock process are condiionally Gaussian, i.e. Brownian moion, wih a volailiy consisen wih he inegraed variance process. We hen show ha he resuling disribuion of he log-sock prices is Lévy-Sable. We also provide a characerisaion of he mos general possible model wihin our class of inegraed variance processes, which is an ineresing resul in is own righ. In addiion o pricing opions when he inegraed variance process and he sock process are independen (as above), we also show how o incorporae a leverage effec, resoring a degree of correlaion beween he wo. The paper is srucured as follows. Secion presens definiions and properies of Lévy-Sable processes. In paricular we show how symmeric Lévy-Sable random variables may be buil as a combinaion of wo independen Lévy-Sable random variables and define Lévy-Sable processes as in [ST94]. Secion 3 discusses he pah properies required o model inegraed variance as a oally skewed o he righ Lévy- Sable process. Secion 4 describes he dynamics of he sock process under boh he physical and risk-neural measure and shows how opion prices are calculaed when he sock reurns or log-sock process follows a Lévy-Sable process. Finally, Secion 5 shows numerical resuls and Secion 6 concludes. Lévy-Sable random variables and processes In his secion we show how o obain any symmeric Lévy-Sable process as a sochasic process whose innovaions are he produc of wo independen Lévy-Sable random variables. The only condiions we require (saed precisely in Proposiion ) are ha one of he independen random variables is symmeric and he oher is oally skewed o he righ. This is a simple, ye very imporan, resul since we can choose a Gaussian random variable as one of he building blocks ogeher wih any oher oally skewed random variable o produce symmeric Lévy-Sable random variables. 4

5 Furhermore, choosing a Gaussian random variable as one of he building blocks of a symmeric random variable will be very convenien since we will be able o reformulae any symmeric Lévy-Sable process as a condiional Brownian moion, condiioned on he oher building block, he oally skewed Lévy-Sable random variable, which in our case will be he model for inegraed variance..1 Lévy-Sable random variables The characerisic funcion of a Lévy-Sable random variable X is given by log E [ { e iθx] κ α θ α {1 iβsign(θ) an(απ/)} + imθ for α 1, Ψ(θ) = κ θ { 1 + iβ sign(θ) log θ } + imθ for α = 1, π where he parameer α (, ] is known as he sabiliy index; κ > is a scaling parameer; β [ 1, 1] is a skewness parameer and m is a locaion parameer [ST94]. If he random variable X has a Lévy-Sable disribuion wih parameers α, κ, β, m we wrie X S α (κ, β, m). I is sraighforward o see ha for he case < α 1 he random variable X does no have any momens, and for he case 1 < α < only he firs momen exiss (he case α = is Gaussian); however, fracional momens E [ X p ] do exis for p < α, see [ST94]. Moreover, given he asympoic behaviour of he ails of he disribuion of a Lévy-Sable random variable i can be shown ha he Laplace ransform E [ e τx] of X exiss only when is disribuion is oally skewed o he righ, ha is β = 1, which we sae in he following proposiion which we use laer. Proposiion 1. The Laplace Transform [ST94]. The Laplace ransform E [ e τx] wih τ of he Lévy-Sable variable X S α (κ, 1, ) wih < α and scale parameer κ > saisfies log E [ e τx] = (3) { κ α τ α sec απ for α 1, κ π τ log τ for α = 1. (4) The exisence of he Laplace ransform of a oally skewed o he righ Lévy-Sable random variable will enable us o show how o price opions as a weighed average 5

6 of he classical Black-Scholes price when he shocks o he sock process follow a Lévy-Sable process. Firs we see ha any symmeric Lévy-Sable random variable can be represened as he produc of a oally skewed wih a symmeric Lévy-Sable variable as shown by he following proposiion. Proposiion. Consrucing Symmeric Variables, (page in [ST94]). Le X S α (κ,, ), Y S α/α ((cos πα ) α α Then he random variable Z = Y 1/α X S α (κ,, ). α, 1, ), wih < α < α, be independen.. Lévy-Sable processes A sochasic process {L, T } is Lévy-Sable if all is finie-dimensional disribuions are Lévy-Sable. A paricular case of Lévy-Sable process, which will be denoed by {L α,β, }, is he Lévy-Sable moion [ST94]. Definiion 1. Lévy-Sable moion. A Lévy-Sable process L α,β is called a Lévy- Sable moion if L α,β = ; L α,β has independen incremens; and L α,β L α,β s ( S α ( s) 1/α, β, ) for any s < < and for some < α and 1 β 1 (ime-homogeneiy of he incremens). Observe ha when α = and β = i is Brownian moion, while when α < 1 and β = 1 (resp. β = 1) he process L α,β suppor on he negaive (resp. posiive) line. has The log-characerisic funcion of a Lévy-Sable moion L α,β is given by [ST94] log E[e iθl ] Ψ (θ) = { κ α θ α {1 iβsign(θ) an(απ/)} + imθ for α 1, κ θ { 1 + iβ π sign(θ) log θ } + imθ for α = 1. (5) Proposiion can be exended o processes; hence we may use Brownian moion as one of he building blocks o obain symmeric Lévy-Sable processes (see proposiion 3.8.1, page 143 in [ST94]). 6

7 3 Sochasic Volailiy wih Lévy-Sable Shocks In modelling inegraed variance as a building block here are wo properies ha inegraed variance Y,T = σ s ds should have: I should be coninuous and increasing in T ; I should be ime-consisen in ha Y,T = for all τ T. σ s ds = τ σ s ds + τ σ s ds = Y,τ + Y τ,t (6) As moivaed in he inroducion, we seek a model in which he shocks o he sock process are Lévy-Sable. If we assume ha he reurns process is given by ds S = µd + σ dw so ha S T = e where µ is a consan and dw R 1 µ(t ) T σsds+ R T σ sdw s, he incremen of Brownian moion, we migh be emped, based on Proposiion, o model volailiy by assuming ha he inegraed variance is given by Y,T = σ sds = dl α/,1 s. (7) Noe ha dl α/,1 is he incremen of a posiive Lévy-Sable moion (because α/ < 1 so ha (7) is an increasing process. This seems a reasonable choice since E [e iθ R ] T σ sdw s = e 1 sec(απ/4)(t ) θ α α/ hence he shocks o he process would be symmeric Lévy-Sable by Proposiion. Unforunaely his model for inegraed variance is inconsisen since on he lefhand side of (7) we have he inegraed variance σ sds which is, by consrucion, a coninuous process. However, on he righ-hand side, we have he nonnegaive Lévy- Sable moion dl α/,1 s which is by consrucion a purely disconinuous process. 7

8 Despie hese difficulies, we do no abandon he idea of inegraing agains a Lévy- Sable moion. Insead, we discuss a way of consrucing a process for he inegraed variance ha is Lévy-Sable bu wih coninuous pahs in T. If he purely disconinuous process dl α/,1 s can be modified o f(s, T )dl α/,1 s, for a suiable deerminisic funcion f(s, T ), he jumps can be damped and he resuling process made coninuous and increasing in T. Specifically, we require ha f(s, T ) > for s < T and ha f(s, T ) as s T, so he las jumps of he process ge smoohed ou. (For a general discussion of he pah behaviour of processes of he ype f(s, T )dl α/,1 s, and more general Lévy-Sable sochasic inegrals, see [ST94].) We now give condiions under which he sochasic inegral in he righ-hand side of equaion (8) below, given by f(s, T )dl α/,1 s, is coninuous in T, denoing he class of funcions f(s, T ) for which his is rue by F. Proposiion 3. Le f(s, T ) be coninuous in T wih f(t, T ) =, and assume in addiion ha, for each T, f(s, T )/ s := f 1 (s, T ) is coninuous on an inerval s < T <. Then he process X,T = T belonging o (s, T ]. f(s, T ) dl α/,1 s is coninuous in T for any Proof. Inegraing by pars [Pro9], and using f(t, T ) =, f(s, T )dl α/,1 s = f(, T )L α/,1 f 1 (s, T )L α/,1 s ds. The firs erm is coninuous in T by assumpion on f(, T ), as is fixed. Evaluaing he second erm a T + ɛ and T and subracing gives +ɛ f 1 (s, T + ɛ)l α/,1 s ds = f 1 (s, T )Ls α/,1 ds (f 1 (s, T + ɛ) f 1 (s, T )) L α/,1 s ds + Boh erms on he righ clearly end o zero wih ɛ. +ɛ T f 1 (s, T + ɛ)l α/,1 s ds. 8

9 Since we are ineresed in pricing opions where he underlying sochasic componen is driven by a symmeric Lévy-Sable process we would like o specify a kernel f(s, T ) so he finie-dimensional disribuion of inegraed variance is oally skewed o he righ Lévy-Sable. We propose as a model for inegraed variance Y,T = σ sds = h(, T )σ + f(s, T )dl α/,1 s (8) for suiable posiive funcions h(, T ) and f(s, T ). We assume ha f(t, T ) = for all o damp he Lévy-Sable jumps, and ha h(, ) = for consisency when T =, and for he same reason we also need o ake h(, T )/ T T = = 1; his is shown below. For < T (resp. s < T ) we require ha h(, T ) > (resp. f(s, T ) > o ensure ha Y,T is sricly posiive and properly random. Furher condiions on f and h which specify heir general form are given in Proposiion 4 below. For example, in our model we may choose h(, T ) = 1 ( ) 1 e γ(t ) γ and f(s, T ) = 1 ( ) 1 e γ(t s), γ (9) for γ > in (8) o obain, as a paricular case, he OU-ype model for inegraed variance firs inroduced by Barndorff-Nielsen and Shephard [BNS1] where he incremens in (8) are driven by a general non-negaive Lévy process L. (Noe, however, ha in general, he funcions h(, T ) and f(s, T ) do no depend only on he lag T (resp. T s) as one migh expec. Their mos general form is given below.) Before proceeding, we noe an imporan poin concerning unis. The inegraed variance is dimensionless (ha is, as a pure number i has no unis). Hence he funcion h(, T ) mus have he dimensions of ime, and since he Lévy process L α/,1 scales as ime o he power /α, he funcion f(s, T ) mus have dimensions of ime o he power /α. This disincion only maers, of course, if we change he uni of ime: in (9), f(s, T ) conains an implici dimensional consan, equal o 1 in he ime unis of he model, o make he dimensions correc. Proposiion 4. Suppose ha he funcions f(s, T ) and h(, T ) are wice differeniable in heir second argumen and once differeniable in heir firs argumen, wih f(s, T ) > for all s < T, while f(t, T ) =, and h(, T ) > for all < T, while h(, ) =. Then he process Y,T = σ sds = h(, T )σ + 9 f(s, T )dl α/,1 s (1)

10 is non-negaive, coninuous and increasing in T, and saisfies he consisency condiion Y,T = Y,τ + Y τ,t if and only if f(s, T ) and h(, T ) are non-negaive and ake he form h(, T ) = H(T ) H(), f(s, T ) = F (s) (H(T ) H(s)) (11) H () where H( ) is a sricly monoonic, differeniable funcion wih derivaive H, and F ( ) is coninuous and posiive (resp. negaive) if H( ) is increasing (resp. decreasing). Proof. We use subscrips 1 (resp. ) on h(, ) and f(, ) o denoe differeniaion wih respec o (wr) he firs (resp. second) argumen, wih an obvious exension o higher derivaives. Suppose ha, for τ >, τ σ sds = h(, τ)σ + τ f(s, τ)dl s, (1) where L denoes a non-negaive Lévy process (including L α/,1 This is clearly s posiive process wih our assumpions. as a special case). Differeniaing wr τ and using f(τ, τ) =, στ = h (, τ)σ + Noe ha his immediaely implies ha τ h (, ) = 1, as saed above. f (s, τ)dl s. (13) Since τ σ s ds = h(τ, T )σ τ + τ f(s, T )dl s, (14) 1

11 we have τ σ sds + τ σsds = h(, τ)σ + h(τ, T )στ + f(s, τ)dl s + f(s, T )dl s τ ( τ ) = h(, τ)σ + h(τ, T ) h (, τ)σ + f (s, τ)dl s τ τ + f(s, τ)dl s + f(s, T )dl s τ = (h(, τ) + h(τ, T )h (, τ)) σ + τ (f(s, τ) + h(τ, T )f (s, τ)) dl s + τ f(s, T )dl s. Wriing he lef-hand side as σ s ds, using (1) and noing ha he pah is arbirary, he consisency condiion (6) is me if and only if for all s, τ (, T ). h(, T ) = h(, τ) + h(τ, T )h (, τ), (15) f(s, T ) = f(s, τ) + h(τ, T )f (s, τ) (16) We characerise f and h from he funcional equaions (15) and (16) by a separaion of variables echnique, beginning wih h. Firs differeniae (15) wr τ o give which is rearranged o = h (, τ) + h 1 (τ, T )h (, τ) + h(τ, T )h (, τ), h (, τ) h (, τ) = 1 + h 1(τ, T ). h(τ, T ) The lef-hand side of his equaion is a funcion of and τ, he righ-hand side is a funcion of τ and T, so boh mus be equal o an arbirary funcion of τ alone. Seing he lef-hand side equal o his funcion, we have an ordinary differenial equaion in τ for h(, τ), whose mos general soluion saisfying h(, ) = and h (, ) = 1 is indeed h(, τ) = H(τ) H(), (17) H () for an arbirary non-consan funcion H( ). (The same resul can be obained by differeniaing (15) wih respec o T wice.) 11

12 As h(, τ) > and is bounded, a simple argumen by conradicion shows ha, for each τ, H() H(τ) eiher increases or decreases as τ increases; i canno have a urning poin and H( ) is herefore monoonic. o Conversely, direc subsiuion shows ha (17) saisfies (15). The proof for f is similar: differeniaion of (16) wr τ and rearrangemen leads f (s, τ) f (s, τ) = 1 + h 1(τ, T ) h(τ, T ) from which boh sides are equal o an arbirary funcion of τ; solving he resuling ordinary differenial equaion in τ for f(s, τ), wih he condiion f(s, s) =, shows ha f(s, τ) = F (s) (G(τ) G(s)) for arbirary F ( ) and G( ), he laer being differeniable. Subsiuion back ino (16) shows ha G( ) = H( ) as required. The sign of F ( ) clearly follows from (11) given ha h is monoonic. The converse is shown by direc subsiuion. Two possible choices for f(s, T ) and h(, T ) are 1 f(s, T ) = T s and h(, T ) = T, s, T, (18) f(s, T ) = 1 e γ((t +c)n (s+c) n ) γn(s + c) n 1 and h(, T ) = 1 e γ((t +c)n (+c)n) γn( + c) n 1, (19) for s, T and 1 n < where γ is a posiive consan ha can be seen as a damping facor which we can choose freely, and c is consan. Boh choices saisfy he γ(t +c)n addiiviy condiion (6); for example (19) is obained by assuming H(T ) = e and F (s) = 1/H (s) in Proposiion 4. ha Henceforh we ake H( ) > wihou loss of generaliy, and we furher assume 1 ds <, for T <, () H (s) 1 Alhough hese funcions are apparenly he same, as remarked above, here is a dimensional consan muliplying hem which would change if he ime unis were changed. 1

13 which is a condiion we will require below o price insrumens under he risk-neural measure. I simply amouns o saying ha H () >, namely ha he ime is no special (recall ha H( ) canno have urning poins for > ). 3.1 Illusraion We now illusrae he differen building blocks needed o obain he inegraed variance process described above. Firs we simulae a oally skewed o he righ Lévy- Sable moion; hen we ge he spo variance process, by choosing an appropriae kernel; hen we produce he inegraed variance process. We focus on kernels of he inegraed variance of he form (19). The solid line in he wo boom graphs of Figure 1 represens he case wih n = 1, c =.1, =, T 1, σ =, γ = 5, which is a sandard OU-ype process as in [BNS1] wih a wo-week mean-reversion period. In he same figure he doed lines represen he case n = 1., T = 1 and γ = 5. 4 Model dynamics and opion prices We now urn o models of he asse price evoluion and he pricing of vanilla opions. Secion 4.1 looks a a basic model where he shocks o he reurns or log-sock process are symmeric; Secion 4. exends i o a model where shocks can also be asymmeric. Finally, Secion 4.3 shows how o price vanilla opions when he shocks o he underlying sock process follow a Lévy-Sable process for α > 1 and 1 β 1. Given he naure of he model, here is no unique equivalen maringale measure (EMM). In line wih mos of he Lévy process lieraure we choose an EMM ha is srucure-preserving since, among oher feaures (see [CT4]), ransform mehods for pricing are sraighforward o implemen; his is discussed a he end of Secion

14 6 Levy Sable Moion, α =.8, β = Spo variance 1.5 σ 1.5 Figure 1: Simulaed inegraed variance wih kernel f(s, T ) = ( 1 e 5((T +c)n (s+c) n ) ) wih c =.1, n = 1, T = 1, solid line, and c =.1, σ s ds..1 Inegraed variance n = 1., T = 1, doed line. In boh cases = and σ =. 4.1 Modelling reurns Time As poined ou in he inroducion we can model eiher reurns or log-sock-prices; when shocks are symmeric we can ake eiher roue. For example, if we believe ha he shocks o he reurns process follow a Lévy-Sable disribuion, we assume ha in he physical measure P ds = µd + σ dw (1) S σ sds = h(, )σ + f(s, )dl α/,1 s, () where dw denoes he incremen of he sandard Brownian moion, h(, ) and f(, ) saisfy he condiions in Proposiion 4, f(, ) F and µ is a consan. In he following proposiion we show he disribuion of he sock process. 14

15 Proposiion 5. Le he sock process follow (1) and he inegraed variance process follow (). Assume furher ha W and L α/,1 are independen, hen he log-sock process (1) is he sum of wo independen processes: a symmeric Lévy-Sable process and a Gaussian process. Proof. Firs noe ha he sochasic componen of he log-sock process is given by U, = σ s dw s. (3) Now we calculae he characerisic funcion of he random process U,. We have E[e iθu, ] = E [e iθ R ] σsdws ; and by independence of σ and W, σ s dw s is a zero-mean Normal variable whose variance is he random variable σ s ds. Thus he characerisic funcion of σ s dw s is given by E[e iθu, ] = [ E exp [ 1 θ = exp ]] σsds [ 1 ] [ h(, )σθ E exp [ 1 ]] θ f(s, )dl α/,1 s. Furher, using (5) we have ha ( ( f(s, )dlα/,1 s S α/ f(s, )α/ ds and using Proposiion 1 we wrie E[e iθu, ] = exp [ 1 ] h(, )σθ [ = exp [ E exp [ 1 θ 1 h(, )σ θ 1 απ sec α/ 4 ]] f(s, )dl α/,1 s f(s, ) α/ ds θ α ]. ) ) /α, 1,, This is clearly he characerisic funcion of he sum of a Gaussian process and an independen symmeric Lévy-Sable process wih index α. Noe ha we migh also sipulae ha our deparure poin is he risk-neural dynamics for he sock process and ha our model is as above wih µ replaced wih r: ds S = rd + σ dw Q (4) 15

16 wih σ sds as in (), dw Q being he incremens of he sandard Brownian moion. However, we need no specify he risk-neural dynamics as a saring poin since i is possible o posulae he physical dynamics and hen choose an EMM. We discuss he relaion beween he measures P and Q below for a model ha also allows for asymmeric Lévy-Sable shocks and he symmeric case hen becomes a paricular case. We also noe ha he sochasic inegral σ s dw s can be seen as a ime-changed Brownian moion [KS]. In his case he inegraed variance σ sds represens he ime-change and i is sraighforward o show ha σ s dw s d = W ˆT,T where ˆT,T = σ sds. 4. Modelling Log-Sock Prices Financial daa suggess ha reurns are skewed raher han symmeric, see for example [KL76], [CLM97], [CW3]. For insance, since he sock marke crash of 1987, he US sock index opions marke has shown a pronounced skewed implied volailiy (volailiy smirk) which indicaes ha, under he risk-neural measure, log-reurns have a negaively skewed disribuion. The symmeric model above can be exended o allow he dynamics of he logsock process o follow an asymmeric Lévy-Sable process. In sochasic volailiy models one way o inroduce skewness in he log-sock process is o correlae he random shocks of he volailiy process o he shocks of he sock process. I is ypical in he lieraure o assume ha he Brownian moion of he sock process, say dw, is correlaed wih he Brownian moion of he volailiy process, say dz. Thus dw dz = ρd and we can wrie Z = ρw + 1 ρ Z, where Z is independen of W. The correlaion parameer ρ is also known in he lieraure as he leverage effec and empirical sudies sugges ha ρ < [FPS]. In our case he noion of correlaion does no apply because for Lévy-Sable random variables, as given ha 16

17 momens of second and higher order do no exis, nor do correlaions. However, we may also include a leverage effec via a parameer l o produce skewness in he sock reurns. Hence o allow for asymmeric Lévy-Sable shocks, under he physical measure we assume ha log(s T /S ) = µ(t ) + σ sds = h(, T )σ + σ s dw s + l σ d L α, 1 s (5) f(s, T )dl α/,1 s. (6) Here dw denoes he incremen of he sandard Brownian moion independen of boh d L α, 1 and dl α/,1 and we noe ha d L α, 1, independen of dl α/,1, is oally skewed o he lef and ha 1 α <. Moreover, µ and σ are consans, f(, T ) and h(, T ) saisfy he condiions in Proposiion 4 wih f(, T ) F and he leverage parameer l. In appendix A we show ha he shocks o he price process are asymmeric Lévy-Sable. Before proceeding we discuss he connecion in his model beween he dynamics of he sock price under he physical measure P and he risk-neural measure Q. Recall ha a probabiliy measure Q is called an EMM if i is equivalen o he physical probabiliy P and he discouned price process is a maringale. I is sraighforward o see ha in he model proposed here he se of EMMs is no unique, hence we mus moivae he choice of a paricular EMM. Le us focus on he model wih no leverage (i.e. l = ). Based on Girsanov s heorem (see [KS88]), we assume ha he risk-neural dynamics of he model are obained via he Radon-Nikodym derivaive R (r µ Z = e 1 σ s) 1 σs dw s 1 R (r µ 1 σ s) 1 σs ds. (7) To be able o apply Girsanov s heorem we need o check wo condiions. 3 Firs, Noe ha here we model log-sock prices since we canno include a similar leverage effec in equaion (1) because his allows negaive prices due o he jumps of he incremens of he Lévy- Sable moion d L α, 1. 3 See secion 3.5 in [KS88]. 17

18 we mus verify ha [ ( P r µ 1 ) ] 1 σ s ds < = 1 for T <, (8) σs and second, ha Z is a maringale and E[Z ] = 1. (9) [ ] T Since r µ is a consan he firs condiion is saisfied if P σ sds < = 1 [ ] T 1 and P ds < = 1 for T <. To show he firs, noe ha X σs,t := ( ( ) ) /α T T f(s, T )dlα/,1 s S α/ f(s, T )α/ ds, 1, ; herefore P [X,T < ] = 1 for all T because he cdf of X(T ) inegraes o 1. To show he second, we use (13) and () o show ha 1 ds is bounded above: σs hus P [ 1 σ s ds 1 σ = H () σ 1 h (, s) ds ] 1 ds < = 1 for T <. σs 1 ds < for T < ; H (s) To verify he maringale condiion i is sraighforward o check, using he independence beween L α/,1 and W, ha [ [ R (r µ E[Z ] = E E e 1 σ s) 1 σs dw s 1 R ]] (r µ 1 σ s) 1 σs ds σ s, s = 1. Moreover i is simple o calculae E[Z F u ] = Z u (for < u ) and using he Radon-Nikodym derivaive, E [S Z ] = S e r. Therefore, by Girsanov s heorem, W Q = W ( r µ 1 ) 1 σ s ds, σ s and he risk-neural dynamics of he sock, wih l =, saisfy ds S σ sds = 1 λ = rd + σ dw Q ( 1 e λ(t ) ) σ ( ) 1 e λ(t ) dl s. λ

19 The inclusion of he leverage is sraighforward in his seing, hence he riskneural dynamics of he model (5) and (6) follows log(s T /S ) = r(t ) 1 where W Q ˆL α, 1 σ sds = h(, T )σ + and L α/,1 σsds + l α σ α sec απ (T ) + σ s dws Q + l σ dˆl α, 1 s (3) f(s, T )dl α/,1 s (31) is he sandard Brownian moion independen of he Lévy-Sable moions (also independen from each oher) and r is he (consan) risk-free rae. This is he mos general model ha we consider; noe ha if l = we obain he risk-neural dynamics for he case when he reurns or log-sock process follows a symmeric Lévy-Sable process under P. 4.3 Opion Pricing wih Lévy-Sable Volailiy As moivaed in he inroducion by equaions (1) and (), he price of a vanilla opion, using he EMM Q, is given by he ieraed expecaions ( [E [E [V Qσ Q BS S e l σ R )] T dˆl α, 1 s,, K, Y,T, T V (S, ) = E QˆLα, 1 ]] ˆL α, 1, σ ˆL α, 1,(3) where Y,T = 1 T σ T sds and V BS is he Black-Scholes value for a European opion. Noe ha if we le h(, T ) = f(, T ) = for all, l = 1 and 1 < α < hen he model reduces o log(s T /S ) = µ(t ) + σ dˆl α, 1 s, which is he Finie Momen Log-Sable (FMLS) model of [CW3]. Proposiion 6. I is possible o exend he resuls above o price European call and pu opions when he skewness coefficien β [, 1]. Proof. Using pu-call inversion [McC96], we have by no-arbirage ha European call 19

20 and pu opions are relaed by 4 C(S, ; K, T, α, β) = SKP (S 1, ; K 1, T, α, β). As an example, we can use he approach above o derive closed-form soluions for opion prices when he random shocks o he price process are disribued according o a Cauchy Lévy-Sable process, α = 1 and β = in (3), (31), so ha opion prices are given by V (S, ) = f(s, T ) 1/ ds ) 1 V BS (S,, K, Y,T, T e 3/ y R T f(s,t ) 1/ «ds /y T dy, (T ) π where Y,T = 1 T σ T sds. To see his, firs we noe ha he combinaion of a Gaussian random variable, he Brownian moion in (3), and a Lévy-Smirnov S 1/ (κ, 1, ) random variable, he process followed by he inegraed variance in (31), resuls in a Cauchy random variable S 1 (κ,, ). This can be seen by calculaing he convoluion of heir respecive pdf s. Now, recall ha he pdf for a Lévy-Smirnov random variable S 1/ (κ, 1, ) is given by (κ/π) 1/ x 3/ e κ/x wih suppor (, ); hence he disribuion of he average inegraed variance is given by Y,T 1 ( T ( f(s, T )dl α/,1 1 T ) s S 1/ f(s, T ) ds) 1/, 1,, T (T ) and he value of he opion is as required. 5 Numerical illusraion: Lévy-Sable Opion Prices In his secion we show how vanilla opion prices can be calculaed according o he above derivaions. One roue is o calculae he expeced value of he Black- Scholes formula weighed by he sochasic volailiy componen and he leverage 4 Noe ha using pu-call inversion allows us o obain pu prices when he log-sock price follows a posiively skewed Lévy-Sable process, based on call prices where he underlying log-sock price follows a negaively skewed Lévy-Sable process. Furhermore, pu-call-pariy allows us o obain call prices when he skewness parameer 1 β.

21 effec. Anoher roue o price vanilla opions for sock prices ha follow a geomeric Lévy-Sable processes is o compue he opion value as an inegral in Fourier space, using Complex Fourier Transform echniques [Lew1], [CM99]. We use he Black-Scholes model as a benchmark o compare he opion prices obained when he reurns follow a Lévy-Sable process. Our resuls are consisen wih he findings in [HW87] where he Black-Scholes model underprices in- and ouof-he-money call opion prices and overprices a-he-money opions. 5.1 Opion Prices for Symmeric Lévy-Sable log-sock Prices We firs obain opion prices and implied volailiies when he log-sock prices follow a symmeric Lévy-Sable process. Recall ha, under he risk-neural measure Q, and assuming, for simpliciy, ha σ =, he sock price and variance process are given by S T = S e r(t ) 1 R T σ sds+ R T σ sdw Q s, σ sds = f(s, T )dl α/,1 s. The firs sep we ake is o calculae he characerisic funcion of he process Z,T = 1 σ sds + σ s dw Q s. Proposiion 7. The characerisic funcion of Z,T is given by [ E Q [e iξz,t ] = exp 1 ( απ ) (iξ sec ) ] + ξ α/ f(s, T ) /α ds, (33) α/ 4 where ξ = ξ r + iξ i and 1 ξ i. Moreover, he characerisic funcion is analyic in he srip 1 < ξ i <. 1

22 Proof. The characerisic funcion is given by E [ [ ] Q e iξz,t = E Q [E Q exp [ 1 iξ σsds + iξ σ s dws Q = E [exp [ Q 1 iξ σsds 1 ]] ξ σsds [ = E [exp Q 1 ( ) ]] iξ + ξ f(s, T )dl α/,1 s [ = exp 1 ( απ sec α/ 4 ) (iξ + ξ ) α/ ] ]] σs, s ] f(s, T ) α/ ds. The las sep is possible since he expeced value exiss if ξ is resriced so ha ξ r ξ i + ξ i, by consideraion of he penulimae line. The region where his is rue conains he srip 1 ξ i. Finally, i is sraighforward o observe ha he characerisic funcion is analyic in his srip. To price call opions we use he Fourier inversion formula: C(x, ) = e x 1 π e r(t ) K iξ i + iξ i e iξx K iξ ξ iξ e(t )Ψ( ξ) dξ (34) where x = log S, < ξ i < 1, and Ψ(ξ) is he characerisic funcion of he process log S T. In comparing hese prices wih Black-Scholes prices, we have o decide how o choose he relevan parameers of he wo models. In fac, he only parameer ha we mus examine carefully is he scaling parameer of he Lévy-Sable process; we op for one ha can be relaed o he sandard deviaion used when he classical Black- Scholes model is used. One approach, as in [HPR99], is o mach a given percenile of he Normal and a symmeric Lévy-Sable disribuion. For example, if we wan o mach he firs and hird quarile of a Brownian moion wih sandard deviaion σ BS =. o a symmeric Lévy-Sable moion κdl α, wih characerisic exponen α = 1.7, we would require he scaling parameer κ =.141. We have chosen hese parameers so ha for opions wih 3 monhs o expiry hese quariles mach. ( Moreover, in he examples below, we use he kernel f(s, T ) = ) e 5(T s),

23 monh 3 monhs 6 monhs Srike Price Figure : Difference beween Lévy-Sable and Black-Scholes call opion prices for differen expiry daes: one, hree and six monhs. In he Black-Scholes annual volailiy is σ BS =. and α = 1.7. which is as in (19) wih n = 1, where for illusraive purposes we have assumed mean-reversion over a wo week period, i.e. γ = 5. Figure shows he difference beween European call opions when he sock reurns are disribued according o a symmeric Lévy-Sable moion wih α = 1.7 and when reurns follow a Brownian moion wih annual volailiy σ BS =.. For ou-ofhe-money call opions he Lévy-Sable call prices are higher han he Black-Scholes and for a-he-money opions Black-Scholes delivers higher prices. These resuls are a direc consequence of he heavier ails under he Lévy-Sable case. 5. Opion Prices for Asymmeric Lévy-Sable log-sock Prices We now obain opion prices and implied volailiies when here is a negaive leverage effec, i.e. log-sock prices follow an asymmeric Lévy-Sable process. Recall ha, 3

24 monh 3 monhs 6 monhs 3 Implied Volailiy (%) Srike Price Figure 3: Black-Scholes implied volailiy for he Lévy-Sable call opion prices when reurns follow a symmeric Lévy-Sable moion wih α = 1.7, β = and hree expiry daes: one, hree and six monhs. under he risk-neural measure Q, he sock price and variance process are given by S T = S e r(t ) 1 σ sds = R T f(s, T )dl α/,1 s. where for simpliciy we have assumed σ = in (6). σs ds+(t )lα σ α sec απ +R T σ s dws Q +l σ R T dˆl α, 1 s, We proceed as above and calculae he characerisic funcion of he process Z l,t = 1 σ sds + σ s dw Q s + l σ dˆl α, 1 s. Proposiion 8. The characerisic funcion of Z l,t is given by [ E Q [e Zl,T ] = exp 1 ( απ ) (iξ sec ) + ξ α/ f(s, T ) α/ ds + (T )(iξl σ) α sec πα α/ 4 where 1 ξ i, ξ = ξ r + iξ r, and is analyic in he srip 1 < ξ i <. ],(35) 4

25 Proof. The proof is very similar o he one above. I suffices o noe ha for ξ i [ E Q e iξ R ] [ T dˆl α, 1 s E Q e iξ R ] T dˆl α, 1 s = E [e Q ξ R T ] i dˆl α, 1 s <. [ Moreover, for ξ i < we have ha E Q e iξ R ] T α, 1 d ˆL s is analyic, i.e. d [ dξ EQ e iξ R T ] dˆl α, 1 s = [i EQ <. dˆl s α, 1 e iξ R T ] dˆl α, 1 s Puing hese resuls ogeher wih he resuls from Proposiion 7 we ge he desired resul. The requiremen 1 < ξ i < arises because dˆl α, 1 lef, so we need ξ i >. is oally skewed o he We use he same f(s, T ) as above and include a leverage parameer l = 1 and σ =.15 so ha reurns follow a negaively skewed process wih β(, T ) =.5 when here is 3 monhs o expiry. Figure 4 shows he difference beween Lévy-Sable and Black-Scholes call opion prices for differen expiry daes. In he Black-Scholes case annual volailiy is σ BS =.. Finally, Figure 5 shows he corresponding implied volailiy. The negaive skewness inroduced produces a hump for call prices wih srike below 1. This is financially inuiive since relaive o he Black-Scholes he risk-neural probabiliy of he call opion ending ou-of-he-money is subsanially higher in he Lévy-Sable case. 6 Conclusion The GCLT provides a very srong heoreical foundaion o argue ha he limiing disribuion of sock reurns or log-sock prices follows a Lévy-Sable process. We 5

26 1.8 1 monh 3 monhs 6 monhs Srike Price Figure 4: Difference beween Lévy-Sable and Black-Scholes call opion prices for differen expiry daes: one, hree and six monhs. In he Black-Scholes annual volailiy is σ BS =., α = 1.7 and σ =.15. have shown how o model sock reurns and log-sock prices where he sochasic componen is Lévy-Sable disribued covering he whole range of skewness β [ 1, 1]. We showed ha European-syle opion prices are sraighforward o calculae using ransform mehods and we compare hem o Black-Scholes prices where we obain he expeced volailiy smile. 6

27 monh 3 monhs 6 monhs 3 Implied Volailiy (%) Srike Price Figure 5: Black-Scholes implied volailiy for he Lévy-Sable call opion prices when reurns follow a symmeric Lévy-Sable moion wih α = 1.7 and σ =.15 and hree expiry daes: one, hree and six monhs. A Appendix Suppose ha he sock process, as assumed above in secion 4., follows log(s T /S ) = µ(t ) + σ sds = h(, T )σ + σ s dw s + l σ f(s, T )dl α/,1 s, d L α, 1 s under P where dw denoes he incremen of he sandard Brownian moion independen of boh d L α, 1 and dl α/,1. Then i is sraighforward o verify ha he shocks o he above log-sock process under he measure P are he sum of wo independen processes: hose of a Gaussian componen and hose of a Lévy-Sable process wih negaive skewness β ( 1, ]. Le G(, T ) = f(s, T ) α/ ds and, for simpliciy in he calculaions, assume ha σ = (so we focus only on he asymmeric Lévy process). Now consider he process U l,t = σ s dw s + l σ d L α, 1 s. 7

28 The log-characerisic funcion of U,T l is given by [ ] log E e iθu,t l [ [ ( )]] = log E exp iθ σ s dw s + l σ d L α, 1 s = 1 ) { G(T, ) θ α + ( σl) α (T ) θ α = ( απ sec ( α/ 4 1 ( απ sec α/ 4 { 1 1 sec ( απ α/ 4 ) G(, T ) + (T )l α σ α ) θ α 1 + isign(θ) an (T )l ) α σ α ( απ ) } isign(θ) an G(, T ) + (T )lα σ α This is obviously he characerisic funcion of a skewed Lévy-Sable process wih (ime-dependen) skewness parameer (T )l α σ α β(, T ) = 1 sec ( ) ( 1, ]. απ α/ 4 G(, T ) + (T )lα σ α Moreover, when l = we obain β = and β 1 as l. Noe ha he inegraed variance does no have a finie firs momen since α/ < 1. However, in he case of he leverage effec E[ d L α, 1 s ] < since 1 < α <. d L α, 1 s is firs momen exiss, i.e. ( απ )} 8

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