EQUIVALENCE OF FLOATING AND FIXED STIKE ASIAN AND LOOKBACK OPTIONS ENST EBELEIN AND ANTONIS PAPAPANTOLEON Abstract. We prove a symmetry relatioship betwee floatig-strike ad fixed-strike Asia optios for assets drive by geeral Lévy processes usig a chage of uméraire ad the characteristic triplet of the dual process. We apply the same techique to prove a similar relatioship betwee floatig-strike ad fixed-strike lookback optios. 1. Itroductio The aim of this paper is to prove a useful symmetry betwee floatig ad fixed strike Asia ad floatig ad fixed strike lookback optios for Lévy drive assets. We exted the results of Hederso ad Wojakowski (22 i two directios; firstly, by cosiderig a geeral Lévy process as the drivig process of the uderlyig ad secodly by applyig the same techique to prove a symmetry result for lookback optios. A chage of uméraire ad a represetatio for the characteristic triplet of the dual of a Lévy process are the mai tools used for the proof. Moreover, statioarity of the icremets plays a crucial role. Lévy processes have attracted much iterest for fiacial applicatios lately, sice they exhibit certai features of the market that diffusio models caot capture, both i the real ad i the risk-eutral world. For the merits of Lévy modellig i fiace ad some further applicatios see Eberlei (21 ad refereces therei. Carr et al. (22 also provide empirical evidece supportig the use of pure jump Lévy processes for fiacial modellig. There are several symmetry results kow i optio pricig theory, relatig various types of optios ad payoffs. These results become more importat whe cosiderig exotic optios, sice closed-form solutios might ot exist for certai payoffs. This is the case for Asia optios, although there are more results available for the fixed tha the floatig strike case. Symmetries are eve more importat whe we depart from Browia motio models ad cosider more geeral drivig processes, such as Lévy processes. I that case, closed form solutios are ot available for several types of payoffs, such as lookback or Asia optios, or are available oly for oe type of payoff, most probably fixed strike optios, as i Borovkov ad Novikov (22 or Behamou (22. Key words ad phrases. Asia optios, lookback optios, Lévy processes, chage of uméraire, statioary icremets, symmetry. The secod amed author ackowledges the fiacial support provided through the Europea Commuity s Huma Potetial Programme uder cotract HPN-CT-2-1 DYNSTOCH. 1
2 ENST EBELEIN AND ANTONIS PAPAPANTOLEON Moreover, the cost i computatioal time of calculatig a joit desity could be sigificatly higher tha that of a sigle oe. Hece, we aim to uify the treatmet of floatig ad fixed strike Asia ad lookback optios i order to be able to trasfer kowledge from oe case to the other. The results of Hederso ad Wojakowski have bee geeralized to forwardstart Asia optios ad discrete averagig by Vamaele et al. (26. Similar symmetry results have bee obtaied by Hooglad ad Neuma (2 usig local scale ivariace. The same chage of uméraire has bee applied to Asia optios by Večeř (22 for a Browia motio model ad Večeř ad Xu (24 for a special semimartigale model, i order to obtai a oe-dimesioal PDE (PIDE respectively for both fixed ad floatig strike Asia optios. Nielse ad Sadma (1999 describe a aalogous chage of uméraire for a Browia motio model with stochastic iterest rates. This chage of uméraire has also bee applied by Adrease (1998 to derive oe dimesioal PDEs for floatig ad fixed strike lookback optios. 2. Model ad Payoffs Let (Ω, F, F, IP be a complete stochastic basis, i.e. the filtratio F (F t t + satisfies the usual coditios. We model the asset price process as a expoetial Lévy process S t S exp L t (2.1 where the Lévy process L satisfies Assumptio (M, which is give below. I that case, L is a special semimartigale ad has the caoical decompositio (cf. Jacod ad Shiryaev 23, II.2.38 t L t bt + σw t + x(µ L ν L (ds, dx (2.2 where the drift term b equals the expectatio of L 1 ad ca be writte as b r δ σ2 2 (e x 1 xλ(dx. (2.3 Here r is the (domestic risk-free iterest rate, δ the cotiuous divided yield (or foreig iterest rate, σ the diffusio coefficiet ad W a stadard Browia motio uder IP. µ L is the radom measure of jumps of the process L ad ν L (dt, dx λ(dxdt is the compesator of the jump measure µ L, where λ is the Lévy measure of L 1. The Lévy process L has the Lévy triplet (b, σ 2, λ. We assume that IP is a risk eutral measure, i.e. the asset price has mea rate of retur µ r δ ad the auxiliary process Ŝt e δt S t, oce discouted, is a martigale uder IP. I geeral, markets modelled by expoetial Lévy processes, as defied i (2.1 (2.3, are icomplete ad there exists a large class of risk eutral measures. We refer to Eberlei ad Jacod (1997 for a characterizatio of the class of equivalet martigale measures. Moreover, ote that fiiteess of IE[Ŝt] is esured by Assumptio (M.
FLOATING AND FIXED STIKE ASIAN AND LOOKBACK OPTIONS 3 Assumptio (M. The Lévy measure λ of the distributio of L 1 is assumed to satisfy the followig itegrability coditios: x λ(dx < ad xe x λ(dx <. {x< 1} {x>1} emark 2.1 (Assumptio (M. For large x ad ε >, we have that x < e εx hece, we could merge the above assumptios ad use the followig stroger coditio: Assume there exists a costat M > 1, such that exp(uxλ(dx <, u < M; (2.4 { x >1} sice g(x exp(x is submultiplicative, we see that usig Theorem 25.3 i Sato (1999, (2.4 is equivalet to IE [exp(ul 1 ] <, u < M. We deote by λ the (o-egative Lévy measure defied by λ([a, b] : λ([ b, a] (2.5 for a, b, a < b. Thus, λ is the mirror image of the origial measure with respect to the vertical axis. Wheever we use the symbol i frot of a Lévy measure, we will refer to the Lévy measure defied as above. Next we provide a useful lemma, which describes the characteristic triplet of the dual of a Lévy process i terms of the characteristic triplet of the origial process. Lemma 2.2 (dual characteristics. Let L be a Lévy process with Lévy triplet (b, c, λ. The L : L is agai a Lévy process with Lévy triplet (b, c, λ, where b b, c c ad λ λ. Proof. From the Lévy-Khitchie represetatio we kow that ϕ Lt (u IE [ e iult] { ( exp t ibu c 2 u2 + (e iux 1 iuxλ(dx }. We get immediately ϕ Lt (u ϕ Lt ( u { ( exp t ib( u c 2 u2 + { ( exp t i( bu c 2 u2 + } (e i( ux 1 i( uxλ(dx (e iu( x 1 iu( xλ(dx }. Hece, we ca coclude that L is also a Lévy process ad has characteristics b b, c c ad λ λ. Defie as Σ D the arithmetic average of the process S either cotiuously or discretely observed durig a time iterval of legth D. More precisely, let T 1 < T 2 < < T be equidistat time poits such that D T T 1 ; the, for a cotiuously observed process we defie Σ D 1 D S u du, whereas for a discretely observed process we defie istead Σ D 1 i. T T 1
4 ENST EBELEIN AND ANTONIS PAPAPANTOLEON Optio type Asia Payoff Lookback payoff Fixed Strike call (Σ D K + (M D K + Fixed Strike put (K Σ D + (K N D + Floatig Strike call (θ Σ D + (θ N D + Floatig Strike put (Σ D θ + (M D θ + Table 2.1. Types of payoffs for Asia ad Lookback optios Similarly, defie N D ad M D to be the miimum ad maximum of the process S either cotiuously or discretely observed durig a time iterval of legth D, that is, i the discrete case we defie N D mi T1 T i T i ad M D max T1 T i T i. There exist fixed ad floatig strike Asia ad lookback optios. The payoff of fixed strike optios depeds o the differece betwee a average or a extreme value of the uderlyig ad a fixed strike. The payoff of the floatig strike optio depeds o the differece betwee a average or a extreme value of the uderlyig ad the value of the asset at maturity; hece, i that case, the average or the extreme value plays the role of the strike. More geerally, oe ca cosider θ istead of the value at maturity, for some costat θ +. The differet types of payoffs of the Asia ad lookback optio are summarized i Table 2.1, where x + max{x, }. Accordig to the duratio of the averagig period, there exist differet variats of the Asia optio. Let [, T ] be a time iterval, where the optio starts at time t [, T ad matures at time T. If the averagig starts at time T + > t, we have a forward-start optio, at time t we have the stadard optio ad if the averagig starts at time T < t we have a i-progress Asia optio. I-progress Asia optios ca be re-writte i terms of stadard Asia optios, see Večeř (22, but floatig strike optios become a mixture of floatig ad fixed strike Asia optios, because of the additioal term that correspods to the averagig up to time t. Here, we cocetrate o forward-start Asia optios, treatig stadard optios as a special case of them. A variat of floatig strike lookback optios are partial lookback optios, with payoff ( αn D + ad (αm D + for call ad put optios respectively, where the costat α 1, respectively α (, 1], deotes the degree of partiality. By arbitrage argumets, the price of a optio is equal to its discouted expected payoff uder a equivalet martigale measure. We itroduce the followig otatio for the floatig strike forward-start Asia call optio V c (θ, Σ D, r; b, σ 2, λ; T 1, T V flc e rt IE [ (θ Σ D +] where T 1 ad T deote the first ad last value of the equidistat time poits of a iterval of legth D; the asset is modelled as a expoetial Lévy process accordig to (2.1 (2.3 ad the optio starts at time ad matures at T. For the fixed strike forward-start Asia put optio we set V p (K, Σ D, r; b, σ 2, λ; T 1, T V fxp e rt IE [ (K Σ D +].
FLOATING AND FIXED STIKE ASIAN AND LOOKBACK OPTIONS 5 Similar otatio will be used for the other types of Asia optios. 3. A equivalece result for Asia optios I this sectio we state ad prove the mai result that shows a equivalece relatioship betwee floatig ad fixed strike Asia optios. Theorem 3.1. Assumig that the asset price evolves as a expoetial Lévy process accordig to equatios (2.1 (2.3, we ca relate the floatig ad fixed strike forward-start Asia optio via the followig symmetry: V c ( θst, Σ D, r; b, σ 2, λ; T 1, T V p ( θs, Σ D, δ; b, σ 2, fλ;, T T 1 V p ( ΣD, θ, r; b, σ 2, λ; T 1, T V c ( ΣD, θs, δ; b, σ 2, fλ;, T T 1 where D T T 1, b δ r σ2 2 (e x 1 + xe x λ(dx ad f(x e x. Proof. We will prove the result for simplicity ad without loss of geerality i the case of discrete averagig for the floatig strike call. The case of cotiuous averagig ad the floatig strike put ca be proved i a aalogous way. The price of the floatig strike optio expressed i uits of the uméraire yields Ṽ flc : V flc S e δt IE e rt IE [ (θ Σ D +] S [ e rt (θ Σ D + ] e δt. (3.1 S Defie a ew measure ĨP via its ado-nikodym derivative dĩp e rt dip e δt η T (3.2 S FT ad usig S as the uméraire, the valuatio problem (3.1 becomes [ Ṽ flc e δt ĨE (θ Σ D +]. (3.3 Here Σ D : Σ D 1 i 1 ad S is defied as i : i. Because the measures IP ad ĨP are related via the adapted ad positive loc desity process η, we immediately coclude that ĨP IP ad we ca apply Girsaov s theorem for semimartigales; we refer to Jacod ad Shiryaev (23, III.3.24. The desity process correspodig to that chage of measure ca be represeted i the usual form [ ] d ĨP η t IE dip F t { exp σw t + e rt S t e δt S t ( σ x(µ L ν L 2 (ds, dx 2 + i } (e x 1 xλ(dx t
6 ENST EBELEIN AND ANTONIS PAPAPANTOLEON therefore we ca idetify the tuple (β, Y of predictable processes (fuctios β(t 1 ad Y (t, x exp(x that characterize the chage of measure. Combiig Girsaov s theorem with Theorem II.4.15 ad Corollary II.4.19 i Jacod ad Shiryaev (23, we deduce that a Lévy process (PIIS remais a Lévy process uder the measure ĨP, because the processes β ad Y are determiistic ad Y is idepedet of t. As a cosequece of Girsaov s theorem for semimartigales we ifer that W t W t σt is a ĨP-Browia motio ad νl Y ν L is the ĨP compesator of the radom measure of jumps µ L. Whece, we defie λ(dx e x λ(dx. Note that this chage of measure ca also be iterpreted as a Esscher trasformatio; we refer to Shiryaev (1999 for more o the Esscher trasformatio ad Eberlei ad Keller (1995 for a applicatio i Lévy processes ad fiace. Therefore, we have that i i exp exp exp { ( r δ σ2 2 + T i { ( r δ + σ2 2 + + T i { [( r δ + σ2 2 + + T i (e x 1 xλ(dx (T i T + σ (W Ti W T x(µ L ν L (ds, dx T x(µ L ν L (ds, dx (e x 1 + x λ(dx (T i T + σ ( WTi W T x(µ L ν L (ds, dx T x(µ L ν L (ds, dx (e x 1 + x λ(dx T i + σ W Ti ] x(µ L ν L (ds, dx [ ( r δ + σ2 2 + T + (e x 1 + x λ(dx T + σ W T x(µ L ν L (ds, dx] }. } }
FLOATING AND FIXED STIKE ASIAN AND LOOKBACK OPTIONS 7 We ow defie the Lévy process L via where t L t bt + σ W t + x(µ L ν L (ds, dx (3.4 σ b 2 r δ + 2 + (e x 1 + x λ(dx. (3.5 Notice that usig Assumptio (M ad Theorem 25.3 i Sato (1999, we ca easily deduce that ĨE[ L 1 ] <, hece, L is a ĨP-special semimartigale. The characteristic triplet of L is ( b, σ 2, λ ad the part without drift of L is a martigale uder the measure ĨP. Sice L is a Lévy process, the dual process defied as L : L is also a Lévy process ad from Lemma 2.2 we deduce that its Lévy triplet is ( b, σ 2, λ. This simplifies the expressio for S i exp { LTi L } T d exp { L T Ti where statioarity of L is used for the last equatio ad d deotes equality i law. As a result we have that Σ D 1 i d 1 } } exp { L T Ti. eversig the time idex via the substitutio T j T T j+1 we get Σ D d 1 j1 exp { L btj} : Σ D. Hece, we ca coclude the proof Ṽ flc e δt ĨE [(θ Σ D +] e δt ĨE [(θ Σ D +] where, i the last expressio we cosider the expectatio with respect to ĨP, uder which the Lévy process L has the triplet ( b, σ 2, λ. Notice that e rt S t oce discouted at the rate δ, is a ĨP-martigale. emark 3.2. The equivalece results do ot hold i the case of i-progress Asia optios, because of the additioal term created i floatig strike optios from averagig up to time t, whe the optio starts.
8 ENST EBELEIN AND ANTONIS PAPAPANTOLEON emark 3.3. The equivalece results hold i the case of a optio o the geometric or harmoic average, that is whe the average is of the form respectively. ( 1 Γ D i ad A D 1 i (3.6 Proof. It suffices to otice that, if we apply the same chage of uméraire as i (3.1, the resultig averagig term is of the same form as i (3.6 ad we ca defie a radom variable S i a aalogous way. We have for the geometric ad harmoic average respectively Γ D Γ ( D S 1 ( T i ( i 1 S 1 Ti ad à D A D 1 ei P 1 i i i where i. The remaiig part follows alog the same lies as the proof of Theorem 3.1. 4. A equivalece result for Lookback optios I this sectio we use the techiques applied i the previous sectio to prove a equivalece relatioship betwee floatig ad fixed strike lookback optios. We itroduce the followig otatio for the floatig strike forward-start lookback call optio V c (θ, N D, r; b, σ 2, λ; T 1, T V flc e rt IE [ (θ N D +] where T 1 ad T deote the first ad last value of the equidistat time poits of a iterval of legth D; the asset is modelled as a expoetial Lévy process accordig to (2.1 (2.3 ad the optio starts at time ad matures at T. For the fixed strike forward-start lookback put optio we set V p (K, N D, r; b, σ 2, λ; T 1, T V fxp e rt IE [ (K N D +] Similar otatio will be used for the other types of lookback optios. Theorem 4.1. Assumig that the asset price evolves as a expoetial Lévy process accordig to equatios (2.1 (2.3, we ca relate the floatig ad
FLOATING AND FIXED STIKE ASIAN AND LOOKBACK OPTIONS 9 fixed strike forward-start lookback optio via the followig symmetry: V c (θ, N D, r; b, σ 2, λ; T 1, T V p (θs, N D, δ; b, σ 2, fλ;, T T 1 V p (M D, θ, r; b, σ 2, λ; T 1, T V p (M D, θs, δ; b, σ 2, fλ;, T T 1 where D T T 1, b δ r σ2 2 (e x 1 + xe x λ(dx ad f(x e x. Proof. We cosider a floatig strike lookback put optio ad the case of a floatig strike lookback call ca be proved i a aalogous way. Applyig the same chage of uméraire as i (3.1, the valuatio problem is equivalet to the oe i (3.3 [ Ṽ flp e δt ĨE ( M D θ +]. Here M D : M D max T 1 T i T i max T 1 T i T i where S is defied as i i. Followig the steps of the proof of Theorem 3.1 we complete the result. emark 4.2. Because the pricig fuctio is homogeeous of degree oe, i.e. ϱie [ (M T K +] IE [ (ϱm T ϱk +] where ϱ +, for a suitable choice of α ad ϱ we ca apply the symmetry result of Theorem 4.1 for pricig partial lookback optios whe a method for fixed strike optios is available. efereces Adrease, J. (1998. The pricig of discretely sampled Asia ad lookback optios: a chage of umeraire approach. J. Comput. Fiace 2 (1, 5 3. Behamou, E. (22. Fast Fourier trasform for discrete Asia optios. J. Comput. Fiace 6 (1, 49 68. Borovkov, K. ad A. Novikov (22. O a ew approach to calculatig expectatios for optio pricig. J. Appl. Probab. 39, 889 895. Carr, P., H. Gema, D. B. Mada, ad M. Yor (22. The fie structure of asset returs: a empirical ivestigatio. J. Busiess 75, 35 332. Eberlei, E. (21. Applicatio of geeralized hyperbolic Lévy motios to fiace. I O. E. Bardorff-Nielse, T. Mikosch, ad S. I. esick (Eds., Lévy processes: theory ad applicatios, pp. 319 336. Birkhäuser. Eberlei, E. ad J. Jacod (1997. O the rage of optios prices. Fiace Stoch. 1, 131 14. Eberlei, E. ad U. Keller (1995. Hyperbolic distributios i fiace. Beroulli 1, 281 299. Hederso, V. ad. Wojakowski (22. O the equivalece of floatig ad fixed-strike Asia optios. J. Appl. Probab. 39, 391 394.
1 ENST EBELEIN AND ANTONIS PAPAPANTOLEON Hooglad, J. ad C. D. Neuma (2. Asias ad cash divideds: exploitig symmetries i pricig theory. Techical report, CWI. Jacod, J. ad A. N. Shiryaev (23. Limit theorems for stochastic processes (2d ed.. Spriger. Nielse, J. A. ad K. Sadma (1999. Pricig of Asia exchage rate optios uder stochastic iterest rates as a sum of optios. Discussio Paper No. B-431, Uiversität Bo. Sato, K.-I. (1999. Lévy processes ad ifiitely divisible distributios. Cambridge Uiversity Press. Shiryaev, A. N. (1999. Essetials of stochastic fiace: facts, models, theory. World Scietific. Vamaele, M., G. Deelstra, J. Liiev, J. Dhaee, ad M. J. Goovaerts (26. Bouds for the price of discrete arithmetic Asia optios. J. Comput. Appl. Math. 185, 51 9. Večeř, J. (22. Uified Asia pricig. isk 15 (6, 113 116. Večeř, J. ad M. Xu (24. Pricig Asia optios i a semimartigale model. Quat. Fiace 4 (2, 17 175. Departmet of Mathematical Stochastics, Uiversity of Freiburg, D 7914 Freiburg, Germay E-mail address: {eberlei,papapa}@stochastik.ui-freiburg.de UL: http://www.stochastik.ui-freiburg.de/~{eberlei,papapa}