No Fear of Jumps. 1 Introduction

Transcription

1 No Fear of Jumps Y. d Hallun chool of Computer cence, Unversty of Waterloo, Waterloo ON, Canada (e-mal: ydhallu@elora.uwaterloo.ca). D.M. Pooley ITO 33 A, 39 rue Lhomond, 755 Pars France (e-mal: davd@to33.com). P.A. Forsyth chool of Computer cence, Unversty of Waterloo, Waterloo ON, Canada (e-mal: paforsyt@elora.uwaterloo.ca). Abstract Jump dffuson based models have recently ncreased n popularty. In ths artcle, we develop robust and effcent technques for the numercal soluton of opton prcng models where the underlyng process s a jump dffuson process. The numercal technques can be appled to a varety of contngent clam valuatons. Numercal examples for European, Amercan and Parsan optons are provded. 1 Introducton In 1973, the Black-choles model revolutonzed dervatve prcng (Black and choles, 1973). Usng only a volatlty and an nterest rate, Robert Black and Myron choles developed an arbtrage free prcng formula that does not requre knowledge of nvestor belefs about the underlyng stock s expected return. However, over the years practtoners have recognzed the lmtatons of the Black-choles model. In partcular, the constant volatlty assumpton s nsuffcent to capture the smle or skew that s exhbted by the mpled volatltes of traded fnancal optons. To better capture these volatlty profles, numerous avenues of research have been explored whch ether extend the Black-choles model or explore completely new approaches. Among these extensve works, the jump dffuson model (Merton, 1976) and the stochastc volatlty model (whch could nclude jumps as well) (Bates, 1996; cott, 1997; Baksh et al., 1997) appear to be the most popular among practtoners. Unfortunately, a large porton of the lterature devoted to these approaches s lmted to analytcal or quas-analytcal solutons for vanlla optons. Very few of these methods can be extended to prce exotc or path-dependent optons. For these more complcated scenaros, numercal partal dfferental equaton technques must be used. The objectve of ths paper s to present a robust and effcent numercal method for solvng the partal ntegro dfferental equaton (PIDE) whch arses from the jump dffuson model. We lmt ourselves to prcng optons under the jump dffuson model, but ths framework s also applcable to credt rsk models or more complex valuaton models such as stochastc volatlty wth jumps. In the latter case, one smply has to solve a two dmensonal PIDE problem, and apply the technques presented below for the jump dffuson part n the stock drecton. A major advantage of the methods ntroduced here s that they are easly added to exstng numercal opton prcng software. In partcular, software that uses an mplct approach for valung Amercan optons can be easly modfed to prce Amercan optons wth jump dffuson. The ttle of ths paper s obvously based on the very readable artcle Fear of Jumps by Lews (22). Ths artcle was mostly analytcal n nature, and reled on an equlbrum based approach to opton prcng. In contrast, the artcle presented here has a numercal focus for prcng optons under jump dffuson. Further, we attempt to convnce the reader that addng a jump component to prcng software can be approached wth no fear. 62 Wlmott magazne

2 Alternatvely, ths paper could have been enttled Fear of No Jumps, as our examples are ntended to show that a jump component adds essental features to a prcng model. Wthout these features, one should be concerned about the accuracy and stablty of the prcng framework. Our technque s smlar n some respects to Zhang (1997), though less constraned n terms of stablty restrctons. Our method also offers a hgher rate of convergence than Zhang s. mlar comments apply f we compare our approach to that of Andersen and Andreasen (2), at least n the case of Amercan optons. In ths artcle, the PIDE presented by (Merton, 1976; Andersen and Andreasen, 2) s studed exclusvely. Whle t s true that Merton s assumpton about jump rsk beng dversfable does not hold for ndex based optons, and n ths case one must use an equlbrum based method (Lews, 22) or a mean varance hedgng approach (Ayache et al., 24), the PIDEs resultng n ether case are essentally dentcal. Consequently, the numercal technques presented here can be appled. Ths artcle s organzed as follows. In secton 2, the numercal method for solvng the opton prcng PIDE whch results from a jump dffuson model s presented. In secton 3, a wde varety of numercal examples of exotc, path-dependent contracts are presented. In partcular, we nclude numercal examples for Amercan, and Parsan optons. Fnally, secton 4 contans concludng remarks. 2 Mathematcal Model Ths secton provdes an overvew of the mathematcal modelng ssues that arse n a jump dffuson framework. The presentaton and notaton closely follows that of d Hallun et al. (23). However, partcular attenton s pad here to the practcal ssues that arse n a numercal mplementaton. Further, snce the goal of ths paper s somewhat llustratve, several proofs and techncal detals have been omtted. The reader s referred to d Hallun et al. (23) and the references theren for a complete treatment of the theory of opton prcng n a jump dffuson framework. In the usual (no jumps) Black-choles model for opton prcng (Black and choles, 1973; Merton, 1976), the underlyng asset prce evolves accordng to d = µdt + σ dz, (2.1) where µ s the (real) drft rate, σ s the volatlty, and dz s the ncrement of a Gauss-Wener process. Let V(, t) be the value of a contngent clam that depends on the underlyng asset and tme t. By appealng to the prncple of no-arbtrage, a partal dfferental equaton (PDE) for the value of V can be derved: V τ = 1 2 σ 2 2 V + rv rv, (2.2) where τ = T t s the tme remanng untl expry T, and r s the contnuously compounded rsk-free nterest rate. Equaton (2.2) s smply a second order parabolc PDE of one space dmenson and one tme dmenson. Ths equaton has been the subject of countless studes, and s wellunderstood from a varety of vewponts (fnancal, mathematcal, numercal). Lettng LV = 1 2 σ 2 2 V + rv rv (2.3) equaton (2.2) can be wrtten n the smple form V τ = LV. (2.4) It s assumed that the reader s famlar wth the numercal soluton of PDEs of the form (2.4). oftware for ths problem s easly wrtten, and offthe-shelf mplementatons are readly avalable. Nevertheless, the process specfed by equaton (2.1) s not suffcent to explan observed market behavour (Baksh and Cao, 22). In realty, stock prces have been observed to have large nstantaneous jumps. uch behavour can be modeled by the rsk-neutral process (Merton, 1976) d = (r λκ) dt + σ dz + (η 1)dq, (2.5) where dq s a Posson process (ndependent of the Brownan moton), and η 1 s an mpulse functon producng a jump from to η. If λ s the arrval ntensty of the Posson process, then dq = wth probablty 1 λdt, and dq = 1 wth probablty λdt. The expected jump sze can be denoted by κ = E[η 1], where E s the expectaton operator. As s well known, the far prce of a contngent clam V(, t) under a process of the form (2.5) s gven by the followng partal ntegro dfferental equaton (PIDE): V τ = 1 2 σ 2 2 V + (r λκ) V rv + λ V(η)g(η)dη λv. (2.6) In equaton (2.6), g(η) s the probablty densty functon of the jump ampltude η. The probablty densty functon s assumed to have the usual dstrbuton propertes, such as η, g(η) and g(η)dη = 1. Lettng ˆLV = 1 2 σ 2 2 V + (r λκ)v (r + λ)v, (2.7) equaton (2.6) can be wrtten as V τ = ˆLV + λ V(η)g(η)dη. (2.8) As wth LV, the behavour of ˆLV s well understood. Further, t should be straghtforward to modfy any reasonably desgned software that can handle numercally LV to compute ˆLV. Of a more dffcult nature s the ntegral term n equaton (2.8). The obvous approach for the numercal computaton of the ntegral term s to use standard numercal ntegraton methods such as mpson s rule or Gaussan quadrature. Unfortunately, for a numercal grd of sze n, these technques are O(n 2 ). For real-tme prcng software, and especally for calbraton routnes, qucker algorthms are desrable. ^ Wlmott magazne 63

3 To ths end, the ntegral term of equaton (2.8) should be computed n a way that s effcent (better than O(n 2 )), robust, flexble (can be used wth nonlnear prcng models, and/or exotc optons), easly added to exstng opton prcng software. All of these propertes are satsfed f the ntegral term s evaluated by FFTs, thereby only requrng O(n log n) operatons per tmestep, the ntegral term s appled mplctly, thereby ncreasng stablty and allowng the possblty of second order convergence. The FFT evaluaton of the ntegral and the mplct treatment of the resultng terms wll be dscussed separately below. Followng these, an extenson to Amercan optons wll be provded, as well as a bref descrpton of credt rsk. Examples whch use the technques descrbed below are provded n secton 3. It should be noted that n some cases, the ntegral term can be evaluated drectly n O(n) tme usng fast Gauss transform (FGT) technques (Greengard and tran, 1991). Whle ths technque works for the case where jump szes are lognormally dstrbuted, t s not clear f t works for more general dstrbutons. Furthermore, numercal experments show that for any practcal grd sze the FFT approach for evaluatng the ntegral term s faster than the FGT method. (Note that the ntegral needs only to be evaluated wth an accuracy consstent wth the dscretzaton of the PDE). 2.1 FFT evaluaton Before the ntegral term of equaton (2.8) can be evaluated by FFTs, t must be manpulated nto the form of a correlaton ntegral. Ths process s descrbed n secton Once ths process s done, at least two numercal ssues reman. Frst, standard FFT algorthms requre an equally spaced grd, whereas an effcent PDE grd wll be unequally spaced. Interpolaton must be used to move from one grd to the other. econd, snce the nput functons to the FFT routnes wll be non-perodc, wrap-around polluton can negatvely affect the soluton. These numercal ssues are dscussed n secton Manpulaton Ignorng the leadng λ, the ntegral term n equaton (2.8) s I() = V(η)g(η)dη. (2.9) The goal s to turn ths expresson nto a correlaton product whch can be evaluated by FFT technques. Lettng x = log() and applyng the change of varable y = log(η), we obtan I = + V(x + y) f ( y)dy, (2.1) where f ( y) = g(e y )e y and V( y) = V(e y ). The f ( y) term can be nterpreted as the probablty densty of a jump of sze y = log η. Convenently, equaton (2.1) corresponds to the correlaton product V( y) f ( y). In dscrete form, equaton (2.1) becomes I = j=n/2 j= N/2+1 where I = I( x), V j = V( j x), and V +j fj y + O ( ( y) 2), (2.11) x j+ x/2 f j = f ( j y) = 1 x x j x/2 f (x)dx (2.12) It has been assumed that y = x. Assumng that f s real (a safe assumpton for fnancal applcatons), the dscrete correlaton of equaton (2.11) can be evaluated usng FFTs snce ( (FFT( I = IFFT V) )( FFT( f ) ) ) (2.13) where ( ) denotes the complex conjugate. For effcency, FFT( f ) can be pre-computed and stored. Durng each tmestep (or each teraton of an teratve method), an FFT and an nverse FFT must be computed Numercal ssues A typcal grd for the dscretzaton of ˆLV n equaton (2.8) wll be unequally spaced n coordnates. For example, small mesh spacng wll be used near strkes or barrers, wth large mesh spacng elsewhere. However, the dscrete form of the correlaton ntegral (2.11) requres an equally spaced grd n log() coordnates. It s hghly unlkely that these two grds are fully compatble. Hence, values must be nterpolated between the two grds. In partcular, values of V on the unequally spaced grd must be nterpolated onto an equally spaced log() grd. The computaton of equaton (2.13) can then be performed 1. Fnally, the resultng equally spaced V data needs to be nterpolated back onto the unequally spaced grd. The overall process s summarzed n algorthm (1). If lnear or Algorthm 1 Method for computng the ntegral term of equaton (2.8) by FFTs. Interpolate the dscrete values of V onto an equally spaced log() grd. Ths generates the requred values of V j. Carry out the FFT on the V data Compute the correlaton n the frequency doman (wth pre-computed FFT( f) values), usng equaton (2.13). Invert the FFT of the correlaton Interpolate the dscrete values of I(x ) back onto the orgnal grd 64 Wlmott magazne

4 hgher order nterpolaton s used, algorthm (1) s second order correct. Ths s consstent wth the dscretzaton error n the PDE and the mdpont rule used to evaluate the ntegral n equaton (2.11). For the actual FFT evaluaton, standard algorthms assume perodc nput data. If the nput data s not perodc (as wth the current applcaton), then the dscrete Fourer transform s effectvely appled to the perodc extenson of the nput functons. Ths can lead to undesrable wrap around polluton, whch manfests tself wth erroneous values n the soluton. To avod wrap around effects, the doman of the ntegral n equaton (2.8) can be extended to the left and rght by amounts y and y +. The ntegral then becomes I ext = ymax + y + ymn y V(x + y) f ( y)dy, (2.14) where ymax = log(max), y mn = log( mn ), and [ mn, max] are selected approprately. Unknown values n the range [ ymax, ymax + y + ] can be obtaned by lnear extrapolaton. Ths assumes that the far feld behavour of the opton prcng problem s lnear. Values n the range [ y mn y, y mn ] can be obtaned from nterpolaton on the orgnal grd, assumng an = grd pont has been mantaned. Once the FFT has been performed n the extended doman, values n the extensons are dscarded. Because of the extenson, values n the orgnal doman wll have been less affected by wrap-around polluton. 2.2 Implct evaluaton We now look at the numercal evaluaton of equaton (2.8). Let n denote the dscrete form of the ntegral evaluated at tmestep n usng data V n (one can thnk of as an applcaton of algorthm (1)). To solve equaton (2.8), the ˆLV term must also be dscretzed. Ths can be done by any standard method, such as fnte dfferences, fnte volumes, ( or fnte ) n elements. Let the dscrete form of ˆLV at tmestep n be gven by ˆLV. A general dscretzed form of equaton (2.8) can then be wrtten as V n+1 where V n τ ) n+1 ( n = (1 θ)( ˆLV + θ ˆLV) + (1 θ J )λ n+1 + θ J λ n, (2.15) θ s a tme-weghtng parameter for ˆL θ = s fully mplct θ = 1/2 s Crank-Ncolson θ = 1 s fully explct θ J s tme-weghtng for the jump term θ J ={, 1/2, 1}. Let M denote the dscretzaton matrx stencl such that ( n [MV] n = ˆLV). (2.16) Equaton (2.15) becomes [I τ (1 θ)m]v n+1 = [I + τ θm]v n + (1 θ J )λ τ n+1 + θ J λ τ n. (2.17) For standard PDE dscretzaton technques, the matrx M n equaton (2.17) s trdagonal. Trdagonal systems are quck and easy to solve. However, an mplct treatment of the jump term (θ J = 1) causes to lead to a hghly undesrable dense matrx (all nodal values are coupled n equaton (2.1)). On the other hand, a fully explct treatment of the jump term s easy to adapt to exstng code, snce only the rght hand sde vector needs to be updated. However, whle stll stable, only frst order convergence s possble. To allow for an mplct treatment of jumps, a fxed pont teraton method must be used. A descrpton of the method s gven n algorthm (2). At teraton k known data s used to construct the jump term. nce only the rght hand sde s affected, a smple trdagonal system needs to be solved at each teraton. n+1 Algorthm 2 Fxed pont teraton. Let (V n+1 ) = V n Let ˆV k = (V n+1 ) k Let ˆ k = ( n+1 ) k Construct vector n usng algorthm (1) for k =, 1, 2,...untl convergence do Construct vector ˆ k usng algorthm (1) olve [I (1 θ)m] ˆV k+1 = [I + θm]v n + (1 θ J )λ ˆ k + λθ J n ˆV f max k+1 ˆV k < tolerance then max(1, ˆV k+1 ) qut end f end for Under some farly mld assumptons - that the dscretzaton of ˆL form an M-matrx, the probablty densty functon has certan standard propertes, the nterpolaton weghts are postve, and that r and λ are postve - t can be proven that algorthm (2) s globally convergent (d Hallun et al., 23). Further, the error at each teraton s reduced by approxmately (1 θ)λ τ, ndcatng convergence n a small number of teratons (.e. for typcal values, 3 teratons are suffcent). 2.3 Amercan optons Amercan optons can be solved by a smple penalty approach. Detals of the penalty approach can be found n (Forsyth and Vetzal, 22). Further detals wth regards to jump dffuson models can be found n (d Hallun et al., 23). Brefly, the penalty approach nvolves addng a penalty term ^ Wlmott magazne 65

5 to the prcng PDE. Equaton (2.8) then becomes V τ = ˆLV + λ V(η)g(η)dη + ρ max(v V, ). (2.18) In the lmt as ρ, the soluton satsfes V V. The Amercan constrant s enforced by settng V to the payoff of the opton. In the dscrete equatons, ρ s set ndependently at each node. If the value at a node drops below V (the payoff), then ρ s set to a large number. Ths essentally adds an extra source term to the PDE, thereby ncreasng the value at the partcular node. If the value at a node s greater than V, then ρ s set to zero, and the regular PDE s solved. Ths can also be thought of as constrant swtchng. Wherever the value drops below the V threshold, the constrant s swtched on and appled. If the value s above the threshold, the constrant s swtched off. As wth the evaluaton of the ntegral term, the penalty constrant can be appled explctly or mplctly. An explct evaluaton smply uses data at the prevous tmestep to determne when the constrant s actvated. An mplct evaluaton could use a fxed pont teraton (or other nonlnear solvng method) to apply the constrant usng data at the current tmestep. If the jump term s already beng evaluated usng an teratve method, lttle or no extra cost s ncurred by the penalty method. Convergence of the penalty approach for Amercan optons n a jump dffuson framework was proven n (d Hallun et al., 23). 2.4 Credt rsk Untl ths pont, jumps n stock prce assocated wth the jump dffuson model have been assumed to occur for arbtrary exceptonal events. However, a specal jump n asset level occurs n the case of bankruptcy. In prcng corporate and convertble bonds, t s of nterest to determne the rsk adjusted hazard rate of bankruptcy. If t s assumed that the stock prce of a frm jumps to zero on default, then λ h can be nterpreted as the rsk adjusted hazard rate of bankruptcy (or default n the case of bonds). In ths case, the PDE satsfed by vanlla puts/calls n the presence of a sngle jump to bankruptcy s gven by V τ = 1 2 σ 2 2 V + (r + λ h )V (r + λ h )V + λ h V(,τ). (2.19) Equaton (2.19) can be derved by hedgng arguments, or by settng κ to 1 and the jump probablty densty functon g(η) to the delta functon δ() (1 at η =, zero elsewhere) n the usual Merton jump dffuson model. It s usually assumed that λ h = λ h (, t), wth λ h (, t) beng determned by calbraton to observed market prces for vanlla optons and credt nstruments. nce opton prces are usually avalable for a range of strkes, more nformaton s provded about default rates than s usually avalable from smply examnng credt nstruments. Note that equaton (2.19) suggests that default rsk has an effect on the prcng of vanlla optons. As well, f the possblty of a sngle jump to bankruptcy s assumed, then a hedgng portfolo consstng of the opton, an underlyng asset, and an addtonal opton can be constructed whch elmnates both the dffuson rsk (a delta hedge) as well as the jump rsk (snce the jump has only one possble outcome). 3 Results The examples of ths secton are ntended to compare the regular Black- choles model and the jump-dffuson model. To ensure a consstent bass for comparson, the followng procedure s used: 1. Gven some jump dffuson parameters, compute the (numercal) atthe-money prce V jump of European put opton. 2. Usng a constant volatlty Black-choles model, determne the mpled volatlty σ mpled whch matches the opton prce to the jump dffuson value V jump at the strke K. 3. Value the opton usng a constant volatlty model (no jumps) usng the mpled volatlty σ mpled estmated n tep 2. The frst example prces a European put opton wth and wthout jumps. Parameters are provded n Table 1. Results are shown n Fgure 1. The mpled volatlty value for the Black-choles model s By constructon, the prces of the Black-choles model and the Merton jump model are equal at the strke prce. In-the-money values are larger for the Black-choles model, but only slghtly. Of nterest s the fact that the jump model prces deep out-of-the-money optons sgnfcantly hgher. Ths reflects the fact that a jump event can dramatcally change the moneyness of an opton to a much larger extent than a smple dffuson only model. The delta and gamma plots for the two models are smlar, although the jump model plots show greater varaton. Ths ndcates that a delta hedge of the jump model may need more frequent rebalancng. Nevertheless, jumps ntroduce market ncompleteness, and smple delta hedgng wll defntely fal. Optmal hedgng n ncomplete markets s preferred (Henrotte, 22; Ayache et al., 24). In any case, hedgng wll requre accurate delta and gamma nformaton. It s essental that the numercal scheme produce smooth delta and gamma values. TABLE 1. INPUT DATA UED TO VALUE VARI- OU OPTION UNDER THE LOGNORMAL JUMP DIFFUION PROCE. THEE PARAMETER ARE APPROXIMATELY THE AME A THOE REPORTED IN (ANDEREN AND ANDREAEN, 2) UING EUROPEAN CALL OPTION ON THE &P 5 TOCK INDEX IN APRIL OF 1999 volatlty: σ.15 rsk-free rate: r.5 jump standard devaton: γ.45 jump mean: µ mean.9 jump ntensty: λ.1 tme to expry: T.25 strke: K Wlmott magazne

6 The second example s a repeat of the frst, except that an Amercan put opton s prced nstead of a European put opton. The mpled volatlty value used s the same as n the prevous example: σ mpled = Results are smlar, except that delta values now reach and reman at 1 for low stock prces, whle gamma values jump to zero. Ths jump to zero occurs at the free boundary between the earlyexercse regon and the regular prcng regon. The early exercse regon s further to the rght for the, ndcatng that jumps cause an ncrease n the probablty that the opton should be exercsed early V.5 V s.5 Black-choles model Black-choles model.6.25 trke trke (a) Prce (V) (b) Delta (V s ) V ss 2 Black-choles model 1 trke (c) Gamma (V ss ). Fgure 1: Put opton prce (V), delta (V ) and gamma (V ). The nput data s contaned n Table 1 ^ Wlmott magazne 67

7 V V s.5 Black-choles model.5.6 Black-choles model.7.25 trke.8 trke (a) Prce (V) (b) Delta (V s ) V ss 3 Black-choles model 2 1 trke (c) Gamma (V ss ). Fgure 2: Amercan put opton prce (V), delta (V ) and gamma (V ). The nput data s contaned n Table 1 The last example s for a Parsan knock-out call opton. The partcular case consdered here s an up-and-out call wth daly dscrete observaton dates. Ths contract ceases to have value f s above a specfed barrer level for a specfed number of consecutve montorng dates. Ths can be valued by solvng a set of one-dmensonal problems whch exchange nformaton at montorng dates (Vetzal and Forsyth, 1999). Base parameters are the same as n Table 1. The knock-out barrer s placed at = 1.2, whle the number of consecutve days above the barrer untl knock-out s 68 Wlmott magazne

8 Black-choles model.2 Black-choles model V.5 V s trke.8 trke (a) Prce (V). (b) Delta (V s ) V ss 5 Black-choles model 1 trke (c) Gamma (V ss ). Fgure 3: Parsan knock-out call opton (V), delta (V ) and gamma (V ) wth dscrete daly observaton dates wth and wthout jumps. The barrer s set at = 1.2 and the number of consecutve daly observatons to knock-out s 1. The nput data s contaned n Table 1 set to 1. The mpled volatlty value s It s nterestng to note that the gves smaller prces for stock values below the strke and above the barrer. Ths s somewhat n contradcton to the put optons, for whch deep out-of-the-money prces were hgher for the jump model. Nevertheless, the dfferences are small, and the delta and gamma plots show the far feld behavour to be qute smlar. ^ Wlmott magazne 69

9 The greatest prce dfference occurs between the strke and barrer levels. Presumably a jump n ths regon hdes the effect of the (upper) barrer, whereas a pure dffuson model wll have ts value decreased by the barrer. However, t s dffcult to ntutvely predct the effect of jumps on prces. For convex payoffs, jumps ncrease the value of an opton. For non-convex payoffs, as s the case for the Parsan knock-out call, t s not clear what effect jumps wll have on the prce. 4 Concluson Ths artcle has demonstrated the numercal evaluaton of the PIDE resultng from the Merton jump-dffuson model n opton prcng. The ntegral term of the prcng equaton was evaluated usng effcent FFT technques. The ssues of nterpolaton between unequally spaced PDE grds and equally spaced FFT grds, as well as wrap-around polluton effects, were brefly dscussed. A fxed pont teraton method was used to obtan an mplct tmesteppng method wthout resortng to a full dense matrx solve. Extensons to Amercan optons and credt rsk were also mentoned. Perhaps the bggest advantage of the technques descrbed n ths paper s the ease wth whch they can be added to an exstng exotc opton prcng lbrary. All that s requred s that a functon be added to the lbrary whch, gven the current vector of dscrete opton prces, returns the vector value of the correlaton ntegral. Ths vector s then added to the rght hand sde of the fxed pont teraton. Ths method can even be appled to any jump sze probablty densty functon. The numercal examples showed the effect of jumps on varous opton values. For European and Amercan put optons, the jump dffuson model ncreases deep out-of-the-money prces. Changes to the hedgng parameters delta and gamma were also noted. The stablty of the methods was alluded to by the smooth delta and gamma plots. An example of a Parsan knock-out opton was also provded. An mportant ssue not addressed n ths paper s hedgng jump dffuson models. nce the market s ncomplete, smple delta hedgng can gve large errors. In ths case optmal hedgng n ncomplete markets must be used (Ayache et al., 24). FOOTNOTE & REFERENCE 1. Methods exst for computng an FFT on unequally spaced data. However, these methods do not appear to be more effcent than the straght forward approach suggested here. Andersen, L. and J. Andreasen (2). Jump-dffuson processes: Volatlty smle fttng and numercal methods for opton prcng. Revew of Dervatves Research 4, Ayache, E., P. Henrotte,. Nassar and X. Wang (24, January). Can anyone solve the smle problem? To appear n Wlmott magazne. Baksh, G. and C. Cao (22). Rsk-neutral kurtoss, jumps, and opton prcng: Evdence from 1 most actvely traded frms on the CBOE. Workng paper, mth chool of Busness, Unversty of Maryland. Baksh, G., C. Cao, and Z. Chen (1997). Emprcal performance of alternatve opton prcng models. Journal of Fnance 52, Bates, D.. (1996). Jumps and stochastc volatlty: Exchange rate processes mplct n Deutsche mark optons. Revew of Fnancal tudes 9, Black, F. and M. choles (1973). The prcng of optons and corporate labltes. Journal of Poltcal Economy 81, d Hallun, Y., P. A. Forsyth and G. Labahn (23). A penalty method for Amercan optons wth jump dffuson processes. ydhallu/jump_amer.pdf, accepted to be publshed n Numersche Mathematk. d Hallun, Y., P. A. Forsyth, and K. R. Vetzal (23). Robust numercal methods for contngent clams under jump dffuson processes. ydhallu/jump.pdf, submtted to IMA Journal of Numercal Analyss. Forsyth, P. A. and K. R. Vetzal (22). Quadratc convergence of a penalty method for valung Amercan optons. IAM Journal on centfc Computaton 23, Greengard, L. and J. tran (1991). The fast Gauss transform. IAM Journal on centfc and tatstcal Computng 12, Henrotte, P. (22, July). Dynamc mean varance analyss. Workng paper. Lews, A. (22, December). Fear of jumps. In Wlmott, pp Wley. Merton, R. C. (1976). Opton prcng when underlyng stock returns are dscontnuous. Journal of Fnancal Economcs 3, cott, L. O. (1997). Prcng stock optons n a jump-dffuson model wth stochastc volatlty and nterest rates: Applcatons of Fourer nverson methods. Mathematcal Fnance 7, Vetzal, K. R. and P. A. Forsyth (1999). Dscrete Parsan and delayed barrer optons: A general numercal approach. Advances n Futures and Optons Research 1, Zhang, X. L. (1997). Numercal analyss of Amercan opton prcng n a jump-dffuson model. Mathematcs of Operatons Research 22, W 7 Wlmott magazne