Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Pricing

Transcription

1 Jump-Diffusion Processes: Volailiy Smile Fiing and Numerical Mehods for Pricing Leif Andersen and Jesper Andreasen General Re Financial Producs Firs Version: November 4, 998 This Version: May 6, 000 Summary This paper discusses exensions of he implied diffusion approach of Dupire (994) o asse processes wih Poisson umps. We show ha his exension yields imporan model improvemens, paricularly in he dynamics of he implied volailiy surface. The paper derives a forward PIDE (Parial Inegro-Differenial Equaion) and demonsraes how his equaion can be used o fi he model o European opion prices. For numerical pricing of general coningen claims, we develop an ADI finie difference mehod ha is shown o be uncondiionally sable and, if combined wih Fas Fourier Transform mehods, compuaionally efficien. The paper conains several deailed examples from he S&P500 marke. Key Words: Jump-Diffusion Process, Local Time, Forward Equaion, Volailiy Smile, ADI Finie Difference Mehod, Fas Fourier Transform JEL Classificaion: G3, C4, C63, D5 MSC Classifiaion: 60J60, 60J75, 60J55, 45K05, 4A99, 65N06

2 . Inroducion. The sandard Black-Scholes (973) assumpion of log-normal sock diffusion wih consan volailiy is, as all marke paricipans are keenly aware of, flawed. To equae he Black-Scholes formula wih quoed prices of European calls and pus, i is hus generally necessary o use differen volailiies (so-called implied volailiies) for differen opion srikes (K) and mauriies (T). The phenomenon is ofen referred o as he volailiy skew or smile (depending on he shape of he mapping of implied volailiy as a funcion of K and T) and exiss in all maor sock index markes oday. Before he crash of 987, he S&P500 volailiies, for insance, formed a smile paern, where deeply in- or ou-of-hemoney opions were characerized by higher volailiies han a-he-money opions. The pos-crash shape of S&P500 implied volailiies, on he oher hand, more resembles a skew (or sneer ), where implied volailiies decrease monoonically wih increasing srikes. Typically, he seepness of he skew decreases wih increasing opion mauriies. The exisence of he skew is ofen aribued o fear of large downward marke movemens (someimes known as crash-o-phobia ). Exensions of he Black-Scholes model ha capure he exisence of volailiy smiles can, broadly speaking, be grouped in hree approaches. In he sochasic volailiy approach (see, for insance, Heson (993), Sein and Sein (99), and Hull and Whie (987)), he volailiy of he sock is assumed o be a mean revering diffusion process, ypically correlaed wih he sock process iself. Depending on he correlaion and he parameers of he volailiy process, a variey of volailiy skews and smiles can be generaed in his model. Empirical evidence from ime-series analysis generally shows some evidence of sochasic volailiy in sock prices (see e.g. Andersen e al (999) for a review and many references). However, in order o generae implied Black-Scholes volailiy skews in a sochasic volailiy model ha are consisen wih hose observed in raded opions, ofen unrealisically high negaive correlaion beween he sock index and volailiy is required. Also, from a compuaional perspecive, sochasic volailiy models are complicaed o handle as hey are rue muli-facor models; as such, one would ypically need muli-dimensional laices o evaluae, say, American opions. We also noice ha sochasic volailiy models do no allow for perfec opion hedging by dynamic posiions in he sock and he money marke accoun (which in absence of oher raded conracs form an incomplee marke). Anoher approach, originally suggesed by Meron (976), generaes volailiy skews and smiles by adding disconinuous (Poisson) umps o he Black-Scholes diffusion dynamics. Again, by choosing he parameers of he ump process appropriaely, his socalled ump-diffusion model can generae a muliude of volailiy smiles and skews. In paricular, by seing he mean of he ump process o be negaive, seep shor-erm skews (which are ypical in pracice) are easily capured in his framework. Indeed, several auhors (e.g. Das and Foresi (996), Baes (996), and Bakshi e al (997)) poin ou he imporance of a ump componen when pricing opions close o mauriy. Like sochasic volailiy models, ump-diffusion models are challenging o handle numerically (an issue we shall spend considerable ime on in his paper) and resul in socks and bonds forming

3 an incomplee marke. Some papers dealing wih eiher empirical or heoreical issues relaed o ump-diffusion models include Ai-Sahalia e al (998), Andersen e al (999), Ball and Torous (985), Baes (99), Duffie e al (998), and Lauren and Leisen (998). The hird approach o volailiy smile modeling reains he pure one-facor diffusion framework of he Black-Scholes model, bu exends i by allowing he sock volailiy be a deerminisic funcion of ime and he sock price. This so-called deerminisic volailiy funcion (DVF) approach was pioneered by Dupire (994), Derman and Kani (994), and Rubinsein (994), and has subsequenly been exended or improved by many auhors, including Andersen and Broheron-Racliffe (998), Andreasen (997), Lagnado and Osher (997), Brown and Tof (996), Jackwerh (996), Chriss (996), and many ohers. The DVF approach has enoyed a cerain populariy wih praciioners, a leas parly because of is simpliciy and he fac ha i convenienly reains he marke compleeness of he Black-Scholes model. Moreover, he exisence of a forward equaion ha describes he evoluion of call opion prices as funcions of mauriy and srike makes i possible o express he unknown volailiy funcion direcly in erms of known opion prices. This again allows for efficien non-parameric fiing of he volailiy funcion and, in principle a leas, a precise fi o quoed marke prices. In conras, sochasic volailiy models and ump-diffusion models are normally parameerized in a few parameers and consequenly subec o fiing errors ha ofen are unaccepably large. While convenien, he DVF model unforunaely suffers from several serious drawbacks. For one, he mechanism by which he volailiy smile is incorporaed is clearly no realisic -- few marke paricipans would seriously aribue he exisence of he volailiy smile solely o ime- and sock-dependen volailiy. Indeed, here is much empirical lieraure reecing DVF models and heir implicaions for hedging and marke compleeness (e.g. Ai-Sahalia e al (998), Andersen e al (999), Buraschi and Jackwerh (998), and Dumas e al (997)). The weak empirical evidence is no surprising, as he DVF framework ypically resuls in srongly non-saionary implied volailiies, ofen predicing ha he skew of implied volailiies will vanish in he near fuure. In pracice, however, volailiy skews appear quie saionary hrough ime. Moreover, o fi DVF models o he ofen quie seep shor-erm skew, he fied implied volailiy funcion mus be conored in quie dramaic (and no very convincing) fashion. This has implicaions no only for he pricing of exoics opions, bu also affecs he hedge parameers for sandard European opions. As discussed in he empirical sudies by Andersen e al (999), Baes (996), and Bakshi e al (997), he mos reasonable model of sock prices would likely include boh sochasic volailiy and ump diffusion (as in he models by Baes (996) and Duffie e al (999)). From he perspecive of he financial engineer, such a model would, however, no necessarily be very aracive as i would be difficul o handle numerically and slow o calibrae accuraely o quoed prices. Raher han working wih such a complee model, his paper more modesly assumes ha sock dynamics can be described by a ump diffusion process where he diffusion volailiy is of he DVF-ype. As we will show, his

4 combines he bes of he wo approaches: ease of modeling seep shor-erm volailiy skews (umps) and accurae fiing o quoed opion prices (DVF diffusion). In paricular, by leing he ump-par of he process dynamics explain a significan par of he volailiy smile/skew, we generally obain a reasonable, sable DVF funcion, wihou he exreme shor-erm variaion ypical of he pure diffusion approach. Empirical suppor for he process used in his paper can be found in Ai-Sahalia e al (998). The res of his paper is organized as follows. In secion we ouline our process assumpions and develop a general forward PIDE describing he evoluion of European call opions as funcions of srike and mauriy. Paying special aenion o he case of lognormal umps, his secion also discusses he applicaions of he forward PIDE o he problem of fiing he sock process o observable opion prices. Secion 3 illusraes he proposed echniques by applying hem o he S&P500 marke. The secion also conains a brief general equilibrium analysis ha provides a link beween he risk-neural and obecive probabiliy measures, allowing us o saniy check our esimaed risk-neural S&P500 process parameers. In secion 4 we urn o he developmen of efficien finie difference mehods ha allow for general opion pricing under he ump-diffusion processes used in his paper. We also discuss he applicaion of Mone Carlo mehods. The pricing algorihms are esed in Secion 5 on boh European and exoic opions. Secion 5 also aemps o quanify he impac of sock price umps on cerain exoic opion conracs. Finally, Secion 6 conains he conclusions of he paper.. Forward equaions for European call opions... General framework. Consider a sock S affeced by wo sources of uncerainy: a sandard onedimensional Brownian moion W(), and a Poisson couning process π( ) wih deerminisic ump inensiy λ( ). Specifically, we will assume ha he risk-neural evoluion of S is given by 3 : b g b g b g ds( ) / S( ) = r( ) q( ) λ( ) m( ) d + σ, S( ) dw( ) + J( ) dπ( ). () where { J( )} 0 is a sequence of posiive, independen sochasic variables wih, a mos, ime-dependen densiy ζ ( ; ). Also in (), σ is a bounded ime- and sae-dependen local volailiy funcion; m is a deerminisic funcion given by m( ) E[ J( ) ]; and r and q are he deerminisic risk-free ineres rae and dividend yield, respecively. We assume ha π, W, and J are all independen. In (), is he usual noaion for he limi ε, ε 0. Under (), he sock price dynamics consis of geomeric Brownian moion wih sae-dependen volailiy, overlaid wih random umps of random magniude S(-J). Noice ha when he ump probabiliy λ approaches 0, () becomes idenical o he diffusion dynamics assumed in mos previous work on volailiy smiles (e.g. Dupire 3

5 (994), Rubinsein (994), Derman and Kani (994), Andersen and Broheron-Racliffe (998), and Andreasen (997)). Using sandard argumens (see e.g. Meron (976)), i is easy o show ha any European-syle coningen claim wrien on S will have a price F = Fb, S( ) g ha saisfies he backward parial inegro-differenial equaion (PIDE) b g F + r( ) q( ) λ( ) m( ) SF + σ (, S) S F + λ( ) E F = r( ) F, () S SS E F(, S) = E F(, J( ) S) F(, S) = F(, Sz) ζ ( z; ) dz F(, S), subec o appropriae boundary condiions for F( T, S). In (), subscrips are used o denoe parial differeniaion (so F equals F /, and so on). In (), r and q can be deduced from quoed sock forwards and bond prices. We wish o derive he remaining erms in () from prices C(, S; T, K) of European call opions 4, spanning all mauriies T and srikes K. To his end, consider he following proposiion: Proposiion : When S evolves according o (), a European call opion C(, S; T, K) saisfies he forward PIDE equaion b g C + q( T) r( T) + λ( T) m( T) KC + σ( T, K) K C + λ'( T) E ' C = q( T) C (4) z 0 T K KK + b g b g subec o C, S( );, K = S( ) K b g, and. In (4), λ'( T) = + m( T) λ( T) E' ' C, S; T, K = ( + m( T)) E J( T) C, S; T, K / J( T) C, S; T, K z = C(, S; T, K / z) ζ '( z; T) dz Cb, S; T, Kg 0 b g b g b g (3) (5) where ζ ' is a Radon-Nikodym ransformed densiy given by ζ z '( z; T) = ζ ( z; T). + m( T) Proof: The proof is based on an applicaion of he Tanaka-Meyer exension of Io's lemma. See Appendix for deails. While no necessary for our purposes, we poin ou ha i is also possible o exend (4) o he case where volailiy is sochasic 5. While our proof of (4) uses he Tanaka formula, independen work of Pappalardo (996) demonsraes ha he forward equaion can also be consruced by inegraing a ump-adused Fokker-Planck equaion 6. 4

6 In is mos general form, equaion (4) conains oo many degrees of freedom o allow for a unique process () consisen wih quoed call opion prices. For pracical applicaions, i is hus necessary o resric, hrough parameerizaion, some of he erms in (4). For insance, we could parameerize he local volailiy funcion (σ ) direcly and aemp o consruc he ump densiy ζ by solving he resuling series of inhomogeneous inegral equaions. As he resuling equaions belong o he class of Fredholm equaions of he firs kind, heir soluion is, however, quie involved and would likely require regularizaion and use of a priori informaion (see e.g. Press e al (99), Chaper 8). Insead, we prefer o parameerize he ump process and imply (non-paramerically) he local volailiy funcion σ. We will discuss his echnique in he following secion... The case of sae-dependen local volailiy and log-normal umps. As in Meron (976), we now assume ha he ump inensiy λ is consan and ha lnj is normally disribued wih consan mean µ and variance γ, such ha E[ J( )] = exp( µ + γ ). We will assume ha he consan parameers µ, γ, λ are all known, eiher from a hisorical analysis, or, as discussed furher in Secion 3, from a besfi procedure applied o quoed opion prices. To convenienly remove he dependence on u r and q, inroduce x( u) = S( u) / F(, u), u >, where F(, u) = S( )exp z [ r( τ) q( τ)] dτ is he ime forward price of S delivered a ime u. We noe ha, by sandard heory, T C(, S( ); T, K) ez r( s) ds + = E x( T) k ( ; T, k), k K / F(, T) F(, T) b g ψ. (6) From Proposiion, (6) and he assumpion of log-normal umps gives he forward PIDE b g µ γz ψ + λ' λ kψ + s( T, k) k ψ + λ' ψ ( ; T, ke ) ϕ( z γ ) dz ψ T k kk F H z I K = 0. (7) where λ λ µ + '= e γ, ϕ is a sandard Gaussian densiy, and s( u, x( u)) σ ( u, S( u)). I is clear ha if we know he funcion C(, S; T, K) and is derivaives for all T and K, hen we can consruc he local volailiy funcion σ direcly from (6)-(7) 7. In realiy, however, only a limied se of call prices C(, S; T, K) is quoed in he marke, making he inverse problem ill-posed. A variey of regularizaion echniques can be applied o overcome his problem, he simples of which involves sufficienly smooh inerpolaion and exrapolaion of known daa (see e.g. Andersen and Broheron-Racliffe (998), and Andreasen (997)). This echnique, which effecively exends he inpu price se o cover all values of K and T, will also be employed in his paper 8. An alernaive approach assumes a specific form of he local volailiy funcion (e.g. a spline as in Jackson e al (999) and Coleman e al (999)) and finds an opimal 9 fi o quoed opion prices by large-scale ieraive mehods. The exisence of a forward equaion () significanly improves he speed of such mehods, as opion prices wih differen srikes and mauriies can be priced in a single finie difference grid 0. Oher ieraive approaches along hese lines can be found, for 5

7 example, in Lagnado and Osher (997) and Avallaneda e al (996). As a general commen, we poin ou ha ieraive mehods ha are feasible in a pure diffusion seing may become prohibiively slow for ump-diffusions due o he presence of inegrals in he forward and backward equaions. To improve speed, one can imagine replacing he nonparameric specificaion of he local volailiy funcion wih a parameric form and combining his wih boosrapping of he forward equaion; we will briefly discuss such a echnique in Secion.3. To proceed, we firs wish o ransform (7) ino an equaion involving implied volailiies raher han call prices. The former is normally much "flaer" as a funcion of K and T han he laer and significanly simplifies inerpolaion and exrapolaion procedures. For he special case of s( u, x( u)) = s$ where $s is a consan, we know from Meron (976) ha ψ (, T, k) = M( ; T, k,$) s A( n) Φ( dn) k B( n) Φ( dn vn), (8) n= 0 n= 0 λ'( T ) n λ( T ) n e [ λ'( T )] e [ λ( T )] A( n) = ; B( n) = ; n! n! k T n vn = s$ ( T ) + nγ ; dn = ln + ( λ λ ')( ) + ( µ + γ ) + v n v n. In (8), Φ denoes he sandard cumulaive normal disribuion funcion. If sbu, x( u) g is no consan, we can use (6) o define an implied Meron volailiy s$( T, k ) (no o be misaken for he usual Black-Scholes implied volailiy) hrough he equaion Mb, T, k,$( s T, k) g= ψ ( ; T, k), (9) where he righ-hand side is observed in he marke. Equaions (7) and (9) allow us o express he local volailiy s( T, k) as a funcion of implied Meron volailiy s$( T, k ) : Proposiion : Defining k = K / F(, T), he local volailiy funcion σ( T, K) s T, k implied Meron volailiy s$( T, k ) in (8)-(9) as follows: s( T, k) = num / den, s$ num = T α A( n) ϕ( d ) + ( λ λ') ks$ + s$ ( T ) = b g is given by he L O n n k T n= 0 NM QP µ γ e c b gh + λ' M, T, ke, s$( k, T) + γ / ( T ) λe JM, T, k / J,$ s T, k / J, 6

8 n n kk k n= 0 den = k T A( n) ϕ( d ) α s$ + s$ wih α n F HG + nγ s$( T, K) ( T ) I KJ. L NM F I F HG K J+ + HG Proof: Follows from inserion of (9) ino (7) and a few manipulaions. α n dn α n( T ) α ndns$ k s$ s$ k T We noice ha when s$( T, k ) is consan and equal o g, say, s$( T, k) = s( T, k) = g, Proposiion reduces o µ γ b g e λe JM, T, k / J, g = λ' M, T, ke, g + γ / ( T ), (0) a resul ha can be verified by direc compuaion and will be useful in he following. Also noice ha when λ 0, Proposiion reduces o I K J O QP s( T, k) = F HG s$ / ( T ) + s$ F HG k s s d T s d s $ kk $ $ k k + $ k T T I K J I KJ which is a known expression for he ump-free case (see Andersen and Broheron- Racliffe (998), or Andreasen (997)). The infinie series in Proposiion are all well-behaved and ypically require he evaluaion of less han 5-6 erms before sufficien accuracy is achieved. For he resul in Proposiion o be useful in pracice, we only need efficien mehods of compuing he inegral erm c b gh c h z λ λ γ ( ω v) γ ( ω v) E JM, T, k / J,$ s T, k / J = ' M, T, e,$( s T, e ) ϕ( v) dv I( T, ω ) µ + γ + ωγ where we have inroduced a variable ω defined by k = e. As he funcion M( ) does no vanish for ω v, we proceed o separae ou he par of he inegrand ha correspond o some (guessed) consan volailiy g. Tha is, we define γ x γ x γ x ξ( x;, T) = M, T, e,$( s T, e ) M, T, e, g and can now wrie c h c h 7

9 z c γ ( ω v) I( T, ω ) = λ' M, T, e, g ϕ( v) dv + λ' ξ( ω v;, T) ϕ( v) dv e µ γ h z = λ' M, T, ke, g + γ / ( T ) + λ' ξ( ω v;, T) ϕ( v) dv z () where we have used (0). We are now lef wih he problem of compuing numerically z c( T, ω ) = ξ( ω v;, T) ϕ( v) dv which can be inerpreed as convoluion ξ * ϕ of he wo funcions ξ and ϕ. This suggess he inroducion of discree Fourier ransform (DFT) mehods. Specifically, assume ha we are ineresed in evaluaing c( T, ω ) as a funcion of ω on an equidisan grid ω i = ω 0 + i, i = 0,,..., N, where N is an even number and some posiive consan. We will assume ha he grid is wide enough o ensure ha ξ( ω ; 0, T ) and ξ( ω N ;, T) are close o zero 3. Wriing ξbω i;, Tg= ξi and ϕ( i ) = ϕ i, he convoluion can be approximaed by N / c( T, ω i ) ξi ϕ = N / + () where we accoun for negaive indices by assuming ha boh ξ and ϕ are periodic wih period N (a necessary assumpion in DFT). The summaion in () is, convenienly, he definiion of he convoluion operaor in he heory of DFT. Using o denoe discree Fourier ransforms, i is well-known ha c i / = ξ i ϕ i where he index i runs over N differen frequencies. ϕ can be consruced analyically (he Fourier ransform of a Gaussian densiy is anoher Gaussian densiy), whereas ξ can be compued efficienly using Fas Fourier Transformaion (FFT). Forming he complex produc of ξ and ϕ and ransforming back by inverse FFT gives us c. Noice ha he algorihm above gives he values of c for all N values of ω on he grid p simulaneously. If N = for some ineger p, FFT is of compuaional order O( N log N ). The algorihm above is hus also of order O( N log N ), a significan improvemen over a direc implemenaion of () ( O( N ) o evaluae c a all N values of ω ). In general, we need o run he algorihm above for differen values of T in some predefined grid. Wih M differen values of T, he oal effor of compuing he convoluion inegrals becomes O( MN log N )..3. A parameric boosrapping echnique. While he approach discussed in he previous secion is very fas, i relies quie heavily on iner- and exrapolaion mehods and on inpu prices being smooh and regular. To make 8

10 his mehod work, i is generally necessary o pre-condiion marke quoes carefully, as will be discussed in deail in he nex secion. Before proceeding o his, we will briefly discuss a more robus boosrapping ha works wih a discree se of opion prices. Suppose in paricular ha we wan hi a range of opion prices wih he mauriies: 0 0 = T < T < K< T N Firs, we choose a disribuion of he umps. As in he previous secion, one could for example assume ha J is log-normal wih bes-fied parameers µ, λ, γ. For each inerval ] Ti, Ti+ ] le he local volailiy be given by σ(, S ) = g ( S ; a i ) where g is some funcion defined by is parameer vecor a i. Saring wih i = 0 we now repeaedly solve (7) over he inerval ] Ti, Ti+ ] for changing values of he parameers a i, unil an opimal fi o he observed opion prices is obained. A good choice for updaing he parameer vecor a i is he Levenberg-Marquard rouine described in Press e al (99). Once he opimal a i is found we proceed o he nex ime sep. If we wish o find a perfec fi o he observed opion prices each parameer vecor a i mus have a dimension ha is a leas as high as he number of quoed opion prices wih mauriy T i+. One would hink ha he updaing and he numerical soluion of he PIDEs would prohibi he pracical applicaion of his algorihm, bu his is no he case. Using he numerical PIDE soluion algorihm ha we presen below we are ypically able o fi o a 0 0 grid of observed opion prices in abou 5 seconds on a Penium PC. 3. Fiing he local volailiy funcion: an example from he S&P500 marke. In his secion we illusrae he heory of he previous secion by an example based on daa from he S&P500 index. We will use he mehod oulined in Secion. and consequenly assume ha umps are log-normal wih consan parameers ( µ, γ, λ). In April 999, he bid-offer implied Black-Scholes volailiies of European call opions on he S&P500 index were as shown in Table. Wih a consan ineres rae of r = 559%. and a consan dividend yield of q = 4%. he bid-offer spreads correspond o bid and offer opion price spreads from mid as given in he second column of Table. 3.. Jump parameer fiing. To deermine he ump parameers, we firs do a bes fi (in a leas-squares sense) of he Meron model (8) o he mid implied Black-Scholes volailiies of Table. The resuling bes-fi parameers are σ = 7. 65%, λ = 8. 90%, µ = %, γ = 4505%.. Measured in erms of implied Black-Scholes volailiies, he oal RMS error in he fi o he opions is Table is 0.04 wih he larges difference for any opion being Ineresingly, he bes-fi coninuous volailiy σ is close o wha one obains by imeseries esimaion on hisorical S&P500 daa (for insance, using he pas 0 year s ime- 9

11 T Table : Bid/Ask implied Black-Scholes volailiies in S&P500, April 999 Srike (K) Bid /Offer Bid Ask Bid Ask Bid Ask Bid Ask Bid Ask Bid Ask Bid Ask Bid Ask Bid Ask Bid Ask Bid Ask Bid Ask Marke bid and offer implied volailiies for he S&P 500 index. Srikes (K) are in percenage of iniial spo and mauriies (T) are measured in years. The second column repors approximae opion price bid and ask spreads from mid in basis poins (/00 of pc) of he spo index value. Volailiies are expressed in percen. Blank cells mean ha he are no observaions for ha paricular mauriy and srike. The ineres rae and dividend yield are 5.59% and.4%, respecively. series of daily S&P500 reurns we ge a hisorical volailiy of approximaely 5.0%). The mean ump in reurn is m = %. This number, and he esimaed ump inensiy λ, are higher han wha one would expec from ime-series daa, and eiher indicae ha he marke currenly perceives he chance of a big crash o be higher han normal or, more likely, ha he ump parameers include significan elemens of risk aversion ( marke price of risk ). Indeed, all parameers above are esimaed in he marke risk-neural measure, and, wih he excepion of he diffusion volailiy, do no generally equal he obecive ( hisorical ) parameers. The relaionship beween he risk-neural and obecive probabiliy measures is governed by he ump-exended Girsanov Theorem (see Borck e al 997). Working in a general equilibrium framework, he link beween he wo 0

12 probabiliy measures can be characerized in erms of he uiliy of a represenaive agen. Secion 3.3 conains a brief analysis along hese lines, and demonsraes ha he parameers esimaed above are acually quie reasonable. 3.. Local volailiy fiing. To consruc a local volailiy surface ha fis he inpu volailiies of Table, we firs conver he bid and offer implied volailiies of Table ino a grid of implied Meron volailiies (as in (9)). We hen generae a smooh surface of hese volailiies ha lies inside he bid and offer spread, and exrapolae o he unobservable corners of he volailiy grid. The smoohing/exrapolaion procedure used is described in deail in Andreasen (997); i involves numerically solving an opimizaion problem wih quadraic obecive funcion and linear consrains. The resuling grid of Meron volailiies is given in Table. As expeced from he relaively igh fi of he consan-volailiy Meron model, he implied Meron volailiies are fairly, hough no perfecly, fla. The Meron volailiies can now be inerpolaed and exrapolaed in ( T, k ) space, as discussed in Andersen and Broheron-Racliffe (998). Here we use a wo-dimensional ensor-spline (Dierckx (995)) which guaranees smoohness in boh T- and k-direcions, and convenienly allows for closed-form compuaions of he derivaives needed in he formula in Proposiion. Figure shows he resuling insananeous local volailiies σ(, S ). The local volailiy surface is, essenially, U-shaped and quie saionary hrough ime. For comparison, Figure shows he local volailiies obained by fiing a pure diffusion (DVF) model o he daa in Table. No surprisingly, Figure shows ha he local volailiies of he DVF model needs o be seep and highly non-saionary in order o fi he S&P500 daa. The impac of his non-saionariy on opion prices will be examined in Secion 5. Table : Smooh Implied Meron Volailiies for S&P500 Srike (K), % of Spo T Smooh implied Meron volailiies for he S&P500 index. Volailiy numbers are in %. Mauriies are in years and srikes are in percenage of iniial spo. Jump parameers are λ = 8. 90%, µ = %, γ = %. The ineres rae and dividend yield are 5.59% and.4%, respecively.

13 Figure : Local Diffusion Volailiies for he S&P500 Index, April % 7% 3% 39% 46% 55% 66% 79% 94% 3% 34% 6% 9% 9% 73% 36% 390% Local volailiies for ump-diffusion model when fied o S&P500 opion prices. Firs axis is fuure spo relaive o curren spo and second axis is ime in years. The local volailiies are generaed on a 50x56 grid. Jump parameers are λ = 8. 90%, µ = %, γ = %. The ineres rae and dividend yield are 5.59% and.4%, respecively General equilibrium analysis. As discussed earlier, he ump parameers lised in Secion 3. are esimaed under he risk-neural probabiliy measure. To gauge wheher he parameers are reasonable, we here briefly wish o demonsrae ha our exreme-appearing parameer values are in fac no inconsisen wih general equilibrium heory. Indeed, i is well-known ha economic heory ha moving from he obecive probabiliy measure o he risk-neural probabiliy measure resuls in higher ump inensiy, lower mean ump, and unchanged coninuous volailiy. The laer is required is, of course, required for he wo probabiliy measures o be equivalen and consiues a necessary condiion for absence of arbirage. One can use he analysis in Naik and Lee (990) o deduce ha if he marke has a represenaive agen ha maximizes expeced addiive separable power uiliy of fuure consumpion, hen he risk-neural parameers of he ump-diffusion model are linked o he hisorical parameers hrough he relaions σ σ λ λ β µ h + γ h =, = e b g ( / ), µ = µ ( β) γ, γ = γ, h h h h h

14 where subscrip h indicaes parameers under he obecive measure, and β is he (consan) relaive risk aversion of he represenaive agen. Obviously, if he represenaive agen is risk-averse and he mean ump is negaive, hen he ump inensiy and he magniude of he mean ump will boh increase under he risk-neural measure. Le us use he resul above in a rough quaniaive analysis. In he 0 h cenury here were wo large umps in he S&P500 index (99 and 987) each of a magniude of approximaely 30%. If we condense his informaion ino a hisorical ump inensiy of abou % and a mean ump of -30% and use our esimae of γ = h γ = (which implies ha µ h = ), hen given our implied risk-neural ump parameers, a bes-fi soluion for he relaive risk-aversion is β * = 339. This level of risk-aversion is by no means excessive and falls well in line wih oher esimaes of he relaive risk-aversion. This bes-fi level relaive risk-aversion correspond o he risk-neural ump-parameers * * * λ = 6. 70%, µ = 4. 6%, m = %. These parameers are quie close o our implied risk-neural ump parameers indicaing ha he bes-fi soluion is indeed a good fi. Naik and Lee s analysis shows ha he equilibrium expeced excess reurn of he sock over he risk-free rae is given by α r = ( β) σ + ( m λ mλ) h h h h Using hisorical parameers of σ h = 05., λ h = 0. 0, mh = and our implied riskneural ump parameers combined wih our bes-fi relaive risk-aversion we ge an expeced excess reurn of around.9%, which again is no inconsisen wih empirical daa. 4. Numerical mehods for opion pricing in he ump-diffusion model. So far we have spen mos of our effors calibraing he ump-diffusion model o he marke for European call opions. For our model o be useful in pracice, we need o consider numerical mehods o efficienly price general coningen claims saisfying he backward PIDE (). A relaed problem is he numerical soluion of he forward equaion (4) or (7), which would ypically be required in ieraive calibraion mehods (see discussion in Secions. and.3). Very lile maerial has been published in he finance lieraure on numerical mehods for PIDEs of he ype occurring in ump-diffusion models. A few excepions include Amin (993), Zhang (993), and Andreasen and Gruenewald (996). The mehods In he sense of minimizing he RMS of relaive errors on he risk-neural parameers λ and µ. 3

15 suggesed in he wo firs papers are essenially mulinomial rees, i.e. explici mehods. Explici mehods generally suffer from insabiliy problems as well as poor convergence in he ime-domain. As is well-known from he finie difference lieraure, implici mehods ypically exhibi beer precision, convergence and sabiliy properies han explici mehods and are preferable for opion pricing problems. The paper by Andreasen and Gruenewald presens such an implici mehod ha solves he pricing PIDE on a single Crank-Nicholson finie difference grid, wih each ime-sep involving he inversion of a non-sparse marix. In he consan-parameer seing of Andreasen and Gruenewald, he marix inversion urns ou o be compuaionally feasible. In our seing wih ime- and price-dependen volailiy funcions, anoher approach is needed. In he following subsecion we describe an accurae and efficien soluion echnique ha can be used under our model assumpions. Figure : DVF Local Volailiies for he S&P500 Index, April % 3% 3% 40% 53% 7% 93% 3% 63% 5% 84% 375% 496% Local volailiies for pure diffusion model when fied o S&P500 opion prices. Firs axis is fuure spo relaive o curren spo and second axis is ime in years. The local volailiies are generaed on a 50x56 grid. The ineres rae and dividend yield are 5.59% and.4%, respecively. 4.. The FFT-ADI finie difference mehod. We firs noice ha afer appropriae logarihmic ransformaions he PIDEs considered so far in his paper ((), (4), and (7)) can all be wrien in he form: 4

16 L NM = F + + r a + b x 0 O + QP z+ F λ ς (, x y) F(, y) dy λf (3) x where 4 ς(, ) is a densiy funcion, and r = r( ), a = a(, x), b = b(, x), λ = λ ( ). Defining D r + a b + x x and using he convoluion operaor * we can wrie (3) in he more compac form 0 = F + DF λf + λς* F (4) Inerpreing F( ) = F(, ) we can discreize (4) in he ime-dimension as follows b g θ C θ C 0 = F( + ) F( ) + D F( ) + ( ) F( + ) + λ( + ς *) θ F( ) + ( θ ) F( + ) J J (5) where θ, θ [ 0, ] are consans. Rearranging (5) yields C J / θ D θ λ( + ς *) F( ) C J = / + ( θ ) D + ( θ ) λ( + ς *) F( + ) C J (6) There are various ways of arranging (6) for numerical soluion. The mos obvious, corresponding o a sandard Crank-Nicolson finie difference scheme, θ C = θ J = /, is no pracically feasible because afer discreizing he x-space ino N poins inversion of a full N N marices is required, a compuaionally cosly procedure of order N 3 per ime sep. Noe ha full marix inversion has o be performed a every sep since he parameers vary in boh ime and sae. The sae-dependen parameers also preclude use of Fourier ransform echniques o solve he inversion problem. Explici schemes, θ C = θ J = 0, are compuaionally feasible bu poenially unsable and suffer from he drawback ha heir convergence in he ime domain are only of O( ), unlike Crank-Nicolson schemes ha have precision of O( ). When using FFT echniques o handle he convoluion inegral, he compuaional order of he explici scheme is O( N log N) per ime sep. Schemes of he ype θ C = /, θ J = 0 are sable and efficien bu accuracy is los due o he asymmeric reamen of he coninuous and ump par. Numerical experimens show ha biases are inroduced in he soluion, paricularly for long daed opions. In our experience, he bes numerical soluion mehod is an ADI (Alernaing Direcions Implici) mehod where each ime-sep in he grid is spli ino wo half-seps. For he firs half-sep we se θ =, θ = 0, which gives us C J 5

17 L NM / O L NM D F( + / ) = λ+ λς * F( + ) QP / QP O (7a) In a discree grid his can be solved by firs compuing he convoluion ς* F( + ) in discree Fourier space, where ς * F( + ) = ς F( + ). If we observe ha ς only needs o be compued once, he compuaional cos associaed wih he convoluion par of (7a) is one FFT and one inverse FFT, i.e. O( N log N). We furher noe ha he discree version of he differenial operaor D is a ri-diagonal marix. Consequenly, once he RHS of (7a) is obained by FFT mehods, hen he sysem (7a) can be solved a a cos of O( N ). Hence, he oal cos of solving (7a) is O( N log N). For he second half-sep we se θ = 0, θ =, whereby L NM O L NM + λ λς * F( ) = + / QP / C J O + QP D F( / ) (7b) Leing y = / + D F( + / ), now ake he Fourier ransform of (7b) o arrive a b / + λg F( ) λ ς F( ) = y F( ) = y / c / + λ λ ς h. (8) We can now ransform he equaion back o obain F( ). All in all his requires one ridiagonal marix muliplicaion, one FFT and one inverse FFT, i.e. a procedure wih a compuaional burden of O( N log N). To formally specify he discree scheme described in (7a-b) we define he operaors δ x f ( x ) = x f ( x + x) f ( x x), δ xx f ( x) = f ( x + x) f ( x) + f ( x x), ( x) D fb xg= r + aδx + b δ xx f ( x); ς * f ( x) = q ( x) f ( x), q ( x) = ς ( x y) dy. zb + / g x b / g x We can hen wrie he discree version of (7a-b) as b g b g / D F + / = / λ+ λς * F +, (9a) 6

18 / + λ λς * F( ) = / + D F( + / ). (9b) The following proposiion describes some imporan properies of he scheme (9a-b). Proposiion 3 The following properies hold for he scheme (9a-b): (i) (ii) (iii) The scheme is uncondiionally sable in he von Neumann sense. For he case of deerminisic parameers, he numerical soluion of he scheme is locally accurae o order O( + x ). If M is he number of ime seps and N is he number of seps in he spaial direcion, he compuaional burden is O( MN log N ). Proof (iii) was shown above. (i) and (ii) are shown in Appendix. 4.. Refinemens While he scheme described by (9a-b) is aracive in ha i is uncondiionally sable and only requires O( N log N) operaions per ime sep, a direc applicaion suffers from cerain drawbacks. Specifically, accurae represenaion of he convoluion inegral will generally require a very wide grid. Since he FFT algorihm only acceps uniform sep lengh in he x-direcion, he precision of he numerical soluion in areas of ineres migh suffer. To overcome his, we here define an algorihm ha assumes lineariy of he opion price ouside a grid of size equal o a number of sandard deviaions of he underlying process. The linear par can convenienly be solved in closed-form 5 whereas he inner grid is solved using he FFT-ADI-algorihm described in (9a-b) above. We spli he funcion in wo pars: 6 On x F = F + F G + H, x ( x, x) x ( x, x) ( x, x), G solves b g. 0 = G + + = G DG λ ς * F + DG λg + λς * G + H If we assume ha H is linear in e x we can wrie x x H(, x) g ( ) e + h ( ) + g ( ) e + h ( ) l l x< x u u x> x where gl, hl, gu, hu are deerminisic funcions. This means ha we can wrie 7

19 x ς* H(, x) g ( ) e + m( ) Pr' x + ln J( ) < x + h ( ) Pr x + ln J( ) < x l x + g ( ) e + m( ) Pr' x + ln J( ) > x + h ( ) Pr x + ln J( ) > x u b g b g l b g b g b g u b g (0) where Pr( ) denoes probabiliy under he disribuion defined by ς and Pr'( ) denoes probabiliy under he disribuion described by he Radon-Nikodym ransformed densiy: ς '(, x) = ς (, x) e / b x + m( ) g. In he Meron (976) case of log-normal umps hese probabiliies can be compued in closed-form as Gaussian disribuion funcions. If he disribuion of he umps is non-parameric, he probabiliies can be calculaed by simple numerical inegraion over he densiies ς, ς '. We now ge he following sysem b g () / D G( + / ) = / λ G( + ) + λς * G( + ) + λς * H( + ), / + λ λς * G( ) = / + D G( + / ) + λς * H( ), where erms of he ype ς*g are handled numerically by FFTs and erms of he ype ς* H are handled by expression (0). The assumpion of lineariy of H is equivalen o saing ha he funcions gl, gu, hl, hu can be obained from he asympoes of he funcion defined by rf = f + r( ) q( ) f b g x () subec o he same boundary condiions as (3). Over a discree ime-sep () has he closed-form soluion c b g h (3) f (, x) = e r f +, x + r( ) q( ) (3) ogeher wih he boundary condiions define gl, gu, hl, hu. The scheme described above scheme can be used for mos applicaions, including barrier opions and opions wih Bermudan or American syle exercise. 4.3 A Numerical Example In his secion we give a quick example of he pracical performance of he mehod ha has been oulined in he previous secion. Table 3 below compares Meron s (976) exac formula for European pus and calls (equaion (8)) wih he prices generaed by he algorihm (9a-b), refined as discussed in he previous secion. To sress he algorihm, he ump parameers have been se o fairly exreme values: r = 0. 05, q = 0. 0, σ = 05., λ = 0., γ = 0. 4, S( 0) = 00, K = 00 The number of ime seps is se o half he number of x-seps. Also, due o he usage of he FFT algorihm, he number of x-seps have been se o muliples of. 8

20 Table 3 also liss CPU imes and he experimenal convergence order of he mehod, he laer compued as he average slope of a log-log plo of absolue error agains he ime sep. I is clear form he able ha he convergence of he algorihm is smooh and approximaely of order in he number of ime- and x-seps a lile higher for shor-daed opions and a lile lower for long-daed opions. This experimenally confirms he second saemen of Proposiion 3. In order o obain accuracy o one basis poin, Table 3 shows ha generally 5 seps in he x-direcion are necessary. CPU ime for such a calculaion is less han second on a 400 MHz Penium PC. Table 3: Prices of European Calls and Pus using ADI-FFT PIDE Solver T = 0.0, µ = -.08 T = 0.0, µ = 0.9 T=, µ = -.08 x-seps Pu Call Pu Call Pu Call CPU Time Closed-form Conv. Order NA T =, µ = 0.9 T = 0, µ = -.08 T=0, µ = 0.9 x-seps Pu Call Pu Call Pu Call CPU Time Closed-form Conv. Order NA European pu and call opion prices of Meron model compued using FFT-ADI mehod wih differen number of sae space seps on he main grid. The number of ime seps is se o half he number of x- seps. CPU imes are in seconds. The process parameers are r = 0. 05, q = 0. 0, σ = 05., λ = 0., γ = 0. 4, S( 0) = 00. 0, K = Mone Carlo simulaion. The finie-difference mehod oulined in he previous secions is primarily useful for opions wih mild pah-dependency (such as American opions and barrier opions), bu is difficul o apply o opions han depend more srongly on he pah of he underlying sock. For such opions, Mone Carlo simulaion mehods are generally useful (see Boyle e al (997) for a good review). Once he mehods in Secion 3 have been applied o deermine he local volailiy funcion, he SDE () can be simulaed direcly in an Euler 9

21 scheme. For each ime sep one would deermine wheher here is a ump or no by randomizing over he ump probabiliy ( λ ) and hen subsequenly randomize over he ump disribuion o deermine he size of he ump. This procedure, however, is compuaionally inferior o oher mehods ha explicily exploi he independence of he umps and he Brownian moion. One such procedure is described below 7. Le { τ } =,, K be he arrival imes of he Poisson process π. We know ha { τ + τ } =,, Kare muually independen wih disribuion given by d i e τ + s Pr τ + τ > s = exp z λ( u) du τ For each pah we wish o simulae we use his o draw he arrival imes for he paricular pah up o our ime horizon ha we are considering. We hen consruc an increasing simulaion ime line ha includes he ump imes and our ime horizon, say { i } i=0, K,. The price process is now simulaed as b ig b ig b g z i i λ 0 x S = F 0, e i, i x( ) = ( u) m( u) du σ k, x( k ) ( k+ k ) i k = 0 + σ, x( ) ε( ) + { τ } ln J( ) k= 0 b g k k k + k k b and { ε( i )} i=0, K is a sequence of independen sandard normal random variables, and { J( τ )} =,, K is a sequence of independen random variables drawn according o he marginal disribuions { ς ( τ ; )} =,, K. The simulaion scheme described by (4) is O( ) convergen o he rue sochasic differenial equaion and ensures ha simulaed sock prices have expecaions equal o heir forwards. Higher order accuracy simulaion schemes can be consruced using he Taylor-expansion echniques described in Kloeden and Plaen (99). 5. Opion Pricing: Numerical Tess. In his secion we will combine he calibraion resuls from Secion 3 wih he pricing algorihm of Secion 4. Firs, we invesigae wha evoluion of he volailiy smile is implied in he model. Second, we price a range of sandard opion conracs. Throughou, we compare he resuls of he ump-diffusion model o he resuls of he DVF model. Boh he DVF and ump-diffusion calibraions were esed for accuracy by numerically pricing all call opions in he inpu se (Table ); in all cases, he compued prices were wihin he bid-offer spreads. g i k = k k (4) 5.. Evoluion of volailiy skew. To illusrae he differences beween he pure diffusion model and he umpdiffusion model, i is illuminaing o invesigae he volailiy skews ha he wo models 0

22 generae a fuure daes and sock price levels. In Table 4 we show he implied Black- Scholes volailiies for -year call opions a differen imes and fuure levels of he underlying index in he ump-diffusion model. Table 4 shows ha he volailiy skew of he ump-diffusion model is surprisingly sable over ime and sock price levels. This is no he case for he fied DVF model, as is obvious from Table 5 below. In paricular, we noice ha he fuure implied volailiy skews of he fied pure diffusion model are highly non-saionary and end o flaen ou as ime progresses. In a few cases, he implied volailiy surface even urns ino a smile or oherwise becomes non-monoonic in he srike. Table 4: Fuure -year S&P500 volailiy skews in ump-diffusion model Srike (K), % of Fuure Spo (S) S() = 70% of oday s spo = = = = = S() = 00% of oday s spo = = = = = S() = 30% of oday s spo = = = = = Fuure implied volailiy skews in he ump-diffusion model fied o S&P daa. The implied Black-Scholes volailiies are for -year opions. The firs column repors he ime in years. For each of he hree sock price levels, srikes are repored in percen of he sock price level Prices of exoic opions. The differen evoluion of he implied volailiy skew in he ump-diffusion and DVF models obviously will have consequences for he pricing of exoic opions ha, unlike European pus and calls, depend on he full dynamics of he sock price, raher han us he disribuion a a single poin in ime. Opions ha fall in his caegory include compound opions and barrier opions, bu also mos near- vanilla conracs, including American/Bermudan opions, forward saring opions, and Asian opions. Table 6 below

23 repor he prices of some of hese conracs in he ump-diffusion and DVF models fied o he S&P500 daa in Secion 3. I is obvious from he resuls in Table 6 ha he ump-diffusion and DVF models reurn significanly differen prices for Bermudan and forward saring opions, and ha his difference grows as he mauriy is increased. The differences in he Asian opion prices are smaller bu sill significan. Table 5: Fuure -year S&P500 volailiy skews in he pure diffusion model Srike (K), % of Fuure Spo (S) S() = 70% of oday s spo = = = = = S() = 00% of oday s spo = = = = = S() = 30% of oday s spo = = = = = Fuure implied volailiy skews in he pure diffusion (DVF) model fied o S&P daa. The implied Black- Scholes volailiies are for -year opions. The firs column repors he ime in years. For each of he hree sock price levels, srikes are repored in percen of he sock price level. 6. Conclusion. This paper has presened a framework for adding Poisson umps o he sandard DVF (Deerminisic Volailiy Funcion) diffusion models of sock price evoluion. We have developed a forward PIDE (Parial Inegro-Differenial Equaion) for he evoluion of call opion prices as funcions of srike and mauriy, and shown how his equaion can be used in an efficien calibraion o marke quoed opion prices. To employ he calibraed model o pricing of various exoic opions, we have developed an efficien ADI (Alernaing Direcion Implici) finie-difference echnique wih aracive sabiliy and convergence properies. Applying our calibraion algorihm o he S&P500 marke resuls in a largely consan diffusion volailiy overlaid wih a significan ump componen. For he S&P500 marke, we find diffusion volailiies around 5-0% and, working in he risk-neural probabiliy measure, a 9% (annual) chance of an index drop averaging around 50%. While