THE IMPORTANCE OF JUMPS IN PRICING EUROPEAN OPTIONS F. Campolongo (1)1*, J. Cariboni (1),(), and W. Schouens () 1. European Commission, Join Research Cenre, Via Fermi 1, Ispra 100, Ialy. K.U.Leuven, U.C.S., W. De Croylaan 54, B-3001 Leuven, Belgium Absrac: The screening mehod proposed by Morris [1] and recenly improved by Campolongo e al. [] has been employed o esimae he imporance of he inclusion of jumps in a model for pricing European opions. Resuls confirm ha, among he sources of unconrollable uncerainy, jumps play a major role and herefore need o be beer invesigaed in order o improve he accuracy of he model predicions. Keywords: opion pricing, Heson model, jumps, sensiiviy analysis, Morris mehod, variance based sensiiviy indices. 1. SETTING THE PROBLEM Imagine an invesor eners ino a European call opion conrac. A European call opion is a conrac ha gives he holder he righ (no he obligaion) o buy an underlying asse by a cerain dae for a cerain price. The price in he conrac is known as he exercise price or srike price; he dae in he conrac is known as he expiraion dae or mauriy. To ener ino an opion conrac here is a cos * Corresponding auhor: F. Campolongo. E-mail: francesca.campolongo@jrc.i; Ph: +39(033)785476; Fax: +39(033)785733 1
corresponding o he righ purchased. This cos, referred o as premium, is he price of he opion a presen day. The opion price is esablished according o he heory of arbirage-free pricing. Classic deerminisic arbirage involves buying an asse a a low price in one marke whils immediaely selling i a a higher price in anoher marke o make a risk-free profi. The heory of arbirage-free prices imposes ha he prices of differen insrumens mus be relaed o one anoher in such a way ha hey offer no arbirage opporuniies. In pracice o price he opion we make use of a model describing he evoluion hrough ime of he underlying asse price and hen impose no arbirage argumens. The risk associaed wih an opion conrac derives from he unknown evoluion of he underlying price on he marke. This risk is no reducible and is an inrinsic feaure of he conrac iself. Apar from his risk, neiher conrollable nor reducible, here is anoher par of risk which is ha deriving from he fac ha he curren opion price is an esimaed quaniy, poenially affeced by an error. If for insance he call opion price is overesimaed, he opion holder faces he risk of losing more money han wha he should (in case of loss). The more accurae he price esimae, he less he risk associaed wih he opion. The curren opion price is calculaed via a mahemaical model (describing he evoluion of he underlying) ha conains a number of inpu variables whose values are affeced by uncerainy. The problem addressed in his paper is ha of quanifying he uncerainy affecing he curren opion price, idenifying is main sources and provide indicaion on how o reduce he conrollable risk. In paricular, we aim o assess he imporance of incorporaing he effecs of jumps in modelling he underlying dynamics. The model chosen for describing he underlying sock price evoluion is he Heson Sochasic Volailiy model (HEST, [3]) wih jumps [4], where he sock price follows he Black-Scholes sochasic differenial equaion SDE in which he volailiy behaves sochasically over ime and jumps are included in he dynamics of he asse. Differen scenarios are assumed, corresponding o differen possible srike prices and imes o mauriy. Resuls show ha jumps drive mos of he uncerainy in he esimaed opion price, hus confirming heir key role in he pricing process. The imporance of jumps is more eviden for higher srike prices.
. THE HESTON STOCHASTIC VOLATILITY MODEL WITH JUMPS In [3] Heson inroduced a sock price model which inroduced sochasic volailiy in he famous Black Scholes [5]. In he Heson model he price pahs of a sock (and he volailiy) are coninuous. Laer, exensions of he Heson model were formulaed ha also allowed for jumps in he sock price pahs. We invesigae here his exension and ask ourself he quesion wheher he inroducion of jumps leads o a significan increase in he variance of he prices of derivaives under his model. We denoe he sock price process by S = { S, 0} and assume ha he sock pays ou a coninuous compound dividend yield of q 0. Moreover, we assume in our marke we have a riskless bank accoun available, paying ou a coninuous ineres rae r. In he Heson Sochasic Volailiy model wih jumps (HESJ) he sock price process is a modelled by a SDE given by: ds d = ( r q λµ ) d + σ dw + J d N S J 0 0 where N = {N, 0} is an independen Poisson process wih inensiy parameer λ>0, i.e. E N ] = λ [. J is he percenage jump size (condiional on a jump occurring) ha is assumed o be log-normally, idenically and independenly disribued over ime, wih uncondiional mean µ J. The sandard deviaion of log( 1+ J ) is σ J : log(1 + J σ J ) ~ N log(1 + µ ), j σ J. The (squared) volailiy follows he classical Cox-Ingersoll-Ross (CIR) process: ~ d σ = k η σ ) d + θσ dw σ 0, ( 0 ~ ~ where W = { W, 0} and W = { W, 0} are wo correlaed sandard Brownian moions such ~ ha Cov[ dw, dw ] = ρd. Finally, J and N are independen, as well as of The characerisic funcion φ ( u, ) is in his case given by: W and W ~. 3
φ( u, ) = E{exp(iu log( S ) S = exp(iu(log S exp( ηκθ exp( σ θ 0 0, σ } = + ( r q) )) (( κ ρθui d) log((1 ge (( κ ρθui d)(1 e exp( λµ ui + λ((1 + µ ) J 0 0 J ui d ) /(1 ge ) /(1 g)))) )) exp( σ ( ui/)( ui -1)) -1)) J d d where d = (( ρθui κ) θ ( ui u g = ( κ ρθui d)/( κ ρθui + d). Pricing of European call opions under his model can be done by he Carr and Madan [6] pricing mehod. This mehod is applicable for classical vanilla opions when he characerisic funcion of he risk-neural sock price process is known. Le α be a posiive consan such ha he α -h momen of he sock price exiss. For all sock price models encounered here, ypically a value of α =0.75 will do fine. Carr and Madan hen showed ha he price C ( K, T ) of a European call opion wih srike K and ime o mauriy T is given by: where exp( α log( K)) C( K, T ) = π + exp( rt ) φ( υ ( α + 1)i), T ) = α + α υ + i(α + 1) υ 0 )) 1/ exp( iυ log( K)) ρ( υ) dυ, exp( rt ) E[exp(i( υ ( α + 1)i) log( S ρ( υ) = α + α υ + i(α + 1) υ Using Fas Fourier Transforms, one can compue wihin a second he complee opion surface on an ordinary compuer., T ))] 3. THE SENSITIVITY ANALYSIS METHODOLOGY A sensiiviy analysis mehod very efficien in idenifying imporan facors in a model wih jus very few model evaluaions is he design proposed by Morris [1]. The mehod requires a oal number of 4
model evaluaions which is a linear funcion of he number of inpu facors involved, and i does no rely on sric assumpions abou he model, such as for insance addiiviy or monooniciy of he model inpu-oupu relaionship. The Morris mehod has several desirable feaures of a sensiiviy analysis mehodology. I is concepually simple, easy o implemen, and is resuls are easily inerpreed. I can cope wih he influence of scale and shape because i is sensiive no only o he effec of he ranges of inpu variaion bu also o he form of heir probabiliy densiy funcion (pdf). I can be regarded as a global mehod (in conras o he derivaive-based ones) as he final sensiiviy measure is obained by averaging a number of local measures (he elemenary effecs), compued a differen poins of he inpu space. In very recen work [] Campolongo and coworkers proposed an improved version of he Morris measure µ, denoed as µ*, which is more effecive in ranking facors in order of imporance. Furhermore, he new measure could be exended o he capabiliy o rea groups of facors as if hey were single facors, anoher desirable propery of a sensiiviy analysis. In his work we employ he new measure µ* o assess he sensiiviy of an opion price o he several uncerain facors deermining is value; and in paricular focus our aenion on he imporance of inroducing jumps in he dynamics of he underlying asse. As a furher confirmaion of he reliabiliy of he new measure, resuls obained according o µ* are also compared wih hose obained hrough he use of he variance based mehods. Variance based mehods have all he desirable properies menioned above and he furher grea advanage of an easy inerpreaion in erms of oupu variance decomposiion. A variance based mehod esimaes he percenage of oupu variance ha each facor is accouning for. However, as a drawback, hese mehods require a compuaional effor ha in some insances may be prohibiive, as he number of model evaluaions required is almos imes higher han ha for he Morris sraegy. The Morris mehod and is improved version The experimenal plan proposed by Morris [1] is composed of individually randomized 'one-facor-a-aime' experimens: he impac of changing one facor a a ime is evaluaed in urn. Each inpu facor may assume a discree number of values, called levels, which are chosen wihin he facor range of variaion. 5
The sensiiviy measures proposed in he work of Morris are based on wha is called an elemenary effec. The elemenary effec for he ih inpu is defined as follows. Le be a predeermined muliple of 1/(p-1). For a given value of x, he elemenary effec of he ih inpu facor is defined as [ y( x,.., x + =, x, x +,.., x ) y(x)] EE (x) 1 i 1 i i 1 k i where x = ( x1, x,..., xk ) is any seleced value in Ω such ha he ransformed poin (x + ei ) - being a vecor of zeros bu wih a uni as is ih componen - is sill in Ω for each index i=1,..,k. The finie disribuion of elemenary effecs associaed wih he ih inpu facor, is obained by randomly sampling differen x from Ω, and is denoed by F i. In Morris [1] wo sensiiviy measures were proposed for each facor: µ, an esimae of he mean of he disribuion F i, and σ, an esimae of he sandard deviaion of e i F i. A high value of µ indicaes an inpu facor wih an imporan overall influence on he oupu. A high value of σ indicaes a facor involved in ineracions wih oher facors or whose effec is non-linear. In Campolongo e al. [] we consider a hird sensiiviy measure, µ*, which is an esimae of he mean of he disribuion, here denoed as G i, of he absolue values of he elemenary effecs. For non monoonic models, he measure µ* performs beer han µ []. In fac, if he disribuion F i conains elemens of opposie sign, which occurs when he model is non-monoonic, when compuing is mean some effecs may cancel each oher ou. Thus a facor which is imporan bu whose effec on he oupu has an oscillaing sign may be erroneously considered as negligible, generaing a mosly undesirable Type II error. The variance based measures Variance based mehods choose as a measure of he main effec of a facor V esimaion of quaniy X i X i on Y, and denoed by ( E ( Y X )) X i V (Y ) i X i on he oupu, an, which is known in he lieraure as he firs order effec of S i. Reasons for his choice are deailed in [7]. 6
Anoher sensiiviy measured based on he variance decomposiion is he oal sensiiviy index, which esimaes he sum of all effecs involving a given facor E ( V ( Y )) X i X i X i V (Y ) X i. S T, i S Ti is esimaed by he quaniy. The oal index is he appropriae measure o use when he goal is ha idenifying irrelevan facors, i.e. hose ha can be fixed o any given value wihin heir range of variaion because hey are non influen on he oal oupu variance. A necessary and sufficien condiion for facor X i o be oally non-influen is ha S Ti = 0. In fac, if facor X i is oally non-influen, hen all he variance is due o i E X, and fixing his vecor resuls in ( Y ) = 0 V X i ( V ( Y X X X ) = 0. The reverse is also rue: if ( Y ) = 0 i i i X i, hen X i is non-influen, so ha S Ti 0. V X i i X, as well as in i X a all fixed poins in he space of Alhough very accurae and reliable, he variance based measures have a disadvanage in heir compuaional cos. These mehods require a number of model evaluaion which is N ( k + ), where k is he number of inpu facors and N is of he order of N = 500, 5000, [7]. When k is large or he model is very ime consuming he required compuaional ime may be unaffordable. 4. THE SENSITIVITY ANALYSIS EXPERIMENT The sensiiviy measure µ* proposed in [] is employed here o assess he imporance of he inroducion of jumps in he dynamics of he sock price. Furhermore, as a furher confirmaion of he reliabiliy of his measure, we compue he variance based indices {S i,s } and compare hem wih he Morris resuls. Noe ha he compuaional cos o esimae he variance based indices in his case is no a problem as he model under examinaion is no excessively ime consuming. The inpu facors seleced for sensiiviy analysis purposes are lised in Table 1 wih he corresponden disribuions. The inpu facors in Table 1 can be disinguished in wo groups: hose whose value can be esimaed wih a cerain degree of confidence by looking a real daa, and herefore represen a source of T i 7
uncerainy ha can be defined as conrollable ; and hose ha canno be checked wih marke daa and are herefore regarded as compleely unconrollable. The firs group consiss of he iniial condiion for he dynamics of he volailiy σ 0, he dividend yield q and he ineres rae r. The remaining inpus, among which he jumps parameers, belong o he second group. The Morris measure µ*, he firs sensiiviy indices S i and oal sensiiviy indices ST i are compued for each inpu facor in 4 differen scenarios, a scenario being deermined by a differen value of he opion srike price and of he ime o mauriy. Seven values of he srike price are considered (srike price =,,, ) o represen siuaions in which he opion is in he money, a he money or ou of he money (he iniial condiion for he sock is fixed a S 0 =). Six differen ime horizons are examined from 0.5 years up o 3 years. The oal number of model execuions o esimae he enire se of he variance based indices {S i,s, i = 1,,...10} is N VB = 4.576. To esimae he revised Morris measure µ*, 4 possible levels T i are considered for each inpu and a oal number of model execuions are performed. In order o compare resuls he Morris µ* and he oal sensiiviy indices S T i, ha normally do no sum up o 1, are rescaled in [0,1] (see Figures 1 and ). In he figures each graph refers o a fixed inpu. Wihin a graph, each do illusraes he inpu imporance in a specific scenario: he bigger he do size he higher he imporance of he inpu in he scenario. Ten differen dos sizes are considered corresponding o en differen classes of imporance, ranging from sensiiviy measure values in [0,0.1] o sensiiviy measure values in [0.9,1]. Fixing a facor, he comparison beween he graph of Figure 1 and ha of Figure allows evaluaing he effeciveness of µ* in measuring he facor imporance in differen scenarios. The Morris design is confirmed o be a good proxy of he oal sensiiviy index (see also []). Minor improvemens in he correspondence of resuls are achieved by increasing he Morris sample size up o N Morris =1.. Resuls highligh ha, overall (i.e. no focusing on a seleced scenario), he mos influenial parameers are: he dividend yield q, he ineres rae r, and he jump parameers λ and σ j. In paricular, q and r are very imporan for low srike prices a all imes o mauriy, while λ and σ j are more relevan a higher 8
srike prices. As expecable, σ 0 is imporan only for low imes o mauriy, especially when he opion is a he money. Noe ha among hese 4 mos imporan facors, q and r belong o he group of hese ha can be considered as conrollable. More ineresing is he role of he jumps parameers λ and σ j ha are he mos relevan among he unconrollable facors. In general, if we resric our aenion o he unconrollable facors, i urns ou ha is he jumps group, i.e. λ, σ j and µ j, which drives he higher amoun of he oal oupu variance in all scenarios. If we compare he relaive imporance of he various facors in differen scenarios, i emerges ha i is raher sable wih respec o shifs in he ime horizon, while i varies subsanially by changing he srike price value. In Figure 4 we represen hree scenarios ha correspond o 3 differen srike prices: we consider an opion in he money (srike = ), a he money (srike = ), and ou of he money (srike = ), wih a ime o mauriy fixed a 1.5 years. In each pie he oal variance of he opion price is apporioned o he conribuions due o he firs order effecs of each inpu (i.e. he Sobol S i ) and o he ineracions of all orders. Since ineracions do no explain a high amoun of he variance hey are summed up in a single erm. The figures sresses ha he imporance of jumps increases considerably wih he srike price. The sum of heir main effecs goes from 4%, when he opion is in he money, o 9% when he opion is a he money, up o 45.3% when he srike price reaches (opion ou he money). When he opion is in he money more han % of he oal variance is due o he conrollable facors q and r, hus leaving few chances o reduce he uncerainy in he opion price by increasing our modelling effor. As he srike price increases he imporance of jumps augmens considerably, making eviden ha heir role in modelling he opion price can no be overlooked: jumps need o be included in he model and heir represenaion should be as much accurae as possible. 9
5. CONCLUSIONS In his work we have employed he revised version of he sensiiviy measure proposed by Morris o esimae he imporance of he inclusion of jumps in a model for pricing European opions: he Heson model. The recenly revised sensiiviy measure confirmed is capabiliy o disinguish beween imporan and negligible inpu facors a low compuaional cos. Concerning he model, he sensiiviy analysis has led o he conclusion ha among he unconrollable facors, i.e. hose ha can no be esimaed from marke daa, jumps play a major role in deermining he opion price, hus sressing he need of including hem in he model formulaion. If we consider all facors, han a low srike prices mos of he uncerainy in he opion price is due o conrollable facors such as q and r. As he opion srike price increases, he imporance of jumps increases considerably: for insance for opion wih srikes and, he imporance of jumps is superior o ha of q and r for all imes o mauriy. This underlines ha an accurae assessmen of he jump process becomes more urgen for ou of he money opions. A final remark is ha, as expecable, a low ime o mauriy he iniial condiion for volailiy needs o be accuraely deermined while is imporance decreases as he ime o mauriy increases. REFERENCES [1] M. D. Morris, Facorial Sampling Plans for Preliminary Compuaional Experimens, Technomerics, 1991, 33, 161-174. [] F. Campolongo, J. Cariboni, and A. Salelli. Sensiiviy analysis: he Morris mehod versus he variance based measures, 003, submied o Technomerics. [3] Heson, S., A closed-form soluion for opions wih sochasic volailiy wih applicaions o bond and currency opions, Review of Financial Sudies, 1993, 6, 37-343. [4] Bakshi, G., Cao, C. and Chen, Z., Empirical Performance of Alernaive Opion Pricing Models, The Journal of Finance, 1997, Vol. LII, No. 5, 003-049. 10
[5] Black, F. and Scholes, M., The pricing of opions and corporae liabiliies, Journal of Poliical Economy, 1973, 81, 637-654. [6] Carr, P. and Madan, D., Opion Valuaion using he Fas Fourier Transform, Journal of Compuaional Finance, 1998,, 61-73. [7] A. Salelli, S. Taranola, F. Campolongo, M. Rao. Sensiiviy Analysis in Pracice. A Guide o Assessing Scienific Models, 004. John Wiley & Sons publishers, Probabiliy and Saisics series. VITAE Francesca Campolongo Jessica Cariboni aained her degree in physics a he Universiy of Milan (Ialy) in 000. Afer wo years working as quan analys in Nexra Invesmen Managemen SgR (Banca Inesa), she obained a gran from he European Commission in January 003 o sar a PhD a he Deparmen of Mahemaics of he Kaholieke Universiei Leuven (Belgium). Wim Schouens FIGURE CAPTIONS Table 1: Disribuions for he inpus of he Heson model. Figure 1: Imporance of he facors in each of he 4 scenarios according o µ* rescaled in [0,1]. The differences in he size of he dos represen he differences in he imporance of he fixed inpu facors in he scenarios. Figure : Imporance of he facors in each of he 4 scenarios according o S Ti rescaled in [0,1]. The differences in he size of he dos represen he differences in he imporance of he fixed inpu facors in he scenarios. Figure 3: Decomposiion of he oal variance of he srike price in hree scenarios. Ineracions of all orders are grouped in a single erm. Table 1 11
Inpu Disribuion Minimum Maximum Inpu Disribuion Minimum Maximum σ 0 Uniform 0.04 0.09 λ Uniform 0 κ η θ ρ Uniform 0 1 µj Uniform -0.1 0.1 Uniform 0.04 0.09 σj Uniform 0 0. Uniform 0. 0.5 r Uniform 0 0.05 Uniform -1 0 q Uniform 0 0.05 Figure 1: Morris r=10 1
σ 0 κ θ η ρ λ µ j σ j r q Figure : Sobol 13
σ 0 η κ ρ θ λ µ j σ j r q Figure 3 14
=, Time o Mauriy = 1.5y =, Time o Mauriy = 1.5y Ineracions: 1.7% σ 0 : 3% Ohers: <1.5% Ineracions: 7.% Ohers: <6% r: 6% σ 0 : 1.1% q: 8% λ: 1.4% µ j < 0.05% σ j :.6% q: 64% r: 17.5% σ : 16.6% j λ: 11.7% µ : 1.3% j =, Time o Mauriy = 1.5y Ohers: <4% ρ: 15.5% q: 8% Ineracions: 1.9% σ j : 5.1% σ 0 : 9.% λ: 17.8% µ j :.4% r: 5.3% 15