Pricing European options on instruments with a constant dividend yield: the randomized discrete-time approach 1

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1 Probability ad Mathematical Statistics ) 417 Pricig Europea optios o istrumets with a costat divided yield: the radomized discrete-time approach 1 by RAFA L WERON Abstract. Due to the well kow fact that market returs are ot ormally distributed, we use geeralized hyperbolic distributios for pricig optios i a radomized discretetime setup. The obtaied formulas ca be used for pricig optios o stock idexes, currecies ad futures cotracts. We test them o optios writte o the Nikkei 225 idex futures ad coclude that a proper calibratio scheme could give us a advatage i the fiacial market Mathematics Subject Classificatio: 90A60, 90A12. Key words ad phrases: optio pricig, divideds, radomizatio, alterative models. 1 Itroductio Oe of the most importat problems i quatitative fiace is to kow the probability distributio of speculative prices returs. I spite of its importace for both theoretical ad practical applicatios the problem is still usolved. The first approach is due to Bachelier [2] who modeled price dyamics as a radom walk. Cosequetly, the distributio of prices was Gaussia. Gaussia models are widely used i fiace as well as i all braches of atural ad social scieces) although the ormal distributio does ot fit fiacial data especially at the tails of the distributio [6, 22]. I other words, the empirical distributios of prices are highly leptokurtic whereas Gaussia is ot. Needless to say, the tails of the price distributios are crucial i the aalysis of fiacial risk. Therefore, obtaiig a reliable distributio has deep cosequeces from a practical poit of view. Oe of the first attempts to explai the appearace of heavy fat, log) tails i fiacial data was made by Madelbrot [17] i 1963, who applied α-stable or Levy-stable) laws [15, 19]. The Levy distributio has bee tested agaist data i a great variety of situatios, however, always with the same result: the tails of the distributio are too fat compared with actual data. I a search for satisfactory descriptive models of fiacial data, large umbers of distributios have bee tried ad may further distributios have bee proposed recetly. Eberlei ad Keller [9, 10] ad Küchler et al. [16] showed that the hyperbolic law fits data from the Germa stock market much better tha the Gaussia oe. Bardorff Nielse [4] proposed the ormal iverse Gaussia NIG) Levy process for modelig stock returs. Such distributios have heavier tails tha the hyperbolic ad 1 Research partially supported by KBN Grat o. PBZ 16/P03/99. 1

2 lighter tha the Levy-stable oes. Tests performed o the DJIA, S&P500, Nikkei ad DAX idexes showed that from a umber of alterative distributios the NIG law best captures ot oly the shape of the empirical distributio aroud the mea but also the typical tail behavior of these idexes [18, 22]. For these reasos we use geeralized hyperbolic distributios a broad class cotaiig hyperbolic ad NIG laws for pricig optios i a radomized discrete-time setup. This paper exteds our earlier results [21] to the case of Europea optios writte o istrumets with a costat divided yield. The obtaied formulas ca be used for pricig optios o stock idexes, currecies ad futures cotracts. The paper is orgaized as follows. I Sectio 2 we itroduce the classical Cox-Ross- Rubistei [7] discrete-time setup. Next we show how the results obtaied for optios o o-divided payig istrumets ca be exteded so that they apply to optios o istrumets payig a kow divided yield Sectio 3). I Sectio 4, followig the reasoig of Rachev ad Rüschedorf [20], we itroduce radomizatio ito the discrete-time setup. Sectio 5 cotais a brief itroductio to geeralized hyperbolic laws. I Sectio 6 we preset the mai results, i.e. we derive the optio pricig formulas. Fially, i Sectio 7 we test them o optios writte o the Nikkei 225 idex futures. 2 The discrete-time setup Let us recall the basic ideas of the Cox-Ross-Rubistei CRR) optio pricig model. Assume that the price S =S k ) of a o-divided payig stock follows a multiplicative biomial process over discrete periods which divide the time iterval [0,t]ito parts of legth h { usk with probability p, S k+h = ds k with probability 1 p, where S 0 > 0adu>1 >dare real costats. Defie U =logu, D =logd ad let ρ d, u) be oe plus the riskless iterest rate over oe period, i.e. B k+1 = ρb k,where B k is the currecy e.g. USD) amout i riskless bods. Uder such assumptios Cox, Ross ad Rubistei [7] determied the fair price of a Europea call optio o a odivided payig stock with strike K ad maturity i steps or equivaletly with time t to maturity): C = S 0 Ψa,, p ) Kρ Ψa,, p), 1) where p = ρ d u d, p = u ρ p, a =1+ log K / log u ), Ψa,, p) =P ɛ S 0 d,i a, d i=1 ad ɛ,i is a sequece of idepedet radom variables with Pɛ,i =1)=p ad Pɛ,i = 0) = 1 p. Thesymbol x deotes here the iteger part of x. Covergece of 1) to the Black-Scholes formula is achieved by settig U = D = σ t/ ad takig ρ satisfyig lim ρ ) =e rt. The limitig formula is give by: C C t = S 0 Φd + ) Ke rt Φd ), 2) 2

3 where d ± = log S 0 K σ2 +r ± )t 2 σ. t 3 Istrumets with a costat divided yield A simple rule eables results obtaied for Europea optios writte o o-divided payig stocks to be exteded so that they apply to Europea optios o istrumets payig a kow divided yield [14, 22]. Cosider the differece betwee a stock that provides a cotiuous divided yield equal to κ per aum ad a similar stock that provides o divideds. The paymet of a divided causes a stock price to drop by a amout equal to the divided. The paymet of a cotiuous divided at rate κ, therefore, causes the growth rate i the stock price to be less tha it would be otherwise by a amout κ. If, with a cotiuous divided yield of κ, the stock price grows from S 0 at time 0 to S t at time t, the i the absece of divideds, it would grow from S 0 at time 0 to S t e κt at time t. Alteratively, i the absece of divideds it would grow from S 0 e κt at time 0 to S t at time t. This leads to a simple rule: Pricig rule: Whe valuig a Europea optio lastig for time t o a istrumet payig a kow divided yield κ, we reduce the curret istrumet price from S 0 to S 0 e κt ad the value the optio as though the istrumet pays o divideds. Why are we usig the word istrumet istead of the word stock? Well, simply because this method ca be also used to value optios o istrumets other tha stocks. I fact, it is used i practice to value optios o stock idexes, currecies, ad futures cotracts: A stock idex ca be treated as a security with a costat divided yield, because it represets a portfolio of a large umber of stocks payig divideds at differet momets of time. The divided yield κ should be set equal to the total aual amout of divideds received by the ower of such a portfolio. A foreig currecy has the property that the holder of the currecy receives a divided yield equal to the risk-free iterest rate i that currecy, i.e. κ = r f. This ca be explaied easily usig the argumet that the holder ca ivest the currecy i a foreig-deomiated bod. The cost of eterig ito a futures cotract is zero, thus the expected gai to the holder of a futures cotract i a risk-eutral world should be zero. This implies that futures cotracts ca be treated as istrumets payig a costat divided yield equal to the risk-free iterest rate, i.e. κ = r. The use of this rule gives us the followig formula for the fair price of a Europea call optio o a divided payig istrumet with strike K ad maturity i steps: C κ = S 0ρ κ Ψa,, p ) Kρ Ψa,, p), 3) 3

4 where ρ κ is oe plus the divided yield over oe period, p = ϱ d u d, p = u ϱ p, a =1+ log K / log u, Ψa,, p) =P S 0 d d ɛ,i a i=1 ad ϱ = ρ/ρ κ is oe plus the riskless iterest rate over oe plus the divided yield over oe period. Covergece of 3) to a Black-Scholes type formula is achieved i the same maer as for optios o o-divided payig stocks; i.e. by settig U = D = σ t/, takig ρ satisfyig lim ρ ) =e rt, ad takig ρ κ such that lim ρ κ ) =eκt. The limitig formula is give by: C κ Ct κ = S 0 e κt Φh + ) Ke rt Φh ), 4) where h ± = log S 0 σ2 +r κ ± )t K 2 σ. t 4 Radomizatio i the discrete-time setup Now, defie the cumulative retur process as log S k k k = ɛ,i U +1 ɛ,i )D) X,i, S 0 i=1 i=1 k =1,...,. 5) The limitig behavior of the sums k i=1 X,i was studied by Rachev ad Rüschedorf [20]. They gave ecessary ad sufficiet coditios for the covergece of these sums to the Gaussia law. Basig o their result, i what follows we assume that for some β ad σ 2 log S d Nβ,σ 2 ). S 0 Moreover, we assume that we obtai Gaussia limits with parameters β, σ 2 ad β, σ 2 for probability measures for which the radom walk logs /S 0 ) exhibits upward movemets with probabilities p ad p, respectively. I the same paper [20] the authors proposed a model with a radom umber of compoets RR). Itroducig N, a positive iteger valued radom variable idepedet of the sequece ɛ,i ), they defied the stock price process that exhibits a radom umber of jumps i the iterval [0,t]as log S t =log S N = X,k, S 0 S 0 k=1 where X,k is defied i 5). Note that X,k ) 1 k are sequeces of iid radom variables ieachseries). If k=1 X,k characteristic fuctio of Z is ϕ Z u) =Ee iuz = d Nβ,σ 2 )ad N 0 N w Y,the N k=1 X,k ) d Z, where the exp {iβzu 1 } 2 zσ2 u 2 df Y z), 6) see Gedeko [12]. This formula shows that ormal variace-mea mixtures ca be obtaied as limitig distributios of sums of idepedet biomial radom variables with a radom umber of compoets. We will use this property i optio pricig. 4,

5 5 Geeralized hyperbolic distributios A radom variable Y has the geeralized iverse Gaussia distributio GIGλ, χ, ψ) ifits Laplace trasform has the form ψ λ/2 K λ χψ +2θ) ) E exp θy)= ψ +2θ) λ/2 K λ. 7) χψ) where K λ ) is a modified Bessel fuctio of the third kid with idex λ. The geeralized hyperbolic distributio is defied as a ormal variace-mea mixture where GIG is the mixig distributio. More precisely, a radom variable Z has the geeralized hyperbolic distributio if Z Y ) Nµ + βy,y ), where Y GIGλ, χ, ψ). This meas that Z GHypλ, χ, ψ, β, µ) ca be represeted i the form Z = µ + βy + YN0, 1) with the characteristic fuctio ϕ Z u) =expiuµ) exp {iβzu 1 } 0 2 zu2 df Y z). 8) For λ = 1 we obtai the hyperbolic distributio itself, see Bardorff Nielse [3] for details. Küchler et al. [16] ad Eberlei ad Keller [9] foud that the hyperbolic distributio provides a excellet fit to the distributios of daily returs, measured o the log scale, of stocks from a umber of leadig Germa eterprises. However, oe desirable feature that the class of hyperbolic distributios lacks is that of beig closed uder covolutio. For λ = 1 we obtai the ormal iverse Gaussia distributio NIG) itroduced by 2 Bardorff Nielse [4]. This law is represeted as a ormal variace-mea mixture where the mixig distributio is the classical iverse Gaussia law, hece its ame. I cotrast to the hyperbolic distributio the NIG is closed uder covolutio. 6 Optio pricig i the radomized discrete-time setup Deote by Ck κ the fair price of a call optio writte o a divided payig uderlyig asset with k movemets X,i util maturity ad by CN κ the ratioal price of a call optio o a similar uderlyig asset with a radom umber of jumps. We defie CN κ as the mea value of the optio prices Ck κ CN κ = Ck κ PN = k). k=1 We refer to this value as the RR price. See also Rejma, Wero ad Wero [21], where a similar approach was used to obtai ratioal prices for Europea optios writte o o-divided payig stocks uder the geeralized hyperbolic model. From the above equatio we obtai the couterpart of 3) C κ N = k=1 S0 ρ k κ Ψa k,k,p ) Kρ k Ψa k,k,p) ) PN = k) = S 0 Eρ N κ Ψa N,N,p ) KEρ N Ψa N,N,p), 9) 5

6 where p = p) adp = p ) were defied i 3) / K a k =1+ log log u) ad S 0 d) k d) k Ψa k,k,p)=p ɛ,i a k ). i=1 The expectatio E is take with respect to N. Formula 9) is complicated from the computatioal poit of view. For this reaso we will look for its limitig case. We will also limit ourselves to the case where the log price of a stock is drive by the geeralized hyperbolic distributio. Theorem 6.1 Let β, σ, β ad σ be the parameters of the Gaussia laws obtaied as limitig distributios of logs /S 0 ) for the probability measures p ad p respectively) cosidered i Sectio 2. Let the iterest rate ad the divided yield satisfy ρ e rt ad ρ κ eκt, respectively. Let N be a positive iteger valued radom variable idepedet of the sequece logs /S 0 ) such that N / d Y,whereY has a geeralized iverse Gaussia distributio. The the limitig RR price of a Europea call optio o a istrumet with a costat divided yield κ, exhibitig a radom umber of jumps util maturity is give by C κ = lim C κ N = S 0 Ee κty K Ee rty f f x; λ, σ 2 χ, ψ +2κt σ 2, β σ 2, 0 ) x; λ, σ 2 χ, ψ +2rt σ 2, β σ 2, 0 ) dx dx, 10) where f ) deotes the desity of the geeralized hyperbolic distributio with the give parameters. Proof: To prove the Theorem we have to fid the limits of the expectatios i 9). From the defiitio of Ψa N,N,p )adx,k we have Eρ N κ Ψa N,N,p ) = = k ) P ɛ,i a k ρ k κ PN = k) k=1 i=1 P ) Xk) 0 ρ k κ PN = k) k=1 P Xk) ) ρ k 0 κ PN = k) Eρ N κ = Eρ N κ k=1 = Eρ N κ P XN ) 0 ), where Xk) = k i=1 X,i a k U D) kd ad N has the distributio PN = k) = ρ k κ PN = k)/eρ N κ. Sice lim ρ κ = eκt,wehave Eρ N κ Ee κty with Y GIGλ, χ, ψ). 6

7 Now, observe that the characteristic fuctio of XN ) has the form E exp [ iu XN )] = PN = k)exp[ iua k U D)+kD) ] E exp [ k ] iu X,i k=1 i=1 = e iuaxu D)+xD) ϕ X,1 u) ) x dp N x) 0 = e iuazu D)+zD) ϕ X,1 u) ) z N dp z), 0 where ϕ X,1 is the characteristic fuctio of the radom variable X,1. To idetify LawY ), the limitig distributio of N /, we first have compute the Laplace trasform of N by Lemma 5.1 of Rachev ad Rüschedorf [20] Ee θ N E exp N θ + tκ + δ ) ) =, Eρ N κ for δ 0. Whe teds to ifiity we obtai the Laplace trasform of Y Ee θ N Y θ+κt) Ee. 11) Ee κty Notice that the above limit is a quotiet of two Laplace trasforms of the geeralized iverse Gaussia distributio at the poits θ + tκ ad tκ. From7)wehave Y θ+tκ) Ee ψ +2tκ)λ/2 K λ χψ +2θ +2tκ) ) = Ee κty ψ +2θ +2tκ) λ/2 K λ χψ +2tκ) ). Hece, Y has the geeralized iverse Gaussia distributio GIGλ, χ, ψ +2tκ). From the defiitio of X,i ad the assumptios of the Theorem we have ϕ X,1 u) ϕ Nβ,σ )u). Moreover, U) D) 0 ad cosequetly a 2 z U D) +zd log K S 0 ). Fially, usig the fact that N d Y, we obtai the followig form of the limitig characteristic fuctio E exp [ iu XN ) ] e iu logk/s 0) ϕnβ,σ )u) ) z dp Y z). 2 Now, usig Corollary 1 of Rejma, Wero ad Wero [21] or equivaletly comparig the above itegral with formulas 6) ad 8)) it is clear that XN ) d Z κ log K S 0, 0 where This lets us write Z κ GHypλ, σ 2 χ, ψ +2tκ σ 2, β σ 2, 0). Eρ N κ Ψa N,N,p ) Ee κty P Z κ log K ) S0 = Ee κty f x; λ, σ 2 χ, ψ +2tκ ) σ 2, β σ 2, 0 dx. 12) 7

8 Idetical argumets hold for the secod expectatio i 9). Thus Eρ N Ψa N,N,p) Ee rty f x; λ, σ 2 χ, ψ +2tr, β ) σ 2 σ, 0 2 Collectig the results 9), 12), ad 13) we complete the proof. dx. 13) 7 Empirical aalysis I our empirical aalysis we use a high-frequecy data set comprisig tick-by-tick prices of Nikkei 225 futures ad Nikkei 225 futures call optios as traded o the Sigapore Iteratioal Moetary Exchage SIMEX) sice Jauary 16th, 1997 util September 2d, The data set is part of the SIMEX 1997 Trade Data CD-ROM kidly supplied by Garliski Fiazhadels GmbH. Note, that sice we are dealig with optios writte o futures cotracts we have to adapt the obtaied earlier formulas. Let logs /S 0 ) be a radom walk o the iterval [0,t] defied i Sectio 2 with t U = D = σ, lim ρ = e rt, lim ρ κ = eκt, p = r κ σ2 ) 2 t 2 σ, such that logs /S 0 ) d N r κ σ2 )t, 2 σ2 t ). Thisimpliesthatp has the followig explicit form p = σ2 r κ + 2 2σ 2 ) t. Moreover, these values of U, D ad p make the o-radomized optio price 3) coverge to the Black-Scholes type formula 4) with a give volatility σ. Now, lettig N / Y GIGλ, χ, ψ) weobtai logs N /S 0 ) d σ2 GHyp λ, σ 2 ψ r κ tχ,, ) 2, 0, σ 2 t σ 2 ad the ratioal price of a optio o a istrumet with a divided yield κ is give by C κ = lim C κ N = S 0 Ee κty f x; λ, σ 2 tχ, ψ +2κt, r κ + 1 ) σ 2 t σ 2 2, 0 dx K Ee rty f x; λ, σ 2 tχ, ψ +2rt, r κ 1 ) σ 2 t σ 2 2, 0 dx. Recall that, whe valuig optios o futures cotracts, we have to make the divided yield equal to the risk-free iterest rate. Thus, substitutig κ = r we get the RR-NIG price C κ=r = lim C κ=r N K Ee rty = S 0 Ee rty f f x; λ, σ 2 tχ, ψ +2rt, 1 ) σ 2 t 2, 0 x; λ, σ 2 tχ, ψ +2rt, 1 ) σ 2 t 2, 0 8 dx, dx

9 where Ee rty is give by formula 7). Now we are i positio to compare the classical Black-Scholes type model to be more precise: the Black model [5] for optios o futures) ad the above result. For the aalysis we chose Nikkei 225 futures ad Nikkei 225 futures optios cotracts with expiry o September 12th, To test the models we decided to pick four tradig days with relatively high volume above 1600 trasactios for the futures ad above 10 for the optios). Our selectio was: Jue 11th, days to maturity), July 11th, days to maturity), August 6th, days to maturity), ad September 2d, days to maturity). To estimate model parameters we used 100 precedig daily logarithmic returs for each of the four selected dates. Both, for the Gaussia ad the NIG distributio we used maximum likelihood estimates. The obtaied parameters are preseted i Table 1. Surprisigly, both distributios passed the Kolmogorov ad Aderso-Darlig goodessof-fit test statistics [1] at the α =0.05 sigificace level with the NIG law givig just a slightly better fit, for more details see [18]. Table 1: Parameter estimates for the SIMEX data. Parameters for: Gaussia µ law σ χ NIG ψ law β µ To coclude which optio pricig model is better we must compare model ad real tradig prices. As the test statistics we use the mea absolute error MAE = 1 N K C real K) C model K), where the sum is over N strike prices K. The results preseted i Table 2 show a slightly better performace of the NIG model. As recet studies [18] suggest, calibratig the RR-NIG model to optio prices istead of the uderlyig futures prices ca improve the model performace cosiderably. Table 2: Mea absolute errors for SIMEX data. Date RR-NIG Black

10 8 Coclusios Recall, that i a complete fiacial market ay cotiget claim is attaiable ad ca be valued o the basis of the uique equivalet martigale measure [13]. Thus the mai objective of cotiget claim valuatio is to fid a equivalet martigale measure. Ufortuately, by cosiderig heavy-tailed distributios, we etered the realm of icomplete models. Istead of a uique equivalet martigale measure typically there is a large class of such measures [8]. Luckily, recet results of Gema et al. [11] justify the use of ormal mea-variace mixture models. Ad the geeralized hyperbolic law is a ormal mea-variace mixture. O the more practical side, the results preseted i the previous Sectio suggest that the RR-NIG model performs oly slightly better tha the classical Black-Scholes type model. This would make the model useless i the real world, sice its slightly smaller errors are offset by much more complicated umerical procedures. However, recet studies [18] suggest, that calibratig the RR-NIG model to optio prices istead of the uderlyig futures prices ca improve the model performace cosiderably. Thus a procedure cosistig of calibratig the RR-NIG model to previous day s optio prices ad the usig it to price today s optios would justify our approach ad give us a advatage i the fiacial market. Refereces [1] R.B. D Agostio, M.A. Stephes, Goodess-of-Fit Techiques, Marcel Dekker, New York, [2] L. Bachelier, Théorie de la spéculatio, Aales Scietifiques de l Ecole Normale Supérieure, III ) [3] O.E. Bardorff-Nielse, Expoetially decreasig distributios for the logarithm of particle size, Proc. Royal Soc. Lodo A ) [4] O.E. Bardorff-Nielse, Processes of the ormal iverse Gaussia type, Fiace & Stochastics ) [5] F. Black, The pricig of commodity cotracts, J. Fiacial Ecoomics ) [6] J.P. Bouchaud, M. Potters, Theory of Fiacial Risk, Cambridge Uiversity Press, [7] J.C. Cox, S.A. Ross, M. Rubistei, Optio pricig: A simplified approach, J. Fiacial Ecoomics ) [8] E. Eberlei, J. Jacod, O the rage of optio prices, Fiace & Stochastics ) [9] E. Eberlei, U. Keller, Hyperbolic distributios i fiace, Beroulli ) [10] E. Eberlei, U. Keller, K. Prause, New isights ito the smile, mispricig ad value at risk: The hyperbolic model, J. Busiess )

11 [11] H. Gema, D. Mada, M. Yor, Time Chages for Lévy Processes, Mathematical Fiace ) [12] B.V. Gedeko, O limit theorems for a radom umber of radom variables, Lecture Notes i Math ) [13] J.M. Harriso, S.R. Pliska, Martigales ad stochastic itegrals i the theory of cotiuous tradig, Stoch. Proc. Appl ) [14] J. Hull, Optios, Futures, ad Other Derivatives, Pretice Hall Iteratioal, Lodo, [15] A. Jaicki, A. Wero, Simulatio ad Chaotic Behavior of α-stable Stochastic Processes, Marcel Dekker, New York, [16] U. Küchler, K. Neuma, M. Sørese, A. Streller, Stock returs ad hyperbolic distributios, Mathematical ad Computer Modellig ) [17] B.B. Madelbrot, The variatio of certai speculative prices, J. Busiess ) [18] A. Misiorek, Alterative optio pricig models ad the volatility smile, Upublished MSc thesis, Wroc law Uiversity of Techology, [19] S. Rachev, S. Mittik, Stable Paretia Models i Fiace, Wiley, [20] S.T. Rachev, L. Rüschedorf, Models for optio prices, Theory Probab. Appl ) [21] A. Rejma, A. Wero, R. Wero, Optio pricig proposals uder the geeralized hyperbolic model, Commu. Statist. Stochastic Models ) [22] A. Wero, R. Wero, Fiacial Egieerig: Derivatives Pricig, Computer Simulatios, Market Statistics, WNT,Warsaw,i Polish, Rafa l Wero Hugo Steihaus Ceter Istitute of Mathematics Wroc law Uiversity of Techology Wroc law, Polad rwero@im.pwr.wroc.pl Received o ; revised versio o