A Commet O he Rate Of Covergece of Discrete ime Cotiget Claims Dietmar P.J. Leise Staford Uiversity, Hoover Istitutio, Staford, CA 945, U.S.A., email: leise@hoover.staford.edu Matthias Reimer WestLB Pamure, Herzogstrasse 5, 47 Düsseldorf, Germay, email: Dr Matthias Reimer@WestLB.DE Abstract I a recet article, Hesto ad Zhou ( rove that the rate of covergece of Euroea call rices is = ad erform a error aalysis at a secific ode close to maturity to suggest that this ca ot be imroved. Leise ad Reimer (996 roved, however, that the rate is ad thereby cofirmed earlier emirical evidece by Broadie ad Detemle (996. I this commet we look i detail at both argumets Hesto ad Zhou ( rovide: First we rove that the deficiecy close to maturity occurs at a ode with a sigle robability of the rate /; it iduces therefore ricig errors at the rate. he we icororate a additioal term ito the exasio they erform to rove that terms of the rate / cacel out ad the rate is. Key words: biomial model, rate of covergece JEL classificatio: G
Itroductio he efficiet ricig of otios is of great ractical imortace: Whe large baskets of otios have to be riced simultaeously seed accuracy trade offs become imortat. he efficiecy of the methods used should be kow ad the be imroved as much as ossible. he study of Euroea call otios is a examle that is of articular iterest sice the kowledge of the closed form cotiuous time solutio rovides a good bechmark ad has the otetial to lead isights ito the comoets of the umerical errors ivolved. he biomial model is a discrete rocess aroximatio i the Black ad Scholes (97 setu that discretize the time iterval [;] ito equidistat stes. Leise ad Reimer (996 roved that the rate is for this model; this cofirmed earlier emirical evidece by Broadie ad Detemle (996. I a recet aer, however, Hesto ad Zhou ( claim i their abstract: We show that the rate of covergece deeds o the smoothess of otio ayoff fuctios, ad is much lower tha commoly believed because otio ayoff fuctios are ofte of all or othig tye ad ot cotiuously differetiable." hey also claim from umerical examles that the errors fluctuate betwee O( t ad O( t. Hesto ad Zhou ( rove that the rate of covergece of Euroea call rices is =. he they erform a error aalysis at a secific ode close to maturity to rove that the error at that ode is /. Both claims are correct, but they fail to roduce the correct rate of covergece, sice they do ot take ito accout iheret offsets i the roblem; i that sese they are icomlete. Aalyzig this i greater detail is the goal of this commet: Regardig their first claim we will argue that this is a sigle evet occurrig with a robability that is of the order /. Regardig their secod claim the followig theorem rovides a exasio aalysis that takes ito accout higher order terms to rove that the term of order / cacels out i the differece of error terms:
heorem For a sequece of iid biomial distributed radom variables X ( i the two values with robabilities, q that take =, resectively, deote X = P X i, μ = E[X], ff = Var(X, ad Z(u = ß exf u g. he, for ay r r μ P [X» r] =N ff + r μ Z + q ff ad, rovided q 5, the error term i this formula is ψ 5 = O + 4 jq j + exf = q g q : PROOF. Johso ad Kotz (969, chater,. 6 with referece to Usesky (97, see also Prohorov ad Rozaov (969. Note that those authors argue i terms of the success/failure robability, i.e. the rocess moves oly i itegers. he biomial stock rice rocess moves i icremets of ff t ;we adjusted mea ad variace accordigly. Review Of he Hesto Zhou Claim he rice C(t; S of a Euroea call with strike rice K ad maturity i a ecoomy with costat (istataeous iterest rate r, where the stock rice follows geometric Browia motio ad does ot ay divideds has bee derived by Black ad Scholes (97 i their classical aer: C(t; S=S N(d K e r( t N (d ff t: (. Here d = l(s=k+(r+ ff ( t ff t ad N ( deotes the cumulative stadard ormal distributio fuctio. Biomial models, first suggested by Cox, Ross, ad Rubistei (979 aroximate the uderlyig stochastic rocess: they discretize the time iterval [;] ito equidistat stes, t = =. wo arameters u ;d eed to be secified the; betwee two dates the stock rice follows a biomial radom variable, either movig u from S to u S or dow to d S. he robability for a u move is uiquely determied by the risk eutrality coditio u S +( d S = e r t S. Already Cox,
Ross, ad Rubistei (979 roved that the biomial otio rice ca be rewritte as ^C(;S =S Φ ; (a Ke r Φ ; (a (. where = u e r t, a = mifj js u j d j Kg ad Φ ; ( deotes the biomial distributio fuctio with degrees of freedom ad a u" robability. While various forms have bee suggested for the arameters u ;d,we stick i this rese- tatio to the choice made by Hesto ad Zhou (: u = ex r ff t + ff t o, ad d = ex r ff t ff t o. Istead of the risk eutrality coditio they imose ==. (his choice for the arameters u ;d ; has bee suggested first by Jarrow ad Rudd (98. Although the robability does ot rereset a arbitrage-free ricig measure, it has become a commo choice i the literature, sice it is coveiet ad iduces errors oly at the rate. Hesto ad Zhou ( rovide two argumets to uderi their claim that the rate of covergece is / i biomial models ( At a ode that is exactly at the strike rice K, the biomial rice ^C ad the Black Scholes rice C are ^C( t ;K= ad C(t t ;K= ψr ff ( t K + O(=; (. Kff ß + O(=: (.4 herefore ^C( t ;K C( t ;K = O(=. Hesto ad Zhou ( argue that otetially this error roagates backward over time; the the rate at date could ot be better tha /. ( Prices calculated coverge; it ca be rove searately for the two distributio terms Φ ; (a N (d ad Φ ; (a N (d, to coclude ^C C. he rate of covergece is of great imortace to evaluate the seed accuracy trade off. Hesto ad Zhou ( look at the rate of covergece searately of both distributio fuctios; they coverge with rate =. herefore they coclude that the rate is /. 4
A Refied Exasio Aalysis: he Rate is Let us ow review the two claims Hesto ad Zhou ( make. We start with the first claim: Nodes that are exactly at the strike are the oly odes where the o smoothess of the ayoff fuctio becomes imortat. his is a sigle evet ad Feller (966 roves that the robability of a sigle evet is of the rate =( q =O( t. herefore the overall error itroduced is O( t. (It could be rove easily that at all other odes close to maturity the error iduced is of rate. However here we are ot iterested i a full claim; we merely wat to exlai that the argumet of Hesto ad Zhou ( does ot hold. Now let us take a look at their secod claim: Before alyig theorem ote that ;q ; therefore = O( t ad for sufficietly high refiemet q 5. Without loss of geerality we will therefore aly theorem ad describe the error term as beig of rate. For each refiemet let us defie two sequeces R ; ;R ; ;::: ad R ; ;R ; ;:::: the radom variables R ;i adot u with robability ad d with comlemetary robability q. he radom variables R ;i differ from them oly i the robability: R ;i adots u with robability ad d with comlemetary robability q. We ca the write the stock rice S ( ad the ricig equatio as herefore ^C(;S =S Prob = S Prob " S exf S ( X at time as = S ex R ;ig K ( X R ;i» l S =K R ;i Ke r Prob " Ke r Prob S exf X R ;i g K ( R ;i» l S =K 5
^C(;S C(;S = S ψ Prob Ke r = S ψ Prob @ Prob 4P Ke r R @ Prob 4 ;i» l S =K R o aly theorem ote that E Var ψ X ψ R ;i = = R ;i = ψr ff ψ herefore, usig theorem : N(d ( R ;i» l S =K N(d ff ;i + r + ff» d 5 N(d A ff P ( R ;i + r ff (ψr ff ff ff q t + ρ ff t + ff q t +» d ff ρ ff ff q t = ff (ψr ff 5 N(d ff A t ff q t Prob P ( R 4 ;i + r ff ff» d ff 5 N(d ff = Z (d +O( t (. q Moreover, = e r t u = ex ff t + ff t o = ( + ff t +O( t so that E Var ψ X R ;i R ;i = = = (ψr ff +( ψr ff ψ herefore, usig theorem : t + ff q t (ψr ff t +( z } ff t t ff q t q ff t + O( ρ q ff ff t +( ρ ff q t ff q t = ψr + ff + O( t 6
Prob 4P R ;i + r + ff ff» d 5 N(d = q Z (d +O( t (. herefore, usig equatios (.,., ^C(;S C(;S = S Z(d Ke r Z(d ff + O( t (. q First ote that usig the defiitio of d follows at t = (d ff = d d ff + ff = d ls =K r so that ad so ( ex (d ff = ex ( (d S Z(d Ke r Z(d ff = S K er his roves that the error terms of rate / i the exasio of theorem cacel exactly out whe takig the differece of the two distributio terms. herefore we are left with terms of rate, i.e. ^C(;S C(;S =O( t : Refereces Black, F., ad M. Scholes (97: he Pricig of Otios ad Cororate Liabilities," Joural of Political Ecoomy, 8, 67 659. Broadie, M., ad J. Detemle (996: America Otio Evaluatio: New Bouds, Aroximatios, ad a Comariso of Existig Methods," he Review of Fiacial Studies, 9, 5. Cox, J. C., S. A. Ross, ad M. Rubistei (979: Otio Pricig: A Simlified Aroach," Joural of Fiacial Ecoomics, 7, 9 6. 7
Feller, W. (966: A Itroductio to Probability heory ad Its Alicatios, vol.. Joh Wiley & Sos. Hesto, S., ad G. Zhou (: O he Rate of Covergece of Discrete ime Cotiget Claims," Mathematical Fiace, (, 5 75. Jarrow, R., ad A. Rudd (98: Otio Pricig. Irwi, Homewood. Johso, N. L., ad S. Kotz (969: Discrete Distributios. Wiley. Leise, D. P., ad M. Reimer (996: Biomial models for otio valuatio Examiig ad imrovig covergece," Alied Mathematical Fiace,, 9 46. Prohorov, Y., ad Y. Rozaov (969: Probability heory. Sriger Verlag. Usesky, J. V. (97: Itroductio to Mathematical Probability. McGraw Hill. 8