Binomial Option Chance

8/12/2019 Binomial Option Chance

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A Synthesis of Binomial OptionPricing iVIodels for LognormaiiyDistributed Assets

Don M Chance

The fmance literature has revealed no fewer than Ialternative versions of the binomial option pricingmod el for options on lognorm aiiy distribute d assets.These models are derived under a variety ofassumptions and in some cases require informationthat is ordinarily unnece ssary to value options. T hispaper provides a review and synthesis of thesemodels, showing their commonalities and differencesand demonstrating how 1 diverse models all producethe samere sultin the limit. Some of the models admitarbitrage with a finite number of time steps and somefail to capture the correct volatility. T his paper alsoexamines the convergence properties of each m odeland finds that none exhibit consistently superiorperformance overthe others. Finally, it demonstrateshow a general model that accepts any arbitrage-freerisk neutral probability will reproduce the Black-Scholes-Merton model in the limit.

Option pricing theory has become one of the mostpowerful tools in economics and finance. The celebratedBlack-Scholes-Merton model not only led to a NobelPrize but completely redefined the financial industry.Its sister model, the binomial or two-state model, hasalso attracted much attention and acclaim., both for itsability to illustrate the essential ideas behind optionpricing theory with a minimum of mathematics and tovalue many complex options.Don M. Chance is a Professor of Finance at Louisiana StateUniversity in Balon Rouge. LA 70803.The author thanks Jim HilHard Boh Brooks, Tung-Hsiao Yang.To m Arnold Adam Schwartz, and an anonymous referee forhetpful comments. A document containing detailed proofs andsupporting results is available on the Journal's web site or canbe obtained by emaiiing the author.

The origins of the binomial model are somewhatunclear . Options folklore has i t that around 1975William Sharpe, latert win a Nobel Prize for his seminalwork on the Capital Asset Pricing Model, suggestedto Mark Rubinstein that option valuation should befeasible under the assumption that the underlyingstock price can change to one of only two possibleoutcomes. Sharpe subsequently developed the ideain the first edition of his textbook.^ Perhaps the best-known and most widely cited original paper on themodel is Cox, R oss, and Rubinstein {1979), but almosts imul t aneous ly , Rend leman and Bar t t e r ( 1979)presented the same model in a sl ightly differentmanner.

Over the years, there has been an extensive bodyof research design ed to improv e tbe model.^ In theliterature the model has appeared in a variety of forms.Anyone at tempt ing to understand the model canbecome bewildered by the array of formulas that allpurport to accomplish the desired result of showinghow to value an option and hedge an option position.These formulas have many similarities but notable

Not s urprisingly , this story do es not appear form ally in theoptions literature but is related by Mark Rubinstein inRiskBooks (2003, p. 581).See Sharpe, Alexander, and Bailey (1998) for the c urrentedition of this book.See . for example, Boyle (1988), Om berg (1988). Tian (19 93).Figlewski and Gao (1999), Baule and Wilkens (2004) (fortrinomials). He (1990) (for multiple state variables), andRogers and Stapleton (1998). Breen (1991). Broadie andDetemple (1997). and Joshi (2007) (for American optionpricing). See also Widdicks, Andricopoulos, Newton, and Duck(2002). Walsh (2003), Johnson, Pawlukicwicz, and Mehta(1997) for various other modifications, and Leisen and Reimer(1996) for a study of the m odel s convergence.


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CH NCE SYNTHESIS OF BINOMI L OPTION PRICING MODELS 39differences. Ano ther source of confusion is that someprese n ta t io ns use oppo s i t e no ta t ion .* But m orefundamentally, the obvious question is how so manydifferent candidates for the inputs of the binomialmodel can exist and how each can technically becorrect.

The objective of this paper is to synthesize thedifferent approaches within a body of uniform notationand provide a coherent treatment of each model. Eachmodel is presented with i ts distinct assumptions.Detailed derivations are om itted but are available in asupplemental docum ent on the journal website or fromthe author.Some would contend that it is wasteful to study amodel that, for European options, in the limit equalsthe Black-Scholes-Me rton model. Use of the binomialmodel, they would argue, serves only a pedagogicalpurpose. But it is difficult to consider the binomialmodel as a method for deriving the values of morecomplex options without knowing how well it worksfor the one scenario in which the true continuous limitis known. An unequivocal b enchmark is rare in finance.For options on lognormally distributed assets, theliterature contains n o less than 11 distinct versions ofthe binomial m odel. Some of the models are improp erlyspecified and can lead to arbitrage profits for a finitenumber of time steps, while some do not capture theexogenous volatility. Several models focus first onfitting the binomial model to the physical process,rather than the risk neutral process, thereby requiringthat the expected return on the stock be known, anunnecessary requirement in arbitrage-free pricing. Ishow that the translation from the physical to the riskneutral process has produced some misleading results.The paper also provides an examinat ion of theconvergence properties of each model and concludeswith a demonstration that an y risk neutral probability(other than one or zero) will correctly price an optionin the limit.My focus is exclusively on models for pricingoptions on lognormally distributed assets and not oninterest rates. Hence, these models can be used foroptions on stocks, indices, currencies, and possiblycom mo dities. Cash flows are ignored on the underlying,but these can be easily added. The paper begins witha brief overview of the model that serves to establishthe notation and terminology.*In some versions, the mean arithmetic return is a while themean log return is fi. In others the opposite notation is used.Although there is no notational standard in the optionsliterature, the inconsistent use of these symbols is a significantcost to comparing the models.

I. Basic Review of the Bino mial M odelThe continuously compounded risk-free rate perannum isr .Consider a risky asset priced atS that canmove up to state '+ fora value ofu or down to state - for a valueof dS.Let there be a call option expiring

in one period with exercise price X. The value of theoption in one period isc_ if the + state occu rs and c.if the - state occurs.A. Deriving the Binomial Model

Now construct a portfolio consisting ofA units ofthe asset and B dollars invested in the risk-free asset.This portfolio rep licates the call option if its outcome sare the same in both states, that is,

The unknowns areB and A. Rearranging to isolateB setting the results equal to each other, and solvingfor Bgives

S u-dSince both values,c and c^, are know n, I substitutefor A in either equation and solve for B. Then, given

knowledge of A,S an d B I obtainc exp( r)

as the value of the option, andK =

(1 )

2)as the risk-neutral probability, sometimes referred toas the pseudo-probabili ty or equivalent martingaleprobability, with h as the period length or time toexpiration, T divided by the number of binomial timeperio ds. A'. Extension to the multiperiod ca se followsand leads to the same result that the option value at anode, given the option values at the next possible twonodes, is given by Equation (I).B. Spe cif icat ion of the Binom ialParameters

At times I will need to work with raw or discretereturns and at others times, I will work w ith continuou sor log returns. Let the concept of return refer to the


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40 JOURNAL OF APPLIED FINANCE SPRING/SUMMER 2008future price dividedbythe current price, or technicallyone plustherateofreturn.Let theexpected priceoneperiod later beE(S^)and theexpected raw returnbeE S^)fS. The true probability ofanupmoveisq.Thus,the per-period expectedrawreturnis

3)The per-period expectedlogreturnis

4 )The varianceoftherawreturnis

5)The varianceofth elogreturnis

5 ^Vl \J. s.6 )

These parameters describe the actual probabili tydis t r ibut ion of the stock return, or the physicalprocess. Option valuation requires transformation ofthe phys ical process to the risk neutral process .Typically,theuser kno ws the v olatility of the log returnas given by the physical process, a value thatmayhave been estimated using historical dataorobtainedas an implied volatility. In any case, 1assume thatvolati l i ty is exogenous and constant, as is usuallyassumed in continuous-time option pricing.II .Fitting the inomial Model

In early research on the binomial model, severalpaper s examined f i t t i ng a binomia l model to acontinuous-time process,andeach provide d differentprescriptionson how to doso. Before exam ining thesemodels , I will review the basic concepts from thecontinuous-time models that are needed to fit thebinomial model.A. Basic Continuous Time Conceptsforthe Binomial Model

The results in this section are from the Black-Scholes-Merton model.Itstartsbyproposing thatthelog return is normally distributed with mean and

variance(f.Given that \Q S^JS) - \n{S^J -stochastic process is proposed as ^),the

whereand o^ are the annuaiized expected returnandvariance, respectively,asgivenbyE[d\n S)] ^fu tandVr[d\n{S)] a^di,anddW^is aWeiner p rocess.Examine now the raw return,dSJS^.LettingG,- ln(5^),then S ^ e^ Needed are the part ial der iv at ives ,dSJdG^ = e^ s and d^SJG^ =e^'- Applying I t 'sLemmatoS.obtains

Noting that dG ^= ^dt + adW^,then dG^ =cf^dt.Substituting these resultsand the partial derivativesobtainsdS .

Define aas theexpected valueofth erawreturnsothatdS . (8 )

an da = /i +tr=/2. The exp ectatio n odSJS, is E[dSJS}^ adtand Var[/5/5J- a^dt. tisevident that the m odelassumesnodifference inthe volatilities of theraw andlogarithmic processesin continuous time. This resultisthe standard assumption and derives from the factthat It's Lemmaisusedtotransform the logprocesstothe raw process. Technically, the variance of theraw processis

Var\ 9)which is adapted to continuous time from Aitchisonand Brown (1957). The difference in the variancedefined ascrdtand inEquation(9)liesin the fact thatthe stochastic process fordSJS^is an approximation.This subtle discrepancy is thesource of someof thedifferences, however small,in the various binomialmodels. 'One final resultisneeded.The expected valueof 5' E q u a t i o n (9) is der ived in Aitchison and B r o w n ( 1 9 5 7 ) byt a k i n g the mo me n t - g e n e r a t i n g f u n c t i o n for the l o g n o r m a ldistr ibution. Thedifference in using Equation(9) in comparisontoCTs quite small over most reasonable ranges of volali l i tyand expected return. For example, with a 10 expeeted return,3 0 volati l i ty, and one day as a rough approximat ion of dt,the difference in instantaneous vola t i l i ty is only .000005.


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CHANCE A SYNTHESIS OF BINOMIAL OPTION PRICING MODELS 41at the horizon Tis given as*

B. Fitting the Binomiai Model to aContinuous-Time Process

Several ofthe papers on the binomial model proceedto fit the model to the continuous-time process byfinding the binomial parameters ,d,an d q that forcethe binomial model mean and variance to equal thecontinuous-time model mean and variance. Thus, inthis approach the binomial model is fit to the physicalprocess. These parameters are then used as thoughthey apply to the risk neutral process when valuingthe option. As shown shortly, this is a dangerous step.

The binomial equations for the physical process are

model to the equations for the physical process is atbest unnecessary and at worst, misleading. Recall that

(10) the Btack-Sch oles-Me rton model requires know ledgeofthe stock price, exercise price, risk-free rate, time toexpiration, and log volatility but not the expectedreturn. Fit t ing the binomial model to the physicalprocess is unnecessary and adds the requirement thatthe expected return be known, thereby eliminating themain advantage of arbitrage-free option pricing overpreference-based pricing.

As is evident from basic option pricing theory, thearbitrage-free and correct price ofthe option is derivedfrom knowledge ofthe volatility with the conditionthat the expected return equals the risk-free rate. Itfollows that correct specification ofthe binomial modelshould require only that these two conditions be met.Le tn be the risk neutral probab ility. The correct m eanspecification is

and(U) KU +{\-n) =e . (14)

This expression is then turned around to isolateK :

oru-d

(12)

( 3 )depending on whether one is fitting the log varianceor raw variance. The volatility defined in the Black-Scholes-Merton model is the log volatility so the logvolatility specification would seem more appropriate.But because the var iance of the raw return i sdeterministically related to the variance ofthe logreturn, fitting the model to the variance of the rawreturn will still give the appropriate values ofwandd

To convert the physical process to the risk-neutralproce ss, a small transformation is needed. The meanraw return a is set to the risk-free rater .Alternatively,the mean log return \i is set to r - 0^/2. But fitting th e*For proof see the appendix in Jarrow and Tumbull (2000, p.112 ) .'One would, however, need to exercise some care. Assume thatthe user knows ihe log variatice. Then the raw variance can bederive d IVoni the right-ha nd side of Equa tion (9), which thenbecomes the right-hand side of Equation (13). If the user knowst he l og va r i a nc e , t he n i l be c ome s i he r i gh t - ha nd s i de o fEquation (12). If the user has empirically estimated the rawand log variances, the former can be used as (he right-handside of Equation (13) and the latter can be used as the right-hand side of Equation (12)- But then Equations (12) and (13)might lead to different values of u and d. because the empiricalrand log stochastic processes are unlikely to conform preciselyto the forms specified by theory.*See Jarrow and Turnbull (2000) for an explanation of thist r a n s f o r m a t i o n .

K (15)Either Equation 14)or (15) is a necessary condition toguarantee the absence of arbitrage.' ' Surprisingly, notall binomial option pricing mod els satisfy Equation (14).Note that this condition is equivalent to, under riskneutrality, forcing the binomial expected raw return.not the expected log return, to equal the continuousrisk-free rate. In other words, the correct value of nshouid come by specifying Equation (14), not

, (16)which comes from adapting Equation (1 1) to the riskneutral measure and setting the log expected return to its risk neutral analog ,r - cfll. Surprisingly, many ofthe binomial m odels in the literature use this im properspecification.'

The no-arbitrage condition is a necessary but notsufficient condition for the binomial model to yield the

' 'The proof is widely known. One simply retaxes this constraintwhereu pon a self-financed portfolio of long stock-sh ort bond(or vice versa as appropriate) always generates posit ive valueeven in the worst outcome. It will follow that there exists ameasure, like n, such that ;ni + (1 - n d = e'^'', which is Equation( 1 4 ) .' A c o r r e c t l oga r i t h mi c s pe c i f i c a t i on o f t he no - a r b i t r a g econdition would involve taking the log of Equation (14). Ifthis modified version of Equation (14) were solved, the modelwould be correc t . 1 found no ins tan ces in the l i te ra ture inwhich this alternative approach is used.


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42 JOURNAL OF APPLIED FINANCE SPRING SUMMER 2008

correct option price. The model m ust also be calibratedto the correct volatility. This constraint is met by usingthe risk-neutral analog of Equation (5),(17)

or Equation (6),/)f7r \-7:) =CT h. 18)

Either condition will suffice because both providethe correct raw or log volatility.C. Convergence of the Binomial Modelto the Black Scholes Merton Model

Threeofthe mo st wide ly cited versionsofthebinomial model. Cox, Ross, and Rubinstein (1979);Rendleman and Bartter (1979); and Jarrow and Rudd(1983), provide proofs that their models converge tothe BSM model when A>oo.Recall that each m odel ischaracterized by formulas for w,d and the probab ility.Hsia (1983) has providedaproof that dem onstratesthat convergence can be shown under less restrictiveassum ption s. For risk n eutral prob ability T, Hs ia's(1983) proof shows that the binomial model co nvergesto the BSM model ifNK-> co as A'^->oo.To meet thisrequirement,0< ;r < 1 is all that is necessary." Thisresult may seem surprising for it suggests that the riskneutral probability can be setatany arbitrary valuesuch as 0.1 or 0.8, In the literature som e versions ofthe binomial mo del constrain the risk neutral proba bilityto /2 and as shown later, all versions of the model haverisk neutral probabilities that converge to V2. But Hsia's(1983) proof shows that an y probability other thanzero or one will lead to convergence. This interestingresult is examined later.D. Alterna tive B inom ial Models

1 now examine the 11 binomial models that haveappeared in the literature.1 . Cox Ross Rubinstein

Cox , Ross, and Rubinstein (1979 ), henceforth C RR.is arguably the seminal article on the model. Their

"The o ther requi rements not noted byHsia (19 83) a re tha tt h e c h o i c e of w,d, and n must force the b inom ial modelv o l a t i l i t y to e q u a l thet r u e v o l a t i l i t y a n d th e me a n mu s tguarantee no arb i t rage .

Equations (2) and (3) (p. 234) show the option value asg iven bymy Eq uat ion (1) wi th the r isk neu tralprobability specified as my Equation (2). CRR thenproceed to examine how their formula behaves when A^-^OO(their p.246-251).They do this by choosing , dan d qso that their mode con verg es in the limit to theexpected value and variance of the physical process.Thus, they solve for u d,an dqusing the p hysicalprocess , my Equat ions (11) and (12) . Note thatEquations (11) and (12) constituteasystem of twoequa t ions and th ree unknowns . CRR proposeasolution while implicitly imposinga necessary thirdcondition, ud 1, an assum ption frequently found inthe literature. Upon obtaining their solution, they thenassume the limiting condition that /r ^ 0. This conditionis necessary so that the correct variance is recovered,though the mean is recovered for any A . The ir solutionsare;

u e a=ewith physical probability

^


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CHANCE A SYNTHESIS OF BINOMIAL OPTION PRICING MODELS 43f i t the binomial model to the physical process,simultaneously deriving the physical probabilityq,andthen substitute the arbitrage-free formula for n as q.Had they imposed the arbitrage-free condition directlyinto the solution, they would have obtained differentformulas, as we will see inanother model.2. Rendleman-Bartterand Jarrow-Rudd-Turnbull

Because of their sim ilarities,discussion of the Rendleman-Bar t t e r (RB) approach i scombined with discussion ofthe Jarrow-Rudd (JR) approach and later appendedwi th the Ja r row-Turnbu l l ( JT ) approach . Theseapproac hes also fit the binomial m odel to the phy sicalprocess. The RB approach specifies the log mean andlog variance E quations (11) and (12), and solves thesetwo equations to obtain:

The so-called binomial modelis really a family of modelsthat, under surprisingly mildconditions, all converge in thelimit to the Black-Schules-Merton model.

u = e \\-q , d = eBecause these formulas do not specify the value of

q, they are too genera to be of use. In risk neu tralizingthe model, RB assume that/ J-r- 2 0)

but again, these formulas are consistent only with aprobability of Yiand risk ne utrality as specified byfu^ r-d^/2. Simply converting the mean is not sufficientto ensure risk neutrality for a finite number of timesteps.'**"A close look al JR shows ihat q is clearly the physicalprobability. On page 187, they constrain q to equal the riskneutral probability, with iheir symbol tor the latter being ^.Bui this constraint is noi upheld in subsequent pages whereuponthey rely on convergence in ihe limit to guarantee the desiredresult that arbitrage is prevented and BSM is obtained. Thispoint has been recognized in a slightly different manner byNawalkha and Chambers (1995, p. 608).


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A number of years later,JTderive thesame modelbul make a much clearer d is t inct ion betweenthephysical and risk neutral processes. They fix7i at itsarbitrage-free value andshow fortheirup anddownparameters that

7 -du-d (21)Like CRR, the correct specification of K ensures thattheir model does not admit arbitrage. But, because theirsolutions foruanddwere obtainedbyspecifyingthelog mean, these solutions are nottechnically correctfor fmite N. The mean constraint ismet, so there mustbe an error somew here, which ha s to be in the variance.Thus, their model does not recover thevariance forfinite A' using therisk neutral pro babilities. Itreturnsthe co r r ec t va r i ance e i the r when thep h y s i c a lprobability isusedor in the limit with the risk neutralprobability converging toVi.

For future reference, this model will becalledtheRBJRT modeland referred only to thelast ve rsionofthe model inwhich the no-arb itrage co nstraintisapplied toobtain K.I have shown that itdoesnotrecover thecorrect v olatility for finite A^.NowIwillconside r a model that fits a binomial tree to the physicalprocess but does prevent arbitrage and recoversthecorrect volatility.

3. hrissChriss's model (1996) specifies the rawmeanand

log varianceofthe physical process. The formerisgiven byqu+{\-q)d=-e \ (22)

an d thelatterby Equation (12). He then assumes thatq =Y2.The solutionsareM =2e ^*- 2eHlThe risk-neutralized analogs are found by

substitutingrfor a:

=2^ri,.: d= (23)

Note that because Chriss' mean specification is theraw mean, transformation to risk neutrality bya=rcorrectly returns the no-arbitrage condition. Equation(15).Thus,forthe Chriss m odel,K= n =ViforallN.an d the model correctly preserves the no-arbitragecondition andrecovers thevolatility for anynumber

of time steps.4 . Trigeorgis

The Trigeorgis 1991 )model transforms the originalprocess intoalog process. That is, letX In5 andspecify the binomial process as thechange in A ,orAA .Thesolutions for the physical processare

Note thatif/i^-0,the Trigeorgis model isthe sam eas the CRR mod el. Trigeorgis then risk neutralizes themodelbyassum ing thatfi r- cH/2. The resu ltsare

J(rh^r-a/2fh , -Jah+{r-a/2fhu= e , d=e > Trige orgis' risk neutral probability comes simply fromsubstitutionof r - d llfor inthe formula for q..thereby obtaining

-a I2)h.7X - + 2Of course this is therisk neutral proxy probability

and isnot given bythe no-a rb i t r age cond i t ion .Therefore,itis not arbitrage -free for fmite A^, thoughitdoes recover the correct volat i l i ty . In thel imi t ,Tr igeorg i s ' s r i sk neu t r a l p roxy p robab i l i t y ,;r*,converges to Aand the arbitrage -free risk neutralprobability, K ,converges toVz, so the Trigeorgis modelis arbitrage-free inthe limit.

5. Wi lmott i ndW ilmott2Wilmott (1998) derives twobinomial models.He

specifies the raw mean and raw variance of the physicalprocess . Equat ions (22) and (17). His first mo del,referred to here as Wil 1, assum esud =.The solutionsfor the physical processare

2 eMM_4The physical probabilityqisfound easily from themean condition.

u-dRisk neutralizing themodel isdone by simply


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CHANCE A SYNTHESIS OF BINOMIAL OPTION PRICING M ODELS 48substitutingrfor a;

-rH=-\e2\1/-r.=-{e ' +2 \

e^

{r-ta-\h\

J

\ -A\e ^ ,and d)d1.This resu lt for C RR is well-known.'** It arises w hen h >(CT//-)\which is likely to occur with low volatility, highinterest rates, and a long time to expiration. Sufficientlylow volatility is unlikely to apply when modeling stockprices, but exchange rate volatility isoften less than0 .1 . Thus, long-term foreign exchange options wherethe interest rate is high canh a v e a r isk neutralprobabili ty greater than on e. For RBJRT,theriskneutral probability can exceed one if/i 4. For very low volatility, asin the foreign exchange market, the time step wouldhave to beextrem ely large. Thus , it would takeexceptionally large volatility and a very small num berof time steps relative to the option maturity for the riskneutral probability to exceed one for the RBJRT model.Of course, if the risk neutral probability exceeds one,a model could still correctly value the option. But,as' ^See, forexample . Carp enter (1998) ,' See, forexample . Chr iss (1996,p.239) ,' 'For example , ii r = , 1 ,CT= ,05.and7 = 5, In that case,wewould requireN>2 0 ,


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A An Init ial Look at Convergencereviously noted, the CRR and RBJRT models use theuan d d formulas from the physical process, which isderived by constraining the log mean, not the rawmea n. It is the raw mean that guarantees no arb itrage.

For the other two desirable conditions that the upfactor ex ceeds one and

If the correct mean and volatility arecaptured, any binomial probabil i tyother than zero or one will producethe Black-Schules-Merton price in thelimit.

the down factor is lessthan one , on ly theRBJRT methodologypermits an up factorthat can be less thano n e . I n t e r e s t i n g l y ,seven of the e levenmodels permit a downfactor greater than one. Only the models of Trigeorgis,Wil 1, and the JK YAB MD model have no anomalies.

These anomalies are interesting but usually occuronly with extreme values of the inputs and/or a smallnumber of time steps relative to the option maturity.They can usually be avoided when actually pricing anoption. The greatest risk they pose is probably whenthe model is used for illustrative purposes.

III Model ComparisonsTable I illustrates an exam ple for valuing a call option

in which the asset is priced at 100, the exerc ise price is100, and the volatility is 30%.^ The continuous risk-free rate is 5% and the option expires in one year. In allcases, I use the probability :Tor ;r* as specified by th eauthors of the respective models. I show the valuesfor 1 ,5,1 0,2 0,3 0,5 0,7 5 and 100 time steps. The correctvalue, as given by the Black-Scholes-M erton formula,is 14.23.At 50 times steps all of the prices are within0.06. At 100 time steps, all ofth e prices are within 0.03.

To further investigate the question of which m odelsperform best, I vary the inputs by letting the volatilitybe 0.10,0.30, and 0.50; the time to expiration be 0.25,1.0, and 4.0; and the moneyness be 10% out-of-the-money, at-the-money, and 10% in-the-money. Theseinputs comprise 27 unique combinations. I examineseveral characterist ics of the convergence of thesemodels to the Black-Scholes-Merton value.

^ The choices of stock price and exer cise price are notparticularly important as long as moneyness is consistent.Standard European options, indeed most options, have valuesthat are linearly homogeneous with respect to the stock priceand exercise price. Thus, after choosing a stock price andexercise price, we could use a scale factor to change to anyother stock price and exercise price, and we would obtain anoption price that differs only by the scale factor.

Let b N)be the value com puted for a given binom ialmodel with A'^time steps and BSM be the true Black-Schoies-Merton value. Binomial models are commonly

descr ibed as convergingin a pattern referred to as odd-even . That is, whenthe number of time stepsis odd (even), the binomialpr ice tends to be above(below) the true price. Wewill call this phenomenonthe odd-even property.Interestingly, my num erical

analyses show that the odd-even phenomenon neveroccurs for any model with out-of-the-money options.For at-the-money options, the odd-even phenomenona lway s occu r s for t he JK YA BM Dl mode l andoccasionally for other models. Odd-even convergencenever occurs for any inputs for JKY RB 2, JKYABM C2,and JKYABMD2c. Thus, the odd-even property is nota consistent phenomenon across models.

Next, I examine whether a model exhibits m onotonieconvergence, dened as

where |e{AO| = IM ^ - BSM |. That is, successive errorsare smal ler . Only the Tr igeorgis model exhibi tsmonotonie convergence and it does so for only one ofthe 27 combinations of inputs examined. Becausemonotonie convergence is virtually non-existent, weexamine a slight variation. Suppose each alternate erroris smaller than the previous one, a phenomenon wecall alternating monotonie convergence, defined as

{N)\2.As i t happens , however , a l t e rna t ing mono ton ieconvergence never occurs.

I then attempt to identify at which step a model isdeemed to have acceptably converged. For a givent ime step , I com pute the average of the cur ren tcomputed price and the previous computed price. Ithen identify the time step at which this average priceis within 0.01 ofthe BSM price with the added criterionthat the difference must remain less than 0.01 throughstep 100. The results are presented in Tables II, III,and IV. One consistent result in all three tables is thatthe RBJRT and Chriss models produce the same results.Further examination shows that tbe values of uan d lare not precisely equal for both models for all values


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CHANCE A SYNTHESIS OFBINOMIAL OPTION PRICING MODELS 9Table I Some Num erical Exam ples ^ iThe tabie contains the binomial option value for various time steps N) for a call option with stock price of 100, exercise priceof 100, volatility of 0.30, risk-free rate of0.05, and time to expiration of one year for each of the 11 binomial models. The riskneutral probability is/j orp* as specified by tbe authors of the models. Tbe Black-Scholes-Merton option value is 14.23.

151020305075100

CRR16 9614 7913 9414 0814 1314 1714 2714 20

RBJRTn oo14 7914 0014 1314 1714 2014 2714 22

ChrJss17 0014 7914 0014 1314 1714 2014 2714 22

Trigeorgis16 9714 7913 9414 0914 1314 1714 2714 20

Win17 7914 9314 0014 1214 1614 1914 2814 21

Wil217 7814 9214 0514 1514 1914 2114 2714 23

JKYABMD116 6914 7413 9214 0714 1314 1714 2614 20

JKYRB217 1714 6914 3914 3614 3314 2914 2514 24

JKYABMC217 2414 7014 4014 3614 3314 2914 251424

JKYABMD2C16 1514 5114 3114 3214 3014 2714 2414 23

JKYABMD316 6514 7313 9714 1114 1614 1914 2614 22

Table II Convergence Time Step for Binom ial Models by MoneynessTbe table sbow s the average time step A at which converg ence is achieved w here the error is defined asHb N) b{N ))/2 - BSM]where b N) is the value computed by the given binomial model for time step /V, BSM is tbe cotrect value of the option ascomputed by the Black-Seboles-Merton model, and convergence is defined as an error of less than 0.01 for all remaining timesteps tbrough 100. The ex ercise price is 100, the risk-fi ee rate 0.05 , the volatilities are is 0.10, 0.30, and 0.5 0, and the times toexpiration are 0.25 , 1.0, and 4.0. Out-of-the-m oney options bave a stock price 10% lower tban the exercise price, and in-the-money options have a stock price 10% higher tban tbe exercise price. These parameters combine to create nine options for eacbmone yness class. A maxim um of 100 time steps is used. For mode ls tbat did not converge by the 100 'time step, a value of 100is inserted.Model Moneyness S/100)

Out-of-the-MoneyCRR 66.86RBJRT 55.57Chriss 55.57Trigeorgis 62.57Will 66.93Wil2 66.64JKYABMDl 59.20JKYRB2 61.67JKYABMC2 61.92JKYABMD2C 61.92JKYABMD3 63.47

At-the-Money40 2554 3054 3031 9148 0859 8550 5160 3960 4763 5556 88

In-the-Money54 2955 0255 0263 6669 0855 066 3 2 457 3458 5863 1763 93


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5 JOURNALOFAPPLIED FINANCE SPRING/SUMMER 2008Table III. Convergence Time Step for Binomial M odeis by Time to ExpirationThe table shows the average time stqj A'at which convergence is achieved where the error is defined as\ b N) +b NA)y2 - BSM|where b{N) is the value computed by the given binomial model for time step N, BSM is the correct value of the option ascomputed by the Black-Scholes-Merton model, and convergence is defined as an error of less than 0.01 for all remaining timesteps through 100. The exe rcise price is 100, the risk-free rate 0.05, the volatilities are is 0.1 0,0 .30 , and 0.50, and the money nessis 10% out-of-the-money, at-the-money, and 10% in-thc-money. The times to expiration are shown in the columns. Theseparameters combine to create nine options for each time to expiration. A maximum of 100 time steps is used. For models that didnot converge by the 100 ' time step, a value of 100 is inserted.

Model Time to Expiration 7^0 25 1 00 4 00

CRR 33.33 54.33 87.89RBJRT 43.33 67.78 79.00Chriss 43.33 67.78 79.00Trigeorgis 33.78 53.33 84.67Will 38.33 72.33 93.33Wil2 48.44 76.67 82.56JKY AB MD l 33.67 53.78 99.22JKY RB2 51.44 74.44 81.44JKYA BMC 2 53.22 74.78 81.44JKYABM D2C 46.78 61.22 .. 100.00JKYA BMD 3 43.44 61.44 97.44

Table IV. Convergence Time Step for B inom ial Models by VolatilityThe table shows the averag e time step A'at which converge nce is achieved w here the error is defined as\{b{N)+h N- ))/2 - BSMwhere (AO is the value com puted by the given binomial model for time step A , BSM is the correct value of the option ascomputed by the Black-Scholes-Merton model, and convergence is defmed as an error of less than 0.01 for all remaining timesteps through 100. The exercise price is 100, the risk-free rate 0.05, the times to expiration are 0.25, 1.00, and 4.00. and themoneyness is 10% out-of-the-money. at-the-money, and 10% in-the-money. The volatilities are shown in the columns. Theseparam eters co mbine to create nine options for each time to expiration. A maximum of 100 time steps is used. For models that didnot converge by the 100 ' time step, a value of 100 is inserted.

Model Volat i l i ty CT)0 ^ 3 0 0 5 0

CRR 40.22 61.89 73.44RBJRT 25.78 68.89 95.44Chriss 25.78 68.89 95.44Trigeorg is 38.00 62.33 71.44Will 40.11 74.67 89.22Wit2 29.78 80.11 97.78JKYABMD 42.78 63.22 80.67JK:YRB2 27.33 82.22 97.78JKYAB MC2 27.33 82.44 99.67JKYAB MD 2e 45.33 67.78 94.78JKYA BM D3 48.33 60.22 93.78


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CH NCE SYNTHESIS OF BINOMI L OPTION PRICING MODELS 51of N but they are very close and become essentiallyequal for fairly small values ofN.

Table II illustrates that forat- the-money options,the Trigeorgis model performs best followed by CRRand Wil1.The wo rst model is JKYA BM D2c, followedby JKYABMC2 andJ K Y R B 2 . For in - the -moneyoptions, the best model isCR R, followedbyChriss-RBJRT with W il2 very close behind. Th e worst is Wil1,followed by JKYABMD3 and Trigeorgis. Forout-of-t h e - m o n e y o p t i o n s , thebest are R B J R T - C h r i s s .followed by JKYAB MD 1. The worst is Wil1,followedby CRR and Wil2.

Table IIIshows that convergen ce isalways fasterwitha shorter time toexpiration, This result shouldnot be surprising. W ith a fixed num ber of time steps,ashorter timetoexpiration mean s that the time stepiss m a l l e r . For the m e d i u m m a t u r i t y , the f as t es tconvergence is achieved by theTrigeorgis mod el,f o l lowed by J K Y A B M D I andC R R . The wor s tperformance is by Wil2, followed by JKYAB MC 2 andJ K Y R B 2 . For the s h o r t e s t m a t u r i t y , the b e s tperformance is by CRR, followed by JKYAB MD 1 andT r i g e o r g i s ; and the w o r s t p e r f o r m a n c e is byJKYABMC2, followed by JKYRB2 and Wil2. For thelongest maturity, the best performance is byRBJRT-Chriss, followed by JKYRB2 and JKYABMC2 (tied).The worst performance is by JKYABMD2c, followedby JKYABMDI andJKYA BMD 3.

TableIV shows that convergenceisalways slowerwith higher volatility. For the lowest volatility,thefastest models are RBJRT-Chriss (tied), followed byJKYRB2 and JKYABMC2 (tied). The slowest modelisJ K Y A B M D 3 , f o l l o w e d by J K Y A B M D 2 c andJKYABMDI. For medium volatility, the fastest modelis JKYABMD3, followed by CRR and Trigeorgis; andthe slowest is JKYABMC2, followed by JKYRB2 andWil2. For the highest volatility, the fastest models areTrigeorgis, followed by CRR and JKYA BM DI; whilethe slowest is JKYABMC2, followed by JKYRB2 andWiI2(tied).

Itisdifficult todraw con clusions about whicharethe fastest and slow est mo dels. Each model finishes inth e top orbottom four at least once. Althoughthetestsare notindependent,we can gain some insightby assigninga simple ranking (1=best, 11=worst)and tally the performance acrossallnine g roupings.CRR has the best performance with the lowest overallscore of36 ,while Trigeorgis isat37, and RBJRT andChriss areat38. The highest scores and, thus, worstperformance are JKYABMC2 at71.5,followed by Wil2at 69.5 and JKYABMD2cat67.5 . These rankings areuseful and could suggest that CRR, Trigeorgis, RBJRT,

and Chriss mightbethe bestset of models, but theyare not sufficient to declareadefinitive winner.

Whether a model conv erges acce ptably can bedefmed by whether the error is within a tolerance for agiven time step.Icalculate the error for the 100''' timestep. These results also reveal noconsistent winneramong the models. Most model values are within fourcentsofthe true value on the lOO time step, and thedifferences are largest with long maturity and/or highvolatility, consistent with myprevio us finding thatshort maturity and low volatility options are the fastestto pr ice .B A More Formal Look at Convergence

One p rob lem wi th ana lyz ing the c o n v e r g e n c eproperties o f a model is that it is difficult to definitivelyidentify when convergence occurs. Visual observationand rules about differences being less thana specifiedvalue are useful but arbitrary.It ispossible, however,touse a more m athematically precise definit ionofconvergence. Leisen and Reimer 1996) (LR) provide adetailed analysis ofthe convergence ofthe CRR andR B J R T m o d e l s u s i n g the n o t i o n of order ofconvergence. Amodel co nverges more rapidlytheh i g h e r the o r d e r of c o n v e r g e n c e . T h e r e f o r e ,determining the order of convergence of these modelsavoids the subjectivity ofthe previous analysis.

Convergence of a binomial model is defined to occurwith orderp ifthere existsaconstantksuch that

Visual examination o fthe errors on a log-Iog graph canreveal theorderofconvergence. LRfurther show ,however, that a better measure of convergence can bederived using the difference between the momentsofthe binomial and continuous-time distributions. Thesemoments are defined as follows:

p N):=7r\nu u-\f+{\-K)\nd{d-\f.The moment s m^(A^)and m^(A ' ) a r e obv ious ly

related to thesecond andthird mo men ts. The thirdterm is referred to as a pseudo-moment . Let p .)represent theorder of convergence of the a b o v emoments and the pseudo-moment. LR show thattbeorder of convergence ofthe binomial seriesis


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52 JOURNAL OF APPLIED FINANCE SPRING/SUMMER 2008

max[lmn (p{m {N)),p{m {N )),p{p{N)))-l].In other words, the order of convergence is theminimum of the orders of convergence of the twomoments and the pseudo-moment minus one with anoverall minimum order of convergence of one. Theyshow that the order of convergence can be derivedmathematically and they do so for the three modelsthey examine. These proofs, however, are quite detailedand cumbersome and, as they note, visual inspectionof these momen ts with a graph is equally effective.

I examine the order of convergence using themoments and pseudo-moments of each of the elevenmodels. Because of the excessive space required, Ipresent the results only for the Chriss model. Figures1, 2, and 3 illustrate various characteristics of theconvergence of the Chriss model for the previouslyused inputs . Because the LR er ror analysis usescomm on logs, I show only the time steps starting with10,

Figure 1 is the option price graphe d against thenumbe r of time steps, with the BSM value representedby the horizontal line. The converg ence is oscillatory,exhibiting the odd-even pattern previously noted.^Figure 2 shows the absolute value of the error, whichexhibits a wavy p attern. The solid line was created byproposing values for k and p such that the error boundalways lies above the absolute value of the error. Thevalue of k is not particularly important, but the valueof p indicates the order of convergence. Here, p = 1.A value of p = 2 would force the bound below the w avyerror line. Thus, the order of convergence is clearlyone. Figure3shows the moments and pseudo-mom entsas def ined by L iesen and Re imer (1996) . Thep s e u d o - m o m e n t s p ( A / ) a n d m^{N) a r e a l m o s tindistinguishable- The heavy solid line is the simpleftinction \fN where p is the order of convergence ofthe moments and pseudo-moments. In this case, p = 2provides the best fit. Therefore, following Theorem Iof Leisen and R eimer (1996 ), the order of convergenceis one, confirming my direct exam ination of the error.These graphs were generated for all of the modelsand all have an order of convergence of one. In thelimit all models produce the correct option value, butof course limit analysis m akeTV essentially infinite. Asshown earlier , seven of the eleven models admitarbitrage with finite A^, but these opportunities vanishin the limit. Also, the values of ;rand K * converge to V2in the limit. These results suggest that a model that^ As previously noted, ihe Ch riss model does not exhibit thisproperty for every case.

correc tly p reve nts a rbitrag e for all A and sets the riskneutral probability ;rat Vi fora ny A might be superior.That model is the Chriss model. And yet, there is noevidence that the Chriss model consistently performsbest for finite N.

C Why the Models ConvergeI have shown that all the models converge, but it is

not clear why. As Hsia s (1983) proof s how s, therequ i r ement fo r convergence i s no t pa r t i cu la r lydemanding, but clearly one cannot arbitrarily chooseformulas for w, d , and n.

As noted, it is possible to prove that all of theformulas for either ;ror ;r*converge toViin the lim it. Iwill exam ine why this result oc curs. F ocusing on ;r, Idivide the models into four categories: 1 )models thata s s u m e TC = V2 ( C h r i s s , W i l 2, J K Y A B M D 3 ) ;2 ) mode l s tha t assume ud = e- ( J K Y R B 2 ,JKYABMC2, JKYABMD2C); 3) models that assumeud = e^ - ^^* (RBJRT); and 4) models that assumeud - 1 (CRR, T r igeorg i s , W i l l , JKYA BMD l) .^^For 1), there is no need to examine the l imitingcondition. For 2), 3) and 4), general convergenc e isshown in the aforementioned supporting document.

Thus, all of the models either have ;ror n* convergeto Vi. The other requirements are that the models returnthe correct mean and volatility in the limit. I will look athow they achieve this result. Re-classify the modelsaccording to their assumptions about the mean. Group(a) includes all models that correctly use the arbitrage-free specification of the raw mean. Equation (14) (CRR ,RBJRT, Chriss, Will , Wil2, JKYABMC2). Group (b)includes all models that correctly use the raw meanspecification but use \+rh instead of e (JK YA BM Dl,JKYABMD2C, and JKYABMD3). Group (c) includesthe models that specify the log mean, Equation (16)(Trigeorgis and JKYRB2). Obviously Group (a) willcorrectly converge to the proper mea n. Grou p (b) willdo so as well, because e is well approximated by 1 +rh in the l imit . Group (c) uses the specification[Equation 16)],Tt \nu+ 1 -K *)\nd=ir - a^/2)h.Usingthe approximation Inw = M - 1 and like wise for d, wehave^Mt is important to understand why RBJRT is classified in thismanner and not in any other group, RB and JR obtain theirsolut ions by se t t ing the physical probabi l i ty q to / i . Theirsolution derives from using the mean of the log process, andthus, is not arbitrage-free. JT then impose the arbitrage-freecond i t i on and , hence , co r r ec t l y u se n for the r isk neut r a lprobabili ty, but this constraint cannot lead to their formulasfor H and d. Their formulas can be obtained only by imposinga third condition, which can be inferred to be the one statedhe re .


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CH NCE SYNTHESIS OF BINOMI L OPTION PRICING MODELS 63Figure 1. Convergence ofthe Chriss Modei tothe Black Schoies Merton iVIodeiThis figure showsthe option price obtainedbytheChrissmodel againsttheBlaek-Sc holes-Me rton model indicatedbythe solid line)fortime steps 10 to 100. The stoek priceistOO,the exercise priceis 100, the risk-free rateis0.05, the timetoexpirationisone year,andthe volatilityis0.30.

14,60

1 19 28 37 46 55 64 73 82 91 100ikneStqe 10-100)

Figure 2. Absoiute Value of the ConvergenceError for the Chriss Modeiand its Order Bound FunctionThis figure shows the absolute value ofthe errorforthe optionprice obtainedby the Chriss model against the Black-Scholes-Merton modelfortime steps 10to100. The stoek prieis 100,the exercise priceis 100, the risk-free rateis0.05,thetimetoexpirationisoneyear,and thevolatility is0.30. Becausetheerror boundislinear in logs,alog-log scale is used. The upperbound is the dark shaded line based on an order of eonvergeneeof one.

Figure 3. Absolute Vaiue of the Moments andPseudo moments for the Chriss Modeiand its Order Bound FunctionThis figure shows the absolute value ofthe error for the secondand third mom ents and the pseudo-mom ent as defined by Leisenand Reimerfor the option price obtainedbythe Chriss modeiagainsttheBlaek-Seholes-Merton modelfor time steps 10to100.The stoek prieis100, the exercise prieis100, the risk-free rateis0.05,thetimetoexpiration isoneyear,andthevolatilityis0.30. Becausetheerror boundislinearinlogs,alog-log scaleisused. The upper boundis the dark shaded linebased on an order of eonvergen ee of 2,which is consistent withorder of convergence ofth e model ofone.

1.0000

10 100

m\

-\ ={r-(j /2)h

This specification isextremely closetothatofGroup b), differing onlyby thevariance termon the RHS,which goestozeroin thelimit.

The next stepis toconsiderthevolatility. Let Group(a) consist of models that correctly specify the logvolatility (CRR, RBJRT, Trigeorgis, JKYRB2, Chriss);(b) consist ofmodels that correctly specify the rawvolatility (Wil 1, Wil2, and JKYABMC2); and (c) consistof models that use an approximation of the rawvolatility, 7-A^e-^' '(e ' ' ' - 1) (JKYABMDl,JKYABMD2C, and JKYABMD3). Group (a) willobviously returnthecorrect logvolatility,andGroup(b) will return the correct rawvolatility. Eitherspecification suffices because constraining the onevolatility automatically constrains the other. Group(c)


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54 JOURNAL OF APPLIED FINANCE SPRING SUMMER 2006can be shown to be based on an accep tab leapproximation by using the Taylor series for theexponential function and assuming / = 0 for all k ofpower2or more.

Hence, all of the models work because in the limitthey al have a binomial probability of'/2, and they allreturn the risk-free rate as the mean and the correctvolatility in the limit. Thus, any model with thesecharacteristics will work. As shown in the next section,however, the constraints are not nearly that severe.

IV A General Binom ial FormulaAs previously noted, Hsia's (1983) proof of the

convergence of the binomial model to the Black-Scholes-Merton model shows that any probability isacceptable provided that u and d return the correctmean and volatility. This result suggests that any valueof the r i sk neu t r a l p robab i l i t y wou ld l ead toconvergence if the correct mean and volatility areupheld. We now p ropose a general binomial m odel witharbitrary n that prohibits arbitrage and recovers thecorrect volatility for all N. Let the mean and variancebe specified as follows:

Of course, these are Equations (14) and (18). Themean equation guarantees no arbitrage profits for allN. Now assume that Kis know n but its valu e is leftunspecified. Solving forw and /gives

While yet one more binomial formula i s notnecessary, this model shows that binomial optionpricing is a remarkably flexible procedure that makesonly minimum demands on its user and the choice ofprobability is not one of them.

V ConclusionThis paper synthesizes the research on binomial

models for European options. It shows that the modelis not a single modei but a family of interpretations ofa d i sc r e t e - t im e p roces s tha t con verg es to thecontinuous Brownian motion process in the limit andaccurately prices options. That there are no less than11 such members of this family may seem surprising.The fact that they all perform equally in the limit, eventhough some admit arbitrage for a finite number oftime steps, is a testament to the extremely generalnature of the Black-Scholes-Merton model and i tsmodest requirements. It would seem that a preferablebinomial modei should prohibit arbitrage for a finitenumber of time steps and recover the correct volatility,and some models fail to meet these requirements. Butperhaps most interesting of the results shown here isthat given H sia's elegantproof the choice of the actualrisk neutral probability is meaningless in the limit,though clearly a risk neutral probability of Aassuresthe fastest convergence.*

M =Tie

neand, of course.

udFor the special case where n^ Vi the equations areequivalent to those of Chriss.These equations tell us that we can arbitrarily set n

to an y value between 0 and 1 and be assured that themodel will converge to the BSM value. This result isobserved in Figure 4. Note that while convergenceappears much smoother and faster with K- Vi th eresults are not much different for pro bab ilities of %andy 4. ForA'^^ 100, a prob ability of gives an optionvalue of 14.27, while a probability ofy4gives an optionvalue of 14.15.The correct BSM value is 14.23."

' ^A get ie ra l formula of th is lype even meat i s tha t ext remeprobabilities, say 0.01 and 0.99, would also correctly price theoption in the l imit . I tested these extrem e value s, how ever ,a nd I he r e s u l t s a r c no t i mpr e s s i ve . Fo r e xa mpl e , w i t h aprob ability of 0.01 I obtain an option value of 13.93. while aproba bili ty of 0.99 gives an option value of 13.01 after 100time steps. Convergenc e is extremeiy erratic and the order ofconv ergen ce i s d i f f icul t to de ter min e . No neth eless , in thelimit , the correct option value is obtained.


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CHANCE A SYNTHESIS OF BINOMIAL OPTION PRICING MODELS 85Figure 4 Convergence of a General BinomialModel that Prohibits Arbitrage andAllows any Probability between Zero and OneThese figures show the value of the option computed from ageneral binomial model that assures the ahsence of arbitrage,recovery of the correct log volatility, and in which the prob-ability can be arbitrarily ch osen as indicated. Tlie stock price is100, the exercise price is 100. the risk-free rate is 0.05, thevolatility is 0.30, and the option expires in one year. Thehorizontal line is the BSM value of 14.23.

Number of Time Steps (N )

21 41 61 8 101 121 141

7C= 3 41817 -

1 21 41 61 81 m i 121 141Number of Time Steps (N )

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