20.Implied Binomial Trees

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Implied Binomial Trees STORMark RubinsteinJournal of Finance, Volume 49, Issue 3, Papers and Proceedings Fifty-Fourth AnnualMeeting of the American Finance Association, Boston, Massachusetts, January 3-5, 1994(Jul., 1994),771-818.

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THE JOURNAL OF FINANCE. VOL. LXIX,NO.3. JULY 1994

Implied Binomial TreesMARK RUBINSTEIN*

ABSTRACTThis article develops a new method for inferring risk-neutral probabilities (orstate-contingent prices) from the simultaneously observed prices of European op-tions. These probabilities are then used to infer a unique fully specifiedrecombiningbinomial tree that is consistent with these probabilities (and, hence, consistent withall the observed option prices). A simple backwards recursive procedure solves forthe entire tree. From the standpoint of the standard binomial option pricing model,which implies a limiting risk-neutral lognormal distribution for the underlyingasset, the approach here provides the natural (and probably the simplest) way togeneralize to arbitrary ending risk-neutral probability distributions.

ONE OF THE CENTRAL IDEAS ofeconomicthought is that, in properly function-ing markets, prices contain valuable information that can be used to make awide variety of economicdecisions.At the simplest level, a farmer learns ofincreased demand (or reduced supply) for his crops by observing increases inprices, which in turn may motivate him to plant more acreage. In financialeconomics, for example, it has been argued that future spot interest rates,predictions of inflation, or even anticipation of turns in the business cycle,can be inferred from current bond prices. The efficacy of such inferencesdepends on four conditions:-A satisfactory model that relates prices to the desired inferred informa-tion,-A model which can be implemented by timely and low-costmethods,

-Correct measurement of the exogenous inputs required by the model,and

-The efficiencyofmarkets.Indeed, given the right model, a fast and low-costmethod ofimplementa-tion, correctly specified inputs, and market efficiency,usually it will not be

possible to obtain a superior estimate of the variable in question by anyother method.In this spirit, financial economists have tried to infer the volatility ofunderlying assets fromthe prices of their associated options. In the classic* University ofCalifornia at Berkeley. Presidential address to the American Finance Associa-

tion, January 1994, Boston, Massachusetts. I would like to give special thanks to Jack Hirsh-leifer, whowhile he has not commented specificallyon this article, nonetheless as my mentor inmy formative years, propelled me in its direction. William Keirstead, while he has been a Ph.D.student at Berkeley, has helped implement the nonlinear programming algorithms described inthis article. I am also grateful for recent conversations with Hua He, John Hull, Hayne Leland,and Alan White, and earlier conversations with RayHawkins and David Shimko.

771


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772 The Journal of Financeexample, the Black-Scholes formula for calls requires measurement of theunderlying asset price and its payout rate, the riskless interest rate, andan associated option price, its striking price, and time-to-expiration.' Theformula can be implemented in a fraction of a second on widely availablelow-cost computers and calculators. In many situations of practical rele-vance, the inputs can be easily measured and the related securities aretraded in highly efficient markets. This model is widely viewed as one ofthe most successful in the social sciences and has perhaps (including itsbinomial extension) the most widely used formula, with embedded proba-bilities, in human history.Despite this success, it is the thesis of this research that not only has the

Black-Scholesformula becomeincreasingly unreliable over time in the verymarkets where one would expect it to be most accurate, but, moreover,attempts by financial economists to extract probabilistic information fromoption prices have been puny in comparison to what is clearly possible.

I. Recent Evidence Concerning S&P 500 Index OptionsThe market for S&P 500 index options on the Chicago Board OptionsExchange provides an arena where the common conditions required for theBlack-Scholes formula would seem to be be best approximated in practice.The market is the second most active options market in the United Statesand has the largest open interest, the underlying is a cash asset rather thana future, the options are European rather than American, the options do nothave the "wildcard" feature, which seriously complicates the valuation of themore active S&P 100 index options, the options can be easily hedged usingS&P 500 index futures, the index payout can be reliably estimated or inferredfrom index futures, unlike bond prices the underlying index can a priori beassumed to followa risk-neutral lognormal process,unlike currency exchangerates the index does not have an obviousnon-competitive trader in its market(i.e., the government), and finally the underlying asset is an index which istherefore less likely to experience jumps than probably any of its componentequities and most other underlying assets such as commodities, currenciesand bonds.In early research on 30 of its component equities using all reported tradesand quotes on their options covering a two-year period during 1976to 1978, Ifound that the Black-Scholesformula seemed to provide reasonably accuratevalues.f Aminimal prediction ofthe Black-Scholes formula is that all optionson the same underlying asset with the same time-to-expiration but withdifferent striking prices should have the same implied volatility. While notstrictly true, the formula passed this test with remarkable fidelity. While Ishowed that biases from the Black-Scholes predictions were statisticallysignificant and there were long periods of time during which another option1See Black and Scholes (1973).2 See Rubinstein (1985).


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Implied Binomial Tree 773model would have worked better, there was no evidence that the biases wereeconomically significant. Moreover, while the alternative model might haveworked better for awhile, it would have performed worse at other times.I used a minimax statistic to measure the economicsignificance of the bias.The idea behind this statistic is to place a lower bound on the performance ofthe formula without having to estimate volatility, either implied or statisti-cal. Here is how it works. Select any two options on the same underlyingasset with the same time-to-expiration, but with different striking prices. Fora given volatility, for each optioncalculate the absolute differencebetween itsmarket price and it corresponding Black-Scholesvalue based on the assumedvolatility ("dollar error"). Record the maximum difference. Now repeat thisprocedure but each time alter the assumed volatility, and span the domain ofvolatilities from zero to infinity. Wewill end up with a function mapping theassumed volatility into the maximum dollar error. The minimax statistic isthe minimum of these errors. We can say then that comparing just these twooptions, for one of them the Black-Scholes formula must have at least thisdollar error, irrespective ofthe volatility.Because the Black-Scholes formula is monotonicly increasing in volatility,

the volatility at which such a minimum is reached always lies between theimplied volatilities of each of the two options and, moreover, will be thevolatility that equalizes the dollar errors for each of the two options. As aresult, the minimax statistic can be computed quite easily. I will call this theminimax dollar error. To correct for the possibility that, other things equal,we might expect a larger dollar error the higher the underlying asset value,the minimax dollar error is scaled to an underlying asset price of 100 bymultiplying it by 100 divided by the concurrent underlying asset price." Anegative sign is appended to the errors if, in the option pair, the higherstriking price option has a lower implied volatility than the lower strikingprice option.Wecouldalsomeasure percentage errors at the volatility that equalizes the

absolute values of the ratio of the dollar error divided by the correspondingoption market price. This, I will call the minimax percentage error.During 1976 to 1978, looking at a variety of pairs of options, minimax

percentage errors were on the order of 2 percent-a figure I would regard assufficiently low to make the Black-Scholes formula a goodworking guide inthe equity options market. (although not of sufficient accuracy to satisfyprofessionals whomake markets in options).More recently, I had occasiontomeasure minimax errors again during 1986 for S&P 500 index options, andagain minimax percentage (as well as dollar) errors were quite low.However,since 1986 there has been a very marked and rapid deterioration for theseoptions. Tables I and II list the minimax percentage and dollar errors bystriking price ranges for S&P 500 index calls with time-to-expiration of 125to 215 days.3 This scaling reflects that the Black-Scholes formula is homogeneous of degree one in theunderlying asset price and the striking price, so that doubling each of these variables, other

things equal, doubles the values ofputs and calls.


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Implied Binomial Tree 7 7 5Table II

Signed Scaled Minimax Dollar ErrorsData are from the S&P 500 index 125 to 215-day maturity calls, 4/2/86 to 8/31/92. Thestriking price range indicates the striking prices of the two options used to construct theminimax statistic. For example -9 percent to +9 percent indicates that the first call wassampled from in-the-money (ITM) options with striking prices between 12 and 6 percent lessthan the concurrent index and the secondcallwas sampled fromout-of-the-money(OTM)optionswith striking prices between 6 and 12 percent more than the concurrent index. In both cases,options were chosen as close as possible to the midpoint of their respective intervals. Thenumbers for each year represent the median signed minimax error from sampling once everytrading day over the year.

Striking PriceRangeITM ATM OTM ITM/OTM S&P 500

Year - 9%to -3% - 3%to +3% +3%to+9% - 9%to +9% low high1986 -0.025 -0.025 -0.007 -0.044 203.49-254.731987 -0.070 -0.056 -0.031 -0.118 223.92-336.771988 -0.251 -0.212 -0.144 -0.551 242.63-283.661989 -0.248 -0.266 -0.191 -0.599 275.31-359.801990 -0.364 -0.382 -0.297 -0.908 295.46-368.951991 -0.371 -0.382 -0.250 -0.887 311.49-417.091992 -0.422 -0.389 -0.221 -0.858 394.50-441.28

the-money puts before the crash and held them during the week of the crashwould have made huge profits: not only did put prices rise because the indexfell by about 20 percent, but put prices rose because implied volatilitiestypically tripled or quadrupled. The market's pricing of index options sincethe crash seems to indicate an increasing "crash-o-phobia," a phenomenonthat we will subsequently document in other ways.The tendency for the graph of implied volatility as a function of striking

price for otherwise identical options to depart from a horizontal line hasbecomepopularly known among market professionals as the "smile." Typicalpre- and postcrash smiles are shown in Figures 1 and 2. The increasedconcern about smiles across options markets generally, and the conferencesand even academic papers concerning them, is rough anecdotal evidence thatsimilar problems with the Black-Scholes formula reported here for S&P 500index options in recent years, may pervade options onmany other underlyingassets.Of course, the estimation of minimax statistics across otherwise identicaloptions with different striking prices is just one way to test whether themarket is pricing options according to the Black-Scholes formula. Apart fromgeneral arbitrage tests such as put-call parity, I consider it the most basictest, since among alternatives it is the easiest to verify. However, it clearlydoes not test all implications of the Black-Scholes formula. One might alsocompare in a similar way otherwise identical options with different time-to-expirations. This can provide useful information, but may not be helpful intesting a slight generalization of the Black-Scholes formula that allows


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The Journal of Finance

....-----------MINIMAX 7. ERROR------------. . . . . . .~11~;ii}7. . " ~ J 1 ~ 1 3 'O O . :3 . : : ~ : ,: ~ ~ ~ : 1 f t~ , ic 2 ~ 1 , ;O O '~ ..... ,. . ~ ~.... . ,. . ;. ; ~ ; .

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0.857 0.884 0.911 0.938 0.965 0.992 1.019 1.046 1.073Striking Price + CUrrent Index~EC

Figure 1. Typical precrash smile. Implied combined volatilities of S&P 500 index options(July 1, 1987;9:00A.M.) .

time-dependent implied volatility. These two cross-section tests can be use-fully supplemented by a third time-series test, which compares the impliedvolatilities measured today with implied volatilities of the same optionsmeasured tomorrow." If the constant-volatility Black-Scholesformula is true,these implied volatilities should be the same. Even the more general formula,allowing for time-dependent volatility, can be tested by looking for variablesother than time that are correlated with changing implied volatility. Alongthese lines, a very interesting working paper by David Shimko reports veryhigh negative correlations during the period 1987to 1989between changes inimplied volatilities on S&P 100 index options and the concurrent return ofthe index-a correlation that should be zero according to the Black-Scholesformula."

4 For the constant volatility Black-Scholes model, these time-series tests are the same as testsbased on hedging using option deltas, since to know an option's delta is the same as knowing thedifference between today's and tomorrow's option prices, conditional on knowing the change inits underlying asset price. Thus, if the Black-Scholes formula produces the correct deltas,time-series changes in implied volatilities must not be significant.5 See David Shimko (1991).


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Implied Binomial Tree 777

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0.830 0.857 0.884 0.911 0.938 0.965 0.992 1.019 1.046 1.073Striking Price ~ Current Index--,UNFigure 2. Typical postcrash smile. Implied combined volatilities of S&P 500 index options

(January 2, 1990; 10;00 A.M.).

r i s k - n e u t r a lp r o b a b i l i t y

P ( 3 }P ( 4 }P ( 2 }P ( l } P ( 5 }

sFigure 3. Risk-neutral probability distribution.

o


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7 7 8 The Journal of FinanceAlthough this article will discuss the use ofthese three types oftests, it will

not investigate what I would call statistical tests. These tests usually takethe form of comparing implied volatility with historically sampled volatility.Since the "true" stochastic process of historical volatility is not known, thesetests are not as convincing to me as the three outlined above.This discussion overlooks one possibility: the Black-Scholes formula is true

but the market for options is inefficient. This would imply that investorsusing the Black-Scholes formula and simply following a strategy of sellinglow striking price index options and buying high striking price index optionsduring the 1988to 1992period should have made considerable profits. WhileI have not tested for this possibility, given my priors concerning marketefficiency and in the face of the large profits that would have been possibleunder this hypothesis, I will suppose that it would be soundly rejected andnot pursue the the matter further, or leave it to skeptics whose priors wouldjustify a different research strategy.The constant volatility Black-Scholes model, as distinguished from its

formula, which can be justified on other grounds," will fail under any of thefollowingfour violations of its assumptions:1. The local volatility of the underlying asset, the riskless interest rate, orthe asset payout rate is a function of the concurrent underlying assetprice or time.

2. The local volatility of the underlying asset, the riskless interest rate, orthe asset payout rate is a function of the prior path of the underlyingasset price.3. The local volatility of the underlying asset, the riskless interest rate, orthe asset payout rate is a function of a state-variable which is not theconcurrent underlying asset price or the prior path of the underlyingasset price; or the underlying asset price, interest rate or payout ratecan experiencejumps in level between successiveopportunities to trade.

4. The market has imperfections such as significant transactions costs,restrictions on short selling, taxes, noncompetitive pricing, etc.

Theviolations ofBlack-Scholesassumptions at each ofthese levels becomesincreasingly serious and difficult to remedy as we move down the list.Although, violations of types 1 and 2 still leave the arbitrage reasoning-theessence of the Black-Scholes argument-intact, type 2 violations lead per-haps to insurmountable computational problems. Violations of type 3 are farmore serious, since they destroy the arbitrage foundations ofthe Black-Scholesmodel and have left researchers so far with two unpalatable alternatives:either an equilibrium model in which investor preferences explicitly enter, orother securities in addition to the underlying and riskless assets must beincluded in the arbitrage strategy. Violations of type 4 are the worst, becausetheir effects are notoriously difficult to model and they typically lead only tobands within which the option price should lie.6 See Rubinstein (1976).


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Implied Binomial Tree 779In this context, here is oneway to think of the contribution of this article. It

will provide a computationally effective way to value options, even in thepresence of violations 1, 2, and 3, and a computationally effective way tohedge options even under violation 1. Other work has dealt with violation 1,most notably the constant elasticity ofvariance diffusion model developedbyJohn Coxand Stephen Ross." But this work begins with a parameterizationof the function relating local volatility to the underlying asset price. Themodel here derives this function (which may depend on time as well) endoge-nously, and it can be thought of as an attempt to exhaust the potential forviolation 1 to explain observed option prices.

II. Implied Ending Risk-Neutral ProbabilitiesThe approach we will use to value and hedge options involves three steps.

First, we must somehowestimate the ending risk-neutral probabilities of theunderlying asset return. The approach emphasized here will be to infer thesefrom the riskless interest rate and concurrent market prices of the underly-ing asset and its associated otherwise identical European options with differ-ent striking prices. Second, we infer a unique, fully specified stochasticprocess of the underlying asset price from these risk-neutral probabilities.Third, armed with this process, we can calculate the value and hedgingparameters of any derivative instrument maturing with or before the Euro-pean options.Longstaffs Method {Amended}Ever since the work of Stephen Ross," it has been well known that in

principle it should be possible to infer state-contingent prices, or their closerelatives, risk-neutral probabilities," from option prices. The first version of arecent working paper by Francis Longstaff describes a way of doing this.l"Here I will describe a somewhat modifiedversion of his method. Let:S = = current price ofunderlying assetC1, C2, C3, C4 = = concurrent prices of associated call options with strikingprices K1 < K2 < K3 < K4, all with the same time-to-ex-pirationS* = = price of underlying asset on the expiration dater n = = one plus the riskless rate ofinterest through the expiration

date= = one plus the payout rate on the underlying asset throughthe expiration date

7 See Cox and Ross (1976).8 See Ross (1976).9 Jacques Dreze (1970) was probably the first to realize the significance of this correspondence

between state-contingent prices and probabilities.10 See Longstaff (1990). His method is quite similar to the first attempt of which I am awaredescribed by Banz and Miller (1978).


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7 8 0 The Journal of FinanceAssume that, conditional on S* being between adjoining striking prices(including 0), all levels of S* have equal risk-neutral probabilities. Alsoassume that there exists a number K5 > K4 such that the probability thatS* > K5 is zero, and that, conditional on S* being between K4 and K5, alllevels of S* have the same risk-neutral probability. Figure 3 depicts thissituation.Taking the market prices S, CI, C2, C3, C4 and the return r" as exogenous,Appendix I derives the followingsolution for PI' P2, P3, P4, P5, and K5:

PI =2[1 - rn(Sa-n - CI)K1I]P2 = 2 [ 1 - PI - rn(CI - C2)(K2 - KI)-l]

P3 = 2 [1 - PI - P2 - rn(C2 - C3)(K3 - K2)-I]P4=2 [ 1 - PI - P2 - P3 - rn(C3 - C4)(K4 - K3)-I]

P5 =1- PI - P2 - P3 - P4K5 =K4 + (2rnC4 -:-P5)

Thus, the implied risk-neutral probabilities can be derived in triangularfashion by solving the first equation for PI' using this value for PI' andsolving the second equation for P2, using these values for PI and P2 andsolvingthe third equation for P3, etc. This reveals an interesting structure ofthe solution. The relation between the underlying asset price, which can bethought of as a payout-protected zero-strike option, and the lowest strikingprice option essentially determines the probability in the lower tail. Theneach successive option contributes information about the probability fromthere to the next striking price. The upper tail probability then followsfromthe constraint that the total probability must be 1.As a test of this method, I calculated risk-neutral implied probabilitiesfrom eleven options assumed to be priced according to the Black-Scholesformula and therefore under a lognormal risk-neutral density function.TableIII compares the Black-Scholes probability in each striking price intervalwith the discrete probabilities derived using the amended Longstaffmethod.Not only are the discrete probabilities quite different than lognormal,jumping from very low to very high over adjoining intervals, even worse,many are negative. Observe that nothing in the amended Longstaffmethodprecludes negative probabilities. Clearly, assuming a uniform distributionbetween strikes and a finite upper bound to the underlying asset price doesnot produce satisfactory results.This method alsohighlights the critical problem:in current optionmarkets,we only observe striking prices at discrete intervals. Moreover, we have

considerableidentification problems in the tails because the distance betweenzero (the "striking price" of the underlying asset) and the lowest optionstriking price is usually quite large, and we have no striking prices abovesome maximum level. To solve this problem, we need to find some way to


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Implied Binomial Tree 7 81Table III

Implied Risk-Neutral Probabilities Using an AmendedLongstaff MethodTime-to-expiration was assumed to be one year, and the volatility used in the Black-Scholes

formula was assumed to be 20 percent. Other variables were S = 100, r" = 1.1, and sn = 1.05.Implied Risk-Neutral Probability

Strike Black-Scholes Call Price Black-Scholes Discrete (P)0 100.00 0.0581 0.009275 27.37 0.0479 0.142780 23.19 0.0663 -0.028385 19.27 0.0826 0.177690 15.69 0.0938 -0.000895 12.51 0.0986 0.1944100 9.78 0.0971 0.0042105 7.49 0.0902 0.1809110 5.62 0.0798 -0.0085115 4.15 0.0676 0.1562120 3.01 0.0552 -0.0338125 2.15 0.1628 0.2062148 or 00 0.00

interpolate between striking prices and extrapolate to provide satisfactorytail probabilities.

Shimko's MethodDouglas Breeden and Robert Litzenberger+' showed that if a continuum of

European options with the same time-to-expiration existed on a single under-lying asset spanning striking prices from zero to infinity, the entire risk-neu-tral probability distribution for that expiration date can be inferred bycalculating the second derivative of each option price with respect to itsstriking price.David Shimko provides a way to implement this idea.P He first plots the

smile and fits a smooth curve to it between the lowest and highest optionstriking prices. This provides him with interpolated Black-Scholes impliedvolatilities. Using the Black-Scholes formula, he inverts the implied volatili-ties, solving for the option price as a continuous function of the striking price.Then, taking the second derivative of this function, he determines the impliedrisk-neutral probability distribution between the lowest and the higheststrike options. Although he uses the Black-Scholes formula, Shimko's methoddoes not require it to be true. He has merely used the formula as a transla-tion device that allows him to interpolate implied volatilities rather than the

11 See Breeden and Litzenberger (1978).12 See Shimko (1993).


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782 The Journal of Financeobserved option prices themselves. He supplies tail probabilities by graftinglognormal distributions onto each of the tails.l"Using S&P 100 index options, Shimko then creates graphs of the risk-neu-tral distribution for various option maturities on selected dates from 1987 to

1989. His method passes the test I applied to the Longstaff method, since ifthe smile is exactly horizontal, he will imply a lognormal risk-neutral proba-bility distribution with the correct volatility.An Optimization MethodHere I propose yet another method. First, we establish a prior guess of the

risk-neutral probabilities. In general, it could be anything; but for workingpurposes, I will suppose that our prior is the result of constructing an n-stepstandard binomial tree using the average of the Black-Scholes implied volatil-ities of the two nearest-the-money call options.l" Denote the nodal underlyingasset prices at the end of the tree from lowest to highest by Sj for j = 0, ... , n.Denote the ending nodal derived risk-neutral probabilities by P ; where it willbe the case that ljP; = 1.For example, if p' is the risk-neutral probability ofan up move over each binomial period, then P; = [n!/j!(n - j)!]p'J(l - v:>.For sufficiently large n, this probability distribution will be approximatelylognormal. Let rand 8 represent, respectively, the riskless interest returnand underlying asset payout return over each binomial period.l" Let Sb (sa)be the current bid (ask) price of the underlying asset and Cib (Cn the bid(ask) price simultaneously observed on European call i =1,... ,m maturingat the end of the tree, assumed not to be protected against payouts. Choosen m.The implied posterior risk-neutral probabilities, Pj' are then the solution to

the following quadratic program:minl.(P - p~)2 subject to:Pj J J J

ljPj = 1 and Pj ~ for j = 0, ... , nSb .::;S .::;sa where S =(8nlPS)/rn

Cib .::; C, .::;Ct where C, = (ljPjmax[O, Sj - KiD/rn for i = 1, ... ,m13 Shimko does not use the information contained in the underlying asset price to help identify

the lower tail. He seems to do this because he is worried about errors that would be created bynonsimultaneity in the reporting of the index due to the familiar problem of lagged trading ofcomponents of the S&P 100 index. Except for rare moments (such as occurred on October 19 to20, 1987), my own cursory research indicates that the error created by this nonsimultaneityshould be very small. For an entire month in 1986, I constructed an index of the average time tothe last trade for the S&P 500, with market value proportions as weights. After the first halfhour of the trading day, the index was typically only about 5 minutes old. I therefore believe thathis method can be improved by treating the underlying asset as just another option, but payoutprotected with a zero striking price.14 This uses the well-known method described in Cox, Ross, and Rubinstein (1979).15 If it is known how the payout return depends on the ending nodal asset prices, then

stochastic payouts can be easily handled by replacing the equation for S with:S = (kjPjS/jp)/rn.


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Implied Binomial Tree 783The Pj are therefore the risk-neutral probabilities, which are, in the leastsquares sense, closest to lognormal that cause the present values of theunderlying asset and all the options calculated with these probabilities to fallbetween their respective bid and ask prices.This technique also passes the earlier test. That is, if all the options are

priced with bid and ask prices surrounding their standard binomial values,then Pj =Pj for all j. In addition, the technique passes a second test. If asolution exists, then the denser the set of options, other things equal, the lesssensitive Pj will be to the prior guess. In the limit, as the number of optionsbecomes increasingly dense on the real line, Pj will become independent ofthe prior.Using NAGnonlinear programming software, for a 200-step tree for S&P

500 index options observed three times a day from 1986 to 1992, in a fewseconds for each time, the algorithm always convergedto a solutionwheneverthere were no general arbitrage opportunities among the underlying asset,riskless asset, and the options-that is, whenever there existed values of Sand {C;lsuch that if transactions to buy and sell could have been effected atthose same prices (without paying the bid-ask spread), no general arbitrageopportunities existed.l"Although I adopted a specific minimization function and a specific prior

guess, the optimization method is quite flexible.As long as a solution existsand the number of probabilities n is greater than the number of options m,the solutionwill depend onthe prior guess and minimization function chosen.The least squares formofthe minimization function isjust one ofa number ofcandidates. For example, one might instead minimize the "goodness of fit"function kiPj - Pj)2/Pj or, as suggested to me by Ray Hawkins, the "maxi-mum entropy" function - kjP}og(P/Pj) or, as Ron Dembo has suggested,the absolute difference function, k)Pj - Pjl.17For most precrash days, there is very little difference if any between therisk-neutral lognormal prior guess and the implied posterior probabilities.Figure 4 provides an illustration of this procedure for a typical postcrash dayJanuary 2, 1990 at 10:00 A.M. in Chicago, using S&P 500 index optionsmaturing in June 1990.Using a 200-step tree, the risk-neutral prior guess is

16 The menu of general arbitrage opportunities is described in Chapter 4 of Cox and Rubin-stein (1985).By far the most frequent arbitrage opportunities present in the data are butterflyspreads.

17 Kendall and Stuart (1979) in Chapter 30 discuss the related problemofmeasurement ofthe"closeness" of an observed frequency distribution to an hypothesized probability distribution,including the "goodness of fit" and. "maximum entropy" criteria. In addition, they discussstrategies for spacing observations. For example, an alternative to spacing the SJ at the end of astandard binomial tree, is to space the ending asset prices, separated by areas of equalprobability. The absolute difference criteria has the advantage that the optimization problem canbe reduced to a linear rather than a quadratic, or more generally, nonlinear program.In independent research, Stutzer (1993) uses the maximum entropy criterion to solve for

state-contingent prices in a manner similar to mine. He provides a three-state example involvinga single option, using exact equality constraints for the current underlying asset and optionprices.


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7. 6.12I 5.78m 5.44f 5.10ed 4.764.424.083.743.403.062.722.382.041.701361.020.680.340.00

The Journal of Finance:~

. . . . . . . . . . . . ~ . . . . . . e !.i.\ ~ :Optimiz.ati on r,ul e: Le.ast_Squ.ares .~f '" ; STR!KE IV: BID: ASK: VALUE: LMULT............ ; "' ,,. ,......... .0 000 349. 16 349. 26 349. 16 .0001"................... j f J, ;......... 2$0.313 109. 47 109. 71' 109. 71 . -.0006 ... . .'" 275 .271 85.66 86.71 86.70 .0000.. .. .. .. .. .. ... .. . ; ; ;- ; \. " ~ :; !6 "O ; :r :; !9 ' b ~ ; 0 "0 "' b 4 ;" o4 b! 3 ; "9 I::; ' ; OOO"O"

. . .: . .: : : : : : : . : : r . . : : : . : : . : . : . : I : : : : : : : : : : : : : : / t : . : : : : : \ : : r : : . . : : . ! j ~ ~ ~ ! !. i ~ ~ ~ I , i ~ j ~ U ~ ~ f . - < ! ~ ~ i j : . j \ : ; t g : H ~ ; ~ t * ~ ; ~ ~ : ! ~ ; ~ t ; g ; : g g g g .................. i ;..i \ i. 355.167 19.32 19.88 19.55 .0000. . . . . . . . . . . . . . . . . . L ! . f . . . : ~ ' . . . : ~.9 ! , .~~.: ;I...;:?,.~.?: !,.(?,r:;:? t..~ ~ .!? ,.t.~ : .9 . 9 .9 . 9 . .~ ~ i 1 ' ; ~ 3,?5 .153. 13.13. 13.75. 13.65 . 0000. . . . . . . . . . . . . . . . . . , ; i ; , . ; 370 .148 10.52 11.15 11.15 -.0047"~ ~ f ~ . . . 375 .145 8.37 9.12 8.96 .0000: : : : : : : : : : : : : : : : : : ; : : : : : : : : : : : : : : : : : ; : : : : " : : 7 : : : : , : : : : : : : : : : : : : : : r : : : : : : ~ ; ~ : ~ ~ ~ : .! : ~ ~ : . : . ! ~ ~ : . ~ ~ :: .~~g~: :

214.4 263.1 311.9 360.6 409.3 458.1 506.8 555.5 604.3 653.0S&P 500 Index

~-N_Prior(Vol=17.17.) ~-N_Implied(Vol=207.)Figure 4. Risk-neutral 164-day probabilities of S&P 500 index options (January 2,

1990;11:00A.MJ.

assumed to be approximately lognormal with an implied volatility of 17percent, equal to the average of the two nearest-the-money options (strikes350 and 355). The risk-neutral implied posterior distribution is slightlybimodal and more highly skewed and kurtotic. The bimodality coming fromthe lower tail ("crash-o-phobia")is quite commonduring the postcrash period.The table on the upper right shows the Black-Scholes implied volatilities(column 2) and the concurrent bid (Sb, (et}) (column 3) and ask (sa, (en)(column 4) prices for each option." The index itself was assumed to have atwo-tick bid-ask spread around the 10:00 A.M. reported level of the index(354.75)after deducting anticipated dividends prior to the options' expirationdate.Using the best fitting risk-neutral probabilities, column 5 reports values(s, {C;l).Note that in each case, the values lie between the correspondingquotes. When a quotation constraint is binding so that the value is equal toeither the bid or the ask, the NAG software also reports the associated

18 The CBOE options market does not perform on command. In particular, the 13puts and 15calls, which were used to construct the bid and ask prices, did not trade simultaneously at 10:00A.M. To surmount this problem, elaborate procedures were followedto adjust nonsimultaneouslyobserved option prices to their probable 10:00 A.M. levels, had they all been quoted at that time.


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Implied Binomial Tree 785Lagrangian multiplier (or shadow price, to use the language of economics),shown in column 6. Positive multipliers indicate that the bid is binding, andnegative multipliers indicate that the ask is binding; and the absolute size ofthe multiplier indicates how important its associated option was, given thequotes of the other options, as a cause of squared differences between priorand posterior probabilities. High absolute multipliers indicate options pricedunder the more extreme departures from risk-neutral lognormality. Thissuggests the first of three potential empirical tests: examine the profitabilityofbuying options with high negative multipliers and selling options with highpositive multipliers. Even though the model is designed to fit the prices of allthe options of a given expiration, information from it can be used to isolatethe most extreme deviations from our prior guess.From this point, by whatever method-Shimko's, optimization, or some

other way-we assume that we have satisfactorily estimated risk-neutralprobabilities Pj associated with asset returns Rj =SjIS.

III. Implied Stochastic ProcessWenow take as exogenous the discretized risk-neutral probability distribu-

tion of the underlying asset returns at some specified time in the future. Forexample, suppose there are three possible ending discrete returns Ro , RI,and R2 where 0 < Ro < RI < R2. Each of these returns has known associ-ated risk-neutral probabilities Po, PI, and P2, where all the probabilities arepositive numbers which sum to one.l?ASSUMPTION 1: The underlying asset return follows a binomial process.ASSUMPTION 2: The binomial tree is recombining.ASSUMPTION 3: The ending nodal values are ordered from lowest to highest.For our example, with only three possible outcomes, we must then have ann =2 step binomial tree:

.--------u X u[u] =R2

1 [ d U..___, uXd[u]=dxu[d]=RI (A)'-------d X d[ d] =Ro

19 If some probabilities are zero, replace them with very small numbers.


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7 8 6 The Journal of FinanceHere the notation indicates that the move at any step can depend on thenode. For example, u[d) is the up move immediately following a down move.Because the tree is assumed to be recombining, u X d[u) = d X u[d).Discussion of Assumption 1The assumption of a binomial process, as a practical matter.i" places the

model, in the continuous-time limit, in the context of a single state-variablediffusion process where both the drift and volatility can be quite generalfunctions of the state variable, the prior path of the state variable, and time.

Discussion of Assumption 2A recombining tree implies path-independence of returns in the sense that

all paths containing the same number of up moves and the same number ofdown moves lead to the same nodal return, irrespective of the ordering of themoves along the paths. This restricts the limiting diffusion to one which doesnot depend on the prior path of the state variable. Itwould seem, in addition,to rule out diffusions that while path-independent, nonetheless must appar-ently be mimicked discretely by a nonrecombining tree. A well-known exam-ple of this is the constant elasticity of variance diffusion process.P However,it has been shown under fairly general conditions that any path-independenttree that is nonrecombining can, by aftjusting the move sizes, be transformedinto a tree that is recombining without changing the limiting form of theresulting diffusion.V To this extent, Assumption 2 is therefore not, as apractical matter, a restriction on the tree but is, rather, a matter of computa-tional convenience.

Discussion of Assumption 3The ordering assumption is equivalent to requiring that after an up (a

down) move, from that point on, the most extreme remaining ending returnin the down (up) direction is dropped from consideration. While this assump-tion will not change the current value of European derivatives maturing atthe end of the tree, it will affect the interior structure of the tree and henceother tree properties such as local volatility and option deltas, as well as thevalue of American options.

20 The other limiting possibility is the binomial jump process discussed in Cox, Ross, andRubinstein (1979).This limiting possibility, however, has such bizarre implications that subse-quent academicwork has shown little interest in it. Ofcourse, in the more general setting of thisarticle, where the binomial move size is itself stochastic, it would be possible for the limitingformto take on a combination of a diffusion and a two-state jump process, where at each instantoftime, it wouldbeknown in advance just which ofthese two processes would be mimicked next.

21 See Coxand Rubinstein (1985).22 See Nelson and Ramaswamy (1990).


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Implied Binomial Tree 7 87Also associate with each move, risk-neutral probabilities, where p[.](l -

p[.]) is the probability of an up (a down)move after the previous sequence ofrealized moves indicated in the brackets:

.----p xp[u] =P2

p X (1- p[u)) + (1- p) xp[d] = P1 (B)

'-----(1-p) X (1-p[d)) = PoBecause the "moveprobabilities" p[.] are risk-neutral:

rI-l =((1- p[.)) X d[.)) + (p[.] X u[.))where r[.] is one plus the riskless rate of interest over the associatedbinomial step, which at this point may be dependent on the previous combi-nation of realized moves.Therefore, each probability must be related to its associated up and down

moves as follows:p[.] = (r[.] - d[.)) 7(u[.] - d[.))

For our example:p = (r - d) 7(u - d)

p[d] =(r[d] - d[d)) 7(u[d] - d[d))p[u] = (r[u] - d[u)) 7(u[u] - d[u)) (C)

PayoutsIf the underlying asset has payouts that do not accrue to the holder of a

derivative, then we may want to measure the returns Ro, R1, and R2exclusiveof payouts and reinterpret the variable r[.] as the ratio of one plusthe riskless interest rate divided by one plus the payout rate over thecorresponding binomial step.Our goals is to infer uniquely the entire tree from the ending nodal returns

(Ro, R1, and R2) and ending nodal risk-neutral probabilities (Po, P1,and P2).Assessing our progress, we must determine 12 unknowns:

d, u, r , p, d[d], u[d], r[d], p[d], d[u], u[u], r[u] and p[u]


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788 The Journal of Financefrom 10 equations, four from equations A, three from equations B, and threefrom equations C. Since the number of unknowns exceeds the number ofequations, we will need to impose other conditions. One possible condition is:ASSUMPTION 4: The interest rate is constant (per unit of time).In our example, this means that r = r[ d) = rIzz];so henceforth we shall

usually refer to one plus the interest rate simply as r.Generalization: This assumption is not required if we know (exogeneousto

the model) the different node-dependent interest rates. These might beinferred from the current prices of default-free bonds of different maturities,from the interest rates implied in futures contracts with different deliverydates, or from a maturity series of otherwise identical European puts andcalls. Should this information be supplied exogenously, in our example, wecan let r * r[d) * r[u).It is noteworthy that so far we have said nothing about the elapsed time for

each move. For example, while the total time of all moves must be equal tothe prespecified time from the beginning to the end of the tree, we can dividethat time up across moves any way we like. For example, we might supposethat the second move is twice as long as the first move. This flexibility mayprove quite useful in some situations. Successivelylengthening the time foreach move may lead to faster convergence to the continuous-time process,because a wider span of ending returns can be reached with fewer steps.23This flexibility also provides a means ofhandling differences between tradingand calendar time-for example, differencesbetween intraweek and weekendvolatility.In this context, I like to think of the interest return r as the "clock"

running the tree. Tying it to a specificinterval or intervals oftime (not simplysaying, as we have so far, that it is the interest return over a binomial step ofindeterminant length) determines the speed of the tree. There is a kind of"equivalence principle" at work here: an individual who can only view thetree cannot distinguish changes in interest returns r[.) from changes in theelapsed time ofdifferent moves.For concreteness, wewill suppose that r[.) isalways measured over the same interval of calendar time.Assumption 4 reduces the problem to 10 equations in 10 unknowns. How-ever, it is easy to show that equations B are not independent of each other.Thus, we will need to add further structure.ASSUMPTION 5: All paths that lead to the same ending node have the samerisk-neutral probability.

23 The recent working paper by Amin and Bodurtha (1993) argues that the difficult numericalproblem of the valuation of path-dependent American options can be efficiently handled by anonrecombining binomial tree where the elapsed time per moveis successively lengthened as theend of the tree is approached.


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Implied Binomial Tree 789This implies we can write down the followingequations B':

p xp[u] =P2 ==Puu(1 - p) X p[ d) = Pl -7 - 2 = = Pdu P X (1 - p[ u)) = Pl -7 - 2 = = Pud

(1- p) X (1- p[d)) = Po = = PddThe variables Pdd, Pdu =Pud, and Puu, then, are probabilities associatedwith single paths through the tree ("path probabilities").Considering equations B' by themselves, since there are 4 equations in 3

unknowns, p, p[d], and p[u], they would seem to be overdetermined. How-ever, using the special structure of their right-hand sides, namely thatPdd + Pdu + Pud + Puu = 1, it is easy to showthat any three of the equationscan be used to derive the fourth.Generalization: This assumption is not required if we know (exogenous to

the model) the different path-dependent probabilities (1 - p) X p[d] =Pduand p X (1 - p[ u)) = Pud' These might be inferred from the current prices ofoptionsmaturing beforethe ending date, fromthe prices ofAmerican options,or from options with path-dependent payoffs. In any case, we continue torequire Pdu + Pud =Pl'This ends the specificationofthe model.Given these equations, the known

ending nodal returns R o ," " R2 and nodal probabilities Po , . ' " P2, we showbelow that it is possible to solve for a unique binomial implied tree: d, u,d[d], u[d], d[u], u[u], and r. Of course, from equations C, we can thenimmediately determine p, p[d], and p[u], shouldwewish to doso.Moreover,we also showthat the solution is consistent with the nonexistence of risklessarbitrage as we work backwards in the tree.Beforewe dothis, note that the standard binomial optionpricing model is a

special case, since all five assumptions hold for that model as well but withthe added requirement that u and d are constant throughout the tree. Animplication of constant move size is that Po = (1 - p)2, P1 = 2p(l - p),P2 =p2. In contrast, the modelhere allows these ending probabilities to takeon arbitrary values and, therefore, represents a significant generalization.

IV. The SolutionThe implied binomial tree can now be solved conveniently by workingbackwards recursively from the end ofthe tree.Here is the general method; it's as simple as One-Two-Three.The unsub-

scripted P variables belowrepresent path probabilities, and the R variablesrepresent nodal values. Say you are working backwards from the end of atree, and you have worked out (P+, R+) and (P-, R-) and want to figure out


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790 The Journal of Financethe prior node (P, R):

[

(P+'R+)

(P,R)

(P-, R-)One: P = P-+ P+Two: p =P+/P

Three: R =[(1 - p)R- + pR+ ]/rThat's it! and you are now ready for the next backwards recursive step.To start everything rolling, go the end of the tree and attach to each nodeits nodal value Rj and nodal probability Pj' Now take each ending nodalprobability and divide it by the number of paths to that node to get the pathprobability, which is in general:

Pj --;- [nl/j!(n - j)!]Also,define the interest return r as the nth root ofthe sum of PjRj, so that:

r" = "i , PR . (with payouts: (r/8)n = "i , PR .)JJJ JJJEach of these three steps makes simple economicsense. The first simplysays that an interior path probability equals the sum of the subsequent path

probabilities that can eminate from it. The second step is the simple proba-bilistic rule that allocates total probability across up and downmoves since:p =P+/P and (l-p) =P-/P

The third step uses the risk-neutral move probability so calculated todetermine the interior nodal value R by setting it equal to the discountedvalue of its risk-neutral expected value at the end of the move.

Interior arbitrageStarting with positive ending path probabilities, step one insures that allinterior path probabilities are also positive, since the sum of two positivenumbers is obviouslypositive. Step two then implies that all moveprobabili-ties p[.] are positive, since they are the ratio of two positive numbers, andmoreover they are less than one since the P+ < P. This fully justifies ourhabit of referring to the p[.] as move "probabilities." A necessary andsufficientconditionfor there to benoriskless arbitrage opportunities betweenthe riskless asset and the underlying asset at any point in the tree is that rI-l(or r[.118[.] with payouts) must always lie between the correspondingvaluesof d[.] and u[.] at each node. Indeed, from equations C, the fact that thecorresponding p[.] qualify as probabilities guarantees that this will be so.


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Implied Binomial Tree 7 9 1Significance of Assumption 3With this solution, it is quite easy to see that Assumption 3, which governs

the ordering of the returns at the end of the tree, affects the structure of thetree. For example, consider a 3-step tree where Po =P1 =P2 =1/3, and wepermute the ordering of Ro, R1, R2 at the end of the tree to R1, Ro, R2.Where before d = [(2/3)Ro + (l/3)R1]jr and u = [(l/3)R1 + (2/3)R2]jr,with the permuted ordering d' = [(2/3)R1 + (1/3)Ro]jr and u' =[(1/3)Ro+ (2/3)R2]jr. Clearly:d < d' < u' < u

so that the local volatility over the first move will be smaller under thepermuted ordering.Some n-Step Tree PropertiesSee Appendix II for a 3-step numerical example. In the third step of thetree, the eight possible move sizes are d[ dd], u[ dd], d[ du], u[ du], d[ ud],

u[ud], d[uu], and u[uu]. For example, d[du] means the downmovefollowinga sequence of first a down move followedby an up move. By assumption,since the tree is recombining, it must be path independent in the sense thatat any node in the tree, for the next move,while its size may depend on thenumber of up or down moves that gave rise to that node, its size must beindependent ofthe ordering of these prior moves.This is an implication ofthereturn path independence we discussed before. In this case, it means opera-tionally that d[du] = d[ud] and u[du] =u[ud].In addition, it is quite easy to verify the followingresult.r"Given only Assumption 1:Ifall ending paths containing the same numbers

ofup and downmoveshave the same ending risk-neutral probability, then allinterior paths containing the same numbers ofup and downmoves also havethe same interior risk-neutral probability.Since, as a special case, the supposition to this result clearly holds underAssumptions 2 and 5, its conclusionwill as well.For example, in a 3-step tree, the equations replacing equations B' would

be:p X p[ u] X p[ uu] =P3 = = Puuu

p xp[u] X (1- p[uu]) =P2 -i- 3==Puudp X (l-p[u]) xp[ud] =P2 -i- 3 =P; ;(l-p) xp[d] xp[du] =P2 -;- 3 ==Pduu

p X (1 - p[u]) X (1 - p[ud]) =P1 -i- 3 = = Pudd

(1)(2)(3)(4)(5)

24One may be tempted to believe the complementary result: given onlyAssumption 1, if allending paths containing the same numbers ofup and downmoves lead to the same ending node,this will be true of interior paths as well (in other words, the tree will be recombining).However,such is not the case.


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792 The Journal of Finance(l-p) xp[d] X (l-p[duD =P1 -7- 3 ==Pdud(l-p) X (l-p[dD xp[dd] =P1 -7- 3 ==Pddu(1 - p) X (1 - p[dD X (1 - p[ddD = P o = = P ddd

(6)(7)(8)

In this case there is only one interior node that can be reached by morethan one path: the middle node at the end of step 2 which is reached by twopaths. We need to show that these path probabilities are equal. That is:

p X (1-p[uD =(1-p) xp[d]To see this, equations (3) and (5) imply p[ud] = P2i(P1 + P2). Equations (4)and (6) imply p[du] = P2i(P1 + P2). Therefore, p[ud] = p[du]. Substitutethis into equations (3) and (4) [or equations (5) and (6)], and we have ourdesired result.This 3-step example also highlights the special role played by the interestrate. In addition to providing a clock, it is also the primary route throughwhich the model communicates with and is bound to the external world. Iconjecture that in any reasonable economy,it is always possible to design asecurity whose stochastic process obeysAssumptions 1and 2. But, given onlythese two assumptions, we are no longer free to impose whatever structurewe wish on the risk-neutral probabilities since:Given only Assumptions 1 and 2: The risk-neutral move probabilities willbe path-independent if and only if the riskless interest rate is path-indepen-dent.This followsimmediately from equations of type C, which provide expres-sions for p[du] and p[ud]:

p[du] =(r[du] - d[duD -7- (u[du] - d[duDp[ ud] =(r[ ud] - d[ udD -7- (u[ ud] - d[ udD.

SinceAssumptions 1 and 2 imply that d[du] = d[ud] and u[du] = u[ud],we must then have p[du] = p[ud] if and only if r[du] = r[ud], the condi-tions which embody the notion of path independence. Since we have justshown that Assumptions 1, 2, and 5 imply p[du] = p[ud], these must alsoimply r[du] =r[ud].As a result of these properties, the qualitative features of the tree that we

have noted for the ending nodes are recursively reproduced as we workbackwards in the tree.This reduction of the solution to a simple recursive procedure is quitefortunate. For example, in a 200-step lattice, we need to determine a total of60,301unknowns: 40,200 potentially different move sizes, 20,100potentially

different move probabilities, and 1 interest rate from 60,301 independentequations, many of which are nonlinear in the unknowns. Despite this, the


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Implied Binomial Tree 793solution procedure is only slightly more time consuming than for a standardbinomial tree with given constant move sizes and move probabilities.r"

v. ExtensionsAsmentioned earlier, Assumptions 4 and 5 can be completelydropped ifwe

have someway ofknowing howone plus the interest rate r[.] varies with theunderlying asset return and time and how the individual ending risk-neutralprobabilities Pj for j =0, ... , n are divvied up among different paths leadingto the same ending node.Node-dependent interest rates

In some situations, we may be willing to infer the time dependence of r[.]from the forward rates implied in the current prices of riskless bonds ofdifferent maturities. For example if Bl and B2 were the current prices ofzerocoupon bonds yielding $1 at the end of steps 1 and 2, respectively, then wecouldpreset r = 1/Bland r[d] = r[ u] = BII2. It is much more difficult tosee where we could obtain reliable information about the dependence of r[.]on the combination of prior moves-that is, to justify by inference from theprices of securities that r[ d ] * - r[ zz,However, suppose we interpret r[.] as one plus the rate of interest divided

by one plus the underlying asset payout rate. Wemay then come to appreci-ate this added flexibility.This gives us a simple way of incorporating into thetree, at minimal computational cost, a payout rate that can be a very generalfunction of the concurrent underlying asset return and time, provided thisfunction can be exogenously specified.Introducing dividends into the standard binomial model by adjusting the

risk-neutral move probabilities can lead to incorrect results when dividendsare highly state or date dependent. For example, for some common equitieswith quarterly dividends, one might try to include the effect of dividends bycalculating the moveprobability as p[.] =((r/8[.]) - d) -7 - (zz- d). However,for accurate trees with a large number ofmoves, with 8[.] sufficiently largeand discrete, it is quite easy for r/8[.] < d over some moves, producingnegative and nonsensical "probabilities." Fortunately, as Bruno Dupire hasmentioned to me, this is not a problem with the implied tree constructedhere, because the move sizes d[.] and u[.] will automatically be adjusted in

25 In his notes on which he has based several talks during the last two years, Hayne Leland atBerkeley has solved a similar problem. He shows that under certain assumptions, given endingand possibly path-dependent desired personal wealth levels, ending subjective (not risk-neutral)probabilities of an investor relative to the market consensus investor can be implied. Itis thenpossible to work backwards from the end of a recombining binomial tree and derive the treestructure of the investor's subjective beliefs. He uses his analysis to answer the puzzlingquestion: who should buy exotic options such as lookbacks and Asians? Although he assumes astandard constant move size binomial tree, his work gave me the faith to pursue the researchreported here-that I should not be daunted by the number ofvariables to be determined in thetree.


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794 The Journal of Financethe recursive procedure to insure that d[.] < (r/o[.]) < u[.] at everymoveinthe tree.So, if we knowhow r[.] and 0[.] depend on the underlying asset and time,we can use these rates in constructing our tree. However, we are notcompletely free to choose them, since, to avoid arbitrage opportunities, theone-plus interest rates (possibly divided by one-plus payout rates) must beindividually positive and jointly satisfy:

1=PdiRo/(r X r[d))) + Pdu(Rd(r X r[d)))+ Pud(Rd(r X r[ u))) + Puu(R2/(r X r[ u)))

This is an obviousgeneralization ofthe above solution for r under constantinterest rates, which, for purposes of comparison, can be restated as:

In words, more generally, each ending return must be discounted by itsassociated path of interest rates before talking risk-neutral expectations.While this generalization extends easily to an n-step tree, to maintain thebenefits of a recombining tree, we cannot go so far as to allow the interestrate structure to be path dependent. In the absence ofAssumption 4 or 5, weno longer have a way of guaranteeing that the interest rate will be pathindependent. Sofor example, in a 3-step tree we must then add the require-ment that r[du] = dud]. .Different Risk-Neutral Path Probabilities at the Same Ending NodeA natural way to infer these probabilities is from standard options with

maturities prior to the ending date. In our 2-step example, suppose throughoptions that mature at the end of step 1,we are able to infer the risk-neutralnodal probabilities (which in this special case are also path probabilities) Pdand Pu at that time, where Pd + Pu = 1. It turns out that this gives us justenough information to infer the individual path probabilities Pdu and Pud,which, since,we have dropped Assumption 5, are no longer assumed to beequal.To see this, in a 2-step tree, over the first step, we must have p =Pu' Wecan now reconsider equations B, which before left the move probabilitiesindeterminant. With this added restriction, they can be solved easily for:

p [ d] = 1 - Po -7 - Pd and p [ u] = P 2 -7 - PuNote that in this case the two path probabilities leading to the middle endingnode at step 2 are not generally equal; that is:

Pdu = (1 - p)p[d] = Pd - Po and Pud = p(l - p[u)) = Pu - P2Of course, jointly they continue to satisfy the requirement that:

Pdu + Pud = (Pd - Po) + (Pu - P2) = 1- Po - P2 = PI


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Implied Binomial Tree 795Extending this example to n-steps, remember that even though it is no longertrue that all paths that lead to the same interior or ending node have thesame probability, the move probabilities must remain independent of theprior path. For example, in a 3-step tree, it remains the case that p[ ud] =p[du]. This followsfrom the assumption that the binomial tree is recombin-ing. As previously noted, a recombining tree implies that d[du] = d[ud],u[ du] =u[ ud]. In addition, tomaintain the benefits of a recombining tree, inthe absence ofAssumptions 4 and 5, we must in their place assume r[du] =r[ud]. We then have by our earlier result: p[ud] =p[ud].To imply all the path probabilities in an n-step tree requires that we knowexogenously all the nodal interior and ending probabilities in the tree. Inturn, these nodal probabilities can be inferred from standard Europeanoptions, provided that their maturities span all nodal dates in the tree.26That is, we would need currently available options maturing at the end ofstep 1, step 2, step 3, etc. Given the nature oftraded options, this is clearly anunrealistic expectation for trees sufficiently fine to be ofpractical value. Thatis why, for application purposes, we will continue to maintain Assumption 5.However, in future research, it should prove useful to investigate methods

of interpolating across the available option maturities to fill in the missingexpiration dates, or to find some way to use the current prices ofAmericanoptions for the same purpose.Even without this, what shorter maturity options that do exist can be putto good purpose. Here is a second potential empirical test: compare theirmarket prices to the value for them that we would infer from the earlyportion of our binomial tree implied from the prices of longer maturityoptions. GivenAssumptions 1 to 4, this gives us a way of separately testingAssumption 5 concerning path-independent risk-neutral probabilities. If thetree successfullypredicts the contemporaneous prices ofthe shorter maturityoptions,Assumption 5 need not concern us.

VI. Volatility and Mean StructureMove (Local) VolatilityKnowing the full binomial tree, at any node we can measure the movevolatility (]"[.]as follows:

fL[.] = = ((1 - p[.]) X logd[.]) + (p[.] X log u[.])(]"2[.] = = ((1 - p[.]) X [logd[.] - fL[.]]2) + (p[.] X [log u[.] - fL[.]]2)Holding fixed the time remaining to the ending node, if limits are taken

properly, as the number of moves increases and the move size goes to zero,the move volatility will approach the instantaneous diffusion volatility, com-26 Note that for there to be no arbitrage opportunities, the interior nodal probabilities derived

from option prices would need to be consistent across dates. In our ,2-step example, we wouldrequire that P d > P o and P u > P 2.


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796 The Journal of Financemonly written as a(S, t ), in the continuous-time literature. In the Black-Scholesmodel the diffusionvolatility is assumed to be constant, or at most afunction of time-either known in advance, or implied in some way fromoption prices. In other models (such as the constant elasticity of the variancediffusionmodel), the diffusionvolatility is assumed to be a known function ofS, where perhaps a free parameter might be inferred from option prices. Bycontrast, not onlydoes the modelhere permit dependenceof a on t as well asS, but the model does not even begin with a specific parameterization ofa(S, 0; instead, using the rules to construct our binomial tree-and despitethe fact that a(S, t ' ) is likely to be a very complexfunction unique for eachdifferent assumed ending return distribution-are]' as an approximation toa(S,O, can nonetheless be easily determined by using the above backwardsrecursive solution procedure.Table IV examines a "typical" day from the postcrash period. Bid (ask)quotes from 13June puts and 15June calls on January 2, 1990at 10:00A.M.

on the S&P 500 index were averaged for each striking price and input intothe quadratic program to estimate risk-neutral probabilities for the Juneexpiration date. These probabilities were then taken as inputs to create animplied binomial tree. Finally, at each node in the tree the local volatility wascalculated using the above equation. Although the annualized local volatilityon January 2 was virtually the same as the annualized global volatility overthe lifeofthe options, the predicted pattern oflocalvolatility over time showsconsiderable variation. Using the model as a lens to peer into the future, wewould expect the local volatility to rise dramatically should the index declinerapidly. For example, if the index fell from 355 to 302 (an 18percent decline)over the next twelve calendar days, the annualized local volatility shouldquadruple from about 20 to 77 percent. But, if the same drop in the indexoccurred over a longer time interval, say three months, then the localvolatility wouldonlydouble.On the other hand, increases in the index shouldbe accompanied by significant decreases in local volatility, although againthis effect tends to be attenuated the longer the increase takes to occur. If,instead, the index remains relatively flat over the next three months, thenlocal volatility should decline to about 15 percent. On the downside, it istempting to conclude that the market has built into option prices a repeat ofthe experience during the crash: a sudden decline in prices led to a tripling oreven quadrupling of Black-Scholes implied volatility, which gradually sub-sided as the market stabilized in the months followingthe crash.These results are consistent with Shimko's time-series empirical observa-

tion that Black-Scholes implied volatility varies strongly and inversely withthe contemporaneous index return. The constant elasticity of variance diffu-sion formula would also predict that local volatility should be inverselycorrelated with a contemporaneous index return. However, since that modelis stationary, it will not predict the time-dependent nature of this relation.Table IV suggests a third and final empirical test: if we really take the

model seriously, we should be able tomarch into the future along the realizedbinomial path and find that options continue to imply what remains of the


28/49

Implied Binomial Tree 797original tree. Ofcourse, the Black-Scholesmodel demonstrably fails this test,and realistically we cannot expect a very closefit here, since the real worldiscertainly much more complex than any model could be. So the interestingquestion to answer is one of comparative models: can predictions of localvolatility be improved by using this approach compared to existing alterna-tives?Global VolatilityRemember that we are looking at local volatilities, not Black-Scholes-im-

plied volatilities that summarize the level of uncertainty over the entire lifeof an option. The very high (low) local volatilities shown in the table couldimply much lower (higher) Black-Scholes-implied volatilities over longerintervals. This is evident from the table, which indicates that the effects ofextreme volatility levels, other things equal, are attenuated as onemovesintothe future. This is an issue that can be resolved within the structure of themodel. As we work backwards in the tree, at each interior node we cancalculate the ending nodal probabilities conditional upon arriving at thatinterior node (see the formula is SectionVII).Using these conditional proba-bilities, at each interior node we then calculate the risk-neutral globalvolatility from that node looking forward to the end of the tree. Table Vdisplays the annualized global volatility structure. Indeed, the sensitivity ofglobal volatility to the level of the index is about half the sensitivity of thelocal volatility. So, for example, if the index falls from 355 to 302 over thenext twelve days, while the local volatility rises from about 20 to 77percent,the global volatility rises from 20 to 38 percent. Similarly, on the upside, ifthe index moves from 355 to 400 in twenty-seven days, the local volatilityfalls from 20 to 4 percent, but the global volatility only falls from 20 to 9percent.It is also possible to calculate a tree of annualized at-the-money Black-Scholes-implied volatilties. To do this, at each interior node calculate theBlack-Scholesvalue of an option (see Section VII) with a striking price setequal to the index level at that node and a time-to-expiration equal to theremaining time to the end of the tree. Then invert the Black-Scholesformulato obtain the implied volatility at each node. Such a tree shows that thecurrent (time 0) implied volatility is 17 percent. If, after twelve days, theindex falls from 355 to 302, the implied volatility rises to 39 percent. On theother hand, if, after twenty-seven days, the index rises from 355 to 400, theimplied volatility falls to 9 percent. So the behavior of the at-the-moneyBlack-Scholes-impliedvolatility is quite similar to the globalvolatility.Consensus MeanIn the diffusion continuous-time limit, the movevolatility calculated fromrisk-neutral probabilities and the movevolatility calculated from the "true"

market-wide (consensus) subjective probabilities convergeto the same num-ber as the move size approaches zero.However,this is not true for the mean.


29/49

798 The Journal of Finance

o

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t - ~ a q M O O ~ t - : C \ l c . c O M t - 1 " " " ' I t q O ' : : l ~ O O O O O ' : : l I 0 1 " " " ' I~ 1 0 1 O c . C c . C t - t - O O O O m m m o o o , . . . . ; , . . . . ; c - . : i c - . : i c v : i ~ 1 " " " ' 1 1 " " " ' 1 1 " " " ' 1 1 " " " ' I . . . . . . t 1 " " " ' l 1 " " " ' l 1 "" " ' 1 1 "" " ' 1~OIOOIOO~OOC\lc.cO~OOC\lt-~~C\!OO~C\I~ t O t O c . C c . C r : . . . : r : . . . : r : . . . : o o o o m m m o o 1 " " " ' l 1 " " " ' l C \ l C \ i c v : i ~

1"" " ' I 1 " " " ' I . . . . . . t 1 " " " ' l 1 " " " ' l . . . . . . t . . . . . . t . . . . . . t

C \ l t - ~ a q C \ l t - 1 " " " ' I ~ O ~ O O C \ l c . c O t q O I O O t - ~ C \ I~ ~ I O I O c . C c . C t - t - O O O O o o m m o o , . . . . ; , . . . . ; c - . : i c - . : i c v : i ~ 1 " " " ' 1 1 " " " ' I 1 " " " ' I 1 " " " ' I . . . . . . t 1 " " " ' l 1 " " " ' 1 1 " " " ' 11 " " " ' I c . c 1 " " " ' l c . c 1 " " " ' l I O O ~ O O C \ l c . c O ~ O ' : : I M O O ~ O t - ~ M~ ~ t O t O c . C c . C ~ r : . . . : r : . . . : o o o o m m m o o , . . . . ; c - . : i c - . : i c v : i ~

, 1"" " ' I , . . . . . j 1 "" " '1 1 "" " ' 1 1 "" " ' 1 1 "" " ' 1

0 ' : : I ~ 0 ' : : I ~ a q o o t - 1 " " " ' I I O m M O O C ' l c . c 1 " " " ' l ~ C \ l 0 ' : : I c . c 1 0 1 Oc v : i ~ ~ I O I O c . C c . C r : . . . : r : . . . : r : . . . : O O o o m m o o , . . . . ; , . . . . ; c - . : i c v : i ~, , 1 "" "' 1 1 "" "' 1 , 1 "" " ' 1 1 "" " ' 1

t - C \ l t - : 1 " " " ' I c . c O l O m o o t - 1 " " " ' I I O O l O m l O ~ a q ~ t - : t -c v : i ~ ~ t O t O c . C c . C c . C r : . . . : r : . . . : o o o o m m m 0 1 " " " ' l 1 " " " ' l C \ l M ~1"" " ' I , . . . . . j , 1 "" " ' 1 1 "" " ' 1 1 "" " ' 1

I O O I 0 0 ' : : I ~ a q C \ l t - 1 " " " ' I I O m M o o M o o ~ ~ a q t - 0 ' : : I 1 " " " ' Ic v : i ~ ~ ~ t O l O c . C c . C r : . . . : r : . . . : r : . . . : O O o o m m o 1 " " " ' l 1 " " " ' l C \ i c v : i t O, 1 " " " ' 1 , , 1 " " " ' 1

M O O M t - C \ l c . c O l O m M t - 1 ' " " ' ! c . c 1 " " " ' l c . c M ~ a q ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ o o o o m m o ~ ~ ~ ~ ~ ~~~~~~~ ~ ~ o ~ ~ ~ ~ ~ ~ o ~ ~ ~ ~ o ~ ~ ~ o~ ~ ~ ~ ~ ~ ~ ~ ~ ~ o o o o o o m o ~ ~ ~ ~ ~ ~~~~~~~ o o ~ ~ ~ ~ ~ ~ o o ~ o o ~ ~ ~ ~ ~ o o ~~ ~ ~ ~ ~ ~ ~ ~ ~ o o o o m o ~ ~ ~ ~ ~ ~~~~~~

~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~ ~ o o c O m o ~ ~ ~ ~ ~ ~~~~~~t - - : ~ ~ ~ ~ o ~ ~ ~ ~ ~ o o o ~~ < : . O ~ ~ ~ o o o o m o ~ C ' i ~ ~ ~~~~~~~

o o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ c O m o ~ C ' i ~ ~ c O~~~~~~~ ~ ~ ~ ~ o o o ~c O m o ~ ~ ~ ~ m~~~~~~

~~~~o~~~~c:i~~~~~


30/49

Implied Binomial Tree

o

~~~~~oo~~~~~o~~~~~~~ o o m ~ ~ ~ o o ~ ~ O O ~ ~ ~ ~ ~ ~ M M~~~~~~~MMM~~~~~~~~M~~M~~~o~m~o~m~MMO~OOO~~~O~~OM~O~M~~~~~~~~~MMM~~~~~~~~~

. . . . . .00 ~~~~M~OO~~~M~O~~~~~~ m ~ M ~ m ~ ~ m M ~ m ~ ~ ~ ~ o o m~~~~~~MMM~~~~~~~~~~m~mOMoo~~~oo~o~m~M~o o m ~ ~ o o ~ ~ o o ~ ~ o o ~ ~ ~ o o m o ~~~~~~MMM~~~~~~~~~~~~~M~~m~~~m~~~~oo~~OOO~~mM~O~~O~~OOO~~~~~~~~MM~~~~~~~~~~~Mm~~~~MO~~~~O~O~O~m ~ ~ o o ~ ~ m ~ ~ O M ~ m ~ ~ ~ ~ ~~~~~MMM~~~~~~~~~~~~O~~~MM~mM~~oooo~m~~O ~ ~ m M ~ ~ ~ O O ~ ~ m ~ ~ ~ ~ m m~~~~MM~~~~~~~~~~~~~~~OO~~~~~~~~~OO~M~~O~~O~OO~~OM~O~~~m~~~~~MMM~~~~~~~~~~~~~~~~~OOO~~~O~M~~~OO~~ ~ o o ~ ~ o ~ m ~ ~ o ~ ~ O O O ~ ~ ~~~~MM~~~~~~~~~~~~~~~~M~~m~~oo~~~m~o~~~ ~ O ~ O O ~ ~ ~ ~ O O ~ ~ O O ~ M ~ ~ O O~~MMM~~~~~~~~~~~~~oo~ooo~~o~~~moo~~~~~~~~~~o~m~~~~~o~~oomo~~MM~~~~~~~~~~~~~OO~o~~~~~~~~o~~~~~~~~~~~~~~~~~~O~~OOO~~~~~~~~~~~~~~~~~OOOOOO

~~~~~~~~~~~~o~q~~~~O~~~OO~~~~OO~~~~~~~~~~~~~~~~~~~~~OOOOOOOO~~~~oo~oo~~~~~~oo~~~~~~~~~O~~~~~~~O~~~OO~~~~~~~~~~~~~OOOOOOOOOO

~~o~~~~~~~~~~~~~~~OO~OO~~~~~~~~~O~~~O~~~~~~~~~~~~~OOOOOOOO~~o~~~~~~~~~~~~~~~~~o ~ m ~ m ~ o o ~ ~ ~ ~ o o ~ ~ ~ m ~ ~~~~~~~~~~~~~OOOOOOOO~~~~o~oo~~~~~~~~~ooo~~~~~~O~O~~~~o~~m~~~~~~~~~~~~~~OOOOOOOO~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~ C ' i ~ o ~ o o ~~~~~~~c:.o~~~~OO

799


31/49

8 0 0 The Journal of Finance

o

~~~~~~ooo~~~~~~~oo~~~~OO~~~~~~~~~~M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~io00

OOMOO~~OM~oo~~~~m~~~m~~oomoo~~~~~~~MMM~~~~~~~~~~~~~~~~~~~~~~~~~~~~. . . . . .00 ~~~O~~~M~OO~~~~OM~OO~~~moo~~~~~~~~MMM~~~~~~~~~~~~~~~~~~~~~~~~~~~

MOOM~~~~~~~~~~~m~~~~~~mmoo~~~~~~~MMM~~~~~~~~~~~~~~~~~MMMMMMMMM

O~M~~~~m~~~~~~OOMM~~~~mmOOOMM~~~~MMM~~~~~~~MMMMMMMMMMMMMMMMMMMOOMOO~~~~~O~~~M~~~~~m~~OOmmOOMMM~~~~MMM~~~~~~MMMMMMMMMMMMMMMMMM~M~O~OOM~OOMM~m~~~M~mMOOoommooo~~~~~~~~MM~~~~~MMMMMMMMMMMMMMMMMM

t-io~m~mM~~~~m~~OO~~~M~m~mOOOOmmOOMM~~~~~M~~~~~~~MMMMMMMMMMMMMMMMM~~~~M~OO~~OOM~~~~~o~m~Mo o o o m m o o o M ~ ~ ~ ~ ~ M M ~ ~ ~ ~ ~ ~MMMMMMMMMMMMMMMMMo~o~mM~o~~m~~oo~~o~~~~o o o o m m m O O ~ M ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~oo~oo~~~~oo~~~~~~~~o~~~~~ o o o o m m o o o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~m~~m~~m~~~~m~~ooo~ o o o o m m m o o o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~

o~m~~~~~~~~~~~~o~~~~o o o o o o m m o o o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~m~~o~~o~m~o~~~~o o m m m o o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~

~oo~~oo~~~~m~~m~oomm m o o o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~o~~~~C()~~~~~~~~o c : i o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ o o~~~~~~~~~~~~~~

m~~~oo~~~~~ooo ~ ~ ~ C ' i ~ ~ ~ ~ ~ o o~~~~~~~~~~~~m~~~~o~C ' i ~ ~ ~ ~ ~ o o m~~~~~~~~

~ooo~~~~~oom~ ~ ~ ~ ~oo'"


32/49

Implied Binomial Tree

o

~oo~~~~~~~~~~~~~~~~~ ~ o o m o ~ ~ ~ ~ ~ ~ o o o o m o o o o~~~~~~~~~~~~~~~~~~~qmoom~~~mommoo~~m~oo~ o o o o m o ~ ~ ~ ~ ~ ~ o o m o o o ~ o~~~~~~~~~~~~~~~MMM

. . . . . .00 ~~M~~OM~~OOOO~MO~~~~~ o o m o ~ ~ ~ ~ ~ ~ o o m o ~ ~ ~ ~ ~~~~~~~~~~~~~MMMMMMoooom~~~~~~~~Mm~~~~~~ o o m ~ ~ ~ ~ ~ ~ o o m o o ~ ~ ~ ~ ~~~~~~~~~~~~MMMMMMM

. . . . . .t- ~~~OO~~~~M~~~~~~qq~OOmO~M~~~OOmO~~~~MM~~~~~~~~~~~MMMMMMMM~oo~~~~ooo~~~~~m~~~~o o m ~ ~ ~ ~ ~ o o m O ~ ~ ~ ~ ~ M M M~~~~~~~~~MMMMMMMMMO~OOMoo~~oomoo~~~~o~~~mo~~~~~oomo~~~~~~~~~~~~~~~~~MMMMMMMMMMOOM~~~m~~M~moo~~ooq~mO~M~~~mO~~~~~~~~~~~~~~~~~MMMMMMMMMM

m~o~~~~~~~q~~mM~oom~~~~~OOO~~MM~~~~~~~~~~~~NMMMMMMMMMMM

~~~MOOM~~mm~~~m~M~~o~~~~oomo~~~~~~~~~~~~~~~~~MMMMMMMMMMM

~~~O~OM~~~~~~Mm~~M~~~~~mo~~~~~~~~~~~~~~~~~M~~~~~~~~~~~

~ ~ ~ ~ ~ ~ o ~ ~ ~ ~ o ~ m ~ ~ ~ ~~ ~ ~ ~ o o m ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ O O O O~~~~~~~~~~~~~~~~~~~ ~ ~ ~ o ~ ~ ~ o o o ~ ~ ~ ~ o ~ ~ ~~ ~ ~ ~ m o ~ ~ ~ ~ ~ ~ ~ ~ O O O O O O O O~~~~~~~~~~~~~~~~~~m ~ ~ o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ o o ~ ~ ~~ ~ ~ o o m ~ ~ ~ ~ ~ ~ ~ ~ o o o o m m m~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~ ~ o ~ o ~ ~ ~ ~ ~ ~ o o ~~ ~ ~ O O O ~ ~ ~ ~ ~ ~ ~ o o o o m m m o~~~~~~~~~~~~~~~~~~

t-. . . . . .~ m ~ ~ ~ ~ ~ ~ o o ~ ~ ~ ~ ~ o o o ~ ~~ ~ ~ m o ~ ~ ~ ~ ~ ~ o o o o m m o o o~~~~~~~~~~~~~~~~~~~ ~ ~ m ~ o o o ~ ~ ~ ~ ~ ~ ~ ~ o o m o o~ ~ o o m ~ ~ ~ ~ ~ ~ o o o o m m o o o o~~~~~~~~~~~~~~~~~~~~o~~~~~~~~~~ ~ m o ~ ~ ~ ~ ~ ~ o o m~~~~~~C'I')~~~~~

801


33/49

8 0 2 The Journal of FinanceThe risk-neutral mean for a singlemove is obviously r. If q[e] (1 - q[eD is theconsensus subjective probability of an up (down) move after the previoussequence of realized moves indicated in the brackets, then the correspondingconsensus movemean is:

m[e] = = [((1- q[e]) X d[e]) + (q[e] X u[e])] X O[e ]where 8[e] is one plus the payout rate over the next binomial step after theprevious sequence of realized moves indicated in the brackets, and we mustbe careful to interpret d[e] and u[e] as only the capital gain portion of theunderlying asset return.In general, not only is m[e] not equal to r in discrete time, but neither willit convergeto r in the continuous-time limit. In fact, if the underlying asset isthe market portfolio, market-wide risk aversion implies that m[e] > rthroughout the entire tree. This is consistent with the common observationthat the subjective probability distribution of ending returns cannot beinferred only fromknowledge ofits risk-neutral distribution.Ending Node-Dependent MeanHowever, in a fully specified utility theory framework, at least for the

marketwide portfolio, knowing the implied risk-neutral binomial tree goes alongway toward a full specificationofthe consensus stochasticmoveprocess.f?To take a very special but classic example, consider a complete marketseconomywith a representative investor whomaximizes the expectedutility ofending wealth with constant relative risk aversion subject to the usualbudget constraint that she invest all her wealth. Since, in this case, herproportional investment choice is invariant to her level of wealth, we canregard her as having a utility function U(8nR), measured in terms ofreturn,and an initial wealth of 1. In brief, she chooses R j by solving the followingLagrangian problemr'"

where Qj> 0 is the subjective probability she attaches to state j (so thatIjQj = 1). The PJirn are often called "state-contingent prices."

27 Arecent complementary article by He and Leland (1993), is also in the context of a singlestate-variable path-independent diffusion process for the market portfolio return and the risklessinterest rate, together with an exogenously specified utility of terminal wealth. Their key resultis a differential equation that the local drift and volatility must follow to be consistent withequilibrium. Knowing one, say the drift, it is then possible to solve the differential equation forthe other, in this case the volatility. In this context, the work here can be viewed as a binomialimplementation of this differential equation by using risk-neutral probabilities inferred frommarket prices as a wedge between the drift and volatility, permitting the solution of either onewithout first knowing the other.

28 Actually, if r, 8, and the Rj are taken as exogenous, the problem amounts to determiningthe equilibrium risk-neutral ending nodal probabilities Pj.


34/49

Implied Binomial Tree 8 0 3Solving the first-order conditions that arise after differentiating with re-

spect to R/

where

Now, returning to our original binomial tree where we assumed that wesomehowknew both the Rj and Pj' we nowhave a formula for converting theending nodal risk-neutral probabilities P, into the ending consensus subjec-tive nodal probabilities Qj.This allows us to say a little more about Assumption 5. Under our utility

theory framework, since the utility function of return does not depend on thepath that leads to the return Rj, and as long as interest rates are not nodedependent (although they may be time dependent), then a necessary andsufficient condition for Assumption 5 (which restricts risk-neutral probabili-ties) is that all paths that lead to the same ending node have the sameconsensus subjective probability.Figure 5, matched to Figure 4, displays the difference between risk-neutral

probabilities and consensus subjective probabilities, assuming logarithmic

I................ d ~~i\ , Opti mizati on rul e: Least Squares .i i iJ'\\ i STRI:KE IV: BID: ASK: VALUE: LMULTI : r ~ \ l ~ ; f f i i ~ : ~ : ~ j ~ i i ; H ' : j m 1, J , \ \ , 330 .204 37.97 38.60 37.99 .0000

""1-1 j r r ' , iii iii I Iii iii iii i i i iii I I i I I I I I I rfl'TrrnliTTr.-,.,;TTrrn'1TTrn,-yTTrrn,-y-214.4 263.1 311.9 360.6 409.3 458.1 506.8 555.5 604.3 653.0SII:P500 Index

~-N_Pr1or(17.17.) ~-N_Implied(207.) ~on_Prior(16.87.) ~on_Implied(17.57.)Figure 5. Consensus 164-day probabilities of S&P 500 index options (January 2, 1990;

11:00A.MJ.

8.0217.6737.3246.9756.626

7. 6.278I 5.929m 5.580r1 5.231ed 4.883P 4.534r0 4.185ba 3.8361 3.488Ir 3.139e 2.790s 2.441

2.0931.7441.3951.0460.6980.3490.000


35/49

8 0 4 The Journal of Financeutility, U(8nR) = 10g(8nR). Although the posterior distributions are shiftedto the right since the market risk premium would then be about 3.3 percent,the qualitative shapes of the distributions are quite similar. Indeed, even ifthe utility functionwere _(8nR)-65, which produces a market risk premiumof 5 percent, the two distributions remain less so, but are still, to the nakedeye, quite similar in shape. This tempts me to suggest that, despite warningsto the contrary, we can justifiably suppose a rough similarity between therisk-neutral probabilities implied in option prices and subjectivebeliefs.Move-Dependent MeanIt is well known, for the special case of logarithmic utility, that over time

our representative investor will followa myopicdecision rule even in the faceofa changing set ofinvestment opportunities. Therefore, as we work forwardin the tree, at each interior node she chooses investments d[.] and u[.] tosolve the followingproblem:

maxt C l - q[.]) X U(8[.]d[.])) + (q[.] X U(8[.]u[.]))subject to (((1 - p[.])/r) X 8[.]d[.]) + ((p[.]jr) X 8[.]u[.]) = 1

where, for our special case, we have U(8[.]d[.]) =10g(8[.]d[.]) andU(8[.]u[.]) = 10g(8[.]u[.]). Sinceutility functions are unique up to increasinglinear transformations, with logarithmic utility we can safety regardU(8[.]d[.]) "" logd[.] and U(8[.]u[.]) "" log U[.] .29 Since A =1 in this case, ateach interior node:

q[.] =p[.] X (u[.]jr) X 8[.]We now have an easy way at each interior node of the tree to convert ourderived risk-neutral move probabilities p[.] into the consensus subjectivemoveprobabilities q[.]. Moreover,this gives us all the information we need tocalculate the consensus mean structure m[.] over all steps in the tree.Table VI, in parallel with Table IV, examines the annualized local consen-

sus mean assuming logarithmic utility. At high mean levels, the localmean isquite nonstationary and very dependent on previous realized return. At lowlevels, the local mean becomes almost insensitive to the index and time as itapproaches the 9.0 percent riskless interest rate that provides a lower bound.Since the local mean is inversely correlated with return, the return itselfshould be highly mean reverting. Of course, given Table IV, this behavior ofthe mean should come as no surprise, since the tradeoff between risk andreturn in equilibrium will typically imply that local consensus mean andvolatility will be positively correlated.

29 Notice that the budget constraint is simply a restatement of equations C defining "risk-neu-tral" probabilities.


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Implied Binomial Tree 805VII. Options

American OptionsAs in a standard binomial tree, the current value of a standard American

call option can be derived by working backwards recursively from the end ofthe tree using the following two rules:

at the end: C[.] =max[O,S[.] -K]in the interior: C[.] =maxj S]] - K, (1 - p[.]) X Cd[.])

+(p[.] X CJ.])) -i- r]where K is the striking price of the option, S[.] = = SR[.], R[.], and C[.] arethe underlying path return and the call value after the previous sequence ofrealized moves indicated in the brackets, and Cd[.] (CJ.D is the call valueafter a following down (up) move.Below, we will also use the notation, C(.)[.] and S(.)[.], more generally to

indicate the call value and underlying asset price after the previous sequenceof realized moves indicated in the brackets and after the following sequenceof moves (.). So for example, Sdu[.] means the underlying asset price observedafter the sequence of moves indicated in the brackets and followed by downmove followed by an up move. Ifthe brackets are omitted, then the values arecurrent, that is, measured at the beginning of the tree.For standard call options, the current option "delta" ("" aC / as) and

"gamma" ("" a2CjaS2) may be read directly from the first steps of thegeneralized binomial tree:a = = (Cu - Cd) -i- 8(Su - Sd)

Md] = = (Cdu - Cdd) --;-8[d](Sdu - Sdd)Mu] = = (Cuu - Cud) --;-8[ u](Suu - Sud)r = = (Mu] - a[d]) -i- 8(Su - Sd)

In a standard binomial tree, one naturally calculates the call option "theta"("" -aCjat) as:

E > = = (Cud - C) --;-2hwhere h is the elapsed time for a single binomial step. This works becaus

Documents

20.Implied Binomial Trees