Estimating Stock Price Volatility in the Black-Scholes-Merton Model

November/December 2011Current risk-free interest rateCumulative normal distribution functionTheoretical option price = pN(d 1) - se -rt N(d 2)Time remaining until expiration, expressed as a % of a yearwhere d 1=p v– – 2s 2 tln( ) + (r + )v √ tVolatility measured by annual standard deviationd 2= d 1-v √ t

A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINESv a l u a t i o n•Estimating Stock Price Volatility in theBlack-Scholes-Merton Model•By Aaron M. Rotkowski, CFA, ASAClosely held companies that issue stock options need to know the fairmarket value of these stock options for a variety of purposes, includingdispute resolution, gift and estate tax compliance, transaction-related,reporting requirements, and others. This article presents analyticalprocedures that can be used to estimate the fair market value of stock optionspayment in a transaction••When options are part of an estatethat is subject to estate tax••The divorce of an executive whoholds optionsissued by companies whose shares arenot listed on a stock exchange or quotedon the NASDAQ.The Black-Scholes-Merton (BSM)option pricing model is perhaps themost widely used option pricing modelused by valuation analysts to estimatethe fair market value of non-tradedstock options issued by closely heldcompanies. I will focus on the stockprice volatility component of the BSMmodel. Stock price volatility is an importantfactor in many option valuationmodels—usually the most importantfactor. This particular input is difficultto estimate, is subject to the judgmentof the valuator, and exerts a significanteffect on the fair market value of thesubject stock option.The BSM option pricing model’spopularity and acceptance have increasedsince it was first introduced inthe early 1970s. It is noteworthy that in1997 the BSM model authors won theNobel Prize for their work in developingthe option pricing model.In this article I will not addressshare-based compensation for financialstatement reporting purposes. Forfinancial statement reporting purposes,share-based compensation is regulatedby FASB’s Accounting StandardsCodification (ASC) 718, formerly SFASNo. 123R.I will first present a list of commonreasons why valuation analysts are askedto estimate the fair market value of closelyheld company stock options. Second,I will discuss analytical procedures thatcan be used to estimate stock price volatilityfor closely held company stock.Reasons to Value StockOptionsEmployer companies and their advisersneed to know the fair marketvalue of stock options in a variety ofsituations, including (but not limitedto) the following:••Transferring ownership of optionsto a third party••Damage suits in which the value ofoptions is at issue, such as a breachof contract suit between an existingor former executive and the issuingcompany••The repurchase of options by the issuingcompany••When options are issued as a form ofEstimating Stock PriceVolatilityThe five basic components of theBSM option pricing model are: (1) optiontime to expiration, (2) option exerciseprice, (3) risk-free interest rate, (4)current price of the underlying stock,and (5) expected volatility of the stockprice over the life of the stock option.The first three items listed are typicallyknown with certainty; those values are notcontroversial in a dispute. However, whenthe company that issues the stock optionis closely held, estimating the fourth andfifth items, stock price and stock volatility,require judgment by the analyst.Estimating the stock price of a closelyheld company is a complicated issue. Forpurposes of this discussion, we shall assumethat the stock price of the subjectclosely held company is known as of theoption valuation date. Therefore, thisdiscussion focuses on the stock pricevolatility estimate in the BSM model.As stock price volatility increases, thelikelihood that a stock's performance willbe extreme (either positive or negative)increases. Since the employee owns a calloption, he or she would be expected tobenefit from price increases, but wouldthe value examinerNovember/December 201113

A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINESTable 1: Stock Option Value Sensitivity to VolatilityOption CharacteristicsStock Price $10.00Exercise Price $12.00Risk-free Rate 0.30%Time To Expiration1 YearExpected Volatility 20% 30% 40% 50% 60% 70%Indicated Stock Option Value 0.22 0.55 0.93 1.32 1.72 2.12be protected from the downside risk thatcould otherwise occur from holding thestock directly. This is because the employeehas nothing to lose (assuming no considerationwas exchanged for the option).Table 1 shows the sensitivity of a hypotheticalstock option to changes inthe expected volatility of the stock price,using the BSM option pricing model.As shown in Table 1, when expectedvolatility increases from 30 to 40 percent,the indicated stock option valueincreases from $0.55 to $0.93, or by 69percent. As expected volatility increases,incremental volatility changes of 10 percenthave less of an effect on the indicatedoption value. That is, an increase inexpected volatility from 20 to 30 percenthas a greater effect on the option valuethan an increase in expected volatilityfrom 60 to 70 percent, all other factorsequal. The example in Table 1 illustratesthe importance of the expected volatilityestimate in the BSM model.The volatility component of theBSM option pricing model is a forward-lookingmeasure. The mostlogical starting place to look for an estimateof future volatility is past volatility.1 Historical stock price volatility is1 Don M. Chance, Analysis of Derivatives for theCFA Program, Baltimore, MD, 2003, pg. 226.a function of the past variability in thereturns on the stock as measured bychanges in the stock price. These historicaldata are generally available forpublicly traded companies. However,when valuing the options of a closelyheld company, reliable historical pricestypically are not available.The remainder of this article willpresent analytical procedures and otherfactors that valuation analysts can considerto estimate the expected stockprice volatility for a closely held company.The analytical procedures and factorsdiscussed in this article are not intendedto provide a set of rules that valuationanalysts must follow to estimate volatilityfor every closely held company. Determiningthe appropriate proceduresto estimate the expected stock pricevolatility for a closely held company (a)should be determined on a case-by-casebasis and (b) requires substantial professionaljudgment.Valuators often estimate the expectedvolatility for a closely held company stockprice by analyzing the volatility of reasonablycomparable publicly traded companies(PTCs). There are several ways toperform this type of guideline companyanalysis. I will present three methods toconduct a guideline PTC volatility analysis.The first procedure in any guidelinecompany analysis is to select a groupof PTCs that are similar to the subjectcompany. A rule of thumb for the numberof guideline companies required fora meaningful comparison is betweenthree and 10 companies. Of course, thecomparability—in terms of line of business,size, geographic diversity, productline diversification, and other factors—ofthe selected guideline PTCs tothe subject company affects how manyguideline PTCs are required for a meaningfulcomparison.If the subject closely held companystock was appraised using a market approachvaluation method, the same companiesthat were used in the market approachvaluation method can be used asthe guideline PTCs to estimate volatilityfor the stock of the subject company.After the valuation analyst selects agroup of guideline PTCs that are sufficientlycomparable to the subject company,the next step is to analyze the volatilityof these companies.The first method of estimating volatilityis to review and analyze the volatilitythat each guideline PTC reports in its SECForm 10-K. Publicly traded companiesthat grant employees some form of sharebasedcompensation (e.g., stock appreciationrights) are required under ASC 718to measure the cost of all share-based paymentplans at fair value. The financial statementfootnotes of PTCs that follow ASC718 typically include a description of howthe fair value of the company’s share-basedcompensation was determined. If the companyused the BSM option pricing model,then it would disclose the expected stockprice volatility. 22 It is important to reiterate that this discussionfocuses on estimating the fair market value of closelyheld company stock options; it is not intended toprovide accounting or valuation advice for the valuationof stock options for financial reporting purposes.14November/December 2011the value examiner

A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINESThe methods that PTCs use to estimatevolatility vary from firm to firm.These estimates may be (a) forward-looking,(b) based on historical trading prices,or (c) a combination of those two factors.For example, the expected stockprice volatility estimated by CDI Corp.“is based on the historical volatility ofthe company’s common stock over theprevious five years.” 3Conversely, Manpower estimatesexpected volatility, “using a weightedaverage of daily historical volatility(weighted 75 percent) of our stock priceover the past five years and impliedvolatility (weighted 25 percent) basedupon exchange traded options for ourcommon stock.” 4A second method to estimate expectedvolatility for a closely held companystock price is to calculate the actual historicalvolatility of the stock prices forthe selected guideline PTCs. This volatilityis backward-looking. Historical volatilitycan be estimated over any periodthe valuation analyst selects. It is oftenappropriate to calculate historical volatilitybased on the same term as the subjectstock option. That is, if the subjectstock option has two years to expiration,then the valuator may want to calculatehistorical stock price volatility for the selectedguideline PTCs over the two yearsprior to the option valuation date.Representative results from the twocomparable company analyses discussedabove are presented in Table 2.This table shows the volatility analysisfor a subject company that providesstaffing services to companies locatedin the western United States. Therefore,the selected guideline PTCs also providestaffing services.3 CDI Corp., SEC Form 10-k for the fiscal yearended December 31, 2010, page 54.4 Manpower, Inc., annual report for the fiscal yearended December 31, 2010, page 57.A third method to analyze the expectedvolatility for a closely held companystock price is to analyze the impliedvolatility in the guideline PTCspublicly traded stock options.Using this method, the valuator canset the option price in the BSM optionpricing model equal to the market priceof the publicly traded stock option andwork backwards to find the impliedvolatility. The implied volatility can beestimated this way because all otherinputs required by the BSM optionpricing model—including the optionvalue—are known. This implied volatilityrepresents the volatility that optiontraders are using to price the publiclytraded stock option. The process usedto infer the implied volatility is an iterativeprocess that involves trial and error.After (a) selecting guideline PTCs,(b) collecting the self-reported stockprice volatility for the selected guidelinePTCs, (c) calculating the historical stockprice volatility for the selected guidelinePTCs, and (d) calculating the impliedvolatility of the guideline PTCs publiclytraded stock options (if available),the valuator has a reasonable basis forestimating a level of volatility that is appropriatefor the subject company andits underlying stock and stock options.This is arguably the most important stepin the comparable company analysis. Itis also the step that requires the mostjudgment on the part of the valuator.Factors to ConsiderThe factors in the following list maybe considered by the valuation analystwhen estimating stock price volatilityfor a closely held company based onguideline PTC data. This list is not comprehensive,and not all of the factorslisted will apply in every engagement.You should use your professional judgmentto estimate stock price volatilityfor a closely held company when usingthe BSM option pricing model.••If the subject company was previouslyappraised (e.g., if the subjectcompany had quarterly or annualvaluations prepared for ESOP administrationpurposes), then youcan compare the historical changein the company stock price with thehistorical change in the stock pricesof the selected guideline PTCs.••Analyze the range of guideline PTCstock price volatility. If there are outliers,you may attempt to understandwhy a company has a particularly highor particularly low stock price volatilityrelative to the other guideline PTCs.••Review guideline PTC volatility fora period of five or 10 years prior tothe valuation date. If there was a significantchange (e.g., merger, significanttransaction, or restructuring)regarding a company, you may berequired to analyze the company orindustry in greater depth to determinewhat caused the change.••Stratify the selected guideline PTCsto produce a smaller group of companiesthat is very similar to thesubject company. For example, if thesubject company operates conventionalgrocery stores, and the guidelinePTCs operate both conventionaland “all natural” grocery stores, youmay want to analyze both (a) the entiregroup of guideline PTCs and (b)a subset of that group that only includesconventional grocers.••Analyze summary statistics fromthe reported volatilities. These summarystatistics may include the high,the low, the median, and the average.••Compare the guideline PTC historicalvolatility with the guidelinethe value examiner November/December 201115

A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINESTable 2: Black-Scholes-Merton Option Pricing ModelSelf-Reported Stock Price Volatility for the Guideline Publicly Traded Companies (%)5-Year 10-Year Volatility [a]Company Name Mean Median Mean Median 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001Barrett BusinessServices60.7 61.5 59.8 60.0 58.8 61.9 61.5 NA NA 59.0 61.0 62.0 58.0 56.0CDI Corp. 44.3 40.5 40.5 37.5 52.5 55.5 40.5 36.0 37.0 35.0 34.0 38.0 36.0 40.0Hudson HighlandGroup57.7 58.0 62.1 59.0 NA NA 58.0 60.0 55.0 55.0 55.0 65.0 73.5 75.0Insperity NA NA 85.2 89.0 NA NA NA NA NA 89.0 90.0 92.0 86.0 69.0Kelly Services 21.3 21.3 28.5 30.0 NA NA NA NA 21.3 27.0 30.0 31.7 31.0 30.0Kforce 73.6 73.0 58.9 50.0 NA NA 75.4 73.0 72.5 50.0 50.0 50.0 50.0 50.0Manpower Group 34.0 30.0 33.9 34.9 41.0 42.0 30.0 27.0 30.0 30.0 39.8 40.1 40.9 17.7On Assignment 59.3 51.7 58.5 58.0 73.3 73.1 51.5 47.1 51.7 58.0 58.0 59.0 58.0 55.0Robert HalfInternational38.5 38.5 49.2 50.2 NA NA NA NA 38.5 47.1 49.4 50.9 55.5 54.0SFN Group 63.2 50.0 59.9 53.5 99.0 78.0 50.0 40.0 49.0 52.0 55.0 59.0 68.0 49.0TrueBlue 48.0 43.5 61.9 56.3 59.6 53.0 43.5 40.5 43.5 50.0 79.0 89.0 79.0 82.0MINIMUM 21.3 21.3 28.5 30.0MAXIMUM 73.6 73.0 85.2 89.0AVERAGE 48.9 45.2 53.8 51.8MEDIAN 48.0 43.5 58.7 51.8[a] Based on the implied and historical volatility of the company’s stock options, as provided in the company’s SEC Forms 10-K.PTC self-reported volatility (whichgenerally is a forward-looking measure).You should seek to understandany material differences between thetwo measures.••Consider the strengths and weaknessesof the guideline PTC self-reportedvolatility. Strengths are: the measureis often forward-looking, and it is estimatedspecifically for use in the BSMoption pricing model. Weaknessesare: little detail is known about howthe volatility was estimated, and it issubject to management judgment.••Consider the strengths and weaknessesof the guideline PTC historicalvolatility. Strengths are: thismeasure is objective and can becalculated for the same term as thesubject option. The primary weaknessof this measure is that it isbackward-looking.••You may consider the operationsand financial situation of the subjectcompany and the selected guidelinePTCs. For example, if the companiesoperate in different regions,have varying amounts of intangibleassets, are capitalized with differentamounts of debt, or are disparate in16November/December 2011the value examiner

A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINESHistorical Stock Price Volatility Estimates for the Guideline Publicly Traded Companies (%)5-Year 10-Year Historical Volatility [b]Company Name Mean Median Mean Median 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001Barrett BusinessServices46.6 41.6 49.8 47.0 37.6 41.6 67.0 39.9 47.0 61.6 53.9 NA NA NACDI Corp. 58.7 48.8 48.2 44.2 48.8 75.5 80.7 40.1 48.2 26.7 35.9 31.4 39.4 55.0Hudson HighlandGroup79.2 66.3 73.3 66.3 60.3 115.1 98.9 55.4 66.3 84.9 32.2 NA NA NAInsperity 50.7 47.4 62.4 50.4 46.5 47.4 65.5 46.7 47.5 53.3 46.2 63.8 124.8 82.4Kelly Services 48.7 55.9 42.3 38.2 55.9 69.0 61.4 34.4 23.0 23.9 27.7 34.3 42.0 51.3Kforce 49.3 44.6 61.7 58.0 41.6 60.5 61.9 37.8 44.6 41.7 55.4 82.9 97.7 92.9Manpower Group 42.1 35.2 37.5 33.5 35.2 54.6 63.8 29.6 27.0 26.4 24.5 31.8 43.5 38.5On Assignment 64.7 50.7 62.6 54.8 50.7 103.4 74.2 48.8 46.4 48.5 50.6 59.0 79.1 65.5Robert HalfInternational39.5 32.3 40.4 36.4 32.3 48.3 55.1 31.0 30.6 33.5 32.3 39.4 49.3 52.4SFN Group 79.1 64.7 65.6 59.2 64.7 123.1 107.5 56.1 44.3 40.3 39.8 54.6 63.1 62.3TrueBlue 56.0 52.7 56.9 50.5 52.7 70.4 65.4 48.4 43.0 40.3 40.4 42.8 74.5 91.6MINIMUM 39.5 32.3 37.5 33.5MAXIMUM 79.2 66.3 73.3 66.3AVERAGE 56.8 49.9 55.1 49.2MEDIAN 53.3 49.8 59.3 50.5[b] Equals the one-year (260 trading days) historical volatility at calendar year-end, per Bloomberg.Sources: SEC Forms 10-K, Bloomberg, and Willamette Management Associates calculations.size, then the stock price volatilityfor the guideline PTCs may be greaterthan or less than what is appropriatefor the subject stock option.The analyses listed above relate primarilyto comparisons between the subjectcompany and the selected guidelinePTCs. However, the valuator shouldalso consider the subject company attributesthat make it unique. For starters,the company is a closely held company(and not a company whose shares arelisted on a national stock exchange orquoted on the NASDAQ).The fact that the company is closelyheld exerts both positive and negativeinfluences when it comes to stock pricevolatility. On one hand, closely held companiesgenerally (although certainly notalways) are smaller, are less geographicallydiversified, are less vertically integrated,and have inferior access to capital, comparedwith their publicly traded peers.These factors typically cause the requiredrate of return on an equity investmentin a closely held company to begreater than the required rate of returnon an equity investment in a publiclytraded company. A greater cost of equi-the value examiner November/December 201117

A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINESTable 3: Daily Index Value ChangeDate (2011)ClosingPriceDowty generally is characterized by greatervolatility of returns.To illustrate this point, let’s considerthe volatility of two investments: onewith a low required rate of return (i.e.,long-term U.S. treasury bonds) and onewith a high required rate of return (i.e.,equity of a start-up biotech company).The volatility of returns for the start-upcompany typically will be greater thanvolatility of returns for the debt investment.That is, the investment with thehigher required rate of return has thehigher expected stock price volatility.Therefore, if (a) the required return onequity is greater for a closely held companythan its publicly traded peers, and(b) a higher required return on equityis associated with a higher expectedvolatility, then the stock price volatilityof a closely held company generallyshould be higher than the stock pricevolatility of its publicly traded peers.However, there are at least fourreasons why this may not necessarilybe true:••Closely held companies typicallyare owned by a small number ofshareholders.Daily Change(Percent)Staffing Companies [a]ClosingPriceDaily Change(Percent)August 12 11,269.02 + 1.1 163.97 - 0.1August 11 11,143.31 + 3.9 164.20 + 5.7August 10 10,719.94 - 4.6 155.40 - 5.3August 9 11,239.77 + 3.7 164.12 + 5.0August 8 10,809.85 - 5.5 156.36 - 8.3August 5 11,444.61 170.46[a] Based on an index comprised of the selected guideline PTCs identified in Table 2.••Shares of closely held company stocktypically trade infrequently.••Closely held companies are not subjectto the same disclosure and reportingrequirements as PTCs.••Publicly traded company stock pricesoften are affected by short termand macroeconomic factors thatgenerally do not exert the same levelof influence on the fair market valueof closely held company stock.Those four related factors cause thevolatility of closely held companies tobe lower than the volatility of publiclytraded companies.To illustrate this point, consider thestock of a closely held staffing company(ABC Co.) whose shares are owned by(a) the ABC founder and (2) the companyemployees through an ESOP. Let’sfurther assume that ABC is appraisedannually for ESOP administration purposes.In the annual ESOP appraisal ofABC, the valuator will consider, amongother factors:••The historical, recent, and projectedresults of ABC Co.••The state of the industry and theeconomy in which ABC operates••The prices of guideline PTCs••How the change in value from thelast appraisal corresponds to theprior three factorsAs a sanity check on the annual stockprice change, the valuator may compare,for example, the annual percent change inthe ABC Co. share price with the percentchange in the ABC aggregate projectedcash flow. This comparison may involvethe values in the current appraisal and thevalues in the prior appraisal.Compare this to the factors that causethe share price of publicly traded companystock to change. The share price ofa PTC stock increases or decreases basedon many of the same factors that cause theshare price of closely held company stockto change. However, a PTC share pricealso is largely affected by macroeconomicnews, short-term factors, and the whimsof non-employee investors.Just look at the results during theweek ending August 12, 2011, of (a) theDow Jones Industrial Average index and(b) a staffing company index that wascreated using the guideline PTCs identifiedin Table 2. The closing price of theDow and a staffing company index duringthat week is presented in Table 3.The change in Dow closing pricefrom August 5 to August 12 in Table 3was a 1.5 percent decrease. The absolutedaily change during that period exceeded3 percent in four out of five days. Inone 15-minute period on August 11, theDow declined by 130 points; on that daythe Dow was up nearly 4 percent.Major news events around thesedates contributed to the index valuefluctuation: the U.S. economy remainedweak and a recession was looming, U.S.debt had recently been downgraded, andthe recent debt ceiling fight was trouble-18November/December 2011the value examiner

A PROFESSIONAL DEVELOPMENT JOURNAL for the CONSULTING DISCIPLINESsome to investors. These negative factorsexplain why stock market investors werepessimistic and/or cautious, but theyalone do not explain the significant volatilitythat occurred during that week.According to Christopher Blum, thechief investment officer of the U.S. BehavioralFinance Group at J.P. Morgan AssetManagement, “Markets have been absolutelyin panic mode. Rumors were goingaround that were wildly fantastic, and themarket sells off on it anyway—you can seehow irrational behavior is.” 5The same dramatic daily swings thatoccurred in the Dow also occurred in astaffing company index. The staffing companyindex analyzed in Table 3 is based onthe guideline PTCs identified in Table 2.As presented in Table 3, the daily indexvalue change for staffing companies wasgreater than or equal to 5 percent in fourof the five days reviewed.Let’s take a step back for a moment,consider the examples above, and makesome common-sense observations.First, the collective outlook regardingthe companies that comprise the Dow didnot decline 5.5 percent Monday, increase3.7 percent Tuesday, decline 4.6 percentWednesday, and then increase 3.9 percenton Thursday. Nor did the staffing companies’outlook change this dramaticallyfrom day to day. There were other factorsat play besides the changing outlook of theindividual companies that comprise theindexes listed on Table 3.Second, the share price of closely heldcompany stock, like that of the made-upABC Co., would probably not change thesame way that the Dow or the staffingcompany index changed between August5 and August 12, 2011.For example, assume that a hypotheticalbuyer of ABC Co. equity had spent5 “Hitting the Panic Button,” Wall Street JournalMarketBeat, April 12, 2011, http://blogs.wsj.com/marketbeat/2011/08/12/hitting-the-panic-button/.months conducting due diligence on ABCand was ready to close the transaction inmid-August 2011. That hypothetical buyerwould probably not be willing to paymore for ABC if the transaction closedon either August 9 or August 11 (dayswhen the stock market was up) than heor she would be willing to pay on eitherAugust 10 or August 12 (days the stockmarket was down). Instead, the purchaseprice would probably be fixed in the daysor even the weeks preceding the closingof the transaction. The purchase pricein this hypothetical transaction may besubject to certain closing adjustments.However, closing adjustments are typicallyrelated to the final target companyfinancial statements, and not to regionalor national economic news. Put anotherway, could a private equity buyer reallydemand a reduced price for closely heldcompany equity because (for example)the monthly housing starts failed to meetanalyst’s expectation?The above example is meant to explainwhy a closely held company’s stockprice may not be affected by the sameshort-term, macroeconomic, or behavioralfactors (at least not to the same degree)that lead to significant volatility inthe stock of PTCs.These common-sense observationssuggest that the stock price of closely heldcompany shares should be less volatilethan PTC stock prices.To reiterate, a higher required rate ofreturn for closely held company equitycompared with PTC equity generally supportsa higher level of volatility for closelyheld company stock. Conversely, the additionalfactors that affect the share price ofPTC stock—which exert less immediateand less significant impact on closely heldcompany stock—supports a lower level ofvolatility for closely held company stockthan PTC stock.ConclusionAfter the valuator has analyzed theguideline PTC volatility and estimatedindustry volatility, how does the valuatorrelate this estimate to the subject closelyheld company for use in the BSM optionpricing model? The answer is: it depends.It depends on the judgment of theanalyst, the outlook for the subject company,and many other factors. If, for example,a subject company is a one-yearoldsoftware provider with no sales andrapid projected growth, then the companystock price volatility may be greaterthan the selected guideline PTCs. If, onthe other hand, the company is 50 yearsold and operates two dozen grocerystores, then that company’s stock pricevolatility may be lower than the selectedguideline PTCs stock price volatility.The point of this discussion is not toprovide a roadmap or formula that youcan follow to estimate stock price volatilityfor closely held companies usingthe BSM option pricing model. Instead,the point is to provide the reader with anumber of factors that can be consideredwhen making the volatility estimate. Byconsidering the factors presented above,you will reach a more supportable conclusionand produce a more supportablenarrative report.VEAaron M. Rotkowski,CFA, ASA, is a manager atWillamette ManagementAssociates, a valuationconsulting, economicanalysis, and financialadvisory services firm.He has over 10 yearsof valuation experience. His practicefocuses on forensic analysis and disputeresolution, business and stock valuations,and valuations for other purposes. E-mail:amrotkowski@willamette.com.the value examiner November/December 201119