Mean-Reverting-Ebit-Based Stock Option Evaluation: Theory and Practice

Transcription

1 Jounal of Applied Finance & aning, vol. 3, no. 5, 03, ISSN: pint vesion, online Scienpess Ltd, 03 Mean-Reveting-bit-ased Stoc Option valuation: Theoy and Pactice Hassan l Ibami Abstact This pape examines the deivation of a capital-stuctue IT-based call option expession with the l Ibami and Nacii [0] equity value as the undelying asset. The model s PD and OD ae simila to lac-scholes but have widely diffeent and non-constant coefficients. An empiical analysis of the new model is conducted to measue its pefomance, using the last close pice of the evaluated stoc options and the lac-scholes values as benchmas. The esults show that the autho s model is obust, wheeas the lac-scholes model oveestimates the stoc options. JL classification numbes: F47,, 3,,, 3, 4. Keywods: OD, PD, Simulation, Stoc Option, Stuctual Models. Intoduction Stuctual models ae seldom used to analyze stoc options. Instead, the liteatue seems to be eplete with eseach on index deivatives. Wheeas these studies seem to be divided between theoy and empiical analysis, we thin that the only stuctual model used fo stoc options analysis is Pucy and Toft [997]. The authos used the Leland [994] equity expession as the undelying asset of thei stoc option value, but applied the lac-scholes stoc option fomula to detemine this expession, a stategy that seems inappopiate given that lac-scholes is not a capital stuctue model. In fact, maet models used to detemine stoc option values fail to conside the accounting infomation included in financial statements. The expessions obtained by using such models do not lead to the detemination of the fai value of deivative financial instuments. That being said, the IT value should be used to detemine the ight value of financial instuments, as advocated by financial theoy. This vaiable should follow a Univesity of Quebec in Monteal. Aticle Info: Received : Mach 9, 03. Revised : May 4, 03. Published online : Septembe, 03

2 36 Hassan l Ibami mean eveting pocess. Accodingly, we use the l Ibami and Nacii [0] model to deive a stoc option s PD and expession. Contay to Pucy and Toft s [997] analysis, we deive this expession by using ito s lemma, Taylo expansion and is neutality of a hedged potfolio. The OD coefficients ae not constant, and the coelation between the stoc option evaluated and its undelying asset is impossible to obtain analytically. Using a Newton-Raphson optimize, we numeically detemine the implied coupon associated with the l Ibami and Nacii [0] equity value. The same implied coupon is used in a simulation to detemine the stoc option s value. An empiical analysis allows us to measue the model s pefomance in evaluating stoc options. As benchmas, we use the last close pice of the listed stoc options and the lac-scholes value fo unlisted options. The analysis will poceed as follows: Section contains a liteatue eview, Section 3 explains the methodology, Section 4 discusses the stoc option expession deivation, and Section 5 pesents the empiical analysis. The conclusion is pesented in Section 6. Liteatue Review As stated peviously, Toft and Pucy [997] wee the only authos to detemine a stoc option value based on stuctual models, using the Leland [994] equity expession as the undelying asset and the lac-scholes [973] fomula fo obtaining a stoc option equation. They waned that financial leveage and debt covenants could invalidate the use of the lac-scholes model due to the implied volatility bias. This bias becomes moe ponounced in the case of a high leveage o potected debt, which educes the out-of-the-money stoc options value. In addition, shot-tem debt poduces additional costs. This finding cooboates Leland s suggestion to use long-tem debt to evaluate the undelying asset. Toft and Pucy [997] used stuctual endogenous volatility when deiving thei stochastic equation and theeby demonstated that stoc pice is a positive function of endogenous volatility, owing to the convexity of the equity-value cuve. This model also assumed an endogenous banuptcy, which causes a bias in the lac-scholes implied volatility in the case of a ponounced shot tem-long tem debt atio. The authos demonstated that potected debt fo companies that pay both coupons and dividends poduces a concave elationship between equity and the value of the company, which weaens the theoy of eal options as an altenative solution fo evaluating equity. In addition, endogenous banuptcy is the only way to obtain a convex elationship between the two paametes. Hence, equity should be consideed as a call option witten on the value of the company, which avoids an oveestimation of the company caused by exogenous banuptcy, tiggeing ponounced volatility. The authos obtained a closed fom solution fo thei PD, which allowed them to detemine the value of a call and put, witten on the Leland [994] equity value, using the lac-scholes fomula. Peais and Ryan [984] explained that the lac-scholes model seves to estimate the value of a stoc option witten on equity, but not its fai value. In the same vein, Constantinides, Jacweth and Peais [005] found that the lac-scholes model is useful only in complete maets with constant volatility and in tansactions that do not follow jump pocesses; hence, the lac-scholes fictionless maet model is only a special case within option picing valuation models.

3 Mean-Reveting-bit-ased Stoc Option valuation: Theoy and Pactice 37 Using linea pogamming to deive the uppe and lowe bounds of Peais and Ryan, Ritchen [985] estimated an option value by using an objective function optimize, and obtained its bounds by maximizing and minimizing the function. The uppe bound thus obtained was simila to Peais and Ryan, but the lowe bound was obtained only in the case of a deep-in-the-money option. Fo his pat, Levy [985] agued that tansaction costs and taxes do not hinde option evaluation unless potfolios ae continuously adjusted by investos. His model thus offes affodable solutions fo discete tansactions and maet fictions. Simila to Peais and Ryan [984] and Ritchen [985], the autho obtained a ange of values containing the equilibium pice. Richen and Kuo [988] added that Peais and Ryan bounds equie moe infomation fo thei use than lac-scholes does, but that unlie the lac-scholes and inomial models, these bounds do not equie consideation of continuous hedge hypotheses, which maes them applicable to both negotiated and non-negotiated contingent claims. The authos also obtained uppe bounds simila to Peais and Ryan by using a multinomial model; howeve, thei lowe bounds wee tighte. Some studies highlighted options valuation empiically, mainly by using equilibium models and maet indexes as undelying assets. ashi, Kapadia and Madan [003] used a sample of 358,85 call and put options quoted in the S&P 00 Index and witten on the 30 most liquid secuities listed on the Chicago Stoc xchange. Coveing the peiod fom Januay 99 to Decembe 995, thei analysis emphasized is neutality distibution sew causes and how they impact options pice. The authos demonstated that investo is avesion influences the distibution s utosis and sewness coefficient. The distibution sew is affected by systematic and specific is. Howeve, they also demonstated that stoc options have less implied volatility than index options because of thei moe negative sewness. As a esult, they concluded that stoc options can be moe accuately evaluated than index options. ollen and Whaley [004] consideed that this implied volatility inceases o deceases accoding to the investo s position in the index, a condition that helps achieve pice equilibium. The authos used bid and as pices of the S&P 500 s implied volatility between June 988 and 000 and the implied volatility of the 0 most active stoc options in the index between Januay 995 and Januay 000. They assumed that investos who pefeed the index would invest in put options while those inclined towad stoc options would choose call options. These assumptions would aid in undestanding the diffeence between the two foms of implied volatility. The esults show that index implied volatility is geate than stoc option volatility, a condition that dives up index option costs and inceases the sewness of thei asymmetic pofits cuve. The authos concluded that the esulting implied volatility bias should not be attibuted solely to the complexity of the unobsevable undelying asset pocess, but also to the abitages limited capacity to absob losses, which pushes pices up. Hence, depending on investo activity, maet maintaines eadjust these pices, a stategy that helps them undestand pice dynamics and establish the is pemium to coect the implied volatility bias. Accoding to Diessen, Maenhout and ilov [009], the options pice bias is attibutable to the coelation is educing the divesification effect. They compaed daily quoted pices of options on S&P 00 stocs with pices quoted between Januay 996 and Decembe 003. They selected these paticula options because they wee maedly continuously active duing that peiod and had a shot expiy and low implied volatility. The authos boe down maet is into systematic and coelation is, and

4 38 Hassan l Ibami demonstated that index options ae moe expensive than stoc options. They concluded that the pice spead can be explained only by coelation is, which is ewaded by a negative is pemium epesenting the divesification cost; as a esult, the CAPM can be used to detemine the fai value of stoc options, but not that of index options, due to coelation is. Duan and Wei [009] easoned that a systematically high is level tigges high implied volatility levels as well as a ponounced tend of implied volatility. The spead between two stoc option pices can be explained only by the systematic is popotion of thei espective undelying assets. To conduct thei study, the authos analyzed daily S&P 00 index options data and the 30 lagest stocs on that index fom Januay 99 to Decembe 995. Simila to ashi, Kapadia and Madan, the authos concluded that the diffeence between eal and is-neutal distibutions was due to systematic is that can be measued by systematic vaiance scaled by total vaiance. The authos also demonstated that implied volatility behavio is explained by systematic is in the sense that a negative is pemium poduces a high implied volatility level, but also high sewness and utosis coefficients. Disageeing with ollen and Whaley, who assumed that net buying pessue influences the slope and level of implied volatility, the authos pointed out that these assetions wee not demonstated and that only index implied volatility wee consideed, athe than all types of options. oyal and Saetto [009] addessed the same issue by investigating the diffeence between ealized and at-the-money implied volatility. They conducted thei study on one-month-expiy daily options data coveing Januay 996 to Decembe 006 in ode to obtain a homogenous sample of liquid options. Afte eliminating illiquid options, non-continuous bid-as data and data in which bid values exceeded as values, they obtained an evaluation sample of 4, 44 stocs used as undelying assets fo 75,67 call and put stoc options. The analysis shows that high etuns wee due to the spead between ealized and implied volatility. The bias obseved in the volatility, accompanied by the utosis and sewness coefficients biases, explains the unfai pice of the options evaluated. The high levels of etun obseved ae not attibutable solely to the coelation between the option and its undelying asset, but also to the misspecification of the models used to mae the estimation. They concluded that both histoical and implied volatility may be used to detemine futue volatility, thans to the petinent and non-contadictoy infomation they contain, if well-specified evaluation models ae used. The body of liteatue eviewed was composed mainly of theoetical and empiical studies. Some eseach used equilibium models, while othes used stuctual models. Most of the studies centeed on options with maet indexes as undelying assets. As well, vey few eseaches used stuctual models to compae stoc and index options. Ou objective was to deive the fai value expession of stoc options using a stuctual model and the l Ibami and Nacii [0] equity as the undelying asset to mae the deivation. This stategy would be convenient fo companies whose state vaiable follows a mean-eveting pocess. As suggested by the foegoing authos, the state vaiable should be the IT. We used the dynamic equation of N to obtain the PD followed by a deivative instument, and then solved the odinay equation to detemine the expession of ou call option. This model avoids the implied volatility poblem and maet speculations, and is used to detemine a stoc s fai value using financial statements and The authos use at-the-money options in ode to avoid the volatility smile.

5 Mean-Reveting-bit-ased Stoc Option valuation: Theoy and Pactice 39 accounting infomation. This pocedue was used in an attempt to establish a bidge between accounting infomation and the maet. Nowadays, the two wolds have become paallel, which explains the poblems of speculation and financial fiascos that quicly esult in company and govenment banuptcies. olatility, the most impotant deteminant of an option, should be measued endogenously by using the model s stuctual expession. This technique avoids the volatility smile and sewness and utosis coefficient poblems, and helps detemine the option s fai value. 3 Methodology Ou analysis begins with a desciption of ou theoetical famewo. The l Ibami and Nacii [0] equity expession was used as the stoc option undelying asset. The PD and OD of the foegoing deivative instuments wee deived, followed by thei expessions. The new stoc options evaluation model was then used to pefom an empiical analysis and measue the model s pefomance. Ou initial sample was composed of the 300 lagest Canadian companies listed on the Toonto Stoc xchange between 006 and 00 and with the potential fo being listed on the Monteal Deivatives xchange, soted by evenue 3. Since the model is useful fo mean-eveting IT companies, the significance level of the companies IT paametes was measued using quately data fo the peiod specified. The final evaluation sample was composed of companies with a 5% mean-eveting IT and speed of mean-evesion significance level. The model allows uses to endogenously geneate stoc options volatility. The empiical study used the neaest expiy of the at-the-money o the neaest to-the-money call options listed on the Monteal Deivatives xchange duing the peiod coinciding with the 00 financial statement dates. The mean spead between these obseved data and the theoetical esults wee calculated numeically by using 0,000 paths simulation. Howeve, the lac-scholes model was used as a benchma fo unlisted stoc options. iven ou belief that the lac-scholes model oveestimates the value of stoc options because of the volatility smile, esults with the model wee compaed to those yielded by ou stuctual model fo the evaluation of the entie sample. As obustness tests, linea egessions and paied-samples tests wee used to compae the theoetical and benchma values. 4 Model As discussed ealie, the l Ibami and Nacii [0] equity expession was used as the undelying asset. Accoding to the authos, this expession may be pesented as follows: 3 Anott, Hsu and Mooe 005 showed that evenue accuately indicates a company s basic chaacteistics and could be used to measue its tue value.

6 30 Hassan l Ibami C C e whee: epesents the mean-evesion speed; epesents the IT mean evesion; epesents asset volatility; epesents the value of the company at banuptcy; τ epesents the tax ate; C epesents the consolidated coupon and; 8 ; v x v x. Howeve, banuptcy is exogenous; hence "" depends only on "". The fist and second deivatives of "" with egad to "" ae detemined as follows: e C and C whee: e e x x 4 e 5 y applying ito s lemma, the following PD is obtained fo equity: dz dt t d 6

7 Mean-Reveting-bit-ased Stoc Option valuation: Theoy and Pactice 3 Assuming pepetuity because of the company s solvency, equation 6 becomes: dz dt d 7 To simplify, let: a 8 and b 9 Substituting these values in equation 7, the latte becomes: dz b dt a d 0 y applying ito s lemma to deivative instument "", which depends on "" and "t", we obtain: bdz dt b a t d Now assuming two financial instuments " " and " " that depend on "" and fulfill the conditions of the following dynamic equation: d = mdt + sdw Assuming a hedged potfolio consisting of s " " assets and s " " assets and consideing pepetuity and is neutality, we obtain: a b 3 y eplacing a and b by thei espective values in expession 3, the latte becomes: 4 Whee: e C 5 and C 6 and

8 3 Hassan l Ibami e e x x 8 e 9 quation 4 epesents the OD fulfilled by a call option witten on the l Ibami and Nacii [0] equity. The coefficients of the expession ae not constant. Only a numeical solution helps estimate the value of the stoc option by using the poposed fomula. 5 mpiical Analysis Among the 300 companies composing ou initial sample, only 6 pesented statistically significant mean-eveting chaacteistics. We used this final sample to conduct ou empiical analysis. Among the 6 companies soted, 30 wee listed on the Monteal Deivatives xchange at thei 00 financial statement date. As a fist step, we compaed the theoetical esults to thei coesponding last close pices. We also used the same sub-sample to pefom a compaison with the lac-scholes model. In the second step, we compaed the theoetical values of the 3 unlisted companies to thei coesponding values calculated by using the lac-scholes model again. Finally, we conducted a thid compaison between the two models by using the entie evaluation sample data. Table below shows the esults of the compaisons. Table : Mean Spead between Theoetical alues and enchmas Mean Spead 4 Stoc Options Numbe of Companies Theoetical alue vs. Last-Close-Pice Theoetical alue vs. &S alue Listed % -3.40% Unlisted 3 n/a -38.0% ntie Sample 6 n/a % 4 Spead=Theoetical alue enchma/enchma

9 Mean-Reveting-bit-ased Stoc Option valuation: Theoy and Pactice 33 As shown in Table, the model estimates the listed call options coectly, but the lac-scholes model oveestimates the same sub-sample by about 3%. As expected, the volatility smile causes this oveestimation, weaening the model s findings. Accoding to the esults, the model oveestimates the value of the 3 unlisted stoc options by about 38% and the entie evaluation sample by about 3%, which confims ou findings about existing stoc options. We sumise that the spead between the esults obtained by using the two models would be moe ponounced fo long tem expiy and deep in-the-money o out-of-the-money stoc options 5. As obustness tests, a paied-samples mean compaison was pefomed by analyzing the diffeence between the theoetical and obseved data fo listed options fist. A second compaison was pefomed with the same sub-sample, using the lac-scholes model as a benchma. The esults obtained by the two models wee compaed fo unlisted companies. Lastly, the two models wee compaed, using all the companies in the evaluation sample. Fo all the compaisons made, the null hypothesis pedicts a lac of significant mean diffeences between the paied-samples, while the complementay hypothesis sees fo significant diffeences between the same paied-samples. A % level of significance was used fo all the compaisons we made. The esults of such compaisons ae given in Table. Table : Paied Samples Test Paied Diffeences 99% Confidence Inteval fo Diffeence Stoc Options Numbe of Companies Theoetical alue vs. Last-Close-Pice Theoetical alue vs. &S alue p-value t-statistic p-value t-statistic Listed 30 7%.6 0.6%.958 Unlisted 3 n/a 0.000% ntie Sample 6 n/a 0.000% The esults pesented in Table, along with those obtained peviously, show that thee is no statistically significant diffeence between the values obtained by ou model and thei coesponding last-close-pices. The paied-samples mean compaison between ou theoetical values and lac-scholes values shows a statistically significant diffeence, as obseved peviously. The compaison between the two models used fo the unlisted companies also cooboates these esults. These conclusions ae consistent with those obtained fo the entie evaluation sample. 5 This assetion is beyond the scope of ou pape.

10 34 Hassan l Ibami The paied-samples tests conducted show that thee is no statistical diffeence between ou esults and the maet values, but that ou model and lac-scholes yield significantly diffeent esults. To measue the magnitude of this diffeence, we pefomed linea egessions of ou model s theoetical values on those obtained by lac-scholes. Table 3 pesents these esults. Table 3: Linea Regession Theoetical alue vs. &S alue Stoc Options Numbe of Companies Coefficient R-squaed Adj. R-squaed p-value t-statistic Listed Unlisted ntie Sample Table 3 shows that all the coefficients ae positive and statistically significant, and have lowe than one values. The R-Squaed and Adjusted R-Squaed ae all high and significant, indicating that, unlie ou model, lac-scholes oveestimates the value of the stoc options fo the thee egessions pefomed. The esults obtained ealie ae theefoe cooboated. 6 Conclusion In this pape we developed a new model of stoc option evaluation. The undelying asset of the stoc option deived was the l Ibami and Nacii [0] equity model. Financial statements fo mean-eveting IT-based companies wee used. The model also uses stuctual data and mean-eveting IT as the state vaiable. The initial sample consisted of the 300 lagest Canadian companies listed on the Toonto Stoc xchange, soted by evenue. The companies with statistically significant mean-eveting paametes wee etained, esulting in a final sample of 6 companies. Only 30 companies in the sample wee listed on the Monteal Deivatives xchange duing the evaluation peiod. Hence, to measue the model s pefomance in evaluating stoc options, its values wee compaed to the stoc options last close pice at the financial statement date. The esults show a vey mino spead between the theoetical and maet values. The same sub-sample was used to measue the spead between the lac-scholes model and ous in ode to chec fo the volatility smile effect. The lac-scholes model oveestimated stoc options values by about 3%, compaed to ou model. The two models wee compaed again by using an unlisted stoc options sub-sample. The esults cooboated those obtained fo the listed stoc options sub-sample, with the lac-scholes model oveestimating values by 38%. Lastly, we pefomed anothe compaison between the two models using the entie data of the evaluation sample. The lac-scholes model oveestimated the stoc options value by

11 Mean-Reveting-bit-ased Stoc Option valuation: Theoy and Pactice 35 about 30%, compaed to ou model s esults. Robustness was tested using paied-sample compaisons. No significant diffeences wee noted between ou theoetical esults and maet values, wheeas significant diffeences wee found between ou model and the lac-scholes model, fo both listed and unlisted companies. As a post hoc test, linea egessions of ou theoetical values wee conducted on the lac-scholes values. All the esults wee significant, leading to the conclusion that the lac-scholes model oveestimates esults because the volatility smile invalidates the model s findings. The model we deived can be used to detemine COs and employees stoc options-based compensation, a pivate company s call option value o how to quote stoc options about to be listed on a Deivatives xchange. It can also be used to measue the efficiency of the stoc options maet by maing it possible to compae the intinsic value of stoc options to the last close pice o quotes. Howeve, the model is useful only fo mean-eveting companies that have IT as a state vaiable. In addition, it pesents a call option expession and does not tae into account fictions. Futhe analysis will help detemine a put option equation using othe state vaiables and taing into consideation maet incompleteness. ACKNOWLDMNTS: I would lie to than Pofesso Anne Fotin, Univesity of Quebec In Monteal, fo pecious comments and pepetual suppot. Refeences [] H. l Ibami and A. Nacii, quity Capital-Stuctue-ased valuation Method, Intenational Jounal of Accounting and Financial Repoting,, 0, [] K.. Toft and. Pucy, Options on Leveaged quity: Theoy and mpiical Tests, Jounal of Finance, 53, 997, [3] H.. Leland, Copoate debt value, bond covenants, and optimal capital stuctue, Jounal of Finance, 494, 994, 3-5. [4] F. lac and M. Scholes, The picing of options and copoate liabilities, Jounal of political economy, 83, 973, [5] S. Peais and P. J. Ryan, Option Picing ounds in Discete Time, Jounal of Finance, 39, 984, [6]. M. Constantinides, J. C. Jacweth, and S. Peais, Option picing: eal and is-neutal distibutions, in I. R. ige and. Linetsy, Financial ngineeing, Handboos in Opeations Reseach and Management Science, lsevie/noth Holland, 005. [7] P. H. Ritchen, On Option Picing ounds, Jounal of Finance, 4040, 985, [8] H. Levy, Uppe and Lowe ounds of Put and Call Option alue: Stochastic Dominance Appoach, Jounal of Finance, 404, 985, [9] P. H. Ritchen and S. Kuo, Option ounds with Finite Revision Oppotunities, Jounal of Finance, 43, 988, [0]. ashi, N. Kapadia and D. Madan, Stoc Retun Chaacteistics, Sew Laws and The Diffeential Picing of Individual quity Options, Review of Financial Studies, 63, 003, 0-43.

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