Guiding the guiders: Foundations of a market-driven theory of disclosure
aa r X i v : . [ q -f i n . T R ] F e b Guiding the guiders:Foundations of a market-driven theory of disclosure
M. Gietzmann, A. J. Ostaszewski, M. H. G. Schr¨oder
Abstract.
A foundational approach is developed for a mathematical the-ory of managerial disclosure in relation to asset pricing; this involves boththe earnings guidance disclosed by firm management and market ‘trackers’pricing the firm’s exposure to quotable risks.
Keywords.
Risk-neutral valuation, asset-price dynamics, earnings guid-ance, optimal censoring, materiality, state observer system.
MSC.
Primary 91G50; Secondary: 91G80; 93E11; 93E35; 60G35; 60G25.
Asset-pricing models make assumptions about how information arrives andis disclosed to its investors (henceforth the market). For assets arising outof a productive activity by a firm, management reports based on internalaudits of the various accounting numbers (accounting variables), preparedin time for the scheduled (publicly pre-specified) dates, are one such source.If, by the nature of their activity, management make more frequent audits(for instance in directing replenishment to a specific level, which enforcesfrequent stock-taking, as in the retail business), then opportunities arise for‘early’ (unscheduled) disclosure. How should this additional information beused to signal the firm’s superior value, i.e. to upgrade its share-price? When(or how) should the market ‘price in’ the absence of early disclosures by a firmto include the possibility that no news is bad news. The answer must rely onan equilibrium between the market’s ability to down-grade the share-priceand the firm’s ability to take advantage of ignorance: hiding some bad newswithin the uncertain cause of absent news, i.e. censoring the information.The accounting literature has usually approached this question by includ-ing a specified (i.e. known in advance), single, ‘additional’ interim reportingdate, ahead of the next scheduled disclosure, and allowing for absence of anearly disclosure by randomizing the possibility that management has held anadditional audit – see [Dye], [JunK]. However, with the early date a datum,1his approach places limits on comparative analysis; for an equally spacedmulti-period model see [EinZ].
In this the companion paper to [GieO] we propose an alternative generalapproach, by instilling more realism into the stylized Black-Scholes model of[GieO]. There the market determination of the (share-)price of the firms ina sector reflects only the discretional information strategically released by afirm (i.e. with anticipation of its price effect), usually in the form of earn-ings guidance , as below, at a stochastic time-point (i.e. at unknown dates inadvance of a subsequent mandatory date of information disclosure). Despitethis highly specific origin for the arrival of information in the market, thatmodel holds considerable advantages, thanks to its continuous-time approachwhich overcomes the limitations of the traditional ‘two-period analysis’ justmentioned. There unspecified moments in time offer an early disclosure. Byway of an example from [GieO], which goes beyond the scope intended here,there is a derivable ‘band-wagon’ effect whereby the introduction of multi-ple sources of information reduces an individual firm’s optimal frequency ofdisclosures by reference to time left to the next mandatory disclosure date.Typically, however, the market responds also to other public sources ofinformation, such as trading in the shares of the firm, and by assessing theexposure of the firm to such economic risks as may be priced by market-quoted options.Here we create a more general framework to include such other, alreadyexisting, market-based information enabling the market to make proper useof this additional information about the firm. This prompts a deeper analy-sis, equivalently a formalization at a foundational level, of the various mech-anisms at work, offered in the Complements Section. For simplicity, weconsider here only the situation where the market’s concern is for a singlefirm, rather than a whole sector. In this we are guided by the clarifying sim-plifications that occur in the case of an isolated (‘single’) firm in the stylizedmodel [GieO].There the firm itself comes to know (‘observes’) its own state V t onlyat discrete, stochastically generated times t , not known to the market; themanager, occupied by a variety of tasks, cannot receive observations exceptwhen these breach agreed thresholds, as reported by personnel delegated2o collect this information, perhaps continuously. This feature of a hiddenobservation scheme enables the firm to bury (hide) ‘bad’ news and only report‘sufficiently good’ news, principally because the manager cannot at any timecredibly claim as absent an observation at that time. That model prices thefirm in periods of silence, i.e. when the firm fails to supply an early reportof information. Key to this is identifying at each time t a value L t such thatif L t happened to coincide with the true and currently observed state V t ,the firm would be indifferent, as regards its market valuation (i.e. given themarket’s information ex ante ), between choosing to disclose or to withholdthe current observed state V t . Such an indifference level L t , determined by theinformation available from before time t, is typically unique. Censoring, i.e.suppression of an observation below this unique L t , draws from the market avaluation of the firm at L t . In fact, L t is the largest possible valuation of thefirm, consistent with the information available from before time t . As suchit is termed the optimal censor of time t. Note that observations above L t that are disclosed cause an upward jump in the firm’s valuation. We shouldemphasize that only truthful disclosures are allowed in the model.The mathematical argument is based on risk-neutral valuation, whichmust incorporate the potential future re-evaluations of firm-value consequentupon future early disclosures.The existence of an indifference pricing process is directly attributableto the firm knowing the market’s filtration F ∗ = {F ∗ t } t and the mechan-ics of how the market performs computations based on past disclosures (inparticular, the probabilities at each instant which the market attaches tothe firm suppressing an observation of its state). Since the firm’s filtration F = {F t } t is an enlargement of F ∗ [Jeu], in that the firm feeds the marketwith information by choosing when to supply its private observations, onemay say that the firm emulates (can simulate) the market. In turn the mar-ket’s calculations are based on the firm’s algorithm, though not on the firm’sup-to-date observations. The indifference price arises from characterizing anotional parametric equilibrium between the two agents: the firm and themarket (we do not differentiate between informed and noisy traders), select-ing parameters in the computation they use to second-guess each other.The paper identifies the mechanisms underlying some fairly general val-uation procedures, allowing the market to form its beliefs from additionalinformation and the firm to exploit the market beliefs by disclosing valuesuperior to that belief, but nevertheless to give a fair view of future dis-closures. As these mechanisms are inspired by the principal argument and3ndings of [GieO], we close here with a summary of that argument (in thesimplified notation used below). Suppose the next mandatory disclosure isat the terminal time 1 and that, under the market’s risk-neutral measure Q, at time t < T ∈ ( t,
1] is τ T . (In [GieO] opportunities to observe the state of thefirm are generated by a Poisson clock.) Based on its information at time T ,the earnings guidance announced at that time by the firm gives its target terminal value as V T = E Q [ V |F T ]; the target and the two filtrations aboveare related to the indifference level L T of time T by the two equations(1 . . a ) V T = τ T E Q [ V | V T ≥ L T , F T ] + (1 − τ T ) L T , (1 . . b ) L T = E Q [ V | N D T ( L T ) , F ∗ T ] . Here
N D T ( L ) is the event that no disclosure occurs at time T, which meansthat either there has been no opportunity to observe V T or else the managerhas observed V T but V T ≤ L. From here, in the context of [GieO], given how F ∗ T is generated from F T via F ∗ T -measurable decisions, one deduces in thelimit as T → t from (1.1.1a) that t L t satisfies a simple ordinary differ-ential equation (involving the instantaneous variance of Q and the Poissonclock’s intensity). Assuming multiplicative scalability, that V T + u = V T ˜ V u with independence of V T and ˜ V u , equations (1.1.1a,b) can be further simpli-fied.In summary, apart from simplifications, this paper’s contributions in-clude: announcements of both sufficiently good and sufficiently bad news(dual, ‘materiality’ aspects in the release of private information), incorpora-tion of current public information in modelling market sentiment, and com-parative statics of early disclosures.The paper is organized as follows. Section 2 contains a preliminary dis-cussion of our modelling aims. Section 3 models the market’s beliefs as tofirm value, based on expected performance indices and is supported by ageometric Brownian (GMB) implementation. This is followed in Section 4by a model of the firm setting its target values; using a benchmark schemeto be followed by the firm in observing its own state, this is shown at its sim-plest to be similar to determining option exercise values, and is supportedby an indicative GMB implementation. This enables us to perform compar-ative statics in Section 5, which we conclude meets a primary objective: toshow how parameter values determine early or delayed voluntary release of4nformation in equilibrium. We comment briefly on the implications of ourapproach in Section 6, thus rounding off the paper’s contribution, and closein Section 7 with Complements indicating a framework for generalizationsand potential variations. An Appendix gives details of well-known GMBformulas needed in the paper. In this section we introduce a model of how a firm F decides to discloseinformation intermittently at times t to the market M about its state V t , voluntarily between legally mandated (mandatory) disclosure dates. Thisinvolves modelling how the market forms beliefs V ∗ t in periods of silenceabout the true current value V t . (We regard the market as dual to the firm,hence the ‘star’ notation here and below.)Our first two tasks are: to model the beliefs of M (in §
3) and then tomodel F ’s choice of ‘equilibrium’ indifference level (in § V t , if any, is not disclosed (as in the Introduction). We willrely on tractable Black-Scholes frameworks, and in the second task we willbe guided by the findings of [GieO]. In § § observation scheme – pos-sibly also intermittent – to its ‘state’ V t (thought of as the income stream).This is modelled as a random time τ not known to M. The firm applies fixeddecision rules by which it determines at time t whether to withhold any obser-vation it may have, or to disclose its information voluntarily (and truthfully)to the market via two items of information: (i) the current value V t , and (ii)the expected state at the terminal date, i.e. the next mandatory disclosuredate. We term the former the declared current value V C and the latter itsdeclared target value
V T (current as at the date of its disclosure). The firm’sintention is to achieve the highest possible market valuation at each point intime; here this is implemented by use of a fixed decision rule , based both onits own private information about its state and on the market’s public belief5bout the state of the firm, which in turn depends on the market’s infor-mation base. We term this, publicly held belief about the value, the marketsentiment . F forms its expectations by reference to a measure Q V T (labelledby the last declared target) under which its (discounted) observation processis a martingale.To model market sentiment, we borrow and amend a concept from con-trol theory, that of a state observer (a.k.a. ‘state estimator’) system; for adiscussion see § V ∗ t is based on the latest disclosed infor-mation and on the current value, or perhaps on the prevailing behaviour, ofsome specified portfolio of traded assets with value S ∗ t (e.g. current value,average value, record value to date). The portfolio, termed the tracker , isviewed by the market as capturing the firm’s exposure to quotable (market-priced) risks. The key property of S ∗ t is that it is priced by a risk-neutral(i.e. market) measure. That is, M forms its expectations by reference toa measure Q ∗ under which the (discounted) tracker process is a martingale.We note that, at each time t, since F has access to M ’s information F ∗ t plusits own observation, i.e. its own filtration {F t } t is an enlargement of {F ∗ t } t , Q ∗ | F ∗ t = Q V T | F ∗ t . As in [GieO], so too here, the link between market sentiment V ∗ and asset S ∗ needs to be determined by equilibrium considerations: if at time t the firmapplies a decision rule h to the observations of S ∗ and V , it will determinean indifference level L (as in the Introduction) which, if F observed that V t = L, would make F indifferent between disclosure or otherwise of V t .While the firm determines its disclosure using a rule h (below) that exploitsany superiority of the observed value over market sentiment, rather than itsexpected terminal value, it complements such a disclosure by supplying themarket with information about the expected terminal value.We take the decision rules for F and consequently also for M (with starrednotation for the latter’s rules) in the (time-independent) form(2 . . h ε,a ( t, x, y ) = ( x − (1+ a ) y ) ε, ( x, y ∈ R , t > a > − mark-up , and ε = ± signature , positive whendeciding a good-news disclosure event at time u, say, when(2 . . V u ≥ (1+ a ) V ∗ u , and respectively negative for a disclosure of bad-news . (A dynamic variantis considered briefly in § Our first task begins at a point in time T with 0 ≤ T ≤ V C and declaredtarget
V T.
We use
V T as a suffix conveniently labelling the various processesstarted at time T .The next time of disclosure, following access to an observation of X T attime T = T + , will occur provided h ( T, V T , V ∗ T ) ≥ . Here the firm F applies its decision rule h taken in general to depend onthe time- T values of V and V ∗ , and perhaps on T itself, a possibility ruledout below to simplify calculations (hence the rules in § T = T + , the firm will declare its current state V T and set a new targetvalue V T = V T + . So the main tasks are to devise a justifiable model for aprocess V ∗ , which we view as ‘shadowing the firm’, and for V T + (in the nextsection). We begin by identifying how to model V ∗ t . This will be determined by twocomponents. Although the entire process is driven by a specified market7ortfolio S ∗ , the tracker , nevertheless, at each non-disclosure time-point t in ( T , T ) a correction term needs to be included to price in the effects ofany possible future disclosure event, say of time τ . We thus aim for a two-component model(3 . . V ∗ t = S ∗ t + ∆ V ∗ t , which requires that we model ∆ V ∗ t . The latter must refer to contingentclaims in regard to whether or not F has attained its target to date; with h ∗ a market decision rule, this requires pricing contingent claims on [ t, τ ]characterized by the instantaneous time- τ pay-off(3 . . S ∗ τ { h ∗ ( τ ,S ∗ τ ,V T ) ≥ } , for τ = τ V T . So this is a ‘securitization’ similar to standard derivative instruments. Itdepends on a ‘random time’ τ = τ V T (with values in ( t, T ]) unknown tothe agent M . Unfortunately, pricing these depends on individual investorattitudes, so on micro level information, typically unobservable. The prag-matic approach is to replace the pricing of these claims with approximations.Assuming a constant risk-free rate r in force, we now make explicit a firstmodelling assumption , that with τ = τ V T (3 . .
3) ∆ V ∗ t := exp( − r ( T − t )) E Q ∗ [ S ∗ τ { h ∗ ( τ ,S ∗ τ ,V T ) ≥ } | F ∗ t ] − S ∗ t ;here Q ∗ is an assumed risk-neutral measure conditional on market informa-tion – conditional at time t on market information F ∗ t (with the expectationon the right assumed meaningful). The formula identifies ∆ V ∗ t as the excessover S ∗ t of the fair value of the effect of a disclosure occurring at time τ . Thefinal step at time t is to pass to an approximation, which make explicit a second modelling assumption , that(3 . . S ∗ τ { h ∗ ( τ ,S ∗ τ ,V T ) ≥ } ≈ E ∗ t { h ∗ ( t, Σ ∗ t,T ,V T ) ≥ } . Here, on the one hand, Σ ∗ t,T is some chosen tracker-performance index overthe entire remaining time interval, for instance(3 . . abc ) Σ ∗ t,T = max { S ∗ u | u ∈ [ t, T ] } , orΣ ∗ t,T = 1 / ( T − t ) R [ t,T ] S ∗ u d u, orΣ ∗ t,T = min { S ∗ u | u ∈ [ t, T } ; E ∗ t is obtained by solving h ∗ ( t, E ∗ t , V T ) = 0. This last has a unique solution, the result of the simpleform of decision rule in § E ∗ t identi-fies a value at-least-as-good-as the true value S ∗ τ at disclosure; in bad-newssituations such an E ∗ t will be at-most-as-bad-as that. We summarize thesemodelling considerations in: Proposition 3.1
The two modelling assumptions (3.1.3), (3.1.4) implythat V ∗ t = ( S ∗ t + ∆ E ∗ t ) Q ∗ ( h ∗ ( t, Σ ∗ t,T , V T ) ≥ | F ∗ t ) exp( − ( T − t ) r ) , where:(i) S ∗ is assumed to start with the value V C at time T ,(ii) ∆ E ∗ t = E ∗ t − S ∗ t , and(iii) Σ ∗ t,T is chosen as in example (3.1.5) above. On the firm side we take the firm’s observation process V to be modelled bya geometric Brownian motion X :(3 . . X s + t = X t exp(( µ V T − σ V T ) s + σ V T W V T,s ) , s ∈ [0 , T − t ];here W V T is a Brownian motion independent of time- t information F t , with σ V T > µ V T ∈ R fixed.Likewise, on the market side we adopt a Black-Scholes model with onerisky security S ∗ , so modelled again by a geometric Brownian motion, whichthus satisfies the requirements of Proposition 3.1 and carries several advan-tages beside: firstly, model-completeness (see for example [MusR, Prop. 8.2.1,p. 302] for the completeness of the multi-dimensional model) and, secondly,the ability of being re-started with value V C at time T . We take the dy-namics in the form(3 . . S ∗ t + u = S ∗ t exp( µ ∗ u + σ ∗ W ∗ u ) , u ∈ [0 , ∞ )with Brownian motion W ∗ independent of time- t information F ∗ t , and twoparameters: volatility σ ∗ > µ ∗ = r − δ − (1 / σ ∗ , for some δ ∈ R .9his modelling choice ensures that the probabilities in the formula ofProposition 3.1 are well-defined (see below), emerging as tail probabilitiesfor good-news decisions, since h ∗ ( t, Σ ∗ , V T ) ≥ h +1 ,a ∗ (Σ ∗ , V T ) ≥ ∗ ≥ (1+ a ∗ ) V T, for Σ ∗ any random variable. It is routine to derive explicit formulae for these,see the next subsections. We focus here on the choices of (3.1.5a,c) of max-and min-tracker-performance index Σ ∗ t,T ( S ∗ ), leaving aside the average valueindex. V ∗ from the running-max approximation We first deal with the running-maxΣ ∗ := max { S ∗ u | u ∈ [ t, T ] } . We note a reversion from bad-news to good-news decisions via: Q ∗ ( h +1 ,a ∗ (Σ ∗ , V T ) ≥
0) + Q ∗ ( h − ,a ∗ (Σ ∗ , V T ) >
0) = 1 . The actual influence of the second term on the first is determined by therelative size of (1+ a ∗ ) V T and S ∗ t , so according to the sign of(3 . . A ∗ = log((1+ a ∗ ) V T /S ∗ t ) . Below Erfc is the complementary error function, for which see subsection A2of the Appendix. Equations (A.7a) and (A.7b) in the Appendix yield thefollowing results.
Proposition 3.2 If A ∗ < , equivalently, S ∗ t > (1+ a ∗ ) V T , then Q ∗ ( h − ,a ∗ (Σ ∗ , V T ) >
0) = 0 , and there is no ‘bad-news’ influence; thus (3 . . Q ∗ ( h +1 ,a ∗ (Σ ∗ , V T ) ≥
0) = 1 , if S ∗ t ≥ (1 + a ∗ ) V T. If A ∗ ≥ , equivalently S ∗ t ≤ (1 + a ∗ ) V T , then ‘bad-news’ carries influencemeasured by (3 . . Q ∗ ( h +1 ,a ∗ (Σ ∗ , V T ) ≥
0) = 1 − Q ∗ ( h − ,a ∗ (Σ ∗ , V T ) > , here (3 . . Q ∗ ( h − ,a ∗ (Σ ∗ , V T ) ≥
0) == 12 Erfc A ∗ t − ( T − t ) µ ∗ σ ∗ p T − t ) ! + 12 exp (cid:18) µ ∗ A ∗ t ( σ ∗ ) (cid:19) Erfc A ∗ t + ( T − t ) µ ∗ σ ∗ p T − t ) ! . To complete the picture, note that the bad news scenario when Σ ∗ is therunning-maximum can be read back from:(3 . . Q ∗ ( h − ,a ∗ (Σ ∗ , V T ) ≥
0) = 0 , if S ∗ t > (1 + a ∗ ) V T, and that for A ∗ > V ∗ from the running-min approximation We now deal with running-minΣ ∗ := min { S ∗ u | u ∈ [ t, T ] } . The good-news and bad-news formulas hold good and, viewed technically,may be derived by replacing µ ∗ by − µ ∗ and A ∗ by − A ∗ = log( S ∗ t / ((1 + a ∗ ) V T )); and switching to probabilities complementary to those in (3.2.1):see the discussion for equation (A8) in Section A2 of the Appendix. Fromthere, we have explicitly:
Proposition 3.3 If S ∗ t ≤ (1+ a ∗ ) V T, then (3 . . Q ∗ ( h +1 ,a ∗ (min { S ∗ u | u ∈ [ t, T ] } , V T ) ≥
0) = 0 , (3 . . Q ∗ ( h − ,a ∗ (min { S ∗ u | u ∈ [ t, T ] } , V T ) ≥
0) = 1 . If S ∗ t > (1+ a ∗ ) V T, then (3 . . Q ∗ ( h +1 ,a ∗ (min { S ∗ u | u ∈ [ t, T ] } , V T ) ≥ − Q ∗ ( h − ,a ∗ (min { S ∗ u | u ∈ [ t, T ] } , V T ) > , where (3 . . Q ∗ ( h − ,a ∗ (min { S ∗ u | u ∈ [ t, T ] } , V T ) ≥
0) == 12 Erfc A ∗ t − ( T − t ) µ ∗ σ ∗ p T − t ) ! −
12 exp (cid:18) µ ∗ A ∗ t ( σ ∗ ) (cid:19) Erfc − A ∗ t + ( T − t ) µ ∗ σ ∗ p T − t ) ! . Setting new targets
The previous section determined how the firm triggers disclosure by referenceto a fixed decision rule h and a model of market sentiment V ∗ t (our proxyfor an observer-system of control theory). This replaces and simplifies thedynamics of the equilibrium approach of [GieO], but comes at the cost oflosing information about the expected terminal firm-value (value at the nextmandatory disclosure date). The model of [GieO] identifies that expected‘terminal value’ as equal to the disclosed value. The modelling in the cur-rent section provides the missing information in the form of a new targetvalue V T + (so ‘plugs’ the gap between the models). The framework hereis, nevertheless, inspired by the equilibrium argument in [GieO], as summa-rized by the concept of an indifference level (see equations (1.1.1a,b) in theIntroduction).If the firm were to use a threshold L to trigger disclosure at some, forthe moment arbitrary, future time moment u in ( t, T ], the firm’s adopteddecision rule, h say, determines disclosure iff h V T,u ( u, X u , L ) ≥ . As onlytruthful disclosures are assumed, this entails a market valuation at the dis-closed level. However, absence of a disclosure entails, for some appropriatelyselected threshold L, as in the model of [GieO], a valuation of L. In sum-mary, if L = L ( u ) = L u is selected appropriately for time u , then the time u valuation of the firm is given by the random variable(4 . . a ) Z V T,u ( L u ) = X u { h V T,u ( u,X u ,L ( u )) ≥ } + L { h V T,u ( u,X u ,L ( u )) < } . Now let τ V T be a random time, with the interpretation that the event τ V T ( u ) = u for u > F observes X u , the complementary eventbeing τ V T ( u ) = 0 . We now modify the random variables in (4.0.1a) by taking into accountthe times of observation and non-observation and define(4 . . b ) Z τ V T
V T,u ( L u ) = Z V T,u ( L ) { τ V T ( u )= u } + L { τ V T ( u )=0 } . Then the expected valuation is Z ( t,T ] E Q V T [ Z τ V T
V T,u ( L u ) | F t ] τ V T (d u ) , denoting here the distribution of τ V T by τ V T again, for notational conve-nience. As in § Q V T is risk-neutral, this should agree12ith X t . Without loss of generality to the analysis of the interval ( t, T ] , wemay agree to resize (rescale) the observation process at time t to unity. Inter-preting the values as discounted to present time t , our modelling assumptionis to seek a constant L = L V T solving(4 . .
2) 1 = Z [ t,T ] E Q V T [ Z τ V T
V T,u ( L V T ) | F t ] τ V T (d u ) . In setting the new target level, this formula relies not on the market filtration F ∗ (so not on future market sentiment), but on fair value computed from thelarger filtration F with which the firm is equipped.Granted the existence of a solution to (4.0.2), a matter addressed in § V T + := L u to correspond to h V T,u . For a tractable implementation of the modelling assumption in formula (4.0.2),we replace the random observation scheme τ by a deterministic one, knownonly to the firm but most certainly not known to the market. This permitsa decomposition(4 . .
1) [ t, T ] = C V T ∪ D
V T ∪ N
V T according as observation extends over continuous intervals, or either at afinite number of (discrete) time moments, or not at all.Then the equation above reduces to1 = Z C V T E Q V T [ Z τ V T
V T,u ( L u ) | F t ] d uT − t (4 . .
2) + X u ∈D V T E Q V T [ Z τ V T
V T,u ( L u ) | F t ] q V T + (cid:18) − vol( C V T ) T − t (cid:19) L V T with q V T = 1 / D V T , effectively the constant probability of discrete moni-toring. 13urthermore, taking h V T,u to be h ε,a V T,u (with a mark-up a V T,u > − Z V T,u into an option part and a non-option part, appropriately corresponding to good-news and bad-news events.For the good-news case ( ε = +1), it can readily be checked that this is(4 . . Z V T,u ( L ) = L +max { X u − (1+ a V T,u ) L, } + a V T,u L { X u ≥ (1+ a V T,u ) L } , and similarly for bad-news ( ε = −
1) :(4 . . Z V T,u ( L ) = L − max { (1+ a V T,u ) L − X u , } + a V T,u L { X u ≤ (1+ a V T,u ) L } . So the ‘optionality’ in Z V T,u ( L ) reduces to a plain vanilla option corrected bya digital option . Turning to the practicalities of options, one way to handlepositions in digital options is to approximate them by plain vanilla positionsusing a selection of slightly amended strikes. From this perspective, theoptionality of Z V T,u ( L ) can be regarded as approximately induced by a plainvanilla call- (respectively put-) option with strikes close to (1+ a V T,u ) L . Inview of its broader role we refer to L V T as the optimal censor (cf. § Proposition 4.1 (Optimal censor optionality):
When a V T,u = 0 forall u, with the additional assumption of only discrete observations ( C V T = ∅ ),the equation (4.1.2) for the optimal censoring thresholds specializes for thegood-news event to (4 . . L V T + q V T X u ∈D V T E Q V T [max { X u − L V T , } | F t ] , and for bad-news (4 . . L V T − q V T X u ∈D V T E Q V T [max { L V T − X u , } | F t ] . L V T for the benchmark observationscheme
This section demonstrates the existence of a target value
V T + := L V T for thebenchmark observation scheme of the preceding subsection as characterizedby equation (4.1.2). The existence theorems splits into two cases according asthe decision rule determines good- or bad-news announcements; in both caseswe analyze the functional form on the right of equation (4.1.2), treating L V T as a free variable, now denoted by L. It is convenient to begin with bad-newsannouncements. 14 .2.1 Bad-news case
The final term in equation (4.1.2), corresponding to non-monitoring, has asimple functional form: it is linear in L. To understand the other contribu-tions, we rewrite the equation in a form which reflects the complementaryconditioning in the two summands of the earlier equation (4.0.1b). This givesrise below to two corresponding functions of L and recasts the characteriza-tion of L V T in the form:(4 . . N V T ( L V T ) + BS V T, ( L V T ) + BS V T, ( L V T ) . The three functions appearing here are defined as follows:(4 . . N V T ( L ) = (cid:18) − vol( C V T ) T − t (cid:19) L, BS V T, ( L ) = Z C V T E Q V T [ X u { X u ≤ (1+ a V T,u ) L } | F t ] d uT − t (4 . . q V T X u ∈D V T E Q V T [ X u { X u ≤ (1+ a V T,u ) L } | F t ] , BS V T, ( L ) /L = Z C V T E Q V T [ { X u > (1+ a V T,u ) L } | F t ] d uT − t (4 . . q V T X u ∈D V T E Q V T [ { X u > (1+ a V T,u ) L } | F t ](with ‘B for bad news’ and ‘S for Black-Scholes’). The relation betweentheir behaviour and consequent existence of a target value is captured in thefollowing result. Proposition 4.2
In bad-news events, for a constant L V T to exist forwhich equation (4 . . a ) holds, the following conditions (1) to (4) are suf-ficient.(1) BS V T, ( L ) and BS V T, are continuous maps on [0 , ∞ ) .(2) BS V T, ( ∞ ) > −∞ . vol( C V T ) = T − t , or BS V T, is unbounded.(4) ≥ BS V T, (0) . Proof.
First consider the behaviour of the function summands as L growslarge. In BS V T, ( L ) the indicator-functions for large L will become those ofthe entire space, i.e. the constant function 1; the summands of BS V T, ( L )should thereby become expressible in terms of the first moments of X asfollows:(4 . . c ) ∞ BS V T, ( L )( ∞ ) = Z C V T E Q V T [ X u | F t ] d uT − t + q V T X u ∈D V T E Q V T [ X u | F t ] . Little can be said about the behaviour for large L of BS V T, ( L ) except forits being non-negative for L non-negative. As a consequence, N V T ( L ) + BS V T, ( L ) + BS V T, ( L ) ≥ N V T ( L ) + BS V T, ( L ) , for any L ≥
0. On inspection from (4.2.1b), the right hand side of thisinequality will grow linearly in L arbitrarily provided vol( C V T ) / ( T − t ) = 1.Situations where vol( C V T ) = T − t amount to monitoring X at all points intime in [ t, T ] except perhaps on an infinite sequence of points; this is contraryto the spirit of this paper’s observation schemes τ V T , and so little will be lostin excluding such schemes. A minor problem arises, when BS V T, ( ∞ ) = −∞ .Also, granting this technicality, the above line of reasoning gives conditionsof unboundedness to the right (one is able to make the right-hand side ofthe inequality (4.2.1a) bigger than any given real by choosing L sufficientlylarge); in particular, in the same way, it gives conditions for making theright-hand side bigger than 1.Assume the functions are continuous in L . An application of the intermediate-value theorem will then establish the existence of L V T provided there is avalue of L for which the right-hand side of (4.2.1a) is smaller than 1. Theremay be no way other than to postulate this, and it is most sensible to do sofor the smallest value L can take, namely 0. We proceed similarly in this case, rewriting the characterizing equation againso as to reflect the relevant conditioning in (4.0.1). The difference here is that16ow a reversal of inequalities in the passage from bad- to good-news decisionsrequires corresponding new function definitions (below). These, alongside theterm N V T ( L ) from (4.2.1b), recast the existence problem to solving for L V T the equation(4 . . N V T ( L V T ) + GS V T, ( L V T ) + GS V T, ( L V T ) . Here (with ‘G for good news’) we define: GS V T, ( L ) = Z C V T E Q V T [ X u { X u ≥ (1+ a V T,u ) L } | F t ] d uT − t (4 . . q V T X u ∈D V T E Q V T [ X u { X u ≥ (1+ a V T,u ) L } | F t ] , GS V T, ( L ) /L = Z C V T E Q V T [ { X u < (1+ a V T,u ) L } | F t ] duT − t (4 . . q V T X u ∈D V T E Q V T [ { X u < (1+ a V T,u ) L } | F t ] . Their behaviour and consequent relation to the existence of a target value isagain captured by a result analogous to Proposition 4.2.
Proposition 4.3
In good-news events, for a constant L V T to exist forwhich equation (4 . . a ) holds, the two conditions (i) and (ii) below are suffi-cient.(i) GS V T, and GS V T, are continuous maps on [0 , ∞ ) .(ii) ≥ GS V T, (0) . Proof.
Mutatis mutandis, the line of reasoning developed for Proposition 4.2now applies. Here, the larger L is, the closer the indicator functions in GS V T, ( L ) will come to the indicator function of the entire space; this trans-lates into GS V T, ( L ) becoming similar to some real GS V T, ( ∞ ) as L growslarge, and this real is positive. Since GS V T, ( L ) ≥ L ≥
0, the lineof reasoning of Section 4.2.1 now establishes without further conditions theunboundedness in L of the right-hand side of (4.2.6a).17 .2.3 Worked Example in the Geometric Brownian case Corresponding to the mark-up decision rules of (2.0.1) there are six expecta-tions appearing in the formulas of section § § L V T . Assume that X follows geometricBrownian motion: X t + s = X t exp (cid:18) ( µ V T − σ V T ) s + σ V T W V T,s (cid:19) s ∈ [0 , ∞ ) , with µ V T ∈ R , σ V T > , and W V T a standard Brownian motion indepen-dent of time t information F t as in (3.2.1). For fixed u = t + s in [ t, T ],these six expectations are provided by standard results on Brownian motion.Corresponding to (4.2.1c) ∞ one has(4 . . E Q V T [ X u | F t ] = X t exp( µ V T s );likewise, corresponding to the pair (4.2.1d), (4.2.6c) and the pair (4.2.1c),(4.2.6b), taking(4 . .
4) ∆ s := log((1 + a V T ) L/X t ) − ( µ V T + σ V T ) sσ V T , one has respectively:(4 . . a ) E Q V T [ { X u ≥ (1+ a V T ) V T } | F t ] = Erfc (cid:0) ∆ s / √ s (cid:1) , (4 . . b ) E Q V T [ { X u ≤ (1+ a V T ) V T } | F t ] = Erfc (cid:0) − ∆ s / √ s (cid:1) , (4 . . a ) E Q V T [ X u { X u ≥ (1+ a V T ) V T } | F t ] = X t exp( µ V T s )Erfc (cid:16) ∆ s − σ V T s √ s (cid:17) , (4 . . b ) E Q V T [ X u { X u ≤ (1+ a V T ) V T } | F t ] = X t exp( µ V T s )Erfc (cid:16) − ∆ s − σ V T s √ s (cid:17) . Here Erfc is again the complementary error function, for which specificallysee Appendix equations (A.7a, b) and (A.8).For periods of continuous monitoring, integrals of these three expressionsneed to be computed over time s . This is unproblematic for (4.2.2), whereit reduces to differencing of the right-hand side across the endpoints of the18onitoring period (and division by µ V T ). For (4.2.5ab) and (4.2.6ab) this willlead to expressions in terms of non-standard special functions: the incompleteBessel functions, given by integrals of the form R [ t,T ] x α exp( − ( A/x + B x )) d x for some real constants A , B ≥ α . Series representations can be derivedfor these integrals; generically in α , the series are in terms of values of theincomplete gamma function, namely(4 . . Z [ t,T ] x α exp( − ( A/x + B x )) dx == ( B ) α +1 X ∞ m =0 ( − m m ! ( B ) m { Γ( − ( α + m + 1) , BT ) − Γ( − ( α + m + 1) , Bt ) } , where the series may be expressed in terms of Erfc only for particular choicesof α (integer or half-integer values). Here we address matters on which [GieO] is silent. Σ Theorem 5.1
With the modelling assumptions of Section 3.1, the follow-ing assertions hold in the framework of Sections 4.1 and 4.2.(1) Time- T disclosure becomes the more likely the smaller are a ∗ and V T .(2) In good-news situations time- T disclosure is the more likely the smalleris r and the bigger are a and E ∗ T .(3) In bad-news situations, time- T disclosure is the more likely the biggeris r and the smaller are a and E ∗ T . Proof.
Working in the general process situation of Sections 4.1 and 4.2,the starting point here are the inequalities holding at any time T in ( T , T ]which trigger a disclosure. In a good-news event this is(5 . . V T ≥ (1 + a ) e − r ( T − T ) E ∗ T Q ∗ (Σ ∗ T,T ≥ (1 + a ∗ ) V T |F ∗ T );19nd for bad-news this is(5 . . V T ≤ (1 + a ) e − r ( T − T ) E ∗ T (1 − Q ∗ (Σ ∗ T,T > (1 + a ∗ ) V T |F ∗ T )) . Treating
V T as a given (computed by the accounts department), the like-lihood of the validity of these inequalities is determined by the size of therespective right-hand side; the good-news event (5.1.1) occurring is the morelikely the bigger is the size of the expression on the right; the bad-news event(5.1.2) becomes the more likely the smaller is the size of the right. The vari-ables on which the validity of these inequalities depend are thus: a and V T ,correspondingly a ∗ and E ∗ T , the interest-rate r, the ‘time to maturity’ T − T (time left to the mandatory disclosure), and the variables beyond these thatenter into the construction of Σ ∗ T,T ; the latter variables include V C via S ∗ T . Remark 5.2
The effects of S ∗ T and T − T on a time- T disclosure decisiondepend on the specific form of the law of Σ ∗ T,T . To justify this assertion assupplementary to Theorem 5.1, we look at the effects of infinitesimal changesin S ∗ T and T − T on the (right-hand sides of ) (5.1.1) and (5.1.2). Grantedfor simplicity the partial differentiability of these probabilities, we have thefollowing two equations in terms of λ ∗ T , the law of Σ ∗ T,T contingent on time- T information F ∗ T : (5 . . ∂ T − T [ e r ( T − T ) Q ∗ (Σ ∗ T,T > (1 + a ∗ ) V T |F ∗ T )]= e r ( T − T ) Z ∞ (1+ a ∗ ) V T { ∂ T − T λ ∗ T − rλ ∗ T } ( u ) d u, (5 . . ∂ S ∗ T [ e r ( T − T ) Q ∗ (Σ ∗ T,T > (1 + a ∗ ) V T |F ∗ T )] = e r ( T − T ) Z ∞ (1+ a ∗ ) V T ∂ S ∗ T λ ∗ T ( u ) d u. We see from these two equations that the signs of the effects depend on theexact form of this law, and so need to be determined on a case by case basis.Suffice it to say that conditions needing to be imposed here in concrete cases of λ ∗ T include conditions that entail that no sign changes occur in the respectiveintegrands on the right-hand sides of these equations. To determine thesesigns explicitly requires concrete choices, at the least for S ∗ and for howexactly S ∗ enters into the definition of Σ ∗ T,T . .2 Comparative statics for changes in S ∗ T We show that the assumption that the process S ∗ follows a Markov processis sufficient for the determination of the effect of S ∗ T on early disclosure. Sowe work with processes with two properties: firstly that, for any time u ≥ . . S ∗ T + u = S ∗ T exp( X ∗ u ) , where S T = V C , where ( X ∗ u ) u ≥ is a process independent of time- T information F ∗ T ; secondlythat, also for arbitrary T ∈ ( T , T ] , (5 . . S ∗ T + u = S ∗ T exp( X ∗ u ) , u ∈ [0 , ∞ ) , where, by abuse of language, ( X ∗ u ) u ≥ (or a suitable version of the process in(4.3.5) denoted by the same same symbol) is a process independent of time- T information F ∗ T . Theorem 5.3
We have ∂ S ∗ T Q ∗ (cid:0) Σ ∗ T,T > (1 + a ∗ ) V T | F ∗ T (cid:1) > in (4 . . under the additional Markovian conditions (5 . . and (5 . . on S ∗ ; so inboth good-news and bad-news situations a time- T disclosure becomes the morelikely the larger is S ∗ T . This conclusion holds more generally for all construc-tions of Σ ∗ T,T that preserve scaling (in the sense that we have Σ ∗ T,T ( S ∗ ) = S ∗ T Σ ∗ T,T (exp( X ∗ )) ). Proof.
The assumption (5.2.1) and (5.2.2) imply that S ∗ is the productof a scaling factor S ∗ T and a ‘standardized version’ of S ∗ . This yields arepresentation for Σ ∗ T,T ( S ∗ ) as a product of two positive factors, the scalingfactor S ∗ T and the random variable that results from the application of therespective construction for Σ ∗ T,T to the standardized version of S ∗ .In (5.1.1) and (5.1.2) the partials w.r.t. S ∗ T of the probability factors willthus be positive so long as we have independence of the normalized versionsof S ∗ from time T -information. (cid:3) We consider here the other parameter mentioned in Remark 5.2 namely T − T ,i.e. the time left to the next mandatory reporting date. At first sight, one21ight expect a proposition asserting that the shorter is this time, the lesslikely are decisions made for an early disclosure. However, on reflection, suchdecisions may well depend on the actual evolution of the market capitaliza-tion of the firm, and market forces may lead to changes in the size of thiscapitalization forcing early disclosure also at dates comparatively close to themandatory date. Therefore, a discussion of the effects of T − T needs to beincorporated in a model framework that includes S ∗ .Here we adopt a standard Black-Scholes modelling for S ∗ , and thereforeconsider now the process X ∗ in Section 5.2 as following scaled Brownianmotion with drift:(5 . . X ∗ u = X ∗ ( µ ∗ , σ ∗ ) u = µ ∗ u + σ ∗ W ∗ u , u ∈ [0 , ∞ ) , where W ∗ is a Q ∗ -Brownian motion independent of time- T information startedwith value 0 at time 0, with parameters σ ∗ > µ ∗ = r − ∆ ∗ − (1 / σ ∗ ) ∈ R . Recall the former parameter is a measure of market volatil-ity, while r − ∆ ∗ is the excess of the (riskless) short rate, r , over the dividendrate of the firm ∆ ∗ as seen by the markets ; in the present context this differ-ence should be viewed as an ‘appreciation rate’ for investments in F (againas seen by the markets).The four effects on V ∗ to consider now are those induced by changes in T − T and also in σ ∗ , r − ∆ ∗ , and r . These four will depend on which of good-news or bad-news situations occurs; they enter via the market proxies, andhence even a qualitative picture will depend on the concrete form of Σ ∗ T,T .We focus on modelling Σ ∗ T,T as the running minimum or maximum of S ∗ , asin equations (3.1.5a) and (3.1.5c), specifically in the good-news situations,so that by equation (5.1.1) we must consider the inequalities(5 . . V T ≥ V ∗• • ∈ { max , min } , with(5 . . V ∗ max := (1 + a ) E ∗ T exp( − r ( T − T )) Q ∗ max , (5 . . Q ∗ max := Q ∗ ( max u ∈ [0 ,T − T ] { X ∗ ( µ ∗ , σ ∗ ) } > A ∗ T ) , (5 . . V ∗ min := (1 + a ) E ∗ T exp( − r ( T − T )) Q ∗ min , . . Q ∗ min := Q ∗ ( min u ∈ [0 ,T − T ] { X ∗ ( µ ∗ , σ ∗ ) } > A ∗ T ) , where we set(5 . . A ∗ T = log((1 + a ∗ ) V T /S ∗ T ) . We give a paradigm discussion of the effects of T − T on the likelihood ofdisclosure decisions, when for the good-news case of (5.3.2) the running max-imum is used in the construction of market proxies according to (5.3.2a,b).These effects turn out to depend on the sign of the mark-up parameter A ∗ T as follows. Theorem 5.4
In the framework of Section 5.3, assume a situation oftime- T non-disclosure of good news. Using the running maximum of S ∗ inthe construction of the market proxy V ∗ , the following two assertions areequivalent.(1) Early disclosure in [ T, T ] is the more likely the nearer is T to T .(2) sign( ∂ T − T V ∗ max ) < .Here the partial derivative in (2) depends on the sign of A T ; for A ∗ T ≤ this is ∂ T − T V ∗ max = − rV ∗ max , while for A ∗ T ≥ this is ∂ T − T V ∗ max = − rV ∗ max + (1 + a ) E ∗ T exp( − r ( T − T )) A ∗ T exp( − η ) σ ∗ p π ( T − T ) , where η = ( A ∗ T − ( T − T ) µ ∗ ) /σ ∗ p T − T ) . To indicate the typical line of reasoning for results like this, start from(5.3.2), observing that (in these good-news situations) disclosure decisionsat some fixed point in time T are the more likely the larger V ∗ T is. Theeffects of some parameter on the likelihood of early disclosure thus translateinto the determination of the corresponding partial derivative of V ∗ T , andso a determination of their qualitative effect reduces to a determination of23he sign of these partials. Early disclosure thus becomes more likely thelarger the relevant parameter, provided the corresponding partial of V ∗ T ispositive, and vice versa. The point of our choice of a geometric Brownianframework is that explicit formulas for the probability factors Q ∗• are availableas standard results in Brownian motion; these are reviewed in Appendix Abelow, with equations (A7a,b) pertinent for the present case of running-maximum performance parameters. Establishing comparatice statics resultstherefore reduces to straightforward partial differentiation of explicitly givenfunctions, albeit of some complexity. Theorem 5.3 collects the results whenthe relevant parameter there is the time left to the next mandatory date. Remark 5.5
Proceeding along the same lines in the same situation, one ob-tains results similar to those of Theorem 5.3 concerning the effects of thevolatility σ ∗ , whereas the effects of r − ∆ ∗ and r are unequivocally unidirec-tional (with the signs of the pertinent partials being equal to minus that of E ∗ T ). Provided E ∗ T > , early disclosure within ( T, T ) is the more likely thesmaller are r − ∆ ∗ and r . Here we note only that if the market proxies are instead constructed usingthe running minimum of S ∗ analogues of Theorem 5.3 and Remark 5.5 againhold and preserve all the conclusions above except for a sign reversal in A ∗ T .This shows how derivation of the effects of T − T on early disclosure requiresthe specifics of a given model. This paper’s approach to asset pricing allows the development of a richer ap-preciation of how voluntary disclosure by firms can affect firm asset valuationin equilibrium. Existing research has typically modelled voluntary disclosureas the choice by firms to make additional voluntary (content) disclosures tothe market at fixed time points. As such this literature does not consider thepossibility that firms may choose not only what to disclose voluntarily butalso when to disclose. Thus voluntary disclosure has at least two dimensions: content and timing . As existing models typically do not consider the latter di-mension, they are not truly dynamic, and hence do not provide the necessary24uilding blocks to develop a realistic empirical model of (‘two dimensional’)voluntary disclosure. Here we have explicitly modelled the joint content-timing interaction, so enabling more realistic formal modelling of problemsfaced by managers of firms: when private news is uncertain, how good doesthat private news have to be before it is in the interests of the firm to issuea voluntary disclosure. The other side of this coin is what materiality stan-dard needs to be followed in managing the voluntary disclosure process. Thecomparative statics derived in the preceding section permit an understandingof how changes in parameter values may explain differences in equilibriumbehaviour between firms – some voluntarily releasing additional informationearly, others not. This meets the challenge of modelling equilibrium asset-pricing with endogenously determined voluntary disclosures, wherein boththe content and the timing of disclosures are rationally chosen, making delayor early release of information in capital markets an equilibrium outcome.
We close with some observations about the potential of the approach aboveespecially with regard to variations on the themes presented and generaliza-tions away from the Brownian framework followed above.
Implicit in our development of a markets-based general modelling frameworkwas the need to pick apart the ‘who does what and how’ into ‘building bricks’,and with these to build a variety of models. We implicitly identified five suchbricks, which in fact are best considered as mechanisms, to borrow a phrasefrom economic theory. These are made explicit here so as to stress both thesensitivity of a model to its assumptions and its adaptability to alternativecontexts.
Mechanism (i). Evolution rules . The perspective adopted above is ratherlike that of a scientist designing experiments and subsequently observing out-comes and evolution. Ingredients thus include design dynamics, start timeand observation times. Thus mechanisms (i) amounts to formal rules for en-coding these three aspects. Real-life features mapped via such ‘experiments’include informational interplay between economic agents and firm-to-market25ommunications. See the summary in Section 7.5 for an explicit illustrationof how this can be further developed.
Mechanism (ii). Decision-rule strategies . The task here is to provide rulesfor triggering ‘events’ (typically, public disclosure of privileged information),and the idea is that these be the consequence of some ‘rule’, i.e. functionalrelation, applied to some observation variables. At a technical level, thismechanism thus amounts to the selection of functional relations subject tothe specification of observation variables. The mechanisms we adopted, start-ing from Section 3, are motivated by the provision of approximations to equilibrium-induced decision rules as derived in [GieO]. There they corre-spond to ‘value-enhancement’ disclosures when observations are ‘sufficientlyhigh’. Whilst outside the scope of the present paper, the argument there maybe dualized to correspond to equilibrium-induced ‘value-erosion’ alerts whenobservations are ‘sufficiently low’, with a resulting notion of endogeneous ‘materiality thresholds’. Such an understanding leads to the following
Conjecture 7.1
First-order approximation of equilibrium decision-rulesbased on the notion of materiality yields decision-rules using functions takingthe form h ε, a ( t , x , y ) = ( x − ( + a t ) y ) ε t for some families of signs ε = ( ε t ) t ∈ [ T ,T ] and mark-ups a = ( a t ) t ∈ [ T ,T ] , andvice versa.Mechanism (iii). Observation processes V . A further fundamental notion isinformational assymetry, the task being to construct a ‘variable’ (or perhapsa vector) with two properties. Firstly, it is capable of observation over timeand is observed over time by the informationally privileged agents of what-ever model is to be constructed (denoted by the symbol F , typifing firms);secondly, the variable is at best partially observable by the remaining agents(denoted by the symbol M ).As to observation variables, we focus on a portfolio view . Continuingto think of F as a firm for a moment, F will not in general observe justa single source of information to set a target, but a portfolio of these, say( X , . . . , X n , . . . , X N ). Formally, the chosen observation process X will be a26unction of the X n ; simple, but typical, functions are linear or multiplicativeforms in the X n as given respectively by X = X Nn =1 a n X n , or X = Y Nn =1 X a n n , with suitable real weights a n ∈ R . Examples of two factor portfolio observation variables . In the paper, V was interpreted as representing the value of the firm F , i.e. a processinternally observed by F . It is natural to complement it by a process thatencodes the external view of the firm’s value, such as provided by the thefirm’s market capitalization, S ∗ . Specializing to portfolios of additive type,the associated observation variable may take the form X = aV + bS ∗ , for some a, b ∈ R . General structure of X apart, the modelling of V and S ∗ isfar from straightforward, and Sections 7.2 to 7.4 below offer an amendmentto the simplified treatment given in the main body of the paper. Mechanism (iv). Observation process proxies V ∗ from state observer systems .The task here is to enable specifically the ‘informationally under-privileged’agents in the models to approximate V . To paraphrase a key idea in thepaper: here S ∗ is seen as a public proxy for V in creating an estimate V ∗ t of V t ; for tractability we made specific approximation assumptions.In so doing, we borrowed an idea from the control theory of an engineeringplant, where one way to deal with imperfect information about the plantis to build a laboratory version (a model) of the plant with accessible full-information of its state at any time (known as a ‘state observer’ system [Rus,Ch. 3], [Son, Ch. 7] – in reality a ‘state estimator’); state-correcting signalsare sent to this model, using plant-based, imperfect, or partial observations,with which to guide the ‘observer system’ (model) into greater agreementwith the plant.Unlike in the engineering context, inclusion into a market-based model ofan observer-style system implies changes to the strategic behaviour of the firmin its decisions to hide certain observations of its state. Indeed, here each side(each of the agents, F or M ) enriches its algorithmic opportunities. In thiscontext, our version of an ‘observer system’ responds to strategic behaviour,27o is far richer. To emphasize this difference when, for example, F was afirm, the observer-style system was termed a proxy-firm . Mechanism (v). Target value processes
V T . Under this heading, the taskis to provide techniques for forecasting values of observation variables (and,significantly, of their proxies), bearing in mind that such items are contingenton market developments as well as on restrictions arising from (production)technologies. In Section 4 we marry accounting analysis with the analysisof Bermudean options: mechanism (v) involved observation levels L t for V t qua strike prices at which the respective agent’s decision rule is indifferentbetween disclosing or suppressing private observation of V t , were V t to takethe value L t (as noted in the introduction). This construction borrows fromthe stylized model of [GieO], where these levels describe the market’s currentview of the value during periods of silence and so provide the basis of currentguidance on its earnings target. As to the observation mechanism, one may ‘drill down’ to the basic structureof profits and address the uncertainty effects created by reporting lags. Thestarting point is a formalization of accounting practice: V u the firm F ’s time- u value is the accrual of an instantaneous variable π w over the period [ T , u ]added to an initially given value V C of time T , so that(7 . . V u = V C + Z [ T ,u ] π w d w, u ∈ [ T , T ] , implicitly assuming w π w to be summable over [ T , T ].The simplest interpretation of V u is incrementing V C by the firm’s actual profit flow rate π w – as it actually arises at each time moment w betweentime- T and time- u . Alternatively, to allow for the possibility of delays inreporting profits (due, say, to reporting delays of costs, as below), we canre-interpret this as the recognized profit flow rate – namely, the value postedin some official ledger.To introduce reporting lags into the model, fix Λ ≥ u , assume the flow π w is certain only for ‘distant’ times w , namely times28arlier than u − Λ, but for times w nearer to u , i.e. in ( u − Λ , u ], π w is uncertain.A further refinement then occurs in the decomposition (7.2.1) created by a deterministic part ∆ ns V u , certain at time- u (with ‘ns’ for non-stochastic),and a part ∆ s V u uncertain at time- u :(7 . . V u = V C + ∆ ns V u + ∆ s V u , u ∈ [ T , T ] , where(7 . . ns V u = Z [ T , max { T ,u − Λ } ] π w d w, (7 . . s V u = Z [max { T ,u − Λ } ,u ] π w d w. We view V ns u = V C +∆ ns V u as an accounting equality , namely, as data held,or stored, by the firm F , and ∆ s V u as a variable that needs to be modelled.Two obvious questions arise: first, how does (a manager) F respond tosuch operational uncertainty. For example, will there be a period in which F is waiting for the time- T accounting information to be corroborated andto be verified as reliable (up to a level of security deemed appropriate for thedecision-making), and how does F then respond to the evolution of marketsentiment during such periods of waiting? Second, is there a correlationbetween market sentiment V ∗ and the degree of operational effectiveness ofthe firm’s accounting department? As a second complement to our discussion of Mechanisms (iii), we provideexamples for modelling firm-value observation processes V concretely. Deterministic
Here the construction of V needs to be linked to the standard functionalforms preferred by the theory of the firm in Economics and Econometrics.We consider Cobb-Douglas technologies, and indicate how to model prof-its derived from a Cobb-Douglas production function ([Var, Ch. 1], [Rom,29h. 2]) corresponding to a single output from two input factors (such ascapital and labour) with respective parameters a, b ≥ a + b < . This yields profits as a function of input prices w = ( w , w ) and outputselling-price p in the form(7 . . π CD ( p, w ) == p / (1 − ( a + b )) (cid:26)(cid:16) a + bκ · c ( w ) (cid:17) ( a + b ) / (1 − ( a + b )) − κ · c ( w ) (cid:16) a + bκ · c ( w ) (cid:17) / (1 − ( a + b )) (cid:27) , for a cost function c ( w ) = ( w a w b ) / ( a + b ) , where κ := (( a/b ) a/ ( a + b ) + ( a/b ) − ( a/ ( a + b ) ) /A / ( a + b ) , with A = (1 − a ) a − /a a . Stochastic Cobb-Douglas profits
To take into account uncertainties in the profit function, assume that uncer-tainty in the output and input price is given by positive stochastic processeson some stochastic basis, say X ( Q ) = (Ω , F , F = ( F u ) u ∈ [ T ,T ] , Q ). As re-gards output, passing to logarithmic prices and so to an exponential priceprocess exp Y , (7.2.1) yields profits of the form(7 . . π w = α exp( Y w ) β = exp( α (log( α ) + βY w ) , w ∈ [ T , T ] , for some fixed α > β ∈ R , and some fixed stochastic process Y on X ( Q ). Inturn this gives the uncertain part of the time- u value of V the representation:(7 . .
3) ∆ s V u = α Z [max { T ,u − Λ } ,u ] exp( βY w ) d w, u ∈ [ T , T ] . Treating input (factor) prices w in similar vein preserves this general formfor the uncertain parts of V in (7.2.2c). In any of these representations, anotable choice for Y is Brownian motion on X ( Q ), and this provides anexplicit illustration of how the modelling above turns V u itself into a randomvariable, given the time- u information (see Section 7.4 for scalable processes,which we suggest as candidate modelling mechanisms).As mentioned, accruals in (7.2.1) can be modelled in (at least) two con-ceptually different ways, depending on whether instantaneous profits π s ortheir accumulated value is taken as a primary variable. The matter of choiceis not just a conceptual but also a practical one, even assuming the clas-sical Cobb-Douglas two-factor production technology above. Neoclassical30conomic theory asserts (cf. § w , w and of the commodity price p taking the form(7 . . π CD ( p, w , w ) = w a/ ( a + b − w b/ ( a + b − p ( a + b − ·· (cid:26)(cid:16) a + bκ (cid:17) ( a + b ) / (1 − ( a + b )) − κ (cid:16) a + bκ (cid:17) / (1 − ( a + b )) (cid:27) , with constants a , b > a + b < . . κ := (( a/b ) b/ ( a + b ) + ( a/b ) − a/ ( a + b ) ) / ((1 − a ) a − /a a ) / ( a + b ) . Assume that non-deterministic profits are the result of fluctuations in anyof these prices, and, for simplicity, staying within the Brownian framework,assume the fluctuations follow geometric Brownian motion. The resultingdynamic for π CD is of the form(7 . . π CD ,T + u = π CD ,T exp( µ π u + σ π W π,u ) , u ∈ [0 , ∞ ) , with driver a Brownian motion W π independent of time- T information F F F,T and with real constants σ π > µ π . Return now to the choice of explicitmodelling variants; according to the choice of accumulated profits or instan-taneous profits, one has respectively(7 . . π s = π CD,T + s , s ∈ [0 , ∞ ) , (7 . . Z [ T,T + s ] π w d w = π CD,T + s , s ∈ [0 , ∞ ) . This last requires for V T + u the integral of geometric Brownian motion, notcovered by the context of Section 4. For the first, the results of Section 4.2do apply, and provide the forecast target V T . As a third complement to Section 7.1, we suggest the use scalable processes for modelling with mechanisms. These processes will be patterned after theexponentials of strong Markov processes S ∗ , which satisfy two equations.Firstly, with T fixed, for any time u ≥ . . S ∗ T + u = S ∗ T exp( X ∗ u ) , where S T = V C , X ∗ u ) u ≥ is independent of time- T information F ∗ T . Secondly, forarbitrary real T in ( T , T ], the representation(7 . . S ∗ T + u = S ∗ T exp( X ∗ u ) , u ∈ [0 , ∞ ) , where, by abuse of language, ( X ∗ u ) u ≥ (or a suitable version of the process in(7.4.1) denoted by the same same symbol) is a process independent of time- T information F ∗ T . We now relax the second condition and define processes S ∗ to be scalable if they are RCLL and satisfy (7.4.1) and (7.4.2), except thatnow (7.4.2) need not necessarily hold for all T in ( T , T ] and instead is to holdnecessarily for all T in some prespecified finite subset T of ( T , T ]. Here weprimarily think of T as containing the endpoints of benchmark observationschemes along the lines of equation (4.1.1). Extending this notion of scalableprocess to allow the sets T to have at most countably many stopping timesshould not, however, pose problems. A characteristic feature of the mechanisms identified in Section 7.1 is thatthey identify the economic agents solely in terms of how they act. In respec-tively Sections 4 and 3, as it happens, the agents F posited by the mechanismsin disclosure situations are indeed interpreted as acting like the manager ofthe firm, and agents M as acting like representative market participants. Forthe wider guidance theme, however, specific market participants will also ‘actout’ the role of agent F within some of the mechanisms. An outline follows.To tell our guidance story we need to single out a distinguished group ofpeople from among the market participants M , whom we shall call analysts .The typical representative member of this group being denoted by A , wecontinue to denote representative market participants as agents M (as inSection 3).The guidance theme then starts at time t with the announcement bymanager F of the current accounting numbers of the firm: V C and its target
V T for the next official reporting date T . These numbers are processed by M as in the disclosure theme, while now A is also assumed to make its owncomputations. For these computations A will be asumed to use mechanism(v) of Section 7.1 (as though in the role of agent F ), and make its owncomputions of the time- T target, possibly based on a re-estimation of V , V ∗ , S and S ∗ ; call the result V T A , and assume that A will announce this32umber to M and F at a time t + ∆ A within [ t, T ]. This announcement willpossibly induce a re-calibration of the price processes S and S ∗ ; the analyst’starget V T A will now be added as a new variable in the decision makingof the manager F . Apart from a possible consequent re-calibration of V ∗ ,the essential difference from the set-up of Section 3 is that now manager F is assumed to use decision rules in four variables, say h ( T, V T , V T ∗ , V T A ).For present outline purposes, we will not make this 4-variable rule explicit,leaving the details to be established elsewhere. Now running the disclosure-mechanisms of Sections 3, 4, and 5 results, mutatis mutandis, in either anearly disclosure at some time τ < T , or a regular one at time T . In bothcases an announcement is made by manager F of new current numbers V C and targets
V T for the next reporting date; these numbers will be announcedsimultaneously to the analysts A and to the market (as represented by agent M ) and the entire activity starts all over again. The details are intended tobe established elsewhere. Appendix: Reductions
This section collects the simplifications arising for general mark-up decisionrules in Brownian contexts. We work with a fixed probability space (Ω , F , P )which is equipped with a filtration F = ( F u ) u ≥ such that the resultingstochastic basis X ( P ) = (Ω , F , F , P ) satisfies the usual conditions (see forexample [JacS, Def. 1.3, p. 2]). The mark-up decision rules are assumed, asabove, in the form h ε,a ( x, y ) = ε ( x − (1 + a ) y ), for fixed parameters ε = ± a > − A.1 Bad-news to good-news reductions:
For Σ a random variable on Ω , measurable with respect to F t , and fixed t > . E P [ { h +1 ,a (Σ ,V T ) ≥ } | F t ] = P ( { Σ ≥ (1 + a ) V T }| F t ) , (A . E P [ { h − ,a (Σ ,V T ) ≥ } | F t ] = 1 − P ( { Σ ≥ (1 + a ) V T }| F t ) . These two yield a reduction of bad-news to good-news disclosures via(A . E P [ { h − ,a (Σ ,V T ) ≥ } | F t ] = 1 − E P [ { h +1 ,a (Σ ,V T ) ≥ } | F t )] , a ) V T , given the continuousprocesses in play here. Note the qualitative consequence that factors influ-encing the relevant probabilities will have opposite effects on good-news andbad-news events; an increase in a factor that leads to an increase of (A.0a)will decrease (A.1) and vice versa.
A.2 Running-minimum to running-maximum reduc-tions:
We collect here the further reductions needed for good-news events when S ∗ is a geometric Brownian motion. With Σ ∗ = Σ ∗ t,T the running maximumor the running minimum of S ∗ on some fixed time interval [ t, T ], let W ∗ bean ( F , P )-Brownian motion on X ( P ) started at 0 at time 0; for σ ∗ > . S ∗ u + t = S ∗ t exp( µ ∗ u + σ ∗ W ∗ u ) , u ∈ [0 , ∞ ) , with µ ∗ = r − δ − ( σ ∗ ) for r, δ ∈ R ; appealing to the strong Markov propertyof Brownian motion, assume also W ∗ to be independent of F t , and expressthe events in terms of W ∗ as follows:(A.3) E P [ { h +1 ,a (max { S ∗ w | w ∈ [ t,T ] } ) ≥ } | F t ] = P ( max u ∈ [0 ,T − t ] { µ ∗ u + σ ∗ W ∗ u } ≥ A ∗ t ) , (A.4) E P [ { h − ,a (min { S ∗ w | w ∈ [ t,T ] } ) ≥ } | F t ] = P ( min u ∈ [0 ,T − t ] { µ ∗ u + σ ∗ W ∗ u } ≤ A ∗ t ) , with(A5) A ∗ T := log((1 + a ∗ ) V T /S ∗ t ) . Setting W ∗∗ := − W ∗ note that(A6) P ( min u ∈ [0 ,T − t ] { µ ∗ u + σ ∗ W ∗ u } ≤ A ∗ t ) = P ( max u ∈ [0 ,T − t ] {− µ ∗ u + σ ∗ W ∗∗ u } ≥ − A ∗ t ) . It is special to the Brownian context that (A.6) provides a reduction ofthe running-minimum event in (A.4) to a running-maximum event in (A.3),since, if W ∗ is Brownian, then so is its negative W ∗∗ (above). An explicitdetermination of the expectation (A.3) can be had from the explicit lawfor the running-maximum of Brownian motion; see e.g. [Fre, (30) Corollary,p. 25]. This relies, in this Brownian context, on the running-maximum always34eing positive on time intervals of positive length; indeed, this follows fromthe fact that the running-maximum of the process is zero on non-positivetime arguments. This is not directly of use here, since the drift µ ∗ is ingeneral non-zero. But an appropriate Girsanov transformation applied tothe measure P will achieve a reduction to the zero-drift case (cf. [RogW, § I.13, eqn. (13.9)]), at the cost, however, of an additional exponential factorin (A.3):(A7a) P ( max u ∈ [0 ,T − t ] { µ ∗ u + σ ∗ W ∗ u } ≥ A ∗ t ) = 1 , unless A ∗ t > , in which case(A7b) P (max u ∈ [0 ,T − t ] { µ ∗ u + σ ∗ W ∗ u } ≥ A ∗ t ) == 12 Erfc A ∗ t − ( T − t ) µ ∗ σ ∗ p T − t ) ! + 12 exp (cid:18) µ ∗ A ∗ t ( σ ∗ ) (cid:19) Erfc A ∗ t + ( T − t ) µ ∗ σ ∗ p T − t ) ! , with Erfc ( z ) := (2 / √ π ) R [ z, ∞ ) exp( − w ) dw , for any complex z , the comple-mentary error function. This result can be established, mutatis mutandis,along the lines of [MusR, Lemma A.18.2, p. 617seq]); for a proof by a reduc-tion to this result, start from the equality P ( max u ∈ [0 ,T − t ] { µ ∗ u + σ ∗ W ∗ u } ≥ A ∗ t ) = 1 − P ( max u ∈ [0 ,T − t ] { µ ∗ u + σ ∗ W ∗ u } ≤ A ∗ t );on the right-hand side we have from [MusR, eq. (A.85)] the equality P ( max u ∈ [0 ,T − t ] { µ ∗ u + σ ∗ W ∗ u } ≤ A ∗ t ) == N (cid:18) A ∗ t − ( T − t ) µ ∗ σ ∗ √ T − t (cid:19) − exp (cid:18) µ ∗ A ∗ t ( σ ∗ ) (cid:19) N (cid:18) − A ∗ t + ( T − t ) µ ∗ σ ∗ √ T − t (cid:19) , if A ∗ t ≥
0, but otherwise this probability is 0; then successively use theidentities 1 = N ( ξ ) + N ( − ξ ) and N ( ξ ) = (1 / − ξ/ √
2) to arrive at(A7a,b).Formulas for the tails of the running-minimum expressions of (A.4) area consequence of (A.7a,b). For this start by passing to the complementaryprobability P ( min u ∈ [0 ,T − t ] { µ ∗ u + σ ∗ W ∗ u } ≥ A ∗ t ) = 1 − P ( min u ∈ [0 ,T − t ] { µ ∗ u + σ ∗ W ∗ u } ≤ A ∗ t );35ow use (A.6) to translate the right-hand side in terms of probabilities forthe running maximum, and apply (A.7a,b) to obtain( A. P ( min u ∈ [0 ,T − t ] { µ ∗ u + σ ∗ W ∗ u } ≥ A ∗ t ) == 12 Erfc + A ∗ t − ( T − t ) µ ∗ σ ∗ p T − t ) ! −
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