Discounting, value and dollar beta matching in flexible cashflow systems. Steinar Ekern, Mark B. Shackleton and Sigbjørn Sødal Autumn 2014 Abstract Discounting, value and dollar beta matching in flexible cashflow systems. The flexibility to switch from one cashflow to another has option value. We develop the betas of these option values and use them in first order conditions for optimal exercise. Because option functions smooth paste to their targets, along with their values at exercise the dollar weighted betas of options must match too. For operational flexibility which allows switching between distinct cashflow modes, we use a discount function representation to relate an option’s value at creation to its end value and to derive option betas. Using three sets of equations, a system is solved for all option and investment sums (strikes) as a function of given investment thresholds and the betas there. Essential option values are placed in vectors and the investment network (graph) is captured by a matrix of discount factors that allow large structures with many operating modes to be tracked. C61, G31. Key Words; Value matching, smooth pasting, dollar beta and rate of return equalisation, cashflow networks and bipartite directed graphs. 1 Introduction When a financial option is exercised, tradeable quantities are exchanged; for instance upon exercise a call option’s value coupled with the strike price is swapped for a stock. This is a simple example of the benefit of flexibility which allows the present value of one cash flow (the certain interest on a strike price) to be converted into another (the uncertain dividend flow on a stock).1 This article follows those in the literature on management of real assets, where there is flexibility to switch between operational cashflow types. Many articles have made the analogy between operational and financial options, where capital investment including running costs, is viewed as a strike price and operational value a stock price.2 These techniques are employed to evaluate capacity and other investment decisions.3 Although less has been written on the acquisition or creation4 of such option flexibility, their worth is also based on an exercise condition at investment. Managers maximise flexibility value by ensuring optimal timing for these investment decisions. After an investment, if further flexibility remains (i.e. the use of one option creates another) optimal timing should be driven by maximisation of the joint value of both initial and subsequent options. When an option is exercised and an investment transition made, we therefore recognise that flexibility is unlikely to end and this article accounts for the simultaneous application and creation of operational options. Furthermore these follow-on options influence the timing of prior exercise; a decision to proceed can not be taken without anticipating follow on flexibility and its optimal decision too. Our valuation procedure tracks sequential flexibility through operating modes, requiring the values at optimal investment 1 See Black and Scholes [4], Merton [17]; also Cox, Ross and Rubinstein [7] for the risk neutral valuation of options. 2 See Myers [18], Brennan and Schwartz [5], McDonald and Siegel [16] and the texts of Dixit and Pindyck [10] and Trigeorgis [24]. Baldwin and Clark [3] with Gamba and Fusari [14] motivate and value the real options of project design using six principles of modularity (splitting, substitution, augmenting, excluding, inversion and porting). Latterly these are valued using multidimensional stochastic processes and methods. 3 Including capacity (Pindyck [19]), land (Capozza and Li [6]), costly reversibility (Abel and Eberly [2] and Eberly and van Mieghem [12]), marginal cost of capital (Abel, Dixit, Eberly and Pindyck [1]), information (Lambrecht and Perraudin [15]), capital structure and debt valuation (Sarkar and Zapatero [20]). 4 Simple calls (and puts where the exchange is reversed) in financial markets are sold by option traders to clients for a premium or fee which is designed to cover the hedging strategy and some profit. In return the client enjoys the financial flexibility described. Operational or real options are not explicitly purchased in this manner but are embedded in ownership of any physical asset. 2 or divestment times, i.e. when the options are applied and created, to be tracked. For an option’s value to be maximised by choice of its decision threshold and time, so called smooth pasting (or tangency) must also hold. At an optimal threshold, the value functions of the option and that of the net payoff must converge (value matching) and at that transition point, do so tangentially (smooth pasting). That is the two value functions must meet and join with equal first derivatives there. When flexibility is exercised at a decision threshold, we make use of the intuition embedded in the first order condition. Take for example an American style call option on a stock (i.e. one that can be exercised early). At the critical threshold that represents the optimal exercise point, the derivatives (slope of functions with respect to the stock price) for both the option and its payoff must match. The first derivative of an option price with respect to its stock is known as delta, so this condition says that upon exercise a call option should have a delta of one (unity being the slope of a call). Options also have an elasticity, defined as this delta (or slope) multiplied by the ratio of the stock price to the option price. Although similar to a profit mark-up (as shown by Dixit, Pindyck and Sødal [11]), we interpret the option elasticity as a (CAPM) beta relative to its underlying investment. For instance a call option that is exercised in the money at a critical price to strike ratio of two, will have a relative beta of two.5 Because it contains a 2:1 levered position, this means that the local rate of return (or drift) is double that of the underlying. Thus the option beta (or local rate of return in Shackleton and Sødal [21]) can be used as a metric in optimal exercise timing. When the use of one option generates the creation of another, we apply value and weighted beta matching to account for their interactions at exercise. In addition to equations that track the value before and after each transition, this is done by representing first order conditions including the beta of options weighted by the dollar value of the options. The other element required to complete our analysis, is a discount factor representation of option values.6 This provides both the means to determine betas and also to link option values that are separated by threshold and time. Section two gives examples of how dollar beta matching is useful in the early exercise of American style put and call options, both individually and when they interact in an operational setting. Section three incorporates 5 In the next section a $4 stock price threshold and an option value of $2 commands $4 of stock with $2 of borrowing implicit in the option hedge; this has a relative beta of 2. 6 See Dixit, Pindyck and Sødal [11] and Sødal [23] for the discount factor approach. 3 interactions with a third threshold and mode, using geometric Brownian to illustrate. Section four uses matrix algebra to solve these systems. Section five introduces a fourth threshold and solves a more advanced problem with two way switching. Sections six and seven discuss conditions for maxima and conclusions. 2 Linking slopes and betas at thresholds 2.1 Stand alone options at individual thresholds For an option ( ) its sensitivity to underlying is known as its delta, ∆ ( ) All assets, not just stocks with respect to the market, have a beta; for options this quantity may be dynamic like delta. The beta (or elasticity) ( ), tracks the sensitivity of an option’s percentage change ( ) ( ) to the return of its stochastic underlying driver (assumed to have unit beta). The beta is closely linked to the delta and a third quantity shown in (1). ∆ ( ) = ( ) ( ) = ( ) ( ) ( ) ( ) = ( ) (1) The last quantity in (1) is the dollar beta, i.e. the regular beta scaled by the dollar value ( ) of the option itself; it measures the dollar impact of the beta, i.e. weighted by its value and is equivalent to the delta scaled by the underlying . Since the delta at early exercise of an option has a particular value (often 1 or −1), we use these values with knowledge of the beta and dollar beta at early exercise to infer other unknowns. To quantify the example in the introduction, take an American style call with value7 ( ) on a stock . The delta of this option is ∆ ( ) = ( ) ; multiplying this by the underlying price gives the dollar beta, then dividing this by the option value yields its beta ( ) = ∆ ( ) ( ). Note that an option’s scale plays a role in its dollar beta but not in its beta. Suppose upon the stochastic price reaching the fixed point 4 = $4 we see this option being exercised; since the delta there is unity ∆ (4 ) = 1 the dollar beta at this point is $4. We will assume that the location of thresholds such as 4 are known (or observed).8 7 Option values and betas are labelled ( ) and ( ) but have subscripts that distinguish their type, for calls, for puts and later for idle, for full. Apart from this labelling, in the GBM case constants are also used for their particular beta values. 8 This threshold 4 = 4 is the highest of all those considered; 1 = 1 the lowest, is motivated next and interim 2 = 2 and 3 = 3 in following sections. 4 Suppose we also know that the beta (4 ) at this threshold is 2; it might have been derived from an uncertainty model or measured by regression. From these facts we can infer (4 ) = $2 ¯ 4 ( ) ¯¯ 4 = (2) = (4 ) = 2 = ¯ ( ) =4 (4 ) 2 Typically it is the strike price that is a given and the exercise threshold 4 (4 ) that is determined as a function of 4 , but here treating the threshold 4 = $4 and option beta (4 ) = 2 as given we intend to solve strike price 4 (4 ) (the reverse problem is a numerical task). At this critical underlying price 4 , the call option value converts into a payoff. With a strike labelled 4 (because it is applied at 4 ) and from 4 = $4 we then can solve for 4 (4 ) = $2 along with (4 ) = $2 (4 ) = 4 − 4 = 2 (4 ) (4 ) = 4 = 4 (3) At this threshold, smooth pasting implies that the slope or delta of the option and its payoff match so that the differential of the first line in (3) at 4 confirms this. However the second line of (3) is a dollar beta representation (from (2)), where the beta (4 ) has been applied on the left to ( ) and the unit beta (of ) to on the right. At the point of exercise, these equations say that the beta of ( ) is double that of the underlying and that the option ($4 of stock net of $2 borrowing) has twice the local rate of return (or drift) of in the CAPM world. As well as a call option to capture revenue once potential value reaches 4 there is an analogous put. This operating argument for puts rests on the flexibility to suspend operations from running to the idle condition. For a stand alone American style put ( ) we consider exercise at 1 = $1 Smooth pasting requires a delta at 1 of −1 and (from an uncertainty model or empirically by regression) if we also know the put’s beta is −1, we can infer the put value itself (1 ) = $1 at exercise ¯ 1 ( ) ¯¯ 1 =− (4) =− (1 ) = −1 = ¯ ( ) =1 (1 ) 1 This is consistent with an exercise price 1 = $2 for the put (labelled 1 because it occurs at 1 ; it could be different to 4 e.g. if (1 ) were less than −1) because the payoff and dollar beta (−1) conditions are given in two lines of (5) (1 ) = 1 − 1 = 1 (1 ) (1 ) = −1 = −1 5 (5) The first line of (5) smooth pastes with a delta of ∆ (1 ) = −1 i.e. the put payoff has unit negative slope. The negative beta indicates a put return rate or local drift of the risk free rate less the market risk premium ($2 on deposit with a $1 liability on the short hedge). The second line of (5) is the dollar beta matching condition, and this is written beneath the value matching condition to indicate that (given (1 ) 1 ) it is as important in the determination of 1 (1 ) as the value condition (first line of 5). Thus if the betas and thresholds of options are known or observable, they can be used to infer option prices and strikes at exercise.9 2.2 Follow on options and two thresholds As suggested, the put may not be financial but operational in nature and therefore not stand alone. Now we represent the present value of revenues in and costs in whilst option value represent the flexibility value of launch or closure decisions. Upon its exercise into an idle state (when the operational value is sacrificed in return for retrieving fixed value 1 from a prior investment), it is possible that the call option to relaunch may be acquired. The condition at the put exercise point must then embrace the acquisition of ( ). On the left hand side of the next equation (6), the put value ( ) is added to the present value of perpetual revenue less a present value cost, 1 = − 1 where is the perpetual cost rate of operations saved on suspension and 1 is a one time and irrecoverable cost that diminishes these savings. These sums are foregone upon suspension but a call option to relaunch ( ) is acquired and this goes on the right of an equation which matches value when = 1 i.e. (1 ) + 1 − 1 = (1 ) (6) This differs from (5) because now more flexibility is present. Until the optimal point = 1 values on left and right are dynamic in and equality only holds at 1 . However up to and including that time, the slopes and betas with respect to can be calculated to portray proximity to the first order condition through betas. Smooth pasting is akin to examining slopes on either side of such value equations. Differentiating equation (6) by (in the limit as it reaches 1 ) and multiplying by represents ( ) = ( ) ( ) the dollar beta 9 These beta values, e.g. = 2 and = −1 later become fixed under GBM and carried with constants that match call and put options. 6 of ( ) on either side of (6).10 Therefore as well as (6), if smooth pasting holds, dollar (or value weighted) betas must match at = 1 in (7) (1 ) (1 ) + 11 = (1 ) (1 ) (7) Now compared to (6), the put value ( ) has been weighted by its beta ( ), the value of operations (foregone) by a beta of 1 and the call value ( ) weighted by its beta ( ) Since its beta is zero, 1 has disappeared. For decisions with follow on flexibility, smooth pasting demands rate of return equalisation across value including the option acquired. Optimal exercise of the put requires that the total dollar betas must match at 1 including options acquired , so this call has an influence through ( ) ( ) at 1 . Now suppose that this system were to be closed within the two thresholds mentioned, 1 4 ; this standard hysteresis has been studied before (Dixit [9]). We apply the same logic at 4 where the call is exercised (not as in (3)) but including recovery of the put we have motivated for use at 1 . Value matching is easy to monitor by inclusion of the put on the right hand side of (3), and smooth pasting via the means of dollar beta matching just described (for both options and the payoffs at the boundary, differentiation and scaling) (4 ) = (4 ) + 4 − 4 (4 ) (4 ) = (4 ) (4 ) + 14 (8) Now hysteresis equations (6-8) give four conditions for the six unknowns (4 ) (1 ) (4 ) (1 ) 4 1 as a function of known thresholds 4 1 and assumed betas (4 ) (4 ) (1 ) (1 ) Two more conditions are required, i.e. six in total. The four conditions presented so far apply at transitions; the two extra required equations link between these transitions. They involve a relationship between (1 ) & (4 ) and another between (4 ) & (1 ) For a given scale, the means to link the relative size of option values at two different transition times (i.e. thresholds) is using discount functions. Discount functions are option solutions with a standardised, unit payoff. They also provide the means to determine ( ) etc. and are described in the next subsection. 10 Taking the beta of the each side of (6), would make sense for the right ( ( )) but not for the left hand side ( ( )+ −). The dollar beta operator however produces the sum of dollar components directly (i.e. (1 ) (1 ) + 11 on the left of 7). This is also important if an option subject to knock out can attain a zero value (due to complementarity, in section 5). The dollar beta operator ( ) is robust to this situation. 7 In solving option problems, a stochastic process for is chosen first and then solutions are customised at boundaries that complete the problem. By referring to dollar betas at thresholds 1 etc., for a given process the aim is include beta specifications in boundary conditions (such as 7) alongside those of value (i.e. jointly in 8). By treating value and dollar beta matching as equally important, we place all conditions in a simultaneous system which allows option values (1 ) (1 ) and corresponding 1 to be solved from 1 (1 ) (1 ) etc. This is done by using standardised (unit payoff) functions to separate the form of the option values ( ) and ( ) from their scales (1 ) (1 ) which are measurements at the boundary of ( ) ( ). The process chosen for determines the form of functions ( ) ( ) and these in turn fix ( ) and ( ). From these functional forms, the boundary conditions require certain fixed values (1 ) (1 ) alone and for given 41 these can be treated as knowns in (8) etc. This means that the remaining unknowns (1 ) (1 ) and 1 etc. can be solved using linear algebra. 2.3 Discount functions The exercise time and life of each option depends on an unknown time taken for its state variable to travel from its current value to a given threshold. For time homogeneous problems, valuation can occur using stochastic stopping time methods. Although the time taken for to evolve from 1 to 4 is unknown, expectations can be formed with respect to and its continuous discount − where11 ( ) = 4 In common with Dixit, Pindyck and Sødal [11], we consider the present value of $1 paid at the boundary 4 From the perspective of = 1 ¿ 4 = ( ) this threshold is far away in both terms of and time . Thus the present value of the future sum is much less than $1 but increases12 toward $1 as approaches 4 . For several stochastic processes, the probability distribution of the stopping time can be evaluated and an expected discount factor can be found analytically (see Table 1 in next section) but numerical methods are also possible in other cases. Excluding changes from cash flows (with present values 4 − 4 ), the payoff to timing flexibility at a threshold is taken as the option value. Values 11 To separate them for subscripts indicating fixed thresholds, indices for stopping times () indicate a value of the uncertain variable at time These are usually omitted for the initial (known) level (0) which is abbreviated to (similarly for ). 12 Note that if the boundary is above then the discount factor increases with (call) but it decreases with if the boundary to be hit is below (put). 8 at the end of an option’s life (e.g. (4 ) or (1 )) assume the role of scaling constants and the functional form of the flexibility value is carried in a multiplier which is labelled as a discount ( 4 ) or ( 1 ). Having previously labelled the call ( ) we relabel it ( ) to indicate the flexibility present when operations are in idle mode . The put ( ) is relabelled13 ( ) since it is active when operations are full, hence . Discount functions are labelled with notation to associate put or call with full or idle states. Also since more than one call or put may present, it allows labels rather than to identify the state in which the option is active. Furthermore a third state ( ) is introduced later that contains an option that is neither uniquely a put nor a call but a combination of ( ) and ( ) and which therefore cannot be labelled by or . 2.4 Call option discount For a unit payoff at the random time conditional on current value the valuation of discount 0 ( 4 ) 6 1 is given by the risk neutral expected14 , risk free discounted time when ( ) = 4 ¯ £ ¤ ( 4 ) = − ¯ ( ) = 4 (9) This fraction becomes 1 at the time when reaches 4 and is stochastic up to that point. Applied to a dollar payoff at the stopping time, from a prior point (and time) it gives the present value of that one off future cashflow at an uncertain time . From the perspective of call ( ) creation at 1 the discount function has fixed value (1 4 ) ; the first argument represents the initial time point and the second the final one when the option is used. The first argument is set to dynamic when a beta formula is required but a prior threshold when the call’s initial fraction of its final value is required (both shown in 10) ( ) = ( 4 ) (4 ) (1 ) = (1 4 ) (4 ) (10) In (10), the first interpretation gives the means to separate an option’s scale (4 ) from its dynamics ( 4 ) whilst the second relates two option scaling values. Rather than one scaling constant, two are needed; one each in 13 Their betas too are labelled with For a discrete outcome case, if there were risk neutral chances of 1/3 for the process stopping at = 1 1/3 of = 10 and 1/3 of it never stopping ( → ∞) with risk free = 5% the expected discount factor is (−05 + −05 + 0)3 = 0519 Discount factors can thus adapt to structures where stopping is never certain or where [ ] is unbounded. 14 9 different value matching and dollar beta equations. However one is determined from the other using discounting (e.g. on the right in 10). Although discounting links (1 ) to (4 ) it only provides one third of the necessary conditions. Value equations (first line of 8) join option values of different types at a threshold e.g. (1 ) and (1 ) and another set of conditions apply there too. The dollar betas (in the second line of 8) must match and these equations result from a derivative scaled by (or a beta and value), both of which come from the dynamic discount factor. With option values portrayed via discounting, the end value (4 ) can be treated as a scaling constant. Therefore differentiating the option ( ) or discount function ( 4 ) (in ) is equivalent and gives the same beta function. Since boundary conditions include both value matching and smooth pasting, from (1) the beta at 1 that is important to rate of return matching can be expressed in ( 4 ) (first line of 11) and evaluated at certain thresholds (in the second line where a similar result holds for (4 ) (4 )). 2.5 ( 4 ) ( ) = = ( ) ( ) ( 4 ) ¯ ( 4 ) ¯¯ (1 ) = ¯ ( 4 ) =1 (11) Put option discount The put depends on a discount factor that increases if falls toward a lower threshold 1 Its discount factor ( 1 ) applies to $1 paid at a random time occurring when falls to 1 ¯ £ ¤ ( 1 ) = − ¯ ( ) = 1 (12) From this representation the put value function ( ) is given as a dynamic fraction of its final value or for (4 ) a static fraction of (1 ) ( ) = ( 1 ) (1 ) (4 ) = (4 1 ) (1 ) (13) The put value and discount functions have negative slope and beta in . Since ( 1 ) 0 differentiating ( ) by and scaling by gives a negative beta; the beta at 1 is necessary for rate of return matching at the point of put usage 1 ( ) ( 1 ) = = ( ) ( ) ( 1 ) ¯ ( 1 ) ¯¯ (1 ) = ¯ ( 1 ) =1 10 (14) To reiterate, discount functions fulfil two useful roles; first they capture the dynamics of values and their betas and second they fix the ratios and scales of options at the time of their creation compared to their use. Even though discount factors do not appear in the value matching condition, their first property is used at that instant in smooth pasting or dollar beta matching (8). Although defined by 4 1 the flexibility system described so far could not have been solved without the discount factors; these provided both the means to determine the betas and also to link beginning and end options.15 Before solving (in section four) we include another possible operating state and intervention point 2 . In order to do this, a third cashflow mode is included. Whilst this is easiest to incorporate using a Geometric Brownian motion, the results of this paper also pertain for other stochastic processes. The advantage of GBM is that the claims ( ) are iso-elastic, i.e. the betas are constant for all . Even if betas are not constant but depend on the diffusion dynamics and the solution to its asset pricing equation provide the means to calculate particular (1 ) and (1 ) etc. 3 Three threshold GBM example So far we have not yet utilised a specific diffusion. We will do this now for a stochastic flow and value that follows a Brownian motion geometrically = = ( − ) + (15) Under this risk neutral diffusion, if the revenue flow has drift − (where is the risk free rate and the yield) = follows the same diffusion.16 This is convenient for a third cashflow mode. 15 Equations (6, 7, 8, 8, 10, 13) represent classical hysteresis across two operational modes, full and idle . Two known thresholds 1 4 the fixed discount factors between them (4 1 ) (1 4 ) (from 11, 14) and the option betas at these points (1 ) (4 ) (1 ) (4 ) (which depend on the process rather than option constants) determine the values of (4 ) (1 ) (4 ) (1 ) and the two strike prices 14 The system has six unknowns and due to the discounting equations, six equations in total; two value matching, two weighted beta matching and two discounting. However, rather than stop here, the flexibility system is first expanded to include a third and later a fourth state operating between overall hysteresis levels 1 and 4 . 16 = Z 0 ∞ [ ()] − = Z ∞ 0 11 h i∞ − = − − = 0 Claims such as discount factors ( ) that are contingent (on reaching certain levels) must satisfy a Bellman equation. This condition is derived from the risk neutral expectation of a local change generating a risk free rate of return over an interval . Applying expectations to the Ito’s lemma expansion17 , we require [( )] = ( ) For a discount claim ( ) that depends on the proximity of to a boundary, an asset pricing equation generates option solutions ( ) ∝ and beta constants = (for the call) or = (for the put) given by ( ) 1 2 2 2 ( ) − ( ) + ( − ) 2 2 s µ ¶2 − 1 1 − 2 − 2 ± = − + 2 2 2 2 0 = (16) This says that the discount factors ( ) ∝ are iso—elastic claims, either depending on a positive or negative power (beta) = 1 or = 0 Their dollar beta is a constant multiple of their value i.e. ( ) = ( ). These claims must also satisfy no—bubble conditions ( ( ) → 0 as → 0 and ( ) → 0 as → ∞); these match to the idle state (the call) and (the put) to . Both discount factors reach a unit value, but at their own boundary; (4 4 ) = 1 and (1 1 ) = 1 These complete the specification of standard unit discount factor claims with their constant betas idle, call 6 4 ( 4 ) = (4 ) full, put > 1 ( 1 ) = (1 ) (17) These forms appear in real options paper with GBM processes; they can be combined to form other discount factors that satisfy boundary conditions at both of 41 (in section 5, terminology labels these discounts that depend on ). 3.1 Other processes and discount factors The same procedure can be applied to other stochastic processes who’s discount factors are available; Table 1 shows standardised discount factors for 17 Unlike risk neutral, using physical or real world (CAPM) expectations [] to determine local expected returns of generates a beta dependent risk premium, i.e. [ ( )] ( ) = ( + ( ) ∗ risk premium) However, for the isoelastic GBM, discounting can occur using CAPM expecations (see Shackleton and Wojakowski [22]). 12 RN diffusion ( − ) + Call/put discount functions ( 4 ) = ( 4 ) : 6 4 ( 1 ) = ( 1 ) : > 1 ( 4 ) = (−4 ) : 6 4 ( 1) = (−1 ) : > 1 (−2+2+22 2 2 ) 1 ( 1 ) = −2+2+22 2 ( 1 2 ) + ( − ) + quadratic for 2 ( − 1) + 2 ( − ) − = 0 2 2 + 2 − = 0 2 ( − 1) 2 − − = 0 Table 1: Discount factors for Geometric, Arithmetic and Mean Reverting processes. Arithmetic and Mean Reverting flows along with the GBM claims just discussed. Whilst these processes have similar solutions to a fundamental quadratic, unlike GBM they are not iso-elastic (constant beta). However the techniques developed here are robust for processes other than GBM (e.g. the mean reversion process used in Sarkar and Zapatero [20]). 3.2 Power flow For a different function of the cashflow driver (which still follows (15)), we now motivate the present value of a third cashflow mode. Before the full state with cashflow and value is launched, suppose that there is operational flexibility to engage with the power of underlying cashflow . This might arise for instance when a test market is possible during which time the economies to scale are different to those in full operations For a GBM flow, an expectation of () (a uncertain cashflow years in the future raised to power (0 1)) is also proportional to the power of the current value ; i.e. [ () ] ∝ Like that of the full flow ( = ), the perpetuity value for a power of cashflows (proportional to ) has a fixed dividend yield18 with respect to its 18 The yield 0 on a perpetuity of the power cashflow arises through Z ∞ − [ () ] = 0 Z 0 0 ∞ " #∞ − 0 1 2 − (−1)+(−)− ( ) 2 = = 0 0 1 = 0 where 0 = − ( − ) − 2 ( − 1) 2 Note that 0 1 implies that 0 lies between and all are positive. The PV of power flows 0 would revert to for = 1 or to a constant 1 for = 0 Finally, note that the beta of the present values of the power of these flows with respect to the driver is = . Although this does not affect the beta of the options themselves, due to the presence of a cashflow with value it is needed in smooth 13 flow Before going to full value , we thus compare a mode of operations engaging in a cashflow with value . Since this occurs with scales to return we label this power state gamma with subscript Options whilst in this mode are labelled ( ) This mode is beneficial for ¿ 1 but detrimental for À 1 so the new cashflow pattern and its present value are scaled to align at a value of = $1 (i.e. when = $1 then the present value of the power cashflow and full cashflow are equal = at = $1) Thus = $1 is also a very rough (without option value) indicator of the desired switching level, the actual switching level is designated 2 so that the call from idle19 operates from 1 to 2 and the call within the new gamma state from 2 to 4 3.3 Three policy points; 1 4 and 2 If 1 has already been encountered and switching out of the full state has occurred, we assess the conditions under which is optimal to switch from idle (zero cashflow but with non—zero call value) to cashflow with value by incurring a PV cost of 2 (the strike, which will be solved for, must be aligned to a perpetual cost rate and a one time cost 2 ). We label option values in the power state ( ) operations continue in this mode after reaches 2 until 4 (when switching to full occurs ( )). Thus the key separation for ( ) in section three is 2 4 . To summarise, rather than waiting to go full at 4 we assume the first call ( ) only takes operations into the power mode described, (cashflow and value ) at 2 . Having started this power mode at 2 we presume continues to 4 ( 2 ) whereupon it is possible to exercise a second call into full production . At 4 this would cause to be relinquished but to be gained when ( ) is used to claim the full flow along with its put (back to idle again). Thus compared to section 2, the call has been split into two separate calls, ( ) and ( ) (hence the need for notation other than ( )). Although we label the value of the second call option in the gamma cashflow mode ( ) this does not mean that ( ) has dependency on ( ( ) 6= nor do options in this state depend on ). If the uncertainty driver is the same and process (15) throughout then ( ) will be another straight call ( ) with beta also given by from (16); i.e. under GBM a call on still has the same beta as a call on .20 pasting and dollar beta matching. 19 In this case, (10) needs revising to (1 ) = (1 2 ) (2 ) 20 That is to say that claims () can either be formulated as functions of or (say where) = Under GBM, ( ) and ( ( )) generate asset pricing equations with 14 Using the discount framework, this extra call also has beta and is used to relate its dynamic or initial value to final (4 ) through = ( ) = ( 4 ) (4 ) (2 ) = (2 4 ) (4 ) (18) When the first call from idle ( ) is used to get into power mode at 2 , the present value of the power cashflow is acquired along with the second call ( ) (to go from power to full mode). If this operational transition requires a present value cost 2 = + 2 (to be solved) then the value matching condition at 2 reflects use of the call from idle into an operational value net of costs plus the option in the next (power) state (2 ) = (2 ) + 2 − 2 (19) In addition to the six conditions in standard hysteresis, note that another two option values and another strike 2 have been introduced. However (18 & 19) gives two of the extra three conditions required maintain solvability. Another condition is given by weighted beta matching at 2 this says that not only must the values either side of (19) match but to do so smoothly i.e. with dollar betas matching too. The betas are known because the option function takes the same form in this region as idle . Taking value matching equation (19) and applying the dollar beta operator (differentiate with respect to and scale by ; alternatively evaluate the beta and apply that to the option value) gives (2 ) = (2 ) + 2 (20) Now we close the cycle in this section; allowing transfer from power mode to full i.e. from present value of revenue to by incurring or saving some cost/benefit 4 . Thus the present value cost 4 must allow for the addition of , the saving of net of a another one time cost or friction 4 ; the net cost being 4 = 4 + − When the power call is used to exit the power mode and enter the full mode this generates a value condition (21). This happens at = 4 when the call exercises into full mode, upon which 4 (the present value of the power flow) is lost (left hand side) along with (4 ) but 4 the full present value, is gained (right hand side) (4 ) + 4 = (4 ) + 4 − 4 (21) 0 betas 0 that are different, but scaled by Their solutions ( ) ∝ and () ∝ 0 0 are equivalent because = 0 ensures that = = This is true for calls or puts. 15 In equation (21), items on the left represent the power state, including the option and net present values of the cashflows there; on the right the full state contains the next option ( ) a put, net present values of cashflows there less the one time friction 4 (this is embedded in 4 and causes the transition to be irreversible). In (21), all fixed present values have been consolidated on the right into one sum 4 . This equation too must match by dollar beta (again where 4 disappears) and since = and = we get (22) (4 ) + 4 = (4 ) + 4 (22) The option (4 ) is then valued as a function of its discounted payoff, the discount factor requiring betas at 41 (4 ) = (4 1 ) (1 ) = (23) Now we have three value matching equations, three discountings, and three weighted beta matchings.21 Given thresholds 1 2 4 present value cashflows 4 etc. and discount factors () (used twice), () and betas, the three value equations, three discounting equations and three weighted beta equations can all be solved for the following variables (1 ) (2 ) (2 ) (4 ) (1 ) (4 ) and 1 2 4 These are most easily summarised after describing them in a network or graph and its associated matrix. 3.4 The three level problem These option value can be placed in a graph or network as in Table 2. The underlying value of full operations is mapped onto the horizontal axis, whilst the vertical axis increases with values . In particular it is important to track the passage of toward each of the three thresholds 421 which are marked and the investment jumps 421 that occur there (upward when calling, incurring a cost and downward when putting, realising a saving). From the full state (in the top row of the network) when reaches 1 from above i.e. moving from the right to the left, full operations are suspended with loss of present value revenue 1 = 1 but also with a saving of present value of operating and transition costs 1 = − 1 In the idle state (on the bottom row), no operational cashflow is present; however there is the call option ( ) there. This is gained at 1 with value (1 ) and is used at 2 where its value is (2 ) 21 From within equations (10 (as amended in footnote 19), 18, 23 for discounting), (7, 20, 22 for dollar betas) and (6, 19, 21 for value matching). 16 (1 ) + 1 ← (4 ) + 4 ↑ −4 (4 ) + 4 Full flow (2 ) + 2 → Power flow ↑ −2 (1 ) → (2 ) Idle no flow 1 threshold 2 threshold 4 threshold ↓ +1 Table 2: Investment graph at three thresholds (horizontal) for three state system (vertical, idle power full). Value matching at investment occurs vertically, diffusion and discounting occurs horizontally. Due to required investment 2 exercise of the call option from idle, lifts value upward. This represents the present value of running costs in the gamma state as well as transition costs 2 i.e. 2 = + 2 . In return for sacrificing (2 ) and committing this present value cost, the call in the power state (2 ) is gained along with the present value of operations in the power mode 2 This cycle completes with the power state (middle row); at 4 this second call option is used with value (4 ). This is combined with a third transition cost 4 which now comprises three items; first the cost saving of stopping power operations secondly the present value cost of engaging in full operations − and finally another one off transaction cost −4 so that 4 = + 4 − (to be solved). Along with the acquisition of the put option in the full state, (4 ) this completes the augmented hysteresis cycle in this section (Ekern [13] presents another extension to switching hysteresis, one with a finite number of transitions that would admit matrix solutions). To reiterate, normally 421 and would have been specified by the characteristics of the assets and their flexibility, but here for given thresholds 421 and betas we will solve them through 421 . The total value at each stage is given by taking the value of these options and adding the present value of operations in all states, i.e. ( ) ( )+ or ( ) + . The vertical dimension is augmented with quantities 421 which determine their separation (this is apparent in the Figure in Section 5 which displays solved and interim values for a four level system). 4 Solving with vectors and matrices With many option constants to consider (this method can encompass arbitrarily large systems and is generalised to a fourth threshold in Section 5), it 17 is useful to collect similar items into vectors and then use matrix and vector equations to represent their linkages. Given the three discount equations, rather than using one common vector of values (say V) a more sensible grouping is to separate option values into one of two vectors U or W Option values at the beginning of their life (on the | right hand side of a value equation) are put into U = [ (4 ) (2 ) (1 )] and22 those at the end of their life (left hand side of a value transition equa| tion) W = [ (4 ) (2 ) (1 )] This allows the value of options at their birth to be expressed succinctly as a matrix multiplication D of values at their use, i.e. the relationship between beginning (left U in 24) of their life compared to the end (right W in 24) U ⎤ = ⎡ D ⎤ ⎡ W ⎤ (4 ) (4 ) 0 0 (4 1 ) ⎦ ⎣ (2 ) ⎦ ⎣ (2 ) ⎦ = ⎣ (2 4 ) 0 0 (1 ) 0 (1 2 ) (1 ) 0 (24) ⎡ Note that the call discount factor () has been used twice, between intervals 1 2 (idle ) and 2 4 (gamma ) whilst with only one put present (in the full state ), () appears once (between 4 and 1 ). Two other vectors capture the present values of operational cash flows before (i.e. preceding on the left of the value transition (25)) option exercise | Z = [4 0 1 ] and after (proceeding, i.e. on the right) the transition at | each level Y = [4 2 0] . These flows are treated as potentially perpetual in nature but their life will depend upon the potential exercise and timing of the next option (when terminated the loss of the flow is accounted by deduction at the next exercise threshold). Thus the present values of cost rates and option strikes are also grouped into vectors, one for the present value of costs in the preceding state K = | [ 0 ] one for the present value of cost in the proceeding state | K = [ − 0] and finally one for the frictions borne at the exer| cise time K =[4 2 1 ] 0 It is these final frictions that ensure that each transition is irreversible (as elements of K go to zero then reversibility could appear). Collectively in (25), these vectors represent all three value matching equa22 The transpose of a vector is indicated by | . 18 tions, one line for each threshold W ⎤ + ⎡ Z − K ⎤ = ⎡ U ⎤ + ⎡ Y − K − K ⎤ (4 ) 4 − (4 ) 4 − − 4 ⎣ (2 ) ⎦ + ⎣ ⎦ = ⎣ (2 ) ⎦ + ⎣ 2 − − 2 ⎦ 0 1 − (1 ) (1 ) −1 (25) This can be made more compact by grouping all fixed (zero beta) costs into one vector containing 421 ; i.e. X = K + K −K to be subtracted | from value after exercise. This X = [4 2 1 ] has positive strike elements for calls and negative for puts. Furthermore subtracting Z from both sides means that the value of options in the preceding state W can be expressed as the sum of the new, proceeding options U, plus a cash flow difference between modes (Y − Z) less a net present value of costs −X Most succinctly we can use the expression for W with U = DW and inputs D Y Z X to solve for W as an optimisation across 421 ⎡ W = U+Y−Z−X max W = [I − D]−1 [Y − Z − X] (26) 421 That is to say, if all costs and rates (K etc.) in X were given then W = U + Y − Z − X can be used with U = DW to eliminate U (note that I is the identity matrix, three by three in this case). Then the end of life option values in W would be found by a maximisation problem in 421 . For fixed X (and K’s) this is a search procedure in three dimensions 421 . However if we assume 421 with the first order conditions, we can calculate W U X. Since vectors Z Y and matrix D are all given as functions of 421 taking these inputs as optimal, we can calculate U W X directly. To do this the dollar betas of operational flows are also required; these are | | given by β Z = [4 0 1 ] and β Y = [4 2 0] which are combined | into β Y − β Z = [4 − 4 2 −1 ] and used in (27) for dollar beta matching.23 Having Z on the right hand side and differentiating the items of (27) line by line in and then multiplying by gives the dollar betas both sides 23 Although only the vector β Y − β Z is required, matrices β β are given by ⎡ ⎤ ⎡ ⎤ 1 0 0 0 0 β = ⎣ 0 0 ⎦ β = ⎣ 0 1 0 ⎦ 0 0 1 0 0 1 19 of a value transition. This is succinctly expressed using two diagonal beta matrices β β defined in (27) β ⎤ ⎡ U β ⎤ ⎡ W ⎤ = ⎡ ⎤ + ⎡β Y − β Z⎤ 0 0 (4 ) 0 0 (4 ) 4 − 4 ⎣ 0 0 ⎦ ⎣ (2 ) ⎦ = ⎣ 0 0 ⎦ ⎣ (2 ) ⎦ + ⎣ ⎦ 2 0 0 (1 ) (1 ) −1 0 0 (27) In order to separate the option constants in W U X from their (known) betas, matrices β β were used. These diagonal matrices operate on the vectors of option values to weight them with their betas. They are evaluated using the option beta at each threshold; the flexibility options are iso-elastic so they contain either or ( and are both calls with the same function and power 1 and is a put with power 0 with factor on ). ⎡ 4.1 Solution In particular, note that the beta matching and discounting equations do not involve the transition costs X This is another reason why it is easier to solve for those at the end. Direct elimination from (24 & 27) gives W U that are consistent with smooth pasting i.e. dollar beta matching W = [βW − βU D]−1 (β Y − β Z) U = D [βW − βU D]−1 (β Y − β Z) X = U − W + Y − Z (28) In the first two lines of (28), an inverse of a matrix combination is applied to a beta weighted difference (in brackets). The expression Y − Z is a difference of present values of revenues before and after each transition (one in each row), however when weighted by their betas β Y − β Z it is akin to a rate of return on this difference and thus acts like a flow in the definitions of W U The matrix β −β D reflects the net beta of the change in flexibility value, β for the immediate W and β D in anticipation of the demise of U at the end of the next stopping time. Thus option solutions W U can be thought of as aperiodic perpetuities on beta weighted cashflows β Y − β Z with factor [βW − βU D]−1 That is to say that their benefit is linked to the cashflow gain carried by β Y − β Z which comes along with random future timing reflected in a perpetuity factor (matrix inverse). As the system cycles through episodes, the expected timing is carried though this inverse [β − β D]−1 . Although idle, power and full states last for different lengths of time, the matrices carry the expected timings and present value of each through the discount factors embedded in D. 20 (1 ) + 1 ↓ +1 (1 ) 1 thresh. ← (2 ) + 2 ↑ −2 → (2 ) 2 thresh. (3 ) + 3 ↓ +3 ↔ (3 ) + 3 3 thresh. ← (4 ) + 4 ↑ −4 → (4 ) + 4 4 thresh. Table 3: Investment graph with four thresholds (horizontal) and three states (vertical). Value matching at investment occurs vertically, diffusion and discounting occurs horizontally. This method can now be extended to encompass two way flexibility with four thresholds. 5 Four thresholds & two way flexibility In this section we illustrate a more complex investment scenario involving the same interim (power) state Now as well as the chance to invest further if things go well there is also the chance to recoup and partially divest from full by rejoining rather than if the situation worsens. This splits the put into two, creating a high level put join and a second lower level put now embedded in (which becomes more complex as a result). The network of investment transitions, is given in Table 3 which is similar to Table 2 other than retreat (like advance) has two stages. The middle stage has reversion on the left (low = 1 rather than the full) and the full state has reversion to the state at an additional level 3 (upward moves occur at even thresholds 24 and downward odd 13 ).24 Since the power state no longer has a unique direction of propagation (up, i.e. a call as in section 3), first we need to construct new appropriate discounts, ones that can accommodate two different diffusion destinations. This is done with discount factors that are subject to knockout features (i.e. where a claim value of zero is attainable). These are combinations of fundamental the simple one way factors already used. These are best explained using growth factors which we describe first. 24 Note that horizontal arrows ↔ indicate diffusions these take time to transit and involve possible movement to left or right in . Vertical arrows ↑ ↓ indicate either investment or divestment and occur instantaneously at a threshold; these are not reversible other than by further forward passage through the network and its graph. 21 5.1 Growth factors and their betas It is more convenient from now on, to use compact notation for elements that are static and non—dynamic, i.e. to use 12 rather than (1 2 ) and 43 rather than (4 3 ) also 4 rather than (4 ) etc. However ( 4 ) and ( ) will be maintained for quantities that are dynamic in and not fixed at a threshold. The two simplest options in Table 3 are the straight call (bottom row) and put (top row) that can be discounted easily and whose dollar betas are already known (these form part of the D matrix required in this section) ∙ ¸ ∙ ¸∙ ¸ ∙ ¸ ∙ ¸∙ ¸ 4 43 3 4 4 43 3 0 0 = = 0 12 0 12 1 2 1 1 2 The first (discount) matrix can be inverted to form a growth factor which expresses end values 3 2 as a function of 4 1 at the beginning (this forms part of a growth matrix G used in 42) ¸ ∙ ¸∙ ¸ ∙ ¸∙ ¸ ∙ 0 0 (43 )−1 4 34 4 3 = = (29) 0 21 2 1 1 0 (12 )−1 This naturally suggests that the inverse discount can be used as a growth factor in relationships such as 3 = (43 )−1 4 = 34 4 where it is now understood that the put growth factor 34 1 from 4 to 3 is the inverse of the discount factor from 4 to 3 Thus the subscripts fixing the thresholds, not only determine which thresholds are under scrutiny but also whether represents a discount or a growth.25 Due to the definition of the beta, growth factors have the same absolute elasticity as their discounts, but with a different sign. 5.2 Top & bottom complementary discounts For the values in Table 3 not yet identified, we require discount factors which have a unit payment at one threshold but are subject to a knockout possibility, becoming strictly worthless at a second attainable threshold. 25 For GBM the call growth factor is 21 = (2 1 ) 1 Not only do we assume that (12 )−1 = 21 i.e. that inverse discount factors are growth factors with the same functional form applied to a growth ratio 2 1 , but also implied are relationships which chain discounts (or growths) together across intermediate thresholds i.e. 12 23 = 13 etc. This is possible due to concatenation of two stopping times; one from 1 to 2 and the other from 2 to 3. Note that have different elasticities so these discount/growth and pass through relationships only hold within common option type and 12 23 cannot be represented as a single discount. 22 In this section, starting from = 3 there are two conversion boundaries of interest 4 3 and 1 3 For conversion at the top i.e. 4 exercise is completed but at 1 flexibility at the top becomes worthless because another conversion occurs there instead. The complementary option used at the bottom causes to be knocked out for zero value. Since is used at the top of the flexible power state, we use notation , also three arguments are required ( 4 1 ) At the bottom ( 1 4 ) is used; this represents a discount factor that knocks in at the bottom but out at the top. Normalised for unit and zero payoffs, in static notation these claims are subject to payoff or boundary conditions. ¸ ∙ ¸ ∙ 1 0 441 414 = (30) 0 1 141 114 For dynamic the first argument in ( 4 1 ) is the current level of the dynamic variable the second a level (above or below ) at which the discount factor becomes one and the third an attainable level at which it becomes zero. Formally these claims are risk neutral expectations (conditioning on current ) but subject to an extra condition on acceptable paths [ ( 4 1 ) ( 1 4 )] (31) ¯ £ − ¯ ¤ − ¯ ¯ ( ) = 4 min 1 ( ) = 1 max 4 = ¸ ∙ [ ( 4 ) ( 1 )] 1 −41 = −14 1 1 − 14 41 We already have two claims ( 4 ) and ( 1 ) (straight call and put discount factors without knockouts) that satisfy single conditions. For 4 > > 1 the claims ( ) ( ) are fixed linear combinations of these that reflect two pathways. For the direct (or successful) path is via a call from to 4 but the indirect (and unsuccessful) path is from to 1 under the put multiplied by the path from 1 to 4 with the call; this must be subtracted (see Darling and Siegert [8] for complementary stopping times, similarly for the direct put path is from to 1 but the indirect call path from via 4 then a put from 1 to 4 must be subtracted). In their valuation, both routes must be scaled (by a matrix determinant) to account for the discounted round trip, i.e. call from 1 to 4 times put from 4 to 1 . Equations in (31), say that a discount factor that pays off $1 at 4 but knocks out and becomes worthless at 1 is a long position (with magnitude (1 − 14 41 )−1 ) in the call ( ) and a short position (14 (1 − 14 41 )−1 ) in the put ( ) 23 As increases, the value of the straight call increases and that of the straight put decreases so ( ) increases in value until it reaches 1 when = 4 . Conversely as decreases, the call decreases and the put increases its short contribution, thus is monotonic in . As reaches the knockout level 1 even though the straight call still has some residual value (14 ) this is eliminated by the negative element from the straight put and the whole claim becomes worthless (for similar reasons, ( ) is monotonic and decreasing in achieving 0,1 at 41 ) 5.3 Dollar betas of and Since the new discount factors are a linear combinations of the straight put and call, their slopes at any point are too. Rescaling the slopes by gives (33) the dollar weight of their betas ( ) ( 4 1 ) and ( ) ( 1 4 ) [ ( ) ( 4 1 ) ( ) ( 1 4 )] ∙ ¸ 1 1 −41 (32) = [ ( 4 ) ( 1 )] −14 1 1 − 14 41 Even under GBM, these are dynamic in ; although the component betas are fixed, the weights ( 4 ) and ( 1 ) change with and in (32) the dollar beta of claims ( ) are not constant weights of and . Furthermore, the top call with knock out is more levered than the straight call having a higher slope and dollar beta ( has a more negative beta than ). In order to value the two way flexible state ( ) and its dollar beta, we develop equations (37) and (39) next; these will need evaluation at all thresholds 4321 . 5.4 Two way flexibility ( ) Within the power state, knock out options allow flexibility to have one of two possible uses; at 1 a value of (1 ) may be realised but at 4 a different value (4 ) will result (from a perspective within 1 4 the option value ( ) at may be less than (1 ) or (4 ) i.e. it is a combination of both put and call through ). From the perspective of a dynamic point from within this range, the current expected value of flexibility is a probabilistic and discount weighted sum of two options, only one of which will be attained ( ) = ( 4 1 ) 4 + ( 1 4 ) 1 24 (33) This dynamic version of the flex value in the power state carries dependency on through ( 4 1 ) and ( 1 4 ) which are the dynamic discount factors subject to knockouts. The payoffs 4 1 at 41 are static. Having determined that the option in the power state has two possible outcomes at two end thresholds, we can focus on the values at the beginning of the option’s life i.e. 3 2 . Since these are static we also represent them with non dynamic notation for items occurring at fixed thresholds, e.g. (2 4 1 ) = 241 etc. We know that the ( ) option is acquired at either of the thresholds 2 or 3 so can specialise the dynamic equation (33) to those two particular values of Using compressed notation the option value in the interior (3 2 ) of the power state can be expressed as a function of the extreme values (4 1 ) ∙ ¸ ∙ ¸∙ ¸ 3 341 314 4 = (34) 2 241 214 1 For 241 in the second row, the first threshold in the subscripts is the level there e.g. = 2 , the second 4 the target or call level and the third e.g. 1 the level at which the flexibility dies or is knocked out. The complementary option in the second row 214 does the opposite (put at 1 but knock out at 4). In order to work out how 41 depend on 32 this last equation must be inverted to produce growth factors comparable to (29). This is easiest to do by decomposing the matrix (34) using the following ∙ ¸ ∙ ¸∙ ¸ 1 341 314 34 31 1 −41 = (35) 241 214 24 21 −14 1 1 − 14 41 The inverses of both matrices are required and applied in reverse order ∙ ¸−1 ¸ ∙ 1 1 41 1 −41 = (36) −14 1 1 1 − 14 41 14 ¸−1 ∙ ¸ ∙ 1 21 −31 34 31 = 24 21 34 21 − 24 31 −24 34 Now 4 1 can be expressed in 2 3 ∙ ¸ ∙ ¸∙ ¸ ¸∙ 1 4 1 41 3 21 −31 = 1 14 1 −24 34 2 34 21 − 24 31 (37) This can be simplified back to a similar form to the standard discounts subject to knock out. Multiplying the denominator in the determinant’s fraction 25 by a growth factor 43 12 (each row of the last equation must also be multiplied by 43 12 to compensate) and using discount and growth concatenation yields a round trip determinant between 32 (resting on 1 − 23 32 = (34 21 − 24 31 ) 43 12 and the production of a new determinant). Applying these and evaluating the matrix product yields the simplification (38) ¸ ∙ ¸∙ ¸ ∙ 1 43 − 42 23 42 − 43 32 3 4 = (38) 1 2 1 − 23 32 13 − 12 23 12 − 13 32 ∙ ¸∙ ¸ 432 423 3 = 132 123 2 Thus not only are two way discount factors easily expressed as a linear combination of standard put and call discounts, but the inverse or growth functions are too26 . The growth elements reflect the value at extreme points 41 as a function of starting from one or other of the interior points 32 , but for these end values the weights are fixed as a function of the internal points (and their round trip determinant (1 − 23 32 )−1 ). The resultant growth matrix contains both positive and negative quantities; these result from the long and short positions present in the knock out discount factors. This formulation also holds true for dynamic ( ) i.e. any within the region, from which the dynamic dollar beta ( ) ( ) is easy to obtain from (38) and apply to 43 ∙ ¸ ∙ ¸∙ ¸ 1 4 4 43 − 42 23 42 − 43 32 3 = 1 1 2 1 − 23 32 13 − 12 23 12 − 13 32 (39) The weighted betas of 4 1 and 3 2 are necessary for smooth pasting and dollar beta matching, the inversion presented here is useful because it allows both the discount and its corresponding growth matrix to be represent dollar betas for matching. Before doing this, the ladder network is formalised. 5.5 A four level ladder network across three stages We can now formalise the system with flexibility to ratchet up or down a “ladder” with at least three investment states; idle full and intermediate 26 In the bottom left corner of the last matrix, 132 would normally imply (for a discount) achieving 3 from a start point of 1 without hitting 2 Because 1 cannot grow to 3 without hitting 2 a similar and literal interpretation is hampered for the growth factor by the relative levels which is why the growth matrix contains negative quantities. 26 power state with elasticity of an underlying flow.27 We use the same linear algebra but vary the matrix and vector contents in order to examine the extended flexibility setup from Table 3. To recap, operational value is again nil in the idle state, varies with in the power state and full (power 1) in the highest state. i.e. non-flex values varying with level 0 and . Starting with the transition at the highest threshold, from power to full, one control occurs at threshold 4 with a payoff net of exercise cost of 4 − 4 − 4 . That is to say at the top threshold on going to full, 4 is gained but 4 is lost along with incremental investment cost 4 Reversion can occur at 3 yielding 3 −3 +3 a reverse payoff including a partial return of fixed investment cost 3 4 (this imposes a restriction on 3 4 ). After this two outcomes are possible. Similar transitions occur at 2 into the power state from the idle and 1 back to the idle from the power. These last two have investment strikes of 2 and 1 (1 2 ) The value matching conditions within this system are shown in (40), with contents specific to this section in Table 3 ⎡ W ⎤ = ⎡ U 4 4 ⎢ 3 ⎢ 3 ⎥ ⎢ ⎢ ⎥ ⎣ 2 ⎦ = ⎣ 2 1 1 ⎤ + ⎡ Y − Z 4 − 4 ⎢ 3 − 3 ⎥ ⎥ + ⎢ ⎣ ⎦ 2 −1 ⎤ − ⎡ X − + 4 ⎢ ⎥ ⎥ − ⎢ − + 3 ⎣ ⎦ + 2 − + 1 ⎤ ⎥ ⎥ ⎦ (40) The network depicted in Table 3 is a mathematical graph, furthermore it is bipartite and directed.28 Note that with two new possible outcomes from the power state, branching of pathways is now possible. From the power state, since reversion to the idle state is possible (at 1 ) as well as elevation to the full state (at 4 ), the discount matrix is populated with six elements, and in particular two rows now contain complementary discount factors with mutual exclusivity, i.e. conditional upon each other not paying off. For example in row three (within the power state) the element 341 represents from the point of view of threshold 3 the discounted value of the chance of paying a dollar at 4 knowing that if 1 is reached the 27 More ladder regions can be considered using different 1 2 and more regions etc. Note that unless two thresholds converge, the up (even) and down (odd) rungs of this ladder are different. However reversibility can be achieved by letting two thresholds become arbitrarily close. 28 Bipartite (see Wilson [25]) in that investment nodes are either beginning U or end W of state and directed in the sense that W maps forward (one to one) to U upon value matching and smooth pasting/dollar beta. Beginning of state values U are linked to W (one to two) by discounting but this also has a backward growth interpretation. 27 opportunity dies (zero value). The second 314 in row two is complementary and pays off at 1 assuming 4 is not hit. Other discount factors are interpreted similarly but simple 43 12 are one way factors in full and idle states respectively. The detailed discounting equations U = DW are thus given in (42) which contains equation (34) in rows two and three D ⎡ U ⎤ = ⎡ 0 43 4 0 0 ⎢ 341 ⎢ 3 ⎥ 0 0 314 ⎢ ⎢ ⎥ ⎣ 2 ⎦ = ⎣ 241 0 0 214 1 0 0 12 0 ⎤ ⎡ W ⎤ 4 ⎥ ⎢ 3 ⎥ ⎥ ⎥ ⎢ ⎦ ⎣ 2 ⎦ 1 G ⎡ W ⎤ = ⎡ 0 432 423 4 0 ⎢ 34 ⎢ 3 ⎥ 0 0 0 ⎢ ⎢ ⎥ ⎣ 2 ⎦ = ⎣ 0 0 0 21 1 0 132 123 0 ⎤ ⎡ U ⎤ 4 ⎥ ⎢ 3 ⎥ ⎥ ⎥ ⎢ ⎦ ⎣ 2 ⎦ 1 (41) We also summarise the inverse of D as G = D−1 (who’s elements, which can be greater than one and positive or negative, are drawn from (29), (38)). This is a matrix that values W as a function of U 5.6 (42) Example Now we have assembled two thirds of the equations required to solve the four level system; in particular from the value matching condition (40), on the right we can represent U as a discounted version of W and on the left we can represent W as a discounted version of U This means that line by line implicit differentiation and scaling () | =41 is carried in the D G matrices alone. These dollar beta components are denoted with primes e.g. D0 , indicating that they have been implicitly differentiated by and scaled by at the boundary in each row.29 These comprise the matrices required to form β and β from first principles (rather than adhoc in (27)) GU = DW + Y − Z − X ⇒ G0 U = D0 W + (Y − Z)0 (43) Finally to retrieve the beta matrices that we desire, the forward and back projections can be unwound for U W to produce a dollar beta matching set 29 To form dollar betas, this says perform a line by line scaled differentiation ()0 of U = DW (and W = GU). Since D and D0 (G G0 ) have no diagonal elements the operation is carried in the first matrix, i.e. U0 = (DW)0 = D0 W and dependence on in line is carried in D etc. alone. These need rescaling by value to form betas. 28 of equations G0 DW = D0 GU + (Y − Z)0 β W = β U + (β Y − β Z) (44) where30 β = G0 D and β = D0 G and the dollar weighted, threshold scaled payoff (Y − Z)0 = [4 − 4 3 − 3 2 −1 ]| is captured as β Y − β Z31 Results for this system under GBM were evaluated32 for 4321 = 4 3 2 1 with elasticity/beta parameters = 2 −1 05 Figure 1 plots the relationship between the flexibility values (on the vertical) and underlying value Using one way factors, the system accommodated uncertain stopping (e.g. in section 3, there was a chance that if remained high the value of flexibility would become small because switching would cease). With two way discount factors, this system also embraces the potential for different scenarios and branching. Not only is switching uncertain, but it is also not clear in which direction switching from the power state will go; different sequences i.e. 2,4,3,1 or 2,1,2,1 etc. are possible. The advantages of this solution system are apparent. It has building blocks in the sense that individual cashflow components and their intertwined flexibilities can be placed individually within a system governed by a common framework and solution method (these could then be analysed using modularity33 ). This is an important aspect of sequential investment. Secondly, although the example has used GBMs, other processes could be used 30 For the numerical example that ⎡ −1 0 0 ⎢ 0 3263 −2842 β = ⎢ ⎣ 0 1895 −2263 0 0 0 follows, the beta matrices ⎤ ⎡ 0 2048 ⎢ 0 0 ⎥ ⎥ β = ⎢ ⎣ 0 0 ⎦ 2 0190 31 are given by 0 −1 0 0 ⎤ 0 −0762 ⎥ 0 0 ⎥ ⎦ 2 0 0 −1048 See footnote 27 to form the comparable four by fours β β . In this example 4321 = 4 3 2 1 were fixed, resultant discount growth and beta | matrices were evaluated and X = [4 : 1369 3 : −1359 2 : 1338 1 : −1304] calculated. If these do not match known or target policy costs 41 , 41 should be varied until 41 matches those required. Alternatively as mentioned earlier for input X arbitrary −1 (non beta matched) values are given by W = [I − D] [Y − Z − X] when 41 = 41 can be varied numerically to find the turning point for W values. Either way the second order conditions should be checked to ensure these values are maxima. 33 Baldwin and Clark (00) [3] and Gamba Fusari (09) [14] motivate and value modularity. The design rests on six principles; splitting, substituting, augmenting, excluding, inverting, porting and the implementation is via multivariate least squares Monte Carlo. These are open to network or graphical investment approaches. 32 29 Investment ladder Idle, power and full switching 6 5 P+Vf (full inc put) P (full only) 4 P^0.5 + Vp (power + flex) Total value 3 incl. flex P^0.5 (power only) Vi (idle inc call) 2 X1=‐1.304 X2=1.338 1 X3=‐1.359 X4=1.369 0 0 1 2 3 4 5 Underlying revenue value P Figure 1: Investment ladder example with 41 = 4 3 2 1 for GBM and power values = 2 −1 05. Table 3 shows the investment graph whilst this plot shows the idle, power and full flex values with their investment quantities 41 (value matching occurs vertically at = 4 3 2 1). 30 by replacing the discount/growth components within D G The merit of the GBM is that its option elasticities are constant which simplifies the formation of β β etc. For 0 1 it also permits the same option solution for all powers 6 Restrictions for maxima There are several different types of restriction on thresholds e.g. 41 in Section 5. Firstly, for GBM (and some other diffusions but not arithmetic Brownian motion) thresholds must be positive. Secondly, in order for the discount functions to make sense, pairs must satisfy inequalities such as 2 1 (4 3 but 3 2 ). This immediately constrains the flex values U W that depend on 41 and the outputs 41 We would typically require flexibility to be an asset (not a liability with a negative value) and therefore solutions should be restricted to W 0 U 0 (vectors of zeros). As a first order condition, smooth pasting is ensured by dollar beta matching, but this only defines the turning point of a unique value set and does not say if that turning point is a maximum, minimum or inflection. Any proposed solution W U should be confirmed as maxima and neither a minima nor saddle point of value. Without constructing more matrices of differentials, the second order conditions required to establish maximisation can be conducted via numerical investigation. For an assumed policy P = 41 , the candidate cost vector X(P) can be calculated and fixed. Then with this fixed X the second order conditions can be checked by examining (26) W = [I − D]−1 [Y − Z − X] when perturbing away from P = 4321 . The effect on values within W (or U) will indicate if the proposed policy and solutions are maxima, minima or saddle points. Bearing in mind that there are multiple thresholds and flex values, as a function of each threshold change we must take care to identify which individual flex values within W U we would expect to have concave behaviour and zero/non zero slopes. Smooth pasting and dollar beta matching row by row, means that thresholds 41 were chosen to equalize slopes across value matching. Although smooth pasting says that pairs of values within each value matching line are optimally co—determined by smoothness, it does not say that the values (in rows of W U) have turning points with respect to their own threshold. For example at 1 in the last row of (40), 1 transforms into 1 less 1 (plus 1 ). For fixed 1 as 1 moves from its local optimum, smooth pasted components will change in a manner that preserves total sensitivity across the transition 1 = 1 − 1 + 1 . When 1 is varied, flex values at other 31 thresholds however (e.g. 4 to 2 inclusive) will display turning points in 1 . Due to the smooth pasting status of each line, value changes must be the same on both sides of (for fixed 42 ). Consequently, it is a flexibility’s sensitivity with respect to thresholds other than its own that should be zero, with concavity for a maximum. A local perturbations analysis of the system in this Section 5 in this manner, confirmed values in Figure 1 as maxima. Given policy P the existence of invertible matrices D G mapping between U W ensures uniqueness of the solution. There are however also restrictions on 4321 that arise from the economics of desired target (i.e. input) values X = [4 1 ]| i.e. investment costs. With an analytic method for mapping from a (restricted) set of 4−1 to solved 4−1 the system can be used numerically via search to find the 4−1 that map to specific investment costs. If a match of X(P) against a target values (for K’s) can be found and it is one that satisfies the numerical maximisation test in this subsection, then the solution can be said to be optimal given the costs X = K + K −K . 7 Conclusion Flexibility to time the launch, suspension or other transformation within a policy of thresholds P are valued as discount claims on an underlying uncertainty process. Their values at the beginning and end of option lives can be separately identified on a mathematical graph or network which then lends itself to discounting. Discounting has been used before with the factors and values appropriate in each flexibility state depending on diffusion dynamics. We extended this approach by capturing these features in a matrix that carries the appropriate diffusion dynamics at each point of the investment network. The rate of return of each flexibility value depends on the elasticity of its discount factor. Not only must discounting hold between flex values but for optimal smooth pasting, their rates of return must match too. This can be captured using beta matrix conditions derived from, but complementary to, those carrying fixed discounts. For fixed thresholds, these conditions allow solution and interpretation of flex values as a series of discounted flexibility payoffs. For a given policy of thresholds P flexibility values at the beginning and end of each state are driven by two considerations; discounting and smooth pasting or dollar beta matching. For given thresholds, the third and most natural condition of value matching only contributes to the solution of fixed investment costs X(P) (which have zero beta). To find policy thresholds as a function of costs P(X) a numerical search must be undertaken but the 32 approach and equations presented here facilitate this task greatly. References [1] Andrew B. Abel, Avinash K. Dixit, Janice C. Eberly, and Robert S. Pindyck. 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