A Top-Down Approach for the Multiple Exercises and Valuation of Employee Stock Options
aa r X i v : . [ q -f i n . P R ] S e p A Top-Down Approach for the Multiple Exercises and Valuationof Employee Stock Options
Tim Leung ∗ Yang Zhou † September 17, 2019
Abstract
We propose a new framework to value employee stock options (ESOs) that capturesmultiple exercises of different quantities over time. We also model the ESO holder’s jobtermination risk and incorporate its impact on the payoffs of both vested and unvestedESOs. Numerical methods based on Fourier transform and finite differences are developedand implemented to solve the associated systems of PDEs. In addition, we introduce a newvaluation method based on maturity randomization that yields analytic formulae for vestedand unvested ESO costs. We examine the cost impact of job termination risk, exerciseintensity, and various contractual features. ∗ Department of Applied Mathematics, University of Washington, Seattle WA 98195. E-mail:[email protected]. Corresponding author. † Department of Applied Mathematics, University of Washington, Seattle WA 98195. E-mail: [email protected]. Introduction
The use of employee stock options (ESOs) as part of compensation is a common practice amonglarge and small companies in the United States. Financial Accounting Standards Board (FASB)requires companies to value these stock options and report the total granting cost. This re-quirement raises the need for valuation methods that can effectively capture the payoff structureand exercise pattern of these stock options.Empirical studies suggest that ESO holders tend to start exercising their options exerciseearly, often soon after the vesting period, and gradually exercise the remaining options overmultiple dates before maturity. Huddart and Lang (1996), Marquardt (2002), and Bettis et al.(2005) point out that, for ESOs with 10 years to maturity, the expected time to exercise is 4 to5 years. Investigating how ESO exercises are spread out over time, Huddart and Lang (1996)show that the mean fraction of options exercised by a typical employee at one time varied from0.18 to 0.72. For more empirial studies, we refer to Huddart and Lang (1996), Bettis et al.(2001), Marquardt (2002), Armstrong et al. (2007), Hallock and Olson (2007), Heron and Lie(2016) and Carpenter et al. (2017). These empirical findings motivate us to consider a valuationmodel that account for multiple exercises of various units of options at different times. As notedby Jain and Subramanian (2004), “the incorporation of multiple-date exercise has importanteconomic and account consequences.”In this paper, we take the firm’s perspective to determine the cost of an ESO grant. AnESO grant commonly involves multiple options with a long maturity. There is also a vestingperiod, during which option exercise is prohibited and job termination leads to forfeiture of theoptions. The key component of our proposed valuation framework is an exogenous jump processthat models the random exercises over time. Within our framework, the employee’s exerciseintensity can be constant or stochastic, and the number of options exercised at each time canbe specified to be deterministic or random. In essence, this top-down approach offers a flexiblesetup to model any exercise pattern. The idea is akin to the top-down approach in credit risk(Giesecke and Goldberg (2011)), where the exogenous jump process represents portfolio losses.Since the ESO payoff depends heavily on when the employee leaves the firm, we also include arandom job termination time and allow the job termination rate to be different during and aftervesting period.The valuation problem leads to the study of the system of partial differential equations(PDEs) associated with the vested and unvested options. In order to compute the ESO costs,we present two numerical methods to solve the PDEs. We discuss the method of fast Fouriertransform (FFT), followed by the finite difference method (FDM). By applying Fourier trans-form, we simplify the original second-order PDEs to ODEs in the constant intensity case andfirst-order PDEs in the stochastic intensity case. The ESO costs are recovered via inverse fastFourier transform. The results from the two methods are illustrated and compared under bothdeterministic and stochastic exercise intensities. Furthermore, we introduce a new valuationmethod based on maturity randomization. The key advantage of this method is that it yieldsanalytic formulae, allowing for instant computation.Using all three numerical methods, we compute the costs and examine the impact of jobtermination risk, exercise intensity, vesting period, and other features. Among our findings,we illustrate the distributions of exercise times under different model specifications, and also See FASB Accounting Standards Codification (ASC) no.718 (formerly, FASB Statement 123R),
Accountingfor Stock-Based Compensation . We begin by describing the ESO payoff structure, and then introduce the stochastic model thatcaptures various sources of randomness. The valuation of both vested and unvested ESOs ispresented. 3 .1 Payoff Structure
The ESO is an early exercisable call option written on the company stock with a long maturity T ranging from 5 to 10 years. In order to maintain the incentive effect of ESOs, the companytypically prohibits the ESO holder (employee) from exercising during a vesting period from thegrant date. During the vesting period, which ranges from 1 to 5 years, the holder’s departurefrom the company, voluntarily or forced, will lead to forfeiture of the option, rending it worth-less. We denote [0 , t v ) as the vesting period, and after the date t v the ESO is vested and free tobe exercised until it expires at time T . The ESO payoff at any time τ is ( S τ − K ) + { t v ≤ τ ≤ T } ,where S τ is the firm’s stock price at time τ and K is the strike price. Upon departure, the em-ployee is supposed to exercise all the remaining options. Figure 1 shows all four payoff scenariosassociated with an ESO. ( ) t v ( ) ( ) T Time S S t o ck P r i c e )-K S T ( )-KK0 Figure 1: ESO payoff structure. From bottom path to top path: (i) The employee leaves the firmduring the vesting period, resulting in forfeiture of the ESO and a zero payoff. (ii) The employeeexercises the vested ESO before maturity due to desire to liquidate or job termination and receivethe payoff ( S τ ( ω ) − K ) + . (iii) The employee exercises the vested ESO before maturity due tojob termination, but receives nothing. (iv) The employee exercises the option at maturity T . The employee’s job termination plays a crucial role in the exercise timing and resulting payoffof the ESOs. We model the job termination time during the vesting period by an exponentialrandom variable ζ ∼ exp( α ), with α ≥
0. When the ESO becomes vested after t v , we modelthe employee’s job termination time by another exponential random variable ξ ∼ exp( β ), with β ≥
0. We assume that ζ and ξ are mutually independent. This approach of modeling job4ermination by an exogenous random variable is also used by Jennergren and Naslund (1993),Carpenter (1998), Carr and Linetsky (2000), Hull and White (2004), Sircar and Xiong (2007),Leung and Sircar (2009b), Carmona et al. (2011), and Leung and Wan (2015), among others.In our model, using two different exponential times allows us to account for the varying level ofjob termination risk during and after the vesting period.An ESO grant typically contains multiple options. Empirical studies show that employeetends to exercise the options gradually over time, rather than exercising all options at once.This motivates us to model the sequential random timing of exercises. In our proposed model,we consider a grant of M units of identical early exercisable ESOs with the same strike price K and expiration date T . These M ESOs are exercisable only after the vesting period [0 , t v ). Forthe vested ESOs, we define the random exercise process L t , for t v ≤ t ≤ T , to be the positivejump process representing the number of ESOs exercised over time. As such, L t is an integerprocess that takes value on [0 , M ]. The corresponding jump times are denoted by the sequence( τ , τ , . . . ), and the frequency of exercises is governed by the jump intensity process ( λ t ) t v ≤ t ≤ T .The jump size for the i th jump of L represents the number of ESOs exercised and is describedby a discrete random variable δ i . The exercise process starts at time t v with L t v = 0. Bydefinition, we have L T ≤ M . This means that the random jump size at any time t must takevalue within [1 , M − L t − ]. Also, as soon as L t reaches the upper bound M , the jump intensity λ t must be set to be zero thereafter. Given that the employee still holds m options, the probabilitymass function of the random jump size is p m,z , P { δ i = z | L τ i − = M − m } . (1)In turn, the expected number of options to be exercised at each exercise time is given by¯ p m , m X z =1 zp m,z , (2)which again depends on the current number of ESOs held.The employee may exercise single or multiple units of ESOs over time. On the date ofexpiration or job termination, any unexercised options must be exercised. Hence, the discountedpayoff from the ESOs over [0 , T ] is a sum of two terms, given by (cid:18) Z T ∧ ξt v e − rt ( S t − K ) + dL t + e − r ( T ∧ ξ ) ( M − L T ∧ ξ )( S T ∧ ξ − K ) + (cid:19) { ζ ≥ t v } . (3)The indicator 1 { ζ ≥ t v } means that the ESO payoff is zero if the employee leaves the firm duringthe vesting period. Example 1 (Unit Exercises)
Suppose L t be a nonhomogeneous Poisson process ( N t ) ≤ t ≤ T with a time-varying jump intensity function λ ( t ) , for ≤ t ≤ T . At each jump time a singleoption is exercised. In Figure 2 we illustrate three possible scenarios. In scenario (i), theemployee exercises 6 out of 10 options one by one, but must exercise 4 remaining options uponjob termination realized at time ξ ( ω ) . In scenario (ii) the employee exercises all 10 optionsone by one before maturity. In scenario (iii), the employee has not exercised all the options bymaturity, so all remaining options are exercised at time T . ) ( ) T Time O p t i on s E x e r c i s ed Figure 2: Three illustrative sample paths of the process for Poisson exercises of 10 ESOs. Frompath top to bottom path: (i) The employee first exercises 6 out of 10 options one by one, butis then forced to exercise 4 remaining options upon job termination realized at time ξ ( ω ). (ii)The employee exercises all the options one by one before expiration and job termination. Thelast option is exercised at τ ( ω ) shown in the plot. (iii) The employee exercises 3 options oneby one before maturity and 7 remaining options at maturity. Example 2 (Block Exercises)
Suppose the employee can exercise one or more options at eachexercise time. As an example, We assume a uniform distribution for the number of options tobe exercised, so we set p m,z = m − for z = 1 , . . . , m . In Figure 3, we illustrate the distributionsof the weighted average exercise time ¯ τ defined by ¯ τ = P Ni =1 δ i ∗ τ i M , (4) where δ i is the number of ESOs exercised at the i th exercise time τ i , and N is the number ofdistinct exercise times before or at time T . For each simulated path, we take an average of thedistinct exercise times weighted by the number of options exercised at each time. With commonparameters M = 20 , t v = 0 , T = 10 , the histograms of ¯ τ correspond to different values of λ and β .With a low job termination rate β and low exercise intensity λ (panel (a) where β = 0 , λ = 0 . ),more options tend to be exercised at maturity. Comparing panel (b) to panel (c), and also panel(b) to panel (d), we see that a higher job termination rate or higher exercise intensity lowers theaverage exercise time and reduces instances of exercising at maturity. Similar patterns can alsobe found in empirical studies (Heron and Lie, 2016, Fig. 3). a) (b)(c) (d) Figure 3: Histograms of weighted average exercise times, as defined in (4), based on 10,000simulated exercise processes for 20 vested ESOs with a 10-year maturity. Panels have differentrates of job termination β and exercise intensity λ . (a): β = 0, λ = 0 .
3; (b): β = 0 . λ = 0 . β = 0 . λ = 0 .
3; (d): β = 0 . λ = 0 .
5. 7 .3 PDEs for ESO Valuation
To value ESOs, we consider a risk-neutral pricing measure Q for all stochastic processes andrandom variables in our model. We model the firm’s stock price process ( S t ) t ≥ by a geometricBrownian motion dS t = ( r − q ) S t dt + σS t dW t , (5)where the positive constants r , q and σ are the interest rate, dividend rate, and volatilityparameter respectively, and W is a standard Brownian motion under Q , independent of theexponentially-distributed job termination times ζ and ξ . Our default assumption for the em-ployee’s exercise intensity is that it is a deterministic function of time, denoted by λ ( t ). We willdiscuss the case with a stochastic exercise intensity in Section 4.At any time t ∈ [ t v , T ], the ESO is vested. The vested ESO cost functions C ( m ) ( t, s ), for m = 1 , , . . . , M , where m is the number of options currently held, are given by the risk-neutralexpectation of discounted future ESO payoffs provided that the employee has not left the firm. C ( m ) ( t, s ) = IE (cid:26) Z T ∧ ξt e − r ( u − t ) ( S u − K ) + dL u + e − r ( T ∧ ξ − t ) ( M − L T ∧ ξ )( S T ∧ ξ − K ) + | S t = s, L t = M − m (cid:27) = IE (cid:26) Z Tt e − ( r + β )( u − t ) ( S u − K ) + dL u + e − ( r + β )( T − t ) ( M − L T )( S T − K ) + + Z Tt βe − ( r + β )( v − t ) ( M − L v )( S v − K ) + dv | S t = s, L t = M − m (cid:27) , (6)for m = 1 , , . . . , M , and ( t, s ) ∈ [ t v , T ] × R + .Next, we define the infinitesimal generator associated with the stock price process S by L · = ( r − q ) s∂ s · + σ s ∂ ss · . (7)We determine the vested ESO costs by solving the following system of PDEs. − ( r + λ ( t ) + β ) C ( m ) + C ( m ) t + L C ( m ) + λ ( t ) m − X z =1 p m,z C ( m − z ) + ( λ ( t )¯ p m + mβ ) ( s − K ) + = 0 , (8)for ( t, s ) ∈ [ t v , T ] × R + and m = 1 , , . . . , M . Here, ¯ p m is the expected number of options exercisedand p m,z is the probability of exercising z options with m options left. The terminal conditionis C ( m ) ( T, s ) = m ( s − K ) + for s ∈ R + .During the vesting period [0 , t v ), the ESO is unvested and is subject to forfeiture if theemployee leaves the firm. We denote the cost of m units of unvested ESO by ˜ C ( m ) ( t, s ). Sinceholding an unvested ESO effectively entitles the holder to obtain a vested ESO at time t v provided the holder is still with the firm. If the ESO holder leaves the firm at any time t ∈ [0 , t v ),the unvested ESO cost is zero. Otherwise, given that ζ > t , the (pre-departure) unvested ESOcost is ˜ C ( m ) ( t, s ) = IE n e − r ( t v − t ) C ( m ) ( t v , S t v ) { ζ ≥ t v } | S t = s o = IE n e − ( r + α )( t v − t ) C ( m ) ( t v , S t v ) | S t = s o . (9)8o determine the unvested ESO cost, we solve the PDE problem − ( r + α ) ˜ C ( m ) + ˜ C ( m ) t + L ˜ C ( m ) = 0 , for ( t, s ) ∈ [0 , t v ) × R + , ˜ C ( m ) ( t v , s ) = C ( m ) ( t v , s ) , for s ∈ R + . (10)Here, C ( m ) ( t v , s ) is the vested ESO cost evaluated at time t v . In this section, we present two numerical methods to solve PDE (8). We first discuss theapplication of fast Fourier transform (FFT) to ESO valuation, followed by the finite differencemethod (FDM). The results from the two methods are compared in Section 3.3.
We first consider the vested ESO ( t ∈ [ t v , T ]). Let x such that s = Ke x , and define the function f ( m ) ( t, x ) = C ( m ) ( t, Ke x ) , ( t, x ) ∈ [ t v , T ] × R , (11)for each m = 1 , . . . , M . The PDE for f ( m ) ( t, x ) is given by − ( r + λ ( t )+ β ) f ( m ) + f ( m ) t + e L f ( m ) + λ ( t ) m − X z =1 p m,z f ( m − z ) +( λ ( t )¯ p m + mβ ) ( Ke x − K ) + = 0 , (12)where e L · = ( r − q − σ ∂ x · + σ ∂ xx · . (13)The terminal condition is f ( m ) ( T, x ) = m ( Ke x − K ) + , for x ∈ R .The Fourier transform of f ( m ) ( t, x ) is defined by F [ f ( m ) ]( t, ω ) = Z ∞−∞ f ( m ) ( t, x ) e − iωx dx, (14)for m = 1 , . . . , M , with angular frequency ω in radians per second. Applying Fourier transformto PDE (12), we obtain an ODE for F [ f ( m ) ]( t, ω ), a function of time t parametrized by ω , foreach m = 1 , . . . , M . Precisely, we have ddt F [ f ( m ) ]( t, ω ) = h ( t, ω ) F [ f ( m ) ]( t, ω ) + ψ ( m ) ( t, ω ) , (15)where h ( t, ω ) = r + λ ( t ) + β − iω ( r − q − σ ω σ , (16) ψ ( m ) ( t, ω ) = − λ ( t ) m − X z =1 p m,z F [ f ( m − z ) ]( t, ω ) − ( λ ( t )¯ p m + mβ ) ϕ ( ω ) , (17) ϕ ( ω ) = F [( Ke x − K ) + ]( ω ) , (18)9ith the terminal condition F [ f ( m ) ]( T, ω ) = mϕ ( ω ). Solving the ODE, we obtain F [ f ( m ) ]( t, ω ) = e − R Tt h ( s,ω ) ds F [ f ( m ) ]( T, ω ) − Z Tt e − R ut h ( s,ω ) ds ψ ( m ) ( u, ω ) du. (19)Accordingly, we can recover the vested ESO cost function by inverse Fourier transform: f ( m ) ( t, x ) = F − [ F [ f ( m ) ]]( t, x ) . (20)for every m = 1 , . . . , M , and ( t, x ) ∈ ( t v , T ) × R . In the literature, Leung and Wan (2015) apply a Fourier time-stepping (FST) method itto compute the cost of an American-style ESO when the company stock is driven by a Levyprocess. This FST method has been applied more broadly by Jackson et al. (2008) to solvepartial-integro differential equations (PIDEs) that arise in options pricing problems.
Remark 3 If λ is a constant, then the Fourier transform in (19) can be simplified as F [ f ( m ) ]( t, ω ) = m − X k =0 F ( m ) k ( ω )( T − t ) k e − ( T − t ) h ( ω ) + F ( m ) ( ω ) , (21) where F ( m ) ( ω ) = 1 h ( ω ) (cid:18) λ m − X z =1 p m,z F ( m − z ) ( ω ) + ( λ ¯ p m + mβ ) ϕ ( ω ) (cid:19) , (22) F ( m ) k ( ω ) = λk m − k X z =1 p m,z F ( m − z ) k − ( ω ) , k = 1 , , . . . , m − , (23) F ( m )0 ( ω ) = F [ f ( m ) ]( T, ω ) − h ( ω ) (cid:18) λ m − X z =1 p m,z F ( m − z ) ( ω ) + ( λ ¯ p m + mβ ) ϕ ( ω ) (cid:19) , (24) h ( ω ) = r + λ + β − iω ( r − q − σ ω σ . (25) In (22) and (24) , ϕ ( ω ) is defined in (18). For numerical implementation, we work with a finite domain [ t v , T ] × [ x min , x max ] withuniform discretization of lengths δt = ( T − t v ) /N t and δx = ( x max − x min ) / ( N x −
1) in thetime-space dimensions. We set δt = 0 . x min = − x max = 10 and N x = 2 . Similarly,we discrete the finite frequency space [ ω min , ω max ] with uniform grid size of δω , where we applythe Nyquist critical frequency that ω max = π/δx and δω = 2 ω max /N x . For j = 0 , . . . , N t , and k = 0 , . . . , N x −
1, we denote t j = t v + jδt , x k = x min + kδx , and ω k = ( kδω, ≤ k ≤ N x / ,kδω − ω max , N x / ≤ k ≤ N x − . (26)Then we numerically compute the discrete Fourier transform F [ f ]( t j , ω k ) ≈ N x − X n =0 f ( t j , x n ) e − iω k x n δx = φ k N x − X n =0 f ( t j , x n ) e − i πkn/N x , (27)10ith φ k = e − iω k x min δx . In (27), we evaluate the sum P N x − n =0 f ( t j , x n ) e − i πkn/N x by applyingthe standard fast Fourier transform (FFT) algorithm. The corresponding Fourier inversion isconducted by inverse FFT, yielding the vested ESO cost f ( t j , x n ). Note that the coefficient φ k will be cancelled in the process.As for the unvested ESO, we define the associated cost function˜ f ( m ) ( t, x ) = ˜ C ( m ) ( t, Ke x ) , (28)for each m = 1 , . . . , M . From PDE (10), we derive the PDE for ˜ f ( m ) ( t, x ) − ( r + α ) ˜ f ( m ) + ˜ f ( m ) t + e L ˜ f ( m ) = 0 , (29)for ( t, x ) ∈ [0 , t v ) × R , with the terminal condition ˜ f ( m ) ( t v , x ) = f ( m ) ( t v , x ), for x ∈ R . As wecan see, once the vested ESO cost is computed, it determines the terminal condition for theunvested ESO problem.Applying Fourier transform to (29), we can derive the ODE for F [ ˜ f ( m ) ]( t, ω ), ddt F [ ˜ f ( m ) ]( t, ω ) = ˜ h ( ω ) F [ ˜ f ( m ) ]( t, ω ) , (30)where ˜ h ( ω ) = r + α − iω ( r − q − σ ω σ , (31)for ( t, ω ) ∈ [0 , t v ) × R , with the terminal condition F [ ˜ f ( m ) ]( t v , ω ) = F [ f ( m ) ]( t v , ω ). We solve theODE to get F [ ˜ f ( m ) ]( t, ω ) = e − ˜ h ( ω )( t v − t ) F [ ˜ f ( m ) ]( t v , ω ) . (32)In turn, we apply inverse Fourier transform to recover the unvested ESO cost:˜ C ( m ) ( t, Ke x ) = ˜ f ( m ) ( t, x ) = F − [ F [ ˜ f ( m ) ]]( t, x ) , (33)for ( t, x ) ∈ [0 , t v ) × R . Again, we apply FFT to numerically compute the Fourier transform anduse inverse FFT to recover the cost function. For comparison, we also compute the ESO costs using a finite difference method. Specifically,we apply the Crank-Nicolson method on a uniform grid. Here we provide an outline with focuson the boundary conditions for our application. For more details, we refer to Wilmott et al.(1995), among other references.As for grid settings, We restrict the domain [ t v , T ] × R + to a finite domain D = { ( t, s ) | t v ≤ t ≤ T, ≤ s ≤ S ∗ } , where S ∗ must be relatively very large such that if the current stockprice S t = S ∗ , then the stock price will be larger than the strike price K over [ t, T ] with greatprobability.To determine the boundary condition at s = S ∗ , we introduce a new function¯ C ( m ) ( t, s ) = IE (cid:26) Z Tt e − ( r + β )( u − t ) ( S u − K ) dL u + e − ( r + β )( T − t ) ( M − L T )( S T − K )+ Z Tt βe − ( r + β )( v − t ) ( M − L v )( S v − K ) dv | S t = s, L t = M − m (cid:27) . m = 1 , . . . , M . When s = S ∗ , we see that C ( m ) ( t, s ) ≈ ¯ C ( m ) ( t, s ). Thus, we can set theboundary condition at s = S ∗ to be C ( m ) ( t, S ∗ ) = ¯ C ( m ) ( t, S ∗ ). By Feynman-Kac formula,¯ C ( m ) ( t, s ) satisfies the PDE − ( r + λ ( t ) + β ) ¯ C ( m ) + ¯ C ( m ) t + L ¯ C ( m ) + λ ( t ) m − X z =1 p m,z ¯ C ( m − z ) + ( λ ( t )¯ p m + mβ ) ( s − K ) = 0 , (34)for m = 1 , . . . , M , and ( t, s ) ∈ ( t v , T ) × R + , with terminal condition ¯ C ( m ) ( T, s ) = m ( s − K ), for s ∈ R + . Then, ¯ C ( m ) ( t, s ) has the ansatz solution¯ C ( m ) ( t, s ) = A m ( t ) s − B m ( t ) K, (35)where A m ( t ) and B m ( t ) satisfy the pair of ODEs respectively, − ( q + λ ( t ) + β ) A m + A ′ m + λ ( t ) m − X z =1 p m,z A m − z + ( λ ( t )¯ p m + mβ ) = 0 , − ( r + λ ( t ) + β ) B m + B ′ m + λ ( t ) m − X z =1 p m,z B m − z + ( λ ( t )¯ p m + mβ ) = 0 , (36)for m = 1 , . . . , M , and t ∈ ( t v , T ), with the terminal condition B m ( T ) = A m ( T ) = m , for m = 1 , . . . , M . We can solve the ODEs (36) analytically, or numerically solve it using thebackward Euler method.Next, we discrete the domain D with uniform grid size of δt = ( T − t v ) /M and δS = S ∗ /N .Then, we apply C ( m ) i,j to denote discrete approximations of C ( m ) ( t i , s j ) where t i = t v + iδt and s j = jδS . The Crank-Nicolson method is applied to solve the PDEs satisfied by C ( m ) , for m =1 , . . . , M . Working backward in time, we obtain the vested ESO costs at time t v , which becomethe terminal condition values for the unvested ESO valuation problem. For the unvested ESOcost, we restrict the domain [0 , t v ] × R + to the finite domain ˜ D = { ( t, s ) | ≤ t ≤ t v , ≤ s ≤ S ∗ } ,where S ∗ is relatively very large such that˜ C ( m ) ( t, S ∗ ) = IE n e − ( r + α )( t v − t ) C ( m ) ( t v , S t v ) | S t = S ∗ o (37) ≈ IE n e − ( r + α )( t v − t ) ( A m ( T − t v ) S t v − B m ( T − t v ) K ) | S t = S ∗ o (38)= e − ( q + α )( t v − t ) A m ( T − t v ) S ∗ − e − ( r + α )( t v − t ) B m ( T − t v ) K. (39)We again apply the Crank-Nicolson method solve the PDEs satisfied by ˜ C ( m ) ( t, s ), for m =1 , . . . , M . Using both FFT and FDM we compute different ESO costs by varying the vesting period t v ,job termination rate α and β , as well as exercise intensity λ . In Table 1, we present the ESOcosts and compare the two numerical methods. It is well known that the call option value isincreasing with respect to its maturity. In a similar spirit if the employee tends to exercise theESO earlier, then a smaller ESO cost is expected. As we can see in Table 1, the ESO costdecreases as exercise intensity λ increases, or as job termination rate α or β increases, holding12ther things constant. On the other hand, the effect of vesting period is not monotone. In ascenario with a high job termination α during the vesting period, the employee is very likelyto leave the firm while the options are unvested, leading to a zero payoff. Consequently, theESO cost is decreasing with respect to t v . This corresponds to the case with α = 1 in Table 1.However, if α is small, then the employee is unlikely to leave the firm and lose the options duringthe vesting period. Therefore, a longer vesting period would effectively make the employee holdthe options for a longer period of time, delaying the exercise. As a result, the ESO cost isincreasing with respect to t v , which is shown in other cases in Table 1.Parameters t v = 0 t v = 2 t v = 4FDM FFT FDM FFT FDM FFT α = 0 . , β = 0 λ = 1 5.4729 5.4753 7.8399 7.8405 8.2845 8.2849 λ = 2 3.7067 3.7101 6.9164 6.9170 7.7054 7.7058 α = 0 . , β = 1 λ = 1 3.2483 3.2522 6.7063 6.7069 7.5746 7.5750 λ = 2 2.7024 2.7069 6.4655 6.4661 7.4253 7.4257 α = 0 , β = 0 . λ = 1 5.0603 5.0629 9.3022 9.3031 12.1510 12.1517 λ = 2 3.5595 3.5630 8.3622 8.3631 11.4298 11.4306 α = 1 , β = 0 . λ = 1 5.0603 5.0629 1.2579 1.2590 0.2219 0.2226 λ = 2 3.5595 3.5630 1.1310 1.1318 0.2087 0.2094Table 1: Vested and unvested ESO costs under different exercise intensities λ and differentjob termination rates α and β , computed using FFT and FDM for comparison. CommonParameters: S = K = 10, r = 5%, q = 1 . σ = 20%, p m,z = 1 /m , M = 5 and T = 10. InFDM: S ∗ = 30, δS = 0 . δt = 0 .
1. In FFT: N x = 2 , x min = −
10 and x max = 10.In Figure 4, we plot the ESO cost as a function of the exercise intensity λ for T = 5, 8 and10. It shows that the ESO is decreasing and convex with respect to λ . An employee with a highexercise intensity tends to exercise the ESOs earlier than those with a lower exercise intensity.Since the call option value increases with maturity, exercising the ESO earlier will result in alower cost. As the exercise intensity increases from zero, the ESO cost tends to decrease fasterthan when the exercise intensity is higher. Moreover, Figure 4 also shows that as λ increases,the ESO costs associated with different maturities T = 5, 8 and 10 get close to each other. Theintuition is that when λ is large, the options will be exercised very early and the maturity willnot have a significant impact on the option values.On the right panel of Figure 4, we plot the option value as a function of stock price S withexercise intensity λ = 0, 1 or 5. It shows that, as λ increases from 0 to 1, the option valuedecreases rapidly. When the exercise intensity is very high, i.e. λ = 5 in the figure, there is ahigh chance of immediate exercise, so the ESO value is seen to be very close to the ESO payoff( S − K ) + .Next, we consider the effect of the total number of ESOs granted. Intuitively we expect thetotal cost to increase as the number of options M increases, but the effect is far from linear. InFigure 5 we see that the average per-unit cost and average time to exercise are increasing as M increases. In other words, under the assumption that the ESOs will be exercised gradually,a larger ESO grant has an indirect effect of delaying exercises, and thus leading to higher ESOcosts. The increasing trends hold for different exercise intensities, but the rate of increasediminishes significantly for large M . Also, the higher the exercise intensity, the lower the per-13nit cost and shorter averaged time to exercise. Exercise Intensity ( ) ES O C o s t T=5T=8T=10
Stock Price (S ) ES O C o s t =0=1=5 Figure 4: Left: ESO cost as a function of employee exercise intensity λ when the maturity T = 5 , S with λ = 0 ,
1, or 5.Parameters: K = 10, r = 5%, q = 1 . σ = 20%, p m,z = 1 /m , M = 5, T = 10, t v = 0 and β = 0 .
1. In FFT: N x = 2 , x min = − x max = 10. Number of Options (M) P e r- U n i t C o s t =0.1=0.2=0.4 Number of Options E x pe c t ed T i m e T o E x e r c i s e =0.1=0.2=0.4 Figure 5: Left: Per-unit ESO cost as a function of number of options granted M with differentexercise intensities λ . Right: With M = 20 options under different exercise intensities λ , wecalculate the average exercise times by Monte-Carlo simulation with 10 simulated paths ofexercise process. Common parameters: S = K = 10, r = 5%, q = 1 . σ = 20%, p m,z = 1 /m , T = 10, t v = 1, β = 0 . α = 0 .
1. In FFT: N x = 2 , x min = − x max = 10.14 Stochastic Exercise Intensity
Now we discuss the stochastic exercise intensity, an extension to the previous model, that λ t = λ ( t, S t ), which is the function not only depends on the time t also depends on the stock price S t . Accordingly, the corresponding vested ESO cost C ( m ) ( t, s ) will satisfy − ( r + λ ( t, s ) + β ) C ( m ) + C ( m ) t + L C ( m ) + λ ( t, s ) m − X z =1 p m,z C ( m − z ) + ( λ ( t, s )¯ p m + mβ ) ( s − K ) + = 0 , (40)for m = 1 , . . . , M , and ( t, s ) ∈ [ t v , T ] × R + , with terminal condition C ( m ) ( T, s ) = m ( s − K ) + ,for s ∈ R + .Since we only discuss the stochastic exercise intensity and the employee will not exercise theoption during the vesting period, the PDE for unvested ESO will remain unchanged. Next, wewill discuss how to numerically solve (40) by FFT.For applying Fourier transform, we use the same notation as in Section 3.1 that f ( m ) ( t, x ) = C ( m ) ( t, Ke x ) , (41)for m = 1 , . . . , M , ( t, x ) ∈ [ t v , T ] × R , and F [ f ( m ) ]( t, ω ) = Z ∞−∞ f ( m ) ( t, x ) e − iωx dx, (42)for m = 1 , . . . , M . In this section, we assume that λ ( t, x ) = A ( t ) − B ( t ) x , for some positivetime-dependent functions A ( t ) and B ( t ). For implementation, we assume B ( t ) be relative small,such that λ ( t, x ) stay positive in the truncated space ( t v , T ) × [ − x max , x max ]. Then, f ( m ) ( t, x )satisfies − ( r + A ( t ) − B ( t ) x + β ) f ( m ) + f ( m ) t + e L f ( m ) + ( A ( t ) − B ( t ) x ) m − X z =1 p m,z f ( m − z ) + (cid:18)(cid:0) A ( t ) − B ( t ) x (cid:1) ¯ p m + mβ (cid:19) ( Ke x − K ) + = 0 , (43)where e L is defined in (13). The terminal condition is f ( m ) ( T, x ) = m ( Ke x − K ) + , for x ∈ [ − x max , x max ].Using (42) and the property of Fourier transform that F [ xf ]( t, ω ) = i∂ ω F [ f ]( t, ω ), we trans-form PDE (43) into ∂∂t F [ f ( m ) ]( t, ω ) + iB ( t ) ∂∂ω F [ f ( m ) ]( t, ω ) − h ( t, ω ) F [ f ( m ) ]( t, ω ) + ψ ( m ) ( t, ω ) = 0 , (44)15here h ( t, ω ) = − ( r − q − σ iω + σ ω r + A ( t ) + β, (45) ψ ( m ) ( t, ω ) = m − X z =1 p m,z F [ λf ( m − z ) ]( t, ω ) + F [( λ ¯ p m + mβ ) ( Ke x − K ) + ]( t, ω ) , (46)for ( t, ω ) ∈ [ t v , T ) × R .Observe that (44) is a first-order PDE with terminal condition that F [ f ( m ) ]( T, ω ) = mϕ ( ω )(see (18)). Therefore, we apply the method of characteristics and get F [ f ( m ) ]( t, ω )= e − R Tt h (cid:0) s,ω − i R ts B ( u ) du (cid:1) ds F [ f ( m ) ] (cid:18) T, ω + i Z Tt B ( u ) du (cid:19) + Z Tt g ( m ) ( τ, ω ; t ) dτ. (47)where g ( m ) ( τ, ω ; t ) = e − R τt h (cid:0) s,ω − i R ts B ( u ) du (cid:1) ds ψ ( m ) (cid:18) τ, ω + i Z τt B ( u ) du (cid:19) . (48)For numerical implementation, we can use the similar method mentioned in Section 3.1. Wecan make the approximation Z Tt g ( m ) ( τ, ω ; t ) dτ ≈ (cid:18) g ( m ) ( t, ω ; t ) + 12 g ( m ) ( T, ω ; t ) + i = N − X i =1 g ( m ) ( t + iδt, ω ; t ) (cid:19) δt, (49)where δt = ( T − t ) /N . The integral R B ( u ) du in (47) and (48) can be approximated similarlyor computed explicitly depending on the choice of B ( t ).Table 2 presents the ESO costs in the cases of constant exercise intensity λ = 0 . λ ( s ) = 0 . − .
02 log( s/K ) under different vesting periods and jobtermination rates. The stochastic intensity specified here can be larger or smaller than theconstant level 0.2 depending on whether the current stock price s is higher or lower than thestrike price K . For each case, we compute the ESO cost using both FFT and FDM. For thelatter, we apply the Crank-Nicolson method on a uniform grid and adopt Neumann conditionat the boundary s = S ∗ (see Section 3.2). As we can see, the costs from the two methods arepractically the same. As the vesting period lengthens, from t v = 1 to t v = 4, the ESO costtends to increase under different exercise intensities and job termination rates. When the jobtermination rate β is zero, the ESO costs with stochastic intensity appear to be higher, but thiseffect is greatly reduced as the job termination rate increases. This is intuitive since a high jobtermination rate means that most ESOs will be exercised or forfeited at the departure time,rather than exercised according to an exercise process over the life of the options.16arameters t v = 1 t v = 2 t v = 4FDM FFT FDM FFT FDM FFT λ = 0 . β = 0 α = 0 12.8052 12.8065 13.7122 13.7134 15.0953 15.0967 α = 0 . β = 0 . α = 0 7.8849 7.8859 9.6380 9.6388 12.4022 12.4029 α = 0 . λ ( s ) = 0 . − . ∗ log( s/K ) β = 0 α = 0 12.8310 12.8379 13.7364 13.7445 15.1130 15.1235 α = 0 . β = 0 . α = 0 7.8895 7.8887 9.6428 9.6423 12.4068 12.4076 α = 0 . λ and stochastic exercise intensity λ ( s ) with differ-ent job termination rates α and β and vesting period t v , computed using FFT and FDM forcomparison. Common parameters: S = K = 10, r = 5%, q = 1 . σ = 20%, p m,z = 1 /m , M = 5, T = 10. In FDM: S ∗ = 30, δS = 0 . δt = 0 .
1. In FFT: N x = 2 , X min = −
10 and X max = 10. 17 Maturity Randomization
In this section, we propose an alternative method to value ESOs. It is an analytical method thatyields an approximation to the original ESO valuation problem discussed in Section 2.3. Thecore idea of this method is to randomize the ESO’s finite maturity by an exponential randomvariable τ ∼ exp( κ ), with κ = 1 /T where T here is original constant maturity. Such a choiceof parameter means that IE [ τ ] = T ; that is, the ESO is expected to expire at time T . Forinstance, if the maturity of the ESOs is 10 years, then the randomized maturity is modeled by τ ∼ exp(0 . First we consider the ESO cost at the end of the vesting period. Provided that the employeeremains at the firm by time t v , the vested ESO has a remaining maturity of length T − t v .Therefore, for the exponentially distributed maturity τ ∼ exp( κ ), one may set κ = 1 / ( T − t v ).At time t v , the vested ESO cost function C ( m ) ( s ) is given by C ( m ) ( s ) = IE (cid:26) Z τ ∧ ξt v e − r ( u − t v ) ( S u − K ) + dL u + e − r ( τ ∧ ξ − t v ) ( M − L τ ∧ ξ )( S τ ∧ ξ − K ) + | S t v = s, L t v = M − m, τ ∧ ξ ≥ t v (cid:27) (50)= IE (cid:26) Z ∞ t v e − ( r + κ + β )( u − t v ) ( S u − K ) + dL u + Z ∞ t v ( κ + β ) e − ( r + κ + β )( u − t v ) ( M − L u )( S u − K ) + du | S t v = s, L t v = M − m (cid:27) (51)= IE (cid:26) Z ∞ e − ( r + κ + β ) u ( S u − K ) + dL u + Z ∞ ( κ + β ) e − ( r + κ + β ) u ( M − L u )( S u − K ) + du | S = s, L = M − m (cid:27) , (52)for m = 1 , . . . , M . From (52), we derive the associated ODE for C ( m ) ( s ). For the convenience,we denote a = − ( r + λ + β + κ ) , a = r − q, a = σ , g m = λ ¯ p m + m ( β + κ ) . (53)Then, we obtain a system of second-order linear ODEs: a C ( m ) + a s dds C ( m ) + a s d ds C ( m ) + λ m − X z =1 p m,z C ( m − z ) + g m ( s − K ) + = 0 , (54)for m = 1 , . . . , M , and s ∈ R + , with the boundary condition C ( m ) (0) = 0.18 roposition 4 The solution to the ODE system (54) is C ( m ) ( s ) = A m s + B m K + m − X n =0 E m,n [ln( sK )] n ( sK ) γ − θ ; if s > K, m − X n =0 F m,n [ln( sK )] n ( sK ) γ + θ ; if ≤ s ≤ K, (55) for m = 1 , . . . , M , where A m = 1 a + a − λ m − X z =1 p m,z A m − z − g m ! ,B m = 1 a − λ m − X z =1 p m,z B m − z + g m ! ,E , = − ( A + B ) K ( γ + θ ) − A K θ ,F , = − ( A + B ) K ( γ − θ ) − A K θ ,E m,m − = − λp m, E m − ,m − ( m − a + 2 a ( γ − θ ) − a ] , for m ≥ ,F m,m − = − λp m, F m − ,m − ( m − a + 2 a ( γ + θ ) − a ] , for m ≥ ,E m,n = − λ P m − nz =1 p m,z E m − z,n − + ( n + 1) na E m,n +1 n [ a + 2 a ( γ − θ ) − a ] , for ≤ n ≤ m − ,F m,n = − λ P m − nz =1 p m,z F m − z,n − + ( n + 1) na F m,n +1 n [ a + 2 a ( γ + θ ) − a ] , for ≤ n ≤ m − ,E m, = − ( A m + B m ) K ( γ + θ ) − A m K + F m, − E m, θ , for m ≥ ,F m, = − ( A m + B m ) K ( γ − θ ) − A m K + F m, − E m, θ , for m ≥ , (56) and γ = 12 − r − qσ , θ = r γ + 2( r + λ + β + κ ) σ . (57) Proof.
We begin by considering the case that the employee only holds a single option. With M = 1, the general solution to ODE (54) is given by C (1) ( s ) = A s + B K + E , ( sK ) γ − θ + ˜ E , ( sK ) γ + θ ; if s > K,F , ( sK ) γ + θ + ˜ F , ( sK ) γ − θ ; if 0 ≤ s ≤ K, (58)where A = − g a + a , B = g a . (59)19y imposing that C (1) ( s ) and dds C (1) ( s ) to be continuous at the strike price K , we consider C (1) ( s ) at s = K and obtain (cid:20) γ − θ γ + θ (cid:21) (cid:20) E , − ˜ F , ˜ E , − F , (cid:21) = − K (cid:20) A + B A (cid:21) (60) ⇒ (cid:20) E , − ˜ F , ˜ E , − F , (cid:21) = − K θ (cid:20) ( γ + θ )( A + B ) − A − ( γ − θ )( A + B ) + A (cid:21) . (61)In addition, since γ − θ <
0, we will have ˜ F , = 0 to guarantee that C (1) (0) = 0. And, when κ → ∞ , the maturity τ → P -a.s., which will lead to C (1) ( s ) → ( s − K ) + . Therefore, we have˜ E , = 0. As a result, we obtain the remaining non-zero coefficients: (cid:20) E , F , (cid:21) = − K θ (cid:20) ( γ + θ )( A + B ) − A ( γ − θ )( A + B ) − A (cid:21) . (62)For M ≥
2, the general solution to ODE (54) is C ( m ) ( s ) = A m s + B m K + m − X n =0 E m,n [ln( sK )] n ( sK ) γ − θ if s > K, m − X n =0 F m,n [ln( sK )] n ( sK ) γ + θ if 0 ≤ s ≤ K. (63)Applying ODE (54), we obtain the relationship between the coefficients of C ( m ) ( s ) and thecoefficients of C ( n ) ( s ), for n ≤ m −
1, as follows: A m = 1 a + a − λ m − X z =1 p m,z A m − z − g m ! ,B m = 1 a − λ m − X z =1 p m,z B m − z + g m ! ,E m,m − = − λp m, E m − ,m − ( m − a + 2 a ( γ − θ ) − a ] ,F m,m − = − λp m, F m − ,m − ( m − a + 2 a ( γ + θ ) − a ] ,E m,n = − λ P m − nz =1 p m,z E m − z,n − + ( n + 1) na E m,n +1 n [ a + 2 a ( γ − θ ) − a ] , for 1 ≤ n ≤ m − ,F m,n = − λ P m − nz =1 p m,z F m − z,n − + ( n + 1) na F m,n +1 n [ a + 2 a ( γ + θ ) − a ] , for 1 ≤ n ≤ m − , (64)for m = 2 , . . . , M .In addition, the continuity of C ( m ) ( s ) and dds C ( m ) ( s ) around strike price K yields that ( A m + B m ) K + E m, = F m, ,A m + ( γ − θ ) E m, K + E m, K = ( γ + θ ) F m, K + F m, K . (65)20earranging, we obtain the remaining coefficients for the solution: E m, = − ( A m + B m ) K ( γ + θ ) − A m K + F m, − E m, θ ,F m, = − ( A m + B m ) K ( γ − θ ) − A m K + F m, − E m, θ . (66) For the unvested ESO, we can model the vesting time t v by the exponential random variable τ v ∼ exp(˜ κ ), where ˜ κ = 1 /t v . Then, the unvested ESO cost at time 0 is given by˜ C ( m ) ( s ) = IE (cid:26) e − ( r + α ) τ v C ( m ) ( S τ v ) (cid:12)(cid:12)(cid:12)(cid:12) S = s (cid:27) (67)= IE (cid:26) Z ∞ ˜ κe − ( r + α +˜ κ ) u C ( m ) ( S u ) du (cid:12)(cid:12)(cid:12)(cid:12) S = s (cid:27) . (68)Then we will can derive the ODE for ˜ C ( m ) ( s ): − ( r + α + ˜ κ ) ˜ C ( m ) + ( r − q ) s dds ˜ C ( m ) + σ s d ds ˜ C ( m ) + ˜ κC ( m ) = 0 for s ∈ R + , ˜ C ( m ) (0) = 0 . (69)Assuming λ + β + κ = α + ˜ κ , we could derive the solution for ˜ C ( m ) from the solution for C ( m ) in (63), which is˜ C ( m ) ( s ) = ˜ A m s + ˜ B m K + m − X n =0 ˜ E m,n [ln( sK )] n ( sK ) γ − θ + ˜ E m ( sK ) ˜ γ − ˜ θ if s > K, m − X n =0 ˜ F m,n [ln( sK )] n ( sK ) γ + θ + ˜ F m ( sK ) ˜ γ +˜ θ if 0 ≤ s ≤ K, (70)where ˜ γ = γ = 12 − r − qσ , ˜ θ = r ˜ γ + 2( r + α + ˜ κ ) σ , ˜ A m = ˜ κA m q + α + ˜ κ , ˜ B m = ˜ κB m r + α + ˜ κ , (71)21nd ˜ E m,m − = − ˜ κE m,m − R , ˜ F m,m − = − ˜ κF m,m − R , ˜ E m,m − = − ˜ κE m,m − + ( m − P ˜ E m,m − R , ˜ F m,m − = − ˜ κF m,m − + ( m − Q ˜ F m,m − R , ˜ E m,n = − κE m,n + 2( n + 1) P ˜ E m,n +1 + σ ( n + 2)( n + 1) ˜ E m,n +2 R for 0 ≤ n ≤ m − , ˜ F m,n = − κF m,n + 2( n + 1) Q ˜ F m,n +1 + σ ( n + 2)( n + 1) ˜ F m,n +2 R for 0 ≤ n ≤ m − , ˜ E m = (˜ γ + ˜ θ ) P − Q θ , ˜ F m = (˜ γ − ˜ θ ) P − Q θ , (72)with R = λ + β + κ − α − ˜ κ,P = r − q + σ (2 γ − θ − ,Q = r − q + σ (2 γ + 2 θ − ,P = ˜ F m, − ˜ E m, − K ˜ A m − K ˜ B m ,Q = ( γ + θ ) ˜ F m, − ( γ − θ ) ˜ E m, − K ˜ A m + ˜ F m, − ˜ E m, . (73)Alternatively, one can use FDM or FFT to calculate the unvested ESO cost without applyingmaturity randomization for the second time.In Figure 6, we show the cost of an unvested ESO, computed by our maturity randomizationmethod, as a function of the initial stock price S , along with the ESO payoff. As expected, theESO cost is increasing convex in S . Comparing the costs corresponding to two different jobtermination rates α ∈ { . , . } during the vesting period, we see that a higher job terminationrate reduces the ESO value. This is intuitive as the employee has a higher chance of leaving thefirm during the vesting period and in turn losing the option entirely.The maturity randomization method delivers an analytical approximation that allows forinstant computation. In Figure 7, we examine errors of this method. As we can see, as theexercise intensity λ or post vesting job termination rate β increases the valuation error decreasesexponentially to less than a penny for each option. This shows that the maturity randomizationmethod can be very accurate and effective for ESO valuation.22 Stock Price (S ) ES O C o s t =0.01=0.1(S -K) + Figure 6: The ESO cost computed using the maturity randomization method, and plotted asa function of stock price S with two different job termination rates α = 0 . , .
1, along withthe ESO payoff function ( S − K ) + for comparison. Parameters: T = 10, t v = 2, κ = 0 . κ = 0 . r = 5%, q = 1 . σ = 20%, λ = 0 . β = 0 . Exercise Intensity ( ) E rr o r (a) Job Termination Rate ( ) E rr o r (b) Figure 7: Plots of the errors of maturity randomization method as a function of exercise intensity λ and job termination rate β respectively in (a) and (b). We fix β = 1 in (a) and λ = 1 in(b). Parameters: S = K = 10, T = 10, t v = 0, κ = 0 .
1, ˜ κ = 0, r = 5%, q = 1 . σ = 20%, p m,z = 1 /m and M = 5. Common parameters: λ = 1 and β = 1.23 Implied Maturity
Given that ESOs are very likely to be exercised prior to expiration, the total cost of an ESOgrant is determined by how long the employee effectively holds the options. For each grant of M options, the exercise times are different and they depend on the valuation model and associatedparameters. Therefore, we introduce the notion of implied maturity to give an intuitive measureof the effective maturity implied by any given valuation model.Like the well-known concept of implied volatility, we use the Black-Scholes option pricingformula. The price of a European call with strike K and maturity T is given by C BS ( S t , T ) = e − q ( T − t ) S t Φ( d ) − e − r ( T − t ) K Φ( d ) , (74)where d = 1 σ √ T − t (cid:20) ln (cid:18) S t K (cid:19) + (cid:18) r + σ (cid:19) ( T − t ) (cid:21) , d = d − σ √ T − t. (75)Next, recall the ESO cost function C ( m ) ( t, s ) under the top-down valuation model in Section 2.Then, the implied maturity for m ESOs is defined to be the maturity parameter ˜ T such that C BS ( S , e T ) = C ( m ) (0 , S ) m (76)holds, with all other parameters held constant. To define implied maturity under another modelonly requires replacing the corresponding cost function on the right-hand side in (76).Through the lens of implied maturity, we can see the model and parameter effects in termsof how long the employee will hold the option under the Black-Scholes model. For example, ifthe exercise intensity λ increases, then the ESOs are more likely to be exercised early, resultingin a lower cost. Since the call option value is increasing in maturity, the implied maturity isexpected to decrease as exercise intensity increases. The plots in Figure 8 confirm this intuition.Moreover, under high exercise intensity all ESOs will be exercised very early and the contractmaturity will play a lesser role on the ESO cost and thus implied maturity. Indeed, Figure 8shows that implied maturities associated with different contract maturities T = 5 , λ increases.Next, we consider the effect of the total number of ESOs granted. Intuitively we expectthe implied maturity to increase as the number of options M increases, but the effect is farfrom linear. In Figure 9, we see that the implied maturity is increasing as M increases. Inother words, under the assumption that the ESOs will be exercised gradually, a larger ESOgrant has an indirect effect of delaying exercises, and thus leading to higher implied maturity.The increasing trends hold for different exercise intensities, but the rate of increase diminishessignificantly for large M . Also, the higher the exercise intensity, the lower the implied maturity.24 Exercise Intensity I m p li ed M a t u r i t y T=5T=8T=10
Exercise Intensity I m p li ed M a t u r i t y T=5T=8T=10
Figure 8: Implied maturity as a function of employee exercise intensity λ when the maturity T = 5 , S = K = 10, r = 5%, q = 1 . σ = 20%, p m,z = 1 /m , M = 5, t v = 0 and β = 0 .
1. In FFT: N x = 2 , x min = − x max = 10. Number of Options (M) I m p li ed M a t u r i t y =0.1=0.2=0.4 Number of Options (M) I m p li ed M a t u r i t y =0.1=0.2=0.4 Figure 9: Implied maturity as a function of number of options granted M with different exerciseintensities λ , computed using FFT or maturity randomization. Left: FFT. Right: Maturityrandomization. Common parameters: S = K = 10, r = 5%, q = 1 . σ = 20%, p m,z = 1 /m , T = 10, t v = 0 and β = 0 .
5. In FFT: N x = 2 , x min = − x max = 10.25 Conclusion
We have studied a new valuation framework that allows the ESO holder to spread out the ex-ercises of different quantities over time, rather than assuming that all options will be exercisedat the same time. The holder’s multiple random exercises are modeled by an exogenous jumpprocess. We illustrate the distribution of multiple-date exercises that are consistent with empir-ical evidence. Additional features included are job termination risk during and after the vestingperiod. For cost computation, we apply a fast Fourier transform method and finite differencemethod to solve the associated systems of PDEs. Moreover, we provide an alternative methodbased on maturity randomization for approximating the ESO cost. Its analytic formulae forvested and unvested ESO costs allow for instant computation. The proposed numerical methodis not only applicable to expensing ESO grants as required by regulators, but also useful forunderstanding the combined effects of exercise intensity and job termination risk on the ESOcost. For future research, there are a number of directions related to our proposed framework.For many companies, risk estimation for large ESO pool is both practically important and chal-lenging. Another related issue concerns the incentive effect and optimal design of ESOs so thatthe firm can better align the employee’s interest over a longer period of time.
References
Armstrong, C. S., Jagolinzer, A. D., and Larcker, D. F. (2007). Timing of employee stock option exercisesand the cost of stock option grants. Working paper, Stanford University.Bettis, J. C., Bizjak, J. M., and Lemmon, M. L. (2001). Managerial ownership, incentive contracting,and the use of zero-cost collars and equity swaps by corporate insiders.
Journal of Financial andQuantitative Analysis , 36:345–370.Bettis, J. C., Bizjak, J. M., and Lemmon, M. L. (2005). Exercise behaviors, valuation, and the incentiveeffects of employee stock options.
Journal of Financial Economics , 76(2):445470.Carmona, J., Le´on, A., and Vaello-Sebasti´a, A. (2011). Pricing executive stock option under employmentshocks.
Journal of Economic Dynamics and Control , 35:97–114.Carpenter, J. (1998). The exercise and valuation of executive stock options.
Journal of Financial Eco-nomics , 48:127–158.Carpenter, J. N., Stanton, R., and Wallace, N. (2017). Estimation of employee stock option exercise ratesand firm cost. Working paper, New York University and U.C. Berkeley.Carr, P. and Linetsky, V. (2000). The valuation of executive stock options in an intensity-based framework.
European Finance Review , 4:211–230.Cvitani´c, J., Wiener, Z., and Zapatero, F. (2008). Analytic pricing of employee stock options.
Review ofFinancial Studies , 21(2):683–724.Giesecke, K. and Goldberg, L. (2011). A top down approach to multi-name credit.
Operations Research ,59(22):283–300.Grasselli, M. and Henderson, V. (2009). Risk aversion and block exercise of executive stock options.
Journal of Economic Dynamics and Control , 33(1):109–127.Hallock, K. and Olson, C. A. (2007). New data for answering old questions regarding employee stockoptions. In Abraham, K. G., Spletzer, J. R., and Harper, M., editors,
Labor in the New Economy ,pages 149 – 180. University of Chicago Press.Heron, R. A. and Lie, E. (2016). Do stock options overcome managerial risk aversion? Evidence fromexercises of executive stock options (ESOs).
Management Science , 63(9):2773–3145. uddart, S. and Lang, M. (1996). Employee stock option exercises: an empirical analysis. Journal ofAccounting and Economics , 21:5–43.Hull, J. and White, A. (2004). How to value employee stock options.
Financial Analysts Journal , 60:114–119.Jackson, K. R., Jaimungal, S., and Surkov, V. (2008). Fourier space time-stepping for option pricing withL´evy models.
Journal of Computational Finance , 12(2):1–29.Jain, A. and Subramanian, A. (2004). The intertemporal exercise and valuation of employee options.
The Accounting Review , 79:705–743.Jennergren, L. and Naslund, B. (1993). A comment on ‘Valuation of stock options and the FASBproposal’.
Accounting Review , 68:179–183.Leung, T. and Sircar, R. (2009a). Accounting for risk aversion, vesting, job termination risk and multipleexercises in valuation of employee stock options.
Mathematical Finance , 19(1):99–128.Leung, T. and Sircar, R. (2009b). Exponential hedging with optimal stopping and application to ESOvaluation.
SIAM Journal of Control and Optimization , 48(3):1422–1451.Leung, T. and Wan, H. (2015). ESO valuation with job termination risk and jumps in stock price.
SIAMJournal on Financial Mathematics , 6(1):487–516.Marquardt, C. (2002). The cost of employee stock option grants: an empirical analysis.
Journal ofAccounting Research , 4:1191–1217.Sircar, R. and Xiong, W. (2007). A general framework for evaluating executive stock options.
Journal ofEconomic Dynamics and Control , 31(7):2317–2349.Wilmott, P., Howison, S., and Dewynne, J. (1995).
The Mathematics of Financial Derivatives . CambridgeUniversity Press.. CambridgeUniversity Press.