Analytic Pricing of Employee Stock Options∗†

Jaksa Cvitanic ‡

Zvi Wiener §

Fernando Zapatero¶

This draft: October 28, 2004.

Abstract

We introduce a model that captures the main properties that characterize employeestock options (ESO), in particular, the likelihood of early voluntary exercise and theobligation to exercise immediately if the employee leaves the firm, except if this hap-pens before options are vested, in which case the options are forfeited. We derive ananalytic formula for the price of the ESO and analyze its properties and sensitivitywith respect to the model parameters.

∗We are grateful to Kevin Murphy for very useful preliminary conversations. Existing errors are our soleresponsibility.

†Patent pending.‡Departments of Mathematics and Economics, USC, 3620 S Vermont Ave, MC 2532, Los Angeles, CA

90089. Ph: (213) 740-3794. Fax: (213) 740-2424. E-mail: cvitanic@math.usc.edu.§School of Business Administration, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem,

91905, Israel. Ph: (972) 2-588-3049. Fax: (972) 2-588-1341. E-mail: mswiener@mscc.huji.ac.il.¶FBE, Marshall School of Business, USC, Los Angeles, CA 90089-1427. Ph: (213) 740-6538. Fax: (213)

740-6650. E-mail: fzapatero@marshall.usc.edu.

Analytic Pricing of Employee Stock Options

We introduce a model that captures the main properties that characterize employee stock

options (ESO), in particular, the likelihood of early voluntary exercise and the obligation to

exercise immediately if the employee leaves the firm, except if this happens before options are

vested, in which case the options are forfeited. We derive an analytic formula for the price

of the ESO and analyze its properties and sensitivity with respect to the model parameters.

1 Introduction

Starting in the mid 80’s, stock options have been a large component of compensation pack-

ages granted by corporations. In some cases, firms have awarded employee stock options

(ESO’s henceforth) to the majority of their payroll, and stock options are almost always an

important part of the executives compensation. For example, in 1996, 39% of the compen-

sation package of CEO’s of corporations in the S&P 500 was in the form of option grants

(see Murphy, 1999). This goes up to 47% in 1999, in which 94% of companies in the S&P

500 granted options to their top executives (see Hall and Murphy, 2002).

In 1995, FASB set a standard that requires firms to expense stock-based compensation at

the moment the compensation was granted (see FASB, 1995). Firms are encouraged to use

the “fair value” in order to compute the value of the compensation, but are allowed to use

the “intrinsic value” – market price of the stock minus strike price – which was customary

until then. Since ESO’s are typically granted at the money, the intrinsic value is zero, which

results in no expense recorded at the time of the grant, and this is probably one of the

reasons that helped their popularity. However, pressure is mounting to force corporations

to consider options as a cost at the moment they are granted. Typically, ESO’s have a

vesting period (up to four years) and they are worthless if the employee leaves the company

or is terminated before vesting. However, the argument goes, they represent a potentially

significant liability for the company, which should be recognized at the moment they are

issued, as required by FASB (1995), but at a cost representative of the liability. As a result

of this pressure, an increasing number of firms has started to expense ESO’s.1

This raises the need to come up with a reasonable valuation for these options. In fact,

the lack of a good pricing criterium has been used as an argument against expensing op-

tions. There are two obvious extremes in broaching this problem. On one hand we have

the approach initially used of computing the intrinsic value, On the other extreme of the

spectrum, there is the possibility to use the Black and Scholes (1973) formula (BS hence-

forth). This formula is very useful for standard traded options and easy to apply. However,

its use to price ESO’s would greatly overestimate, as we will argue next, the true value of

the stock options for the firm. BS applies to plain vanilla traded options. ESO’s differ from

them in several aspects. First, they cannot be traded and/or their risk cannot be diversified

away: the only possible action the options holder can take is to exercise them, after they

are vested. Second, the maturity of standard options is fixed. In the case of ESO’s, the

employee is forced to exercise them if she/he leaves the company before maturity (typically

in a period of up to 90 days after departure), and they have already been vested (otherwise

1FASB (2004a), that proposes amendments to the current accounting standards for share-based compen-sation, imposes the recognition of option compensation as a cost at the moment of the grant. The recentlyissued IFRS 2 (2004) also follows this approach.

1

they are forfeited). Third, employees tend to exercise options early, very often just after

they are vested, forfeiting in this way the time value of the option (ESO’s might have a

maturity of many years, after vesting). All three differences imply that the value of the ESO

is lower than the price resulting from BS.2 For a detailed discussion about the differences

between standard traded options and option grants, see Rubinstein (1995). For a thorough

discussion of the arguments in favor of expensing versus not expensing -plus a broad literary

review of pricing methods- see Chance (2004).

In this paper we try to provide a pricing formula that recognizes the value of the liability

implied by the option grants, but takes into consideration the characteristics of ESO’s de-

scribed above, as opposed to the plain vanilla traded options considered in the BS paradigm.

The paper attempts to be a contribution to the problem firms face when expensing ESO’s,

and therefore takes the point of view of the corporation. It is noteworthy that it would be

possible to approach the problem from the point of view of the employee, and prices would

likely be different. This is the approach taken in some of the papers we discuss below. In

particular, the average employee is a risk-averse investor that, however, is not allowed to

diversify the risk embedded in the options grant, to a large extent (see Jin, 2002). However

the firm (or more properly, its shareholders) can diversify the risk deriving from issuing op-

tion grants to employees. As a result, the option grant will be worth more for the firm than

for the employee. The reason that explains why firms choose this type of compensation (as

opposed to a lower cash payment) is that they expect the employee, through the incentive

effects of the option, to offset (and hopefully gain more) the loss in value that results from

granting the options.3 See also Carpenter (1998) for a discussion of both approaches and an

empirical calibration.

The spirit of our approach is similar to Hull and White (2004). First, we take into account

that a proportion of the options in the grant will mature early, as a result of the employee

leaving the firm or being terminated. The rate of exit (as Hull and White, 2004, call it), is

easy to estimate in practice (Carpenter, 1998, has estimates of this parameter). In order to

incorporate it into our pricing formula, we mimic the approach used in the default bonds

literature (see, for example, Duffie and Singleton, 1999).4 Second we take into consideration

2FASB (2004a) acknowledges that BS is not an appropriate reflection of the fair value of option grants,and suggests the use of binomial trees instead. IFRS 2 (2004) requires the use of a generally acceptedvaluation method that incorporates relevant parameters -as a binomial tree or the model we present in thispaper would do.

3There is a large body of literature that focuses on the incentive effects of ESO’s, but we ignore it in thispaper. See for example, Jensen and Murphy (1990). Palmon, Bar-Yosef, Chen and Venezia (2004) studythe optimality of option grants (with choice of the strike price) versus stock grants.

4Jennergren and Naslund (1993) suggest to use a Poisson process to account for the rate of exit. Theyshow how to adjust BS to incorporate this rate.

2

that employees tend to exercise early, even if they do not leave the firm.5 To capture the

effect of early exercise, we include as a feature of our pricing approach a barrier, such that

when the price of the stock hits the barrier, the option is exercised. The implicit assumption

is that when the option is deep enough in-the-money, the employee will collect its value and

avoid the risk of a possible subsequent drop in price, maybe associated with a termination of

her/his contract. The barrier represents the point at which the employee decides to collect

the payoff and forfeit the remaining time-value of the option. The advantage of our approach

with respect to Hull and White (2004) is that we provide an analytic expression that can be

directly computed, after the rate of exit and barrier are estimated. Hull and White (2004)

rely on numerical methods: they use a binomial tree (as in Cox, Ross and Rubinstein,

1985) to compute the price of the option after the parameter values are estimated. Carr

and Linetsky (2000) develop a pricing formula exclusively based on the rate of exit. They

consider two possible models, depending on whether the rate of exit is given by a constant

intensity parameter (which is larger when the option is in-the-money) or it depends on

how deep in-the-money the option is. They provide some numerical examples, but only

for the case in which the option is already vested. Unlike theirs, our formula does not

require numerical integration, even when vesting is included. The binomial tree approach

of Hull and White (2004) converges very slowly and non-monotonically (which also creates

problems for hedging computations). As an example of the differences among the approaches

discussed so far, consider an option grant with price of the stock 100, strike price 100, time

to maturity 10 years, vesting period 3 years, interest rate 6% and volatility 50 %. The Black

and Scholes price of this option, with maturity 10 years is 69.21; the Black and Scholes price

with maturity 3.25 years (that is, shortly after vesting) is 41.21; the price using a binomial

tree as in Hull and White (2004) with 50 steps is 33.24; the price with the model presented

in this paper is 32.29.6 We also mention the paper by Raupach (2003): he considers a

model very similar to the one we study in this paper but solves the integrals numerically.

He calibrates the model to the data.

An alternative approach is presented in Bulow and Shoven (2004): they propose the use

of BS, but with only 90 days to maturity, so as to reflect the period during which the option

does not expire with certainty. They suggest to upgrade values quarterly: if at the end of the

quarter the employee is still with the company, the time-value of the option corresponding

to another 90 days will be expensed. Their argument is that this approach is consistent

with the way labor costs are typically accounted for, on a monthly basis, and only taking

5Huddart (1994) discusses the optimal early exercise policy of a risk-averse, utility-maximizing optionholder. Huddart and Lang (1996) study this issue empirically and point that, although it is pervasive, theearly exercise rule is not uniform.

6For the binomial tree and the model presented in this paper, we need three additional parameter valuesthat we will describe later in detail. For this example, these parameter values are ÃL = 200; α = 0;λ = 0.15.

3

into consideration the cost for that period (one month) and not the expected present value

of the contract with the employee. Additionally, they suggest several possible directions to

account for vesting. These suggestions range from expensing the options the day they are

granted -regardless of the vesting period- to waiting until they become vested.

As we mentioned above, it is also possible to price option grants from the point of view of

the employee. Lambert, Larcker and Verrechia (1991) price the options for the manager by

computing the “certainty equivalent,” or cash amount that will leave the executive indifferent

in terms of the expected utility between a guaranteed cash payment and the option grant.

This approach is also used by Detemple and Sundaresan (1999), who compute numerically

the optimal exercise policy and certainty equivalent. Hall and Murphy (2002) also study the

incentive effects of option-based compensation. As in the case of Hull and White (2004),

and the formula suggested in this paper, they show that for reasonable parameters of risk-

aversion, the resulting price of the options is between BS and the intrinsic value. Departing

from the certainty equivalent approach, Ingersoll (2003) computes the price of the options

from the point of view of the employee by using the risk-neutral probability that corresponds

to the constrained optimization problem of the employee who cannot trade options at will.

There are several features of option grants (some of them discussed in the literature)

that we ignore in this paper (also ignored in Hull and White, 2004). These are not general

features, plus they greatly complicate computations. First, we do not consider the possibility

of resetting. This is the practice of exchanging the terms of the options grant at some point

before maturity, typically when the stock has dropped in price and options are out-of-the-

money. This is not part of the compensation contract, but it often takes place, mostly as a

way to keep disgruntled employees in the company. Its practice has become so general that

some authors point that it should be an element of the pricing of options (see Acharya, John

and Sundaram, 2000, for a discussion of this issue). Brenner, Sundaran and Yermack (2000)

present a model to price these options and discuss the effects on price of the possibility of

resetting. Stoughton and Wong (2003) study the pricing and resetting of stock options in

a labor-competitive environment. Another feature that is some times incorporated into the

option grant but we do not consider in this paper is reloading. This is the provision by

which more options will be granted when the options of the initial package are exercised. Of

course, this increases the incentives to exercise early. Dybvig and Lowenstein (2003) focus

on this problem. Sircar and Xiong (2004) use a dynamic programming approach to find

the price of ESO’s in a similar setting, both with resetting and reloading features, but in a

model with infinite maturity. Additionally, in their paper, early exercise is always random

(the “exit rate” discussed above). We also ignore the dilution effects of option grants which,

although probably not very important in general, are clearly a factor. Similarly, we do not

consider the possibility of default of the company, which will have a negative effect on the

price of the options, since they will have zero payoff if the firm defaults before the ESO is

4

vested. Finally, in our model we ignore all type of agency considerations. In particular, it

is clear that some employees (mainly executives) could affect with their actions the price

of the stock. Additionally, vesting might affect the timing of the decision to leave the firm

-an employee whose grant is in the money and close to vesting might want to wait- and the

decision of the firm to fire the employee.

We structure this paper as follows. In the next section we explain the assumptions of

our model, introduce the pricing formula and analyze its components and properties. In

section 3 we present some numerical examples and illustrate the usefulness of the formula.

We close the paper with some conclusions.

2 Pricing Model

First we will discuss the assumptions of the model and its economic foundations. Then we

present the model and discuss its features. In the next following section we provide some

examples.

2.1 Foundations

Our approach attempts to capture the following stylized facts:

• Typical options granted as compensation have a long maturity and include a long

vesting period during which the option cannot be exercised and it is forfeited if the

employee leaves the firm (whether the employee decides to leave or is fired).

• If the employee leaves the firm after the option right has vested, the employee must

exercise the option quickly (typically has a period of up to ninety days) and after that

the option is forfeited. This is the case whether the employee decides to leave or is

fired.

• Employees tend to exercise options well before maturity. Often, as soon as options

rights are vested, the employee exercises the option, even when there are still several

years left until maturity. Of course, this is only the case if the option is in the money

and, arguably, it is more likely to happen the deeper in the money the option is and

the shorter is the time left until maturity.

Our objective is to price the option from the point of view of the firm. As we pointed

in the literature review of the previous section, the firm is considerably less constrained

than the option holder with regard to risk diversification. While the employee cannot hedge

in general the risk represented by holding the options (there are some companies that are

5

willing to buy option grants under some conditions), the firm is mostly unconstrained with

respect to the diversification of the risk represented by having a short position in the options.

Therefore, it is reasonable to argue that, while the employee is very risk-averse concerning

the expected payoff implicit in the long position in options, it is safe to assume that the firm

is risk-neutral concerning the potential liability.

Our model intends to compute the expected (risk-neutral) payoff of a call option that

can only be exercised after a vesting period. Then, we assume that there is a barrier such

that, if the barrier is crossed, the option is exercised at that point. This barrier captures

the fact that options are exercised early. We allow the barrier to be decreasing. That would

capture the fact that the employee is more likely to exercise the option, (that is, for a lower

price of the stock), the closer the exogenous maturity. Additionally, we assume that there

is an exogenous exit rate of expiry of the option, that captures the possibility that the

employee will leave the firm (willingly or not). Thus, the maturity of the option is one of

the parameters of our formula, but it is possible that the option will expire before that final

maturity date.

2.2 The Model

As in the Black and Scholes setting, we assume that the stock price follows a lognormal

process,

dSt/St = µdt + σdWt; S0 = s

which, under the risk-neutral pricing measure becomes,

dSt/St = rdt + σdWt

with constant parameters σ, r and µ (we only use this at the end of this section in order to

compute the probabilities of early exercise for different reasons). We denote by s the current

price of the stock. There is another source of uncertainty in the model: a Poisson process

that characterizes the exit rate with a parameter that we explain below. Additionally, in

our model we need the parameters that characterize the option grant: time left until vesting

T0 ≤ T , maturity of the option T and strike price K (typically, K = s). -obviously, it has

to be the case that T0 ≤ T . Finally, we have the parameters that capture:

• The usual exercise patterns of ESO’s. We need two parameters, the level L of the

fictitious barrier at which the employee exercises the option, and the rate α of decay

of that barrier as maturity approaches.

• The constant attrition or exit rate of employees, that we denote by λ and is the

intensity of the Poisson process we mentioned above.

6

In fact, we allow the exit rate λ0 before the vesting period to be different from the exit

rate λ after the vesting.

For analysis purposes, we price the option under four scenarios (that we call cases), that

we denote A, B, C and D. Case A represents the situation in which the ESO is immediately

vested and is exercised only when the underlying hits the optimal barrier, or at maturity. In

case B we assume that the ESO is immediately vested but it is only exercised at the random

arrival time whose intensity we denote by λ, or at maturity. Case C is a combination of the

two previous, the ESO is immediately vested, and will be exercised under the conditions

of case A or the conditions of case B, whichever comes earlier. Case D is like C but with

a vesting period. We now present the price of the ESO in each of the four cases, as an

expectation. The analytic version of the formula and the proof is in the Appendix.

The discounted call option payoff is given by

Ct = e−rt(St −K)+

for t after the vesting period, and zero prior to vesting. Let min(τ, T ) be the time when the

option is exercised or expires, where τ is a random time and T is the maturity. Introduce

the conditional distribution of τ :

Ft := Pt(τ ≤ t)

Here, Pt(·) is the probability conditional on the information available from the stock prices

up to time t. The formula for the expectation of random variable Cmin(τ,T ) is

E[Cmin(τ,T )] = E

[∫ T

0CudFu + CT (1− FT )

](2.1)

Intuitively, the first term corresponds to conditioning on τ = u and integrating over u, while

the second term corresponds to τ = T , which happens with conditional probability 1− FT .

We assume that the option is either exercised by the employee when the stock reaches

level Leαt after the vesting period, or it is exercised/expires due to the employee quitting

or being fired, with intensity λ. If quitting/firing happens before the vesting period, the

employee gets nothing. We now list the option price in all four cases, A, B, C and D, as

defined above.

The explicit formulas for all the cases are given in the appendix, together with the proofs.

Case A: Exercise time as a hitting time of desired level, no vesting period, i.e.,

λ = 0, T0 = 0.

Consider the case in which the option is exercised by the employee the first time the

stock price hits the desired level Lt = Leαt before maturity T , where α is a constant number

such that Lt > K for t ≤ T :

τ = TL := inf{t > 0, St ≥ Lt} = inf{t > 0, Ste−αt ≥ L}

7

Then, the option price is equal to, for s < L,

P1 + P2 := E[e−rT (ST −K(T ))+1{τ>T}] + E[(Le−rατ −Ke−rτ )1{τ≤T}] (2.2)

Here, the term P1 corresponds to the case when the stock never reaches the desired level

Lt = Leαt, while P2 corresponds to the option being exercised when the stock reaches the

desired level.

Case B: Intensity based model for exercise time, no vesting period, i.e.,

L = ∞, T0 = 0

Here we assume, as in Carr and Linetsky (2000), that the option is either exercised

according to an arrival of a process with a given intensity, or expires at maturity. More

precisely, suppose now that the conditional distribution of the exercise time is

F (t) = 1− e−∫ t

0λsds

and that

λt = g(t, St)

for some function g. In other words, conditionally on knowing λ, the exercise time is the

first arrival of a Poisson process with the mean arrival rate 1t

∫ t0 λsds per unit time.

Then the price can be written as

E

[∫ T

0(St −K)+g(t, St)e

−∫ t

0(r+g(u,Su))dudt + (ST −K)+e−

∫ T

0(r+g(t,St))dt

]

The PDE for this is

Vt(t, s)+1

2σ2Vss(t, s)+rVs(t, s)−(r+g(t, s))V (t, s)+g(t, s)(s−K)+ = 0, V (T, s) = (s−K)+ .

In the case when the arrival rate λ is constant the price is

E

[∫ T

0λ(St −K)+e−(r+λ)tdt + (ST −K)+e−(r+λ)T

]

The first term corresponds to expiration/exercise before maturity (at times t, with “proba-

bilities” λe−λt), and the second term to expiration/exercise at maturity.

Case C: A combination of the intensity and the hitting time model for exercise

time, no vesting period, i.e. T0 = 0

8

We now assume that the exercise time is

τ = min(TL, Tλ),

where TL is the first time the stock hits the level Lt = Leαt, and Tλ is a time having intensity

λ. When α = 0, this is also the model of Hull and White (2004), but in a binomial tree

model. Assume that TL and Tλ are conditionally independent. Then, we have

F (t) = Pt(τ ≤ t) = Pt(TL ≤ t) + Pt(Tλ ≤ t)− Pt(TL ≤ t)Pt(Tλ ≤ t)

= 1{TL≤t} + Pt(Tλ ≤ t)− 1{TL≤t}Pt(Tλ ≤ t) = 1{TL≤t} + Pt(Tλ ≤ t)1{TL>t}

= 1− e−λt1{TL>t}. (2.3)

Therefore, by (2.1), the price is equal to

J1 + J2 + J3 = E[(Le−(rα+λ)TL −Ke−(r+λ)TL)1{TL≤T}] (2.4)

+E[∫ T

0λe−(r+λ)t(St −K)+1{TL>t}dt] + E[e−(r+λ)T (ST −K)+1{TL>T}]

Here, J1 corresponds to exercising at the desired level, J2 to being fired/quitting at intensity

λ, and J3 to exercise/expiry at maturity.

Case D: Combined model with a vesting period

Suppose now that there is a vesting period [0, T0], T0 < T , in which the employee may

quit or be fired with intensity λ0, and gets nothing from the option. After the vesting period

the intensity of quitting/being fired is λ, and the employee will exercise when the stock

reaches the desired level Lα(t−T0). We denote by T 0λ the time of quitting/being fired and

T 0L = min{t ∈ [T0, T ], St ≥ Leα(t−T0)}

so that the time of exercise/expiry is

τ = min{T 0L, T 0

λ}As before, we find that

F (t) = 1− e−λ0t1{T 0L>t}, t ≤ T0

and

F (t) = 1− e−λ0T0−λ(t−T0)1{T 0L>t}, t > T0.

Therefore, similarly as (2.4), we get that the price is equal to

K11 + K12 + K2 + K3 =

e(λ−λ0)T0E[(Le−αT0e−(rα+λ)T 0L −Ke−(r+λ)T 0

L)1{T 0L≤T,ST0

<LT0}]

+e(λ−λ0)T0E[e−(r+λ)T0(ST0 −K)+1{ST0≥LT0

}]

+e(λ−λ0)T0E[∫ T

T0

λe−(r+λ)t(St −K)+1{T 0L>t}dt] + e(λ−λ0)T0E[e−(r+λ)T (ST −K)+1{T 0

L>T}]

9

We interpret the previous parameters using Figure 1, which explains the different exercise

possibilities in the model. K11 corresponds to exercising at the desired level after the vesting

period, the segment B in figure 1; K12 to exercising right after the vesting period, the segment

C in figure 1; K2 to being fired/quitting at intensity λ after the vesting period, in region D;

and K3 to exercise/expiry at maturity, segment E in figure 1.

The PDE and the boundary conditions for the price, when the intensity is λ(t, s), are

given by:

For t ≥ T0 and s < Leα(t−T0), we have

Vt(t, s) +1

2σ2Vss(t, s) + rVs(t, s)− (r + λ(t, s))V (t, s) + λ(t, s)(s−K)+ = 0 ,

with boundary conditions

V (t, Leα(t−T0)) = Leα(t−T0) −K ;

V (T, s) = (s−K)+ ;

for t < T0, we have

Vt(t, s) +1

2σ2Vss(t, s) + rVs(t, s)− (r + λ(t, s))V (t, s) + λ(t, s)(s−K)+ = 0 ,

V (T0, s) = s−K, s ≥ L .

There are two straightforward extensions we have not included here. In the first place,

dividends. As in the usual Black and Scholes pricing approach, if the dividends are paid

at a continuous rate q, then we simply replace the stock drift rate r with r − q as the

value for the variable r in the formulas. If there is a fixed and known amount of dividends

paid, then we replace the current stock price value s with s − D(0) where D(0) is the

present value of future dividends. Very often, however, the ESO is adjusted for dividends.

Second, forfeitures: Assume that there is also a possibility that during the vesting period

the employee will quit the firm, and not be able to exercise the option. For example, she

might go work for a competing company, in which case she forfeits the option compensation.

If this event is also modelled by an exponential distribution with intensity rate λf , and if

it is independent of other random variables in the model, as in the credit risk literature we

get the result that we simply have to add λf to the discount rate r.

3 Numerical Examples

In this section we use the formula derived in the appendix to perform some comparative

static analysis and study properties of prices of ESO’s. Although the formula involves

10

several terms, it can be easily computed in standard commercial software.7 An important

advantage of the formula is that it is differentiable with respect to all parameters of the

model. This provides the firm with a powerful tool for hedging purposes, as well as to

study the sensitivity of the price of the option, both to the parameters of the underlying

stock price and to the parameters that characterize the exercise policy and attrition rate.

On the other hand, binomial pricing converges very slowly at a speed that depends on the

parameter values considered. Additionally, the convergence is not uniform. This might be

an important obstacle when computing hedging portfolios.8 We point out the following

Remark 3.1 The price that we get for the employee option is the limit of the binomial tree

procedure of Hull and White (2004), if we use the usual parametrization that results in the

convergence of the Cox, Ross, Rubinstein (1979) price to the Black-Scholes price. The only

addition is that, at each step in the tree, the employee may quit/get fired with probability

λ∆t, where λ is the exit rate.

In all pricing exercises we have considered the four cases described in the previous section:

in case A options are automatically vested and they would be exercised if the underlying hits

the barrier L > K; in case B options are also automatically vested but they are exercised at

some exogenously determined time that happens randomly with intensity λ, the exit rate;

case C combines A and B, so that the exercise happens either when the underlying hits the

barrier or when the randomly determined time arrives, whatever comes first; finally, case

D is the most complete and is equivalent to case C but with a vesting period T0. In all

tables we also include the corresponding Black and Scholes prices (that we denote BS) for

comparison purposes.

In table 2 we present all five of those ESO’s prices as a function of the price of the un-

derlying. We do that for several combination of parameter values of the specific parameters

of the ESO (L and λ) and reasonable values for all other parameters required for option

pricing.

In table 3 we study the effect of the vesting period on the price of the ESO. We analyze

that effect for a combination of parameter values that capture the likelihood that the em-

ployee will exercise early (this likelihood is high when L is low and s is close to it) and the

likelihood that the employee will be fired or decide to leave the firm (this likelihood is high

when λ is high).

In table 4 we study the sensitivity of the price of the ESO to changes in the main pricing

parameters (aside from the price of the underlying) required to price standard options. That

7We used Mathematica 5.0.8In table 1 we show the convergence of a tree for different number of steps for an example with reasonable

parameter values. In figure 2 we show the convergence to the option price for different numbers of steps.We observe that, for all our exercises, the binomial tree always overestimates the price of the ESO.

11

allows us to compare hedging strategies of an ESO with hedging strategies for a standard

call.

From these tables we can derive a number of conclusions:

• Not surprisingly, the Black and Scholes formula greatly overestimates the price of the

ESO, even for cases of a high barrier L (the employee is less likely to exercise early)

and low λ (low probability of early departure). For the parameter values used in our

exercises, the Black and Scholes price is between two and three times as high as that

of the ESO computed with the formula suggested in this paper.

• The derivatives of the price of the ESO with respect to the basic parameters of the

Black and Scholes formula (price of underlying, interest rate, time to expiration and

volatility) have the same signs as those of the Black and Scholes formula, but are

smaller. For example, an increase in the price of the underlying translates into an

increase in the price of the ESO, but the delta is lower than in the Black and Scholes

price because of the barrier and possibility of firing. The same applies to the rho,

theta and vega of the option. In fact, in table 4 we observe that the price of the ESO

is hardly sensitive to changes in interest rate and time to maturity: intuitively, it is

unlikely the option will reach maturity. However, volatility is still an important factor.

• The vesting period affects the price of the ESO in two simultaneous and opposite

ways. It has a negative effect on the price of the option because the employee can be

fired before the option is vested and get nothing (regardless of whether the options

is in or out-of-the-money). However, it has a positive effect on the price of the ESO

because it prevents the employee from exercising the option early. When the price of

the underlying is close to the barrier, volatility is high and interest rate is high (so

that it is very likely the barrier will be reached soon), and additionally the probability

of being fired is low, an increase in the vested period means an increase of the price

of the ESO. However, when the previous parameters take the opposite values (so that

the barrier will not be reached soon, but there is a high probability of getting fired)

an increase in the vesting period reduces the price of the ESO.

Additionally, as we show in the Appendix, we can compute the probability that the

option will be exercised before maturity. Altogether, we compute the probabilities of five

mutually exclusive scenarios that can characterize the life of the ESO. Figure 1 illustrates

these scenarios.

1. First, it is possible that the employee will leave, at rate λ0, the firm before the ESO

is vested, in which case she gets nothing. We denote this probability by P1 and it

corresponds to the region A of figure 1.

12

2. Then, the employee might decide to exercise the option right after vesting. We denote

the probability of this event P2, corresponding to segment B at time T0 in figure 1.

3. If the option is still unexercised after T0, it is possible that the employee might decide

to exercise early if the underlying hits the barrier, corresponding to the segment C in

figure 1. We denote the probability of this event P3.

4. Alternatively, if the option is still unexercised after T0, it is possible that the employee

might leave the firm or be fired before the ESO expires or the barrier is reached. That

can happen at the exogenous rate λ in region D. We denote by P4 the probability of

that event.

5. Finally, if none of the above has happened, the option will attain maturity and be ex-

ercised then, corresponding to segment E in figure 1. We denote by P5 the probability

of this event.

Obviously, it has to be the case that

P1 + P2 + P3 + P4 + P5 = 1.

In the Appendix we derive formulas for these probabilities. We present some results in

table 5. Results there are consistent with those of tables 2-4 and are, overall, intuitive. We

just point out that, as the barrier increases, the probability of early exercise upon vesting

obviously decreases, however, this might increase the probability of early exercise at some

point after vesting.

The importance of probabilities P1, P2, P3, P4 and P5 is the fact that they can be easily

estimated in practice and provide the grounds to calibrate the parameter models L, α and

λ for pricing computations. One of the problems brought up in debates about the fair price

approach to expensing ESO’s (see FASB 2004b) is the lack of a uniform criterium. The

formula introduced in this paper only requires three “subjective” parameters, L, α and λ

(we do not include here volatility, which is a problem even if the Black and Scholes formula

is used). It would be easy to introduce a criterium about computation of the probabilities,

that would pin down the values for L, α and λ.

In addition, we also derive analytic formulas for the expected life of the option, as well

as for the expected stock price at expiry/exercise. These can also be used for calibration

purposes.

4 Conclusions

In this paper we derive an analytic expression for the price of an employee stock option

(ESO). The option has a vesting period. The employee is likely to exercise before maturity,

13

after the option vests. Additionally, it is possible that the employee will be fired or quit,

which happens randomly. Prices of ESO’s are considerably lower than the equivalent Black

and Scholes prices and less sensitive to changes in parameter values, since it is likely that the

option will not reach maturity. Since we price the option from a risk-neutral perspective –

suitable to the firm that grants the option – we find that vesting has two effects: it increases

the price of the option by preventing the employee to exercise early (and forfeit the time-

value of the option), but simultaneously it has a negative effect because the employee could

be fired or leave the firm before vesting, in which case the option is worthless. The model

we describe in the paper could be applied to a firm by considering a ladder of different L

values. This characterizes the optimal voluntary early exercise of the employee. By using a

ladder of barriers, we could account for the diversity of risk-aversion/liquidity needs across

the different employees. Additionally, we point out that, since options often do not reach

maturity, we could simplify some of the previous expressions by considering infinite maturity

(as in Sircar and Xiong, 2004).

14

References

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Stock Options,” Journal of Financial Economics 57 (2000), 65–101.

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a Binomial Approach, Review of Financial Studies 12 (4) (1999), 835–872.

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15

[15] Financial Accounting Standards Board, “Minutes of the FASB Open Roundtable Dis-

cussions on Share-Based Payment held on Palo Alto, California on June 24 2004, and

in Norwalk, Connecticut on June 29, 2004,” (2004b).

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(1994), 207-231.

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FASB Proposal’,” Accounting Review 68 (1993), 179–183.

[23] M. Jensen and K. J. Murphy, “Performance Pay and Top-Management Incentives,”

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New York, 1991.

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tive Compensation,” Journal of Accounting Research 29 (1991), 129–149.

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Options for Effort Averse Executives,” working paper, Rutgers University (2004).

16

[29] P. Raupach, “The Valuation of Employee Stock Options - How Good is the Standard?”

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trial Competition,” working paper, UC Irvine, (2003).

17

Appendix

We first list the explicit formulas for all the cases. The notation used in the formulas is

explained after the formulas are listed.

Case A: The price when stopping at the hitting time:

If s < L then the price is

Pα1 (s) + LP (s, T, yα

−, yα+)−KP (s, T, yα

−, y)

If s ≥ L than the price is s− L.

Case B: The price when stopping at the random time:

I1(s, T, K, r0, b0)+I2(s, T, K, r0, r0, c0)+e−λT sN(K

σ√

T+

√T

σy0

+)−Ke−(r0+λ)T N(K

σ√

T+

√T

σy0−)

Case C: The price in the combined case, no vesting period

e−λT P1(s) + LP (s, T, yα−, cα)−KP (s, T, yα

−, c)

+I1(s, T, K, r0, b0) + I2(s, T, K, r0, r0, c0)− I1(s, T, L, rα, bα)− K

LI2(s, T, L, r0, rα, c)

−(

L

s

) 2rασ2 −1

[I1(

L2

s, T,K, r0, b0) + I2(

L2

s, T, K, r0, r0, c0)− I1(

L2

s, T, L, rα, bα)

−K

LI2(

L2

s, T, L, r0, rα, c)

]

Case D: The price in the combined case with vesting period T0

K11 + K12 + K2 + K3

A Notation for the formulas

The notation used above is explained in the following: First, we denote by N the standard

normal distribution function, and by n = N ′ its density. Moreover, we introduce

Kα(T ) = Ke−αT , rα = r0 − α .

xY =log(Y/s)

σ√

T0

− (r0 − σ2/2)

√T0

σ(A.1)

yα+ = rα +

σ2

2, yα

− = rα − σ2

2, y =

√(yα−)2 + 2σ2r0

Kα(T ) = log(s/Kα(T )), L = log(s/L)

18

bα :=√

(rα + σ2/2)2 + 2σ2λ

cα :=√

(rα − σ2/2)2 + 2σ2(λ + rα), c :=√

(rα − σ2/2)2 + 2σ2(λ + r0)

Pα1 (s) = sN

(Kα(T )

σ√

T+

√T

σyα

+

)−Kα(T )e−rαT N

(Kα(T )

σ√

T+

√T

σyα−

)

−sN

(L

σ√

T+

√T

σyα

+

)+ Kα(T )e−rαT N

(L

σ√

T+

√T

σyα−

)

−(

L

s

) 2rασ2 −1

[L2

sN

(1

σ√

Tlog(

L2

sKα(T )) +

√T

σyα

+

)

−Kα(T )e−rαT N

(1

σ√

Tlog(

L2

sKα(T )) +

√T

σyα−

)

−L2

sN

(− L

σ√

T+

√T

σyα

+

)+ Kα(T )e−rαT N

(− L

σ√

T+

√T

σyα−

)](A.2)

P (s, T, µ, y) =(

L

s

)µ−y

σ2

N

(log(s/L)

σ√

T+

√T

σy

)+

(L

s

)µ+y

σ2

N

(log(s/L)

σ√

T−√

T

σy

)(A.3)

I1(x, T, z, r, b) = x

[1{x>z} +

1

21{x=z} − e−λT N

(log(x/z)

σ√

T+

√T

σ(r + σ2/2)

)]

+x(

z

x

) r−b

σ2 + 12

{r + σ2/2

b

[N

(log(x/z)

σ√

T+

√T

σb

)− 1{x>z} − 1

21{x=z}

]

+1

2

[1− r + σ2/2

b

] [N

(log(x/z)

σ√

T+

√T

σb

)+

(z

x

) 2bσ2

N

(log(x/z)

σ√

T−√

T

σb

)

−[1 +

(z

x

) 2bσ2

]1{x>z} − 1{x=z}

]}

I2(x, T, z, R, r, c) = − λz

λ + R

[1{x>z} +

1

21{x=z} − e−(λ+R)T N

(log(x/z)

σ√

T+

√T

σ(r − σ2/2)

)]

− λz

λ + R

(z

x

) r−c

σ2 − 12

{r − σ2/2

c

[N

(log(x/z)

σ√

T+

√T

σc

)− 1{x>z} − 1

21{x=z}

]

+1

2

[1− r − σ2/2

c

] [N

(log(x/z)

σ√

T+

√T

σc

)+

(z

x

) 2cσ2

N

(log(x/z)

σ√

T−√

T

σb

)

−[1 +

(z

x

) 2cσ2

]1{x>z} − 1{x=z}

]}(A.4)

19

B(a, b, c) :=∫ xL

−∞eaxN(bx + c)n(x)dx = e

a2

2 P (X ≤ xL, Y ≤ c) (A.5)

where (X, Y ) has a bivariate normal distribution with

µX = a, µY = −ab, σ2X = 1, σ2

Y = 1 + b2, ρ = − b√1 + b2

Q =

√T0

T − T0

(A.6)

d0(Y ) =1

σ√

T − T0

[log(s/Y ) + (r0 − σ2/2)T0

](A.7)

d1(Y ) =1

σ√

T − T0

[log(

L2

sY)− (r0 − σ2/2)T0

](A.8)

K11e−(λ−λ0)T0

= Le−αT0e−(rα+λ)T0

(L

s

) yα−−cα

σ2

ecα−yα

−σ2 (r0−σ2/2)T0B

((cα − yα

−)√

T0

σ,Q, d0(L) +

√T − T0

σcα

)

− Ke−(r0+λ)T0

(L

s

) yα−−c

σ2

ec−yα

−σ2 (r0−σ2/2)T0B

((c− yα

−)√

T0

σ,Q, d0(L) +

√T − T0

σc

)

+ Le−αT0e−(rα+λ)T0

(L

s

) yα−+cα

σ2

e−cα+yα

−σ2 (r0−σ2/2)T0B

(−(cα + yα

−)√

T0

σ,Q, d0(L)−

√T − T0

σcα

)

− Ke−(r0+λ)T0

(L

s

) yα−+c

σ2

e−c+yα

−σ2 (r0−σ2/2)T0B

(−(c + yα

−)√

T0

σ, Q, d0(L)−

√T − T0

σc

)

K12

= e(λ−λ0)T0e−λT0 [sN

(√T0

σ(r0 +

σ2

2) +

log(s/L)

σ√

T0

)−Ke−r0T0N

(√T0

σ(r0 − σ2

2) +

log(s/L)

σ√

T0

)]

C0(Y ) = se(r0−σ2/2)T0B

(σ

√T0, Q, d0(Y ) +

√T − T0

σ(rα + σ2/2)

)

−Ke−α(T−T0)e−rα(T−T0)B

(0, Q, d0(Y ) +

√T − T0

σ(rα − σ2/2)

)

C1(X) = L(

L

s

) 2rασ2

e−2rασ2 (r0−σ2/2)T0B

(−2rα

σ

√T0,−Q,−d0(L

2/X) +

√T − T0

σ(rα + σ2/2)

)

− Ke−α(T−T0)(

L

s

) 2rασ2 −1

e−rα(T−T0)−( 2rασ2 −1)(r0−σ2/2)T0

×B

(σ2 − 2rα

σ

√T0,−Q,−d0(L

2/X) +

√T − T0

σyα−

)

20

K3 = e(λ−λ0)T0e−λT−r0T0 [C0(Kα(T − T0))− C0(L)− C1(Kα(T − T0)) + C1(L)]

D1(Y,R, r, b) =∫ xL

−∞I1(p(T0, s, x), T − T0, Y, r, b)n(x)dx

= −se(R−σ2/2)T0−λ(T−T0)B

(σ

√T0, Q, d0(Y ) + (r +

σ2

2)

√T − T0

σ

)

+1

2[1 +

1

b(r +

σ2

2)]s

(Y

s

) r−b

σ2 + 12

e[ b−r

σ2 + 12](R−σ2/2)T0

× B

([b− r

σ+

σ

2

] √T0, Q, d0(Y ) + b

√T − T0

σ

)

+1

2[1− 1

b(r +

σ2

2)]s

(Y

s

) r+b

σ2 + 12

e[− b+r

σ2 + 12](R−σ2/2)T0

× B

([−b + r

σ+

σ

2

] √T0, Q, d0(Y )− b

√T − T0

σ

)

+ se(R−σ2/2)T0

[N(xL − σ

√T0)−N(xY − σ

√T0)

]

− 1

2[1 +

1

b(r +

σ2

2)]s

(Y

s

) r−b

σ2 + 12

e[ b−r

σ2 + 12](R−σ2/2)T0+[ b−r

σ2 + 12]2 σ2

2T0

×[N

(xL −

[b− r

σ+

σ

2

] √T0

)−N

(xY −

[b− r

σ+

σ

2

] √T0

)]

− 1

2[1− 1

b(r +

σ2

2)]s

(Y

s

) r+b

σ2 + 12

e[− b+r

σ2 + 12](R−σ2/2)T0+[− b+r

σ2 + 12]2 σ2

2T0

×[N

(xL −

[−b + r

σ+

σ

2

] √T0

)−N

(xY −

[−b + r

σ+

σ

2

] √T0

)]

D2(Y, R, r, c) =∫ xL

−∞I2(p(T0, s, x), T − T0, Y, R, r, c)n(x)dx

=λY

λ + R

{e−(λ+R)(T−T0)B

(0, Q, d0(Y ) + (r − σ2/2)

√T − T0

σ

)

− 1

2[1 +

1

c(r − σ2/2)]

(Y

s

) r−c

σ2 − 12

e[ c−r

σ2 + 12](R−σ2/2)T0

× B

([c− r

σ+

σ

2

] √T0, Q, d0(Y ) + c

√T − T0

σ

)

− 1

2[1− 1

c(r − σ2/2)]

(Y

s

) r+c

σ2 − 12

e[− c+r

σ2 + 12](R−σ2/2)T0

× B

([−c + r

σ+

σ

2

] √T0, Q, d0(Y )− c

√T − T0

σ

)

− [N(xL)−N(xY )]

21

+1

2[1 +

1

c(r − σ2/2)]

(Y

s

) r−c

σ2 − 12

e[ c−r

σ2 + 12](R−σ2/2)T0+[ c−r

σ2 + 12]2 σ2

2T0

×[N

(xL −

[c− r

σ+

σ

2

] √T0

)−N

(xY −

[c− r

σ+

σ

2

] √T0

)]

+1

2[1− 1

c(r − σ2/2)]

(Y

s

) r+c

σ2 − 12

e[− c+r

σ2 + 12](R−σ2/2)T0+[− c+r

σ2 + 12]2 σ2

2T0

×[N

(xL −

[−c + r

σ+

σ

2

] √T0

)−N

(xY −

[−c + r

σ+

σ

2

] √T0

)]}

G1(Y,R, r, b) =∫ xL

−∞

(L

p(T0, s, x)

) 2rασ2 −1

I1(L2/p(T0, s, x), T − T0, Y, r, b)n(x)dx

= −L(

L

s

) 2rασ2

e−2rασ2 (R−σ2/2)T0−λ(T−T0)B

(−2rα

σ

√T0,−Q, d1(Y ) + (r +

σ2

2)

√T − T0

σ

)

+L

2(rα+b−r)

σ2

2[1 +

1

b(r +

σ2

2)]Y

r−b

σ2 + 12 s

r−b−2rασ2 + 1

2 e[ 12+ r−b−2rα

σ2 ](R−σ2/2)T0

× B

([σ

2+

r − b− 2rα

σ

] √T0,−Q, d1(Y ) + b

√T − T0

σ

)

+L

2(rα−b−r)

σ2

2[1− 1

b(r +

σ2

2)]Y

r+b

σ2 + 12 s

r+b−2rασ2 + 1

2 e[ r+b−2rασ2 + 1

2](R−σ2/2)T0

× B

([r + b− 2rα

σ+

σ

2

] √T0,−Q, d1(Y )− b

√T − T0

σ

)

+ L(

L

s

) 2rασ2

e−2rασ2 (R−σ2/2)T0+

2r2α

σ2 T0N(xmin[L,L2/Y ] +

2rα

σ

√T0

)

− L2(rα+b−r)

σ2

2[1 +

1

b(r +

σ2

2)]Y

r−b

σ2 + 12 s

r−b−2rασ2 + 1

2 e[ 12+ r−b−2rα

σ2 ](R−σ2/2)T0+[ 12+ r−b−2rα

σ2 ]2 σ2

2T0

× N

(xmin[L,L2/Y ] −

[σ

2+

r − b− 2rα

σ

] √T0

)

− L2(rα−b−r)

σ2

2[1− 1

b(r +

σ2

2)]Y

r+b

σ2 + 12 s

r+b−2rασ2 + 1

2 e[ r+b−2rασ2 + 1

2](R−σ2/2)T0+[ r+b−2rα

σ2 + 12]2 σ2

2T0

× N

(xmin[L,L2/Y ] −

[σ

2+

r + b− 2rα

σ

] √T0

)

G2(Y,R, r, c) =∫ xL

−∞

(L

p(T0, s, x)

) 2rασ2 −1

I2(L2/p(T0, s, x), T − T0, Y, R, r, c)n(x)dx

=λY

λ + R

(L

s

) 2rασ2 −1

e(1− 2rασ2 )(R−σ2/2)T0−(λ+R)(T−T0)

× B

((σ − 2rα

σ)√

T0,−Q, d1(Y ) + (r − σ2/2)

√T − T0

σ

)

22

− L2(rα+c−r)

σ2

2[1 +

1

c(r − σ2/2)]Y

r−c

σ2 − 12 s

r−c−2rασ2 + 1

2 e[ 12+ r−c−2rα

σ2 ](R−σ2/2)T0

× B

([σ

2+

r − c− 2rα

σ

] √T0,−Q, d1(Y ) + c

√T − T0

σ

)

− L2(rα−c−r)

σ2

2[1− 1

c(r − σ2/2)]Y

r+c

σ2 − 12 s

r+c−2rασ2 + 1

2 e[ r+c−2rασ2 + 1

2](R−σ2/2)T0

× B

([r + c− 2rα

σ+

σ

2

] √T0,−Q, d1(Y )− c

√T − T0

σ

)

−(

L

s

) 2rασ2 −1

e[1− 2rασ2 ](R−σ2/2)T0+[1− 2rα

σ2 ]2 σ2

2T0N

(xmin[L,L2/Y ] − (σ − 2rα

σ)√

T0

)

+L

2(rα+c−r)

σ2

2[1 +

1

c(r − σ2/2)]Y

r−c

σ2 − 12 s

r−c−2rασ2 + 1

2 e[ 12+ r−c−2rα

σ2 ](R−σ2/2)T0+[ 12+ r−c−2rα

σ2 ]2 σ2

2T0

× N(xmin[L,L2/Y ] −

[σ

2+

r − c− 2rα

σ

] √T0

)

+L

2(rα−c−r)

σ2

2[1− 1

c(r − σ2/2)]Y

r+c

σ2 − 12 s

r+c−2rασ2 + 1

2 e[ r+c−2rασ2 + 1

2](R−σ2/2)T0+[ r+c−2rα

σ2 + 12]2 σ2

2T0

× N(xmin[L,L2/Y ] −

[σ

2+

r + c− 2rα

σ

] √T0

)}

For computational purposes, note that the last five lines disappear when Y = L in

D1, D2, G1, G2.

K2 = e(λ−λ0)T0e−(r+λ)T0

[D1(K, r0, r0, b0) + D2(K, r0, r0, c0)−D1(L, r0, rα, bα)− K

LD2(L, r0, rα, c)

−G1(K, r0, r0, b0)−G2(K, r0, r0, c0) + G1(L, r0, rα, bα) +K

LG2(L, r0, rα, c)

]

B Proofs

Case A: Exercise time as a hitting time

The option price can be written as, for s < L,

P1 + P2 := E[e−rαT (ST e−αT −Kα(T ))+1{τ>T}] + E[(Le−rατ −Ke−rτ )1{τ≤T}] (B.9)

The first term corresponds to the price of the up-and-out call option with strike Kα(T ) and

barrier L, of the stock with drift rα. Using the known formulas (see for example Bjork 1998),

we get, denoting by C(s,K) the call option price with strike K, maturity T and stock price

s,

P1 = P1(s, T, K) = C(s, T, Kα(T ))−D(s, T, Kα(T ), L)

−(

L

s

) 2rασ2 −1

[C(L2

s, T,Kα(T ))−D(

L2

s, T, Kα(T ), L)] (B.10)

23

Here,

C(s, T,K) = sN

(√T

σ(rα +

σ2

2) +

log(s/K)

σ√

T)

)−Ke−rαT N

(√T

σ(rα − σ2

2) +

log(s/K)

σ√

T)

)

D(s, T,K, L) = sN

(√T

σ(rα +

σ2

2) +

log(s/L)

σ√

T)

)−Ke−rαT N

(√T

σ(rα − σ2

2) +

log(s/L)

σ√

T)

)

As for P2, from Karatzas-Shreve (1997), the density of τ is given by

fτ (t) =−L

σ√

2πt3e−

(L+yα−t)2

2σ2t

where

L = log(s/L), yα− = rα − σ2

2.

After some computations, we can write

∫ T

0e−xtfτ (t)dt = e

L(√

(yα−)2+2σ2x−yα

−)

σ2

∫ T

0

−L

σ√

2πt3e−

(L+t√

(yα−)2+2σ2x)2

2σ2t dt

Using the fact that

e−L 2y

σ2 n

(1

σ(√

ty − L√t)

)= n

(1

σ(√

ty +L√t)

)

we can easily check that

∂

∂t

[N

(1

σ(√

ty +L√t)

)+ e−L 2y

σ2 N

(1

σ(−√

ty +L√t)

)]=−L

σt1.5n

(1

σ(√

ty +L√t)

)

Therefore, we get

∫ T

0

−L

σ√

2πt3e−

(L+ty)2

2σ2t dt =∫ T

0

−L

σt1.5n

(1

σ(√

ty +L√t)

)dt (B.11)

= N

(1

σ(√

Ty +L√T

)

)+ e−L 2y

σ2 N

(1

σ(−√

Ty +L√T

)

)

−(1 + e−L 2y

σ2 )1{s>L} − 1{s=L}

Then, we get, for s < L, and noting that e−L = L/s,

P2 = LP (s, T, yα−, yα

+)−KP (s, T, yα−, y) ,

where P is defined in (A.3).

Case B: Intensity based model for exercise time

24

The price is

E

[∫ T

0λ(St −K)+e−(r+λ)tdt + (ST −K)+e−(r+λ)T

]

Expected value of the second term is equal to the product of the Black-Scholes formula with

e−λT . The first term can be written, after taking the expectation inside the integral and

using the Black-Scholes formula, as

λ∫ T

0e−λt[sN(d1(t))−Ke−rtN(d2(t))]dt = I1(s, T, K, r) + I2(s, T,K, r) (B.12)

where N(di(t)) is the usual Black-Scholes notation with time to maturity equal to t. Let us

compute this as two integrals I1, I2. For this we will need the following obvious fact:

1√2π

e−a2+b2t2

2σ2t =1√2π

eabσ2 e−

b2

2σ2t(t+a

b)2 = e

abσ2 n

(1

σ(b√

t +a√t)

)(B.13)

Integrating by parts we get

I1(s, T, K, r) = λ∫ T

0e−λtsN(d1(t))dt

= −s∫ T

0N(d1(t))d(e−λt)

= s

[1{s>K} +

1

21{s=K} − e−λT N(d1(T )) +

∫ T

0e−λtn(d1(t))

t(r + σ2/2)− log(s/K)

2σt3/2dt

]

= s

[1{s>K} +

1

21{s=K} − e−λT N

(K

σ√

T+ (r +

σ2

2)

√T

σ

)+ I3(s, T, K, r)

](B.14)

The last integral, I3(s, T, K, r), can be written as

I3(s, T, K, r) =1

2σ√

2π

(K

s

) rσ2 + 1

2

×∫ T

0

[r + σ2/2

t0.5− log(s/K)

t1.5

]exp

{− 1

2σ2

(log2(s/K)

t+ ((r + σ2/2)2 + 2σ2λ)t

)}dt

Using (B.13) we can write this as

I3(s, T, K) =1

2σ

(K

s

) rσ2 + 1

2∫ T

0

[r + σ2/2

t0.5− log(s/K)

t1.5

]e

Kbσ2 n

(1

σ(b√

t +K√

t)

)dt (B.15)

where b = b0 and we recall that

K := log(s/K), bα :=√

(rα + σ2/2)2 + 2σ2λ (B.16)

We have the following useful observation:

d

dtN

(1

σ(b√

t +K√

t)

)= n

(1

σ(b√

t +K√

t)

) [1

2σ(

b√t− K

t1.5)

](B.17)

25

Thus,

∫ T

0

1

2σ√

tn

(1

σ(b√

t +K√

t)

)dt =

N(

1σ(b√

T + K√T))− 1{K>0} + 1

21{K=0}

b

+∫ T

0

K

2bσt1.5n

(1

σ(b√

t +K√

t)

)dt (B.18)

Using this in (B.15) we get

I3(s, T,K, r, b) =(

K

s

) rσ2 + 1

2

eKbσ2

{r + σ2/2

b

[N

(1

σ(b√

T +K√T

)

)− 1{s>K} − 1

21{s=K}

]

+

[K(r + σ2/2)

b− log(s/K)

] ∫ T

0

1

2σt1.5n

(1

σ(b√

t +K√

t)

)dt

}(B.19)

This is nice, because we can compute the last integral from (B.11) to get

I3(s, T,K, r, b) =(

K

s

) r−b

σ2 + 12

{r + σ2/2

b

[N

(1

σ(b√

T +K√T

)

)− 1{s>K} − 1

21{s=K}

]

+1

2

[1− r + σ2/2

b

] N

(1

σ(b√

T +K√T

)

)+

(K

s

) 2bσ2

N

(1

σ(−b

√T +

K√T

)

)

−1 +

(K

s

) 2bσ2

1{s>K} − 1{s=K}

We now similarly compute I2:

I2(s, T, K, R, r) = −λK∫ T

0e−(λ+R)tN(d2(t))dt

=λK

λ + R

∫ T

0N(d2(t))d(e−(λ+R)t)

= − λK

λ + R

[1{s>K} +

1

21{s=K} − e−(λ+R)T N(d2(T )) (B.20)

+∫ T

0e−(λ+R)tn(d2(t))

t(r − σ2/2)− log(s/K)

2σt3/2dt

]

= − λK

λ + R

[1{s>K} +

1

21{s=K} − e−(λ+R)T N

(K

σ√

T+ (r − σ2

2)

√T

σ

)+ I4(s, T, K, R, r)

]

The last integral, I4(s, T, K, R, r), can be written as

I4(s, T, K,R, r) =1

2σ√

2π

(K

s

) rσ2− 1

2

×∫ T

0

[r − σ2/2

t0.5− log(s/K)

t1.5

]exp

{− 1

2σ2

(log2(s/K)

t+ ((r − σ2/2)2 + 2σ2(λ + R))t

)}dt

26

As above for I3, we get that

I4(s, T, K, R, r, c) =(

K

s

) r−c

σ2 − 12

{r − σ2/2

c

[N

(1

σ(c√

T +K√T

)

)− 1{s>K} − 1

21{s=K}

]

+1

2

[1− r − σ2/2

c

] N

(1

σ(c√

T +K√T

)

)+

(K

s

) 2cσ2

N

(1

σ(−c

√T +

K√T

)

)

−1 +

(K

s

) 2cσ2

1{s>K} − 1{s=K}

(B.21)

where

c :=√

(r − σ2/2)2 + 2σ2(λ + R) (B.22)

This, together with (B.20) gives us a formula for I2.

Case C: The combined case, no vesting

Recall the notation for cα and c. Similarly as above, we get

J1 = LP (s, T, yα−, cα)−KP (s, T, yα

−, c) (B.23)

where P is given in (A.3). Similarly, we have

J3 = e−λT P1(s, T, Kα(T )) (B.24)

where P1 is given in (A.2). In order to compute J2, we need to integrate P1, but this reduces

to integrating call option formulas, which we have done above. Doing this, we get, denoting

P1(t) the value of P1 when time to maturity is t,

J2 = J2(s, T ) = λ∫ T

0e−λtP1(s, t, Ke−αt)dt

= I1(s, T, K, r, b0) + I2(s, T,K, r, c0)− I1(s, T, L, rα, bα)− K

LI2(s, T, L, r, rα, c) (B.25)

−(

L

s

) 2rασ2 −1

[I1(

L2

s, T,K, r, b0) + I2(

L2

s, T,K, r, c0)− I1(

L2

s, T, L, rα, bα)− K

LI2(

L2

s, T, L, r, rα, c)

]

Case D: Combined model with a vesting period

Introduce the notation

p(t, s, x) = se(r−σ2/2)t+σ√

tx

and recall that

xY =log(Y/s)

σ√

T0

− (r − σ2/2)

√T0

σ(B.26)

27

By conditioning on the stock price history up to time T0 and comparing to Case C, we get

K3 = e(λ−λ0)T0e−λT−rT0E[P1(ST0 , T − T0, Kα(T − T0))1{ST0<LT0

}]

= e(λ−λ0)T0e−λT−rT0

∫ xL

−∞P1(p(T0, s, x), T − T0, Kα(T − T0))n(x)dx

K11 = Le−αT0e(λ−λ0)T0e−(rα+λ)T0

∫ xL

−∞P (p(T0, s, x), T − T0, y

α−, cα))n(x)dx

−Ke(λ−λ0)T0e−(r+λ)T0

∫ xL

−∞P (p(T0, s, x), T − T0, y

α−, c))n(x)dx

K12 = e(λ−λ0)T0e−λT0 [sN

(√T0

σ(r +

σ2

2) +

log(s/L)

σ√

T0

)

)−Ke−rT0N

(√T0

σ(r − σ2

2) +

log(s/L)

σ√

T0

)

)]

K2 = e(λ−λ0)T0e−(rα+λ)T0

∫ xL

−∞J2(p(T0, s, x), T − T0)n(x)dx

Note that all these integrals are linear combinations of the integrals of the form∫ xL

−∞eaxN(bx + c)n(x)dx

for some constants a, b, c. Let us express this integral in terms of a bivariate normal distri-

bution function. We have∫ xL

−∞eaxN(bx + c)n(x)dx =

∫ xL

−∞

∫ c

−∞eaxn(bx + y)n(x)dydx

=1

2πe

a2

2

∫ xL

−∞

∫ c

−∞exp

{−1

2(1 + b2)(x− a)2 − 1

2(y + ab)2 − b(y + ab)(x− a)

}dydx

This can be related to the bivariate normal distribution as follows:

B(a, b, c) :=∫ xL

−∞eaxN(bx + c)n(x)dx = e

a2

2 P (X ≤ xL, Y ≤ c) (B.27)

where (X, Y ) has a bivariate normal distribution with

µX = a, µY = −ab, σ2X = 1, σ2

Y = 1 + b2, ρ = − b√1 + b2

We get K11 from (B.27) and (A.3). It is straightforward to get K12. We get K3 from (B.27)

and (A.2). We get K2 from (B.25), (B.27).

C Probabilities

Denote by µ the drift of the stock price process and

fα :=√

(µα−)2 + 2λσ2

28

µα− = µ− α− σ2/2, µα = µ− α.

- P1: Probability of being fired before the end of the vesting period:

P1 = 1− e−λ0T0

- P2: Probability of voluntarily exercising at the end of the vesting period (corresponds

to K12 in Case D):

P2 = P [ST0 > L, T 0λ > T0] = e−λ0T0 [1−N(xL)]

- P 03 : Probability of the option being exercised at the desired level, no vesting period

(corresponds to J1 in Case C):

P 03 (λ, T, s) = P [TL < T, TL < Tλ]

=∫ T

0

∫ ∞

y

−L

σ√

2πy3e− (L+µα

−y)2

2σ2y λe−λxdxdy

=∫ T

0

−L

σ√

2πy3eL

√(µα−)2+2λσ2−µα

−σ2 e

− (µα−)2+2λσ2

2σ2y

(y+ L√

(µα−)2+2λσ2

)2

dy

=(

s

L

)√

(µα−)2+2λσ2−µα

−σ2

N

(√T

σ

√(µα−)2 + 2λσ2 +

L

σ√

T

)

+(

s

L

)−√

(µα−)2+2λσ2+µα

−σ2

N

(−√

T

σ

√(µα−)2 + 2λσ2 +

L

σ√

T

)

-P3: Probability of the option being exercised at the desired level, after the vesting period

(corresponds to K11 in Case D):

P3(λ) = P [T0 < T 0L < T, T 0

L < T 0λ ] = E[ET0(1{T 0

L<T,T 0L<T 0

λ})]

= e−λ0T0

(s

L

) fα−µα−

σ2

efα−µα

−σ2 (µ0−σ2/2)T0B

((fα − µα

−)√

T0

σ, Q, d0(L) +

√T − T0

σfα

)

+e−λ0T0

(s

L

)− fα+µα−

σ2

e−fα+µα

−σ2 (µ0−σ2/2)T0B

(−(fα + µα

−)√

T0

σ,Q, d0(L)−

√T − T0

σfα

)

-P 04 : Probability of being fired between T0 = 0 and T, no vesting period (corresponds to

J2 in Case C):

P 04 (λ, T, s) = P [0 < Tλ < min{T, TL}]

29

=∫ T

0

∫ y

0

−L

σ√

2πy3e− (L+µα

−y)2

2σ2y λe−λxdxdy +∫ ∞

T

∫ T

0

−L

σ√

2πy3e− (L+µα

−y)2

2σ2y λe−λxdxdy

=∫ T

0

−L

σ√

2πy3e− (L+µα

−y)2

2σ2y (1− e−λy)dy +∫ ∞

T

−L

σ√

2πy3e− (L+µα

−y)2

2σ2y (1− e−λT )dy

= P 03 (0, T, s)− P 0

3 (λ, T, s) + [1− e−λT ][1− P 03 (0, T, s)]

= 1− e−λT − P 03 (λ, T, s) + e−λT P 0

3 (0, T, s)

-P4: Probability of being fired between T0 and T, with vesting period (corresponds to

K2 in Case D):

P4 = P [T0 < T 0λ < min{T, T 0

L}]= E[PT0 [T0 < T 0

λ < min{T, T 0L}]] = e(λ−λ0)T0e−λT0E[P 0

4 (λ, T − T0, ST0)]

= e(λ−λ0)T0

{[e−λT0 − e−λT ]N(xL)− P3(λ) + e−λT P3(0)

}.

-P 05 : Probability of arriving to maturity, no vesting period (corresponds to J3 in Case

C):

P 05 = P [Tλ > T, TL > T ] = e−λT [1− P 0

3 (0, T, s)]

-P5: Probability of arriving to maturity, with vesting period (corresponds to K3 in Case

C):

P5 = P [T 0λ > T, T 0

L > T ] = e(λ−λ0)T0e−λT E[1{ST0<L}(1− P 03 (0, T − T0, ST0))]

= e(λ−λ0)T0e−λT [N(xL)− P3(0)].

D Expected value of stock at expiry/exercise

We compute the expected value of the stock at expiry/exercise, E[Sτ ]. This is simply the

value of the option without discounting and with K = 0. We need new notation for this

value. First, in all the formulas from elsewhere that we use, we replace r by µ,

the actual drift of the stock. The corresponding values in this case are:

IS1 (T ) =

λs

µ− λ[e(µ−λ)T − 1]

P S1 = eµT

s−D(s, T, 0, L)−

(L

s

) 2µασ2 −1

(s−D(L2

s, T, 0, L))

yµ− = µα − σ2

2, yS

α =√

(yµ−)2 − 2σ2α

30

cSα =

√(yµ−)2 + 2σ2(λ− α)

P S2 = LP (s, T, yµ

−, ySα)

JS1 = LP (s, T, yµ

−, cSα)

JS3 = e−λT P S

1

Also denote

IS,µ1 (s, T ) = I1(s, T, L, µα, bα(λ = λ− µ))

the value of the old function I1, but with r replaced by µ everywhere and by λ replaced by

λ− µ in the computation of bα. Similarly introduce

DS,µ1 (s, T ) = D1(L, µ0, µα, bα(λ = λ− µ))

GS,µ1 (s, T ) = G1(L, µ0, µα, bα(λ = λ− µ))

JS2 = E

∫ T

0λe(µ−λ)te−µtSt1{Tλ>t}dt

= λ∫ T

0e−λtP S

1 (t)dt

= IS1 (T )−

(L

s

) 2µασ2 −1

IS1 (T )− λ

λ− µIS,µ1 (s, T ) +

(L

s

) 2µασ2 −1 λ

λ− µIS,µ1 (

L2

s, T )

KS12 = e(λ−λ0)T0e−λT0sN

(√T0

σ(µ0 +

σ2

2) +

log(s/L)

σ√

T0

)

KS11e

−(λ−λ0)T0

= Le−λT0

(L

s

) yµ−−cS

α

σ2

ecSα−y

µ−

σ2 (µ0−σ2/2)T0B

((cS

α − yµ−)√

T0

σ,Q, d0(L) +

√T − T0

σcSα

)

+ Le−λT0

(L

s

) yµ−+cS

α

σ2

e−cSα+y

µ−

σ2 (µ0−σ2/2)T0B

(−(cS

α + yµ−)√

T0

σ,Q, d0(L)−

√T − T0

σcSα

)

KS2 = e(λ−λ0)T0e−λT0

∫ xL

0JS

2 dx

= e(λ−λ0)T0e−λT0

[(e(µ−λ)(T−T0) − 1)

λ

µ− λse(µ−σ2/2)T0N(xL − σ

√T0)

−L2µασ2 −1(e(µ−λ)(T−T0) − 1)

λ

µ− λs−

2µασ2 e−

2µασ2 (µ−σ2/2)T0N(xL +

2µα

σ

√T0)

− λ

λ− µDS,µ

1 (s, T ) +λ

λ− µGS,µ

1 (s, T )]

31

K3 = e(λ−λ0)T0e−λT+µ(T−T0)[se(µ−σ2/2)T0N(xL − σ

√T0)

−L2µασ2 −1s−

2µασ2 e−

2µασ2 (µ−σ2/2)T0N(xL +

2µα

σ

√T0)

−C0(L; K = 0, r = µ) + C1(L,K = 0, r = µ)]

The expected value of stock at expiry/exercise is

KS11 + KS

12 + KS2 + KS

3 .

E Expected value of the time to expiry/exercise

For this computation we will need to calculate

BS(a, b, c) :=∫ xL

−∞xeaxN(bx + c)n(x)dx (E.28)

Denote by

B(a, b, c, xL = bxL + c)

the value of the function B(a, b, c) defined in (B.27), but with r replaced by µ and with xL

replaced by bxL + c. We find

BS(a, b, c) = −ea2/2∫ xL

−∞N(bx + c)d

(e−(x−a)2/2

√2π

)+ a

∫ xL

−∞eaxN(bx + c)n(x)dx

= −ea2

2 N(bxL + c)N(xL − a) + ea2

2

∫ xL

−∞bN(x− a)n(bx + c)dx + aB(a, b, c)

= −ea2

2 N(bxL + c)N(xL − a) + ea2

2 B(0,

1

b,−c

b− a, xL = bxL + c

)+ aB(a, b, c)

(E.29)

Let us compute different components of the expected value of the time to exercise, E[τ ].

- E1: The expiry occurs before the vesting period:

E1 = E[τ1{Tλ<T0}] = E[Tλ1{Tλ<T0}] =∫ T0

0λ0e

−λ0tdt

= −e−λ0T0(T0 +1

λ0

) +1

λ0

- E2: Voluntarily exercising at the end of the vesting period:

E2 = E[τ1{ST0>L,T 0

λ>T0}] = T0P (ST0 > L, T 0

λ > T0)

= T0e−λ0T0 [1−N(xL)]

32

- E03 : Exercising at the desired level, no vesting period (we use (B.17) in the computa-

tion):

E03(λ, T, s) = E[TL1{0<TL<T,TL<Tλ}]

=∫ T

0

∫ ∞

y

−yL

σ√

2πy3e− (L+µα

−y)2

2σ2y λe−λxdxdy

=∫ T

0

−L

σ√

2πyeL

√(µα−)2+2λσ2−µα

−σ2 e

− (µα−)2+2λσ2

2σ2y

(y+ L√

(µα−)2+2λσ2

)2

dy

= − L

fα

(s

L

)√

(µα−)2+2λσ2−µα

−σ2

N

(√T

σ

√(µα−)2 + 2λσ2 +

L

σ√

T

)

+L

fα

(s

L

)−√

(µα−)2+2λσ2+µα

−σ2

N

(−√

T

σ

√(µα−)2 + 2λσ2 +

L

σ√

T

)

-E3: Exercising at the desired level, after the vesting period:

E3 = E3(λ, T − T0) = E[T 0L1{T0<T 0

L<T,T 0L<T 0

λ}]

= E[ET0(T0L1{T 0

L<T,T 0L<T 0

λ})] = E[E0

3(λ, T − T0, ST0)]

= − L + (µ0 − σ2/2)T0

fα

(s

L

) fα−µα−

σ2

efα−µα

−σ2 (µ0−σ2/2)T0B

((fα − µα

−)√

T0

σ,Q, d0(L) +

√T − T0

σfα

)

+L + (µ0 − σ2

2)T0

fα

(s

L

)− fα+µα−

σ2

e−fα+µα

−σ2 (µ0−σ2

2)T0B

(−(fα + µα

−)√

T0

σ,Q, d0(L)−

√T − T0

σfα

)

− σ√

T − T0

fα

(s

L

) fα−µα−

σ2

efα−µα

−σ2 (µ0−σ2/2)T0BS

((fα − µα

−)√

T0

σ,Q, d0(L) +

√T − T0

σfα

)

+

√T − T0

fα

(s

L

)− fα+µα−

σ2

e−fα+µα

−σ2 (µ0−σ2/2)T0BS

(−(fα + µα

−)√

T0

σ,Q, d0(L)−

√T − T0

σfα

),

where in the last two terms BS is as defined in (E.29).

-E04 : Being fired/quitting between T0 = 0 and T, no vesting period:

E04(λ, T, s) = E[Tλ1{0<Tλ<min{T,TL}}]

=∫ T

0

∫ y

0

−yL

σ√

2πy3e− (L+µα

−y)2

2σ2y λe−λxdxdy +∫ ∞

T

∫ T

0

−yL

σ√

2πy3e− (L+µα

−y)2

2σ2y λe−λxdxdy

=∫ T

0

−L

σ√

2πye− (L+µα

−y)2

2σ2y (1− e−λy)dy +∫ ∞

T

−L

σ√

2πye− (L+µα

−y)2

2σ2y (1− e−λT )dy

33

= E03(0, T, s)− E0

3(λ, T, s) + [1− e−λT ][− L

|µα−|− E0

3(0, T, s)]

= −(1− e−λT )L

|µα−|− E0

3(λ, T, s) + e−λT E03(0, T, s)

-E4: Being fired/quitting between T0 > 0 and T, with vesting period (note that in the

last term we have T , not T − T0):

E4 = E[T 0λ1{T0<T 0

λ<min{T,T 0

L}}]

= E[ET0 [T0λ1{T0<T 0

λ<min{T,T 0

L}}]]

= E

[e−λT0E0

3(0, T − T0, ST0)− E03(λ, T − T0, ST0) + (e−λT − e−λT0)[

L

|µα−|+ E0

3(0, T, ST0)]

]

= e−λT0E3(0, T − T0)− E3(λ, T − T0, ) + (e−λT − e−λT0)[L + (µ0 − σ2/2)T0

|µα−|+ E3(0, T )]

-E05 : Arriving to maturity, no vesting period:

E05 = TP 0

5

-E5: Arriving to maturity, with vesting period:

E5 = TP5

The expected time to expiry/exercise is

E[τ ] = E1 + E2 + E3 + E4 + E5. (E.30)

34

Table 1

Convergence of the Binomial Tree Approach

We compute the price of the ESO using a binomial tree for different numbers of time steps.

N represents the number of steps and P the price of the ESO according to the binomial

tree approach. Parameter values are s = 100; K = 100; T = 10; T0 = 2; σ = 0.2; r = 0.06.

Additionally, for the exercise barrier we take L = 150, for the rate of increase of the barrier

we take α = 0 and for the probability of leaving the firm, we take λ = 0.04. The price of

the ESO (as obtained using the analytical formula discussed in the paper) is 27.8551.

“True” price is 27.8551

N P

50 29.1894

100 29.0063

250 28.8949

500 28.4249

750 28.1550

1000 28.2934

1250 28.0380

1500 27.9424

1750 27.9404

2000 27.9973

2250 28.0925

2500 28.2135

3000 28.1587

4000 27.9921

5000 28.0327

7500 27.9592

10000 28.0239

20000 27.9003

40000 27.9291

35

Table 2

Prices of ESO’s as function of the underlying

We compute prices of ESO’s for different values of the underlying (s) and for several combi-

nations of λ (exogenous rate of early exercise/firing) and L (early exercise barrier). The other

parameter values of the model are K = 100; T = 10; T0 = 3; σ = 0.2; r = 0.05; α = −0.02.

The table presents the results for cases A,B, C and D as explained in section 2, plus the

Black and Scholes price (denoted by BS).

λ = 0.04

s A B C D BS

L = 125 100 16.1088 38.9753 15.3372 22.7792 45.1930

105 17.7965 43.1893 17.1666 25.7946 49.5629

110 19.5328 47.5258 19.0634 28.9783 54.0074

115 21.3142 51.9622 21.0097 32.3165 58.5170

120 23.1375 56.4807 22.9921 35.7948 63.0836

L = 150 100 26.0510 38.9753 23.9052 26.8375 45.1930

105 28.3449 43.1893 26.3736 29.7539 49.5629

110 30.6666 47.5258 28.9054 32.7967 54.0074

115 33.0132 51.9622 31.4829 35.9611 58.5170

120 35.3827 56.4807 34.0925 39.2417 63.0836

λ = 0.2

s A B C D BS

L = 125 100 16.1088 24.4350 12.9962 13.5253 45.1930

105 17.7965 28.2765 15.2355 15.4457 49.5629

110 19.5328 32.3772 17.6150 17.4690 54.0074

115 21.3142 36.6710 20.0664 19.5855 58.5170

120 23.1375 41.1104 22.5402 21.7856 63.0836

L = 150 100 26.0510 24.4350 18.1005 15.2048 45.1930

105 28.3449 28.2765 20.9915 17.1013 49.5629

110 30.6666 32.3772 24.0660 19.0806 54.0074

115 33.0132 36.6710 27.2567 21.1367 58.5170

120 35.3827 41.1104 30.5150 23.2637 63.0836

36

Table 3

Prices of ESO’s as function of the vesting period

We compute prices of ESO’s for different values of the vesting period (T0) and for several

combinations of the price of underlying (s) and early exercise barrier L versus different

values of λ (exogenous rate of early exercise/firing) and L (early exercise barrier). The

other parameter values of the model are K = 100; T = 10; σ = 0.2; r = 0.05; α = −0.02.

The table presents the results for cases A,B, C and D as explained in section 2, plus the

Black and Scholes price (denoted by BS).

s = 120; L = 125

T0 A B C D BS

λ = 0.04 1 23.1375 56.4807 22.9921 29.2254 63.0836

3 23.1375 56.4807 22.9921 35.7948 63.0836

6 23.1375 56.4807 22.9921 40.7978 63.0836

λ = 0.2 1 23.1375 41.1104 22.5402 24.2800 63.0836

3 23.1375 41.1104 22.5402 21.7856 63.0836

6 23.1375 41.1104 22.5402 15.5121 63.0836

s = 100; L = 150

T0 A B C D BS

λ = 0.04 1 26.0510 38.9753 23.9052 24.5668 45.1930

3 26.0510 38.9753 23.9052 26.8375 45.1930

6 26.0510 38.9753 23.9052 29.5728 45.1930

λ = 0.2 1 26.0510 24.4350 18.1005 17.4525 45.1930

3 26.0510 24.4350 18.1005 15.2048 45.1930

6 26.0510 24.4350 18.1005 11.0176 45.1930

37

Table 4

Other Greeks

We study the sensitivity of ESO’s prices to the changes in other key variables, frequently

the object of hedging in practice: σ, r and T . We fix the values of other parameters at

s = 100; K = 100; T0 = 3; L = 150; λ = 0.06; α = −0.02. The table presents the results for

cases A,B, C and D as explained in section 2, plus the Black and Scholes price (denoted by

BS).

r = 0.05; T = 10

σ A B C D BS

0.15 25.2343 33.4420 21.8876 23.0096 42.0707

0.20 26.0510 36.3808 22.9642 24.9348 45.1930

0.25 26.9338 39.6685 24.0819 27.1726 48.7844

σ = 0.2; r = 0.05

T A B C D BS

9 25.7022 34.7783 22.7666 24.6387 42.3606

10 26.0510 36.3808 22.9642 24.9348 45.1930

11 26.3105 37.8078 23.1026 25.1544 47.8712

σ = 0.2; T = 10

r A B C D BS

0.04 25.1545 33.1454 22.0815 23.6033 41.0272

0.05 26.0510 36.3808 22.9642 24.9348 45.1930

0.06 26.8369 39.5844 23.7651 26.2078 49.2873

38

Table 5

Probabilities of different exercise strategies

We compute the probabilities of different exercise strategies for the ESO as function of

the parameters that characterize this specific model: L, α and λ. Referring to Figure 1, the

probabilities are as follows: P1 is the probability that the employee is fired in region A before

vesting; P2 is the probability that at T0 the employee exercises because the underlying is

in segment B; P3 is the probability that the employee exercises after the option is vested

because the underlying hits the barrier (segment C); P4 is the probability that the employee

leaves the firm or is fired in region D; finally, P5 is the probability that the option reaches

maturity. We fix the values of other parameters at µ = 0.1; s = 100; K = 100; T = 10; T0 =

3; σ = 0.2; r = 0.05 = 3.

α = −0.02; λ = 0.1

L P1 P2 P3 P4 P5

120 0.25918 0.29257 0.39152 0.02856 0.02817

150 0.25918 0.13426 0.46099 0.07328 0.07228

180 0.25918 0.05583 0.44133 0.12266 0.12010

L = 150; λ = 0.1

α P1 P2 P3 P4 P5

0 0.25918 0.13426 0.41533 0.09626 0.094958

-0.02 0.25918 0.13426 0.46099 0.07328 0.07228

-0.04 0.25918 0.13426 0.50528 0.05098 0.05029

L = 150; α = −00.2

λ P1 P2 P3 P4 P5

0.04 0.11308 0.16074 0.53463 0.04678 0.14477

0.1 0.25918 0.13426 0.46099 0.07328 0.07228

0.2 0.45119 0.09946 0.35132 0.07386 0.02417

39

T0

T

D

B

A C

L

D

S

E

Figure 1: This figure illustrates all possible scenarios of expiration of the ESO. The ESO

expires in region A if the employee is fired/quits before vesting. We denote the probability

of this event as P1. Then, if at T0 the price of the underlying is in segment B, the ESO is

exercised. We denote the probability of this event as P2 If the ESO is unexercised after T0,

the ESO is exercised if it hits the barrier, corresponding to segment C, with a probability

that we denote as P3, or if the employee is fired or leaves the firm in region D, with a

probability that we denote P4. If none of the above takes place, the option reaches maturity

and is exercised at the segment E below. We denote the probability of this event by P5.

Obviously, P1 + P2 + P3 + P4 + P5 = 1. This probabilities are computed in Table 5.

40

(A)

50 100 150 200

28

29

30

31

32

True price

1000 2000 3000 4000 5000

27.9

28.1

28.2

28.3

28.4

Number of steps

(B)

True price

Number of steps

Figure 2: The previous plots show the convergence of pricing using a binomial tree to the

price of the ESO. Parameter values are s = 100; K = 100; L = 150; T = 10; T0 = 2; σ =

0.2; r = 0.06; λ = 0.04. In A we plot convergence to the price of the ESO for up to 200 steps

in the tree. In B we plote convergence to the price of the ESO for up to 5000 steps in the

tree. The price of the ESO is 27.8551.

41