Valuation Methods of Executive Stock Options

5.4 An Intensity based model for the firm's cost - Ctivanic, Wiener and Zapatero (2004)

In this part we introduce the job termination risk as parameter in the model.

Indeed the Executive have to exercise her option when she leaves voluntarily or unvoluntarily the firm even if the exercising time is not optimal. The job termination risk could be designed as a Poisson process of parameter À. Thus the expected life for the Executive job is exponentially distributed of parameter À. Suppose also that the arrival rate is constant during the time (this parameter could be estimated by historical data according to the relevant company).

Thus the conditional distribution of the exercise time is:

F(t) = 1 - e^-Àt (65)

Through equation (65) we find that the probability of exercise after job terminataion is f(t) = Àe^-Àt.
And thus by taking the general form of the expected cost equation (62) we get the following proposition:

Proposition 5.3. The expected cost to the firm under intensity based model - Ctivanic, Wiener and Zapatero (2004)

The expected cost to the firm where the ESO holder is suject to a job termination risk of intensity À which following a Poisson process is written as:

" f T #

C(0, s) = E^Q À(St - K)⁺e^-(r+À)tdt + (ST - K)+e-(r+À)T | S0 = s (66)

0

Now by combining this part with the previous part we are able to find an explicit solution by considering the optimal stopping time of exercising and the job termination risk parameter.

5.5 ESO cost to the firm with optimal exercise level and job termination risk - Ctivanic, Wiener and Zapatero (2004)

The cost to the firm here is brought about by the optimal exercise and the job termination risk. But we assume again that there is no vesting period. So that the exercise time could be write as:

r** = min(r^À,r^*) (67)

Where r^À is the time when the Executive leaves her firm (which is exponentially distributing according to our assumption) and r^* is the first time when the Company's stock price hits the critical level.

We suppose that r^* Ir^À (thes two time are conditionally independent).

Then by standard probability calculus we get the following equation:

[rÀ < t]

F (t) := Pt [r^** < t] = Pt [r^* < t] + Pt [r^À < t] - Pt [r^* < t] Pt

[rÀ < t]

= I{ô^**=t} + Pt [r^À < t] - I{ô^**=t}Pt = I{ô^**=t}+ Pt [r^À <t] I{ô^**>t}

= 1 - e^-ÀtI{ô^**>t}

Then the cost to the firm can be decomposed in 3 parts:

1. the cost linked to the event: !!the Executive's optimal boundary is reached by the Company's stock price!!;

2. the cost linked to the event: !!the Executive leaves the firm and have to exercise her options!!;

3. the cost linked to the event: !!the company's stock price have never reached the optimal boundary and the Executive is still in the firm!!

Since we have assumed that each event are mutually independent we can exhibit the expected cost to the firm:

Proposition 5.4. The expected cost to the firm under intensity based model and optimal exercise - Ctivanic, Wiener and Zapatero (2004)

The expected cost to the firm for issuing an ESO and by taking into account the Executive 's optimal exercise boundary and her Job termination risk can be written as follows:

I Z T

_i

C(0, s) = E h (S*e-(rá+ë)ô - Ke^-(r+ë)ô)I{_ô*=T }dt+ E Àe^-(r+ë)t(St - K)⁺I{t>ô^*}dt

0

i+E h e-(r+ë)T (S T - K)⁺I{ô^*>T }

(68)

5.6 ESO cost to the firm: Leung & Sircar (2006)

Through this section we will discuss about the firm's granting costs induced by the executive's exercising behaviour. Indeed, the ESO cost is totally determined by the holder's exercising behaviour. Moreover the company is exposed to the exercising risk and in this section the company is assumed to be risk neutral. In fact, the underlying assumption is that the firm can perfectly hedge the risk by trading freely. Under this assumption we can describe the company stock price as a diffusion process under the risk-neutral probability Q.

dS_u = (r - q)S_udu + iS_udW (69)

u

Where:

· u = t is a time index,

· St = S is the company's stock price at time t,

· r and i are respectively the constant stock's expected return and volatility under the risk-neutral measure Q,

· q is the constant and continuous proportional dividend paid by the stock over the time

· W is a Q-Brownian motion defined on the probability space (Ù, T, (T_u), Q) where T_u is the augmented a-algebra generated by {W, t = u = 0}

In order to understand how the executive's exercising behaviour is, the following set of assumptions are made on the holder which are fully explained in the next section:

· she cannot sell the ESO or perfectly hedge her risk,

· she has a risk-preference described by an utility function U depending on her risk aversion,

· she is subject to employement termination risk which is associated to an employement termination time denoted by ôë. ô^ë is a stopping time assumed to be exponentially distributed with a constant intensity À. Moreover, it is assumed that the job termination intensity is identical under measure P and Q. These assumptions reflect the non-predictability's feature of the employment termination time in the firm and the unpriced risk which is associated to the job termination.

In that way, there are two possibilities:

1. if the stock price reaches the utility-maximizing boundary then the holder exercises her option,

2. otherwise the holder exercises her option at the maturity or after leaving the company in the case where the option is in-the-money.

Subsequently, it can be deduce that the expected cost of issuing an ESO is equal to the no-arbitrage price of the barrier-type call option written on the underlying stock S with a Strike price and a maturity which respectively are K and T. The idea here is that the barrier of the option is simply the executive's optimal exercise boundary.

From the firm perspective, we are face on three possibilities:

1. the expected cost of a vested option which is exercised when the stock price reached the optimal boundary defined by the holder's utility,

2. the expected cost of a vested option which is exercised by the holder who are leaving the company. In that case, the job termination arrives before the stock price reached the optimal boundary and the holder have to exercise her option immediatly,

3. the expected cost of an unvested option which mean that the holder leaves the company before the vesting period ends. In that case, the ESO is forfeited.

In a way is it possible to cut the expected cost in two part:

1. the expected cost of a vested option C(t,S) is descibed on the equation (70),

2. the expected cost of an unvested option C(t,S) is descibed on the equation (71).

Suppose that: · the vesting period is tv, · the job termination time is 7^À, · the optimal stopping time is 7* · ?t = tv the sock price St = S. Then the cost C(t,S) of the vested ESO is given by: iC(t, S) = E h e_r(ô*/ôÀ_t) x (Sô^*/ô^À -- K)^+ | ST = s = E h e_(r+À)(ô*_t) x (S ô* -- K)^+ + R ô* t e_(r+

i (70) À)(u_t) x ë(Su -- K)^+du | ST = s

And the cost C(t,S) of the unvested ESO is:

C(t, S) = E [e_r(tv_t) x C(t_v, Sv)I{ô^À>t_v} | ST = s] (71)

So if we assume that the holder is rational then we can define two regions R1 and R2 where option cannot be exercised optimally:

1. R1={(t,S):t_v <t<T,0<S<S* _t },

2. R2 = {(t,S) :0 <t<t_v,0< S}

Where S* t defined the executive's exercise boundary.

Through equation (70) and by taking into account the infinisetimal generator of S under the risk-neutral measure Q the cost of a vested ESO solves the follwing PDE:

?C ç2 2 s2 ?2C

+ ?s2 +(r--q)s ?C ?s --(r+ë)C+ë(s--K)⁺ =0 (72)

?t

With the following boundary conditions:

? ?

?

C(t,0)=0, t_v <t<T

C(t, s^*) = (s^*(t) -- K)⁺, t_v < t < T C(T,S) = (s--K)⁺, 0<s < s^*(T)

And the cost C(t,S) of the unvested ESO solves:

C(t,S) ç2 2 s2 ?2 C(t, S)

?s2 + (r -- q)s? C(t, S)

+ ?s --(r+ë)C(t,S) =0 (73)

?t

With the following boundary conditions:

~

C(t, 0) = 0, 0 < t < t_v
C(t_v,s)=C(t_v,s), s=0 5.7 The effects of parameters

Symetrically to the Executive's side we are going to discuss about the effects of parameters from the firm side.

We have found that B & S price for the option dominates the fair-value found in our model. It can be interesting to show this property regarding the firm side. Let us introduce this part by the effect of job termination risk intensity.

5.7.1 The job termination risk intensity

It is intuitive that more the Executive is risk-averse less can be the ESO cost. The following proposition gives a formal framework.

Proposition 5.5. Let À1, À2 2 job termination risk intensity parameters such that 0 < À1 < À2. Then the ESO cost relative to À2 is dominated by the ESO cost relative to À1 which is dominated by the B 4 S cost.

Cë2 <Cë 1 <CB&S (74)

Proof. see Appendix.

5.7.2 The vesting period

Ituitively the vesting period restrains the Executive's risk averse behaviour. Indeed more the lenght of the vesting period less it is costly for the Executive to wait a little more before exercising her option. The extreme case is that the lenght of the vesting period is until the ESO maturity and in this such case the ESO is no more no less an European call option. So we have to prove that higher is the lenght of the vesting period higher the expected cost to th firm. Formally speaking we have the following proposition.

Proposition 5.6. Leung 4 Sircar

Let À = q = 0, then the ESO expected cost to the firm is non-decreasing with respect to the lenght of the vesting period.

Moreover the cost is dominated by the Black-Scholes price of an European Call written on the company stock with the same strike and maturity.

Proof. see Appendix.

In fact the firm is confronted to the following trade-off: maintain the incentives effects of the ESO to the Executive by imposing the highest possible vesting period but minimize the cost of issuing it by reducing the lenght of the vesting period.