Valuation Methods of Executive Stock Options

4 The Executive's Optimal Exercise Policy: Leung & Sircar Approach (2006)

4.1 Settings

In this paper, the agent is endowed by 1 unit of ESO with a vesting period of t_v years. She lives in the Economy described previously and has preferences modelled by an exponential utility.

The main difference with the general framework is that the holder is subject to a job termination risk with a constant intensity: À which is exponentially distributed.

Recall: The Company Stock Price and the Trading Strategies under the historical probability P evolve according to the following diffusion processes:

? ???????

???????

~ dS_u = (v-q)S_udu+çS_udW_u ~ dXè

St = S

u = {è_u(u - r) + rX} du + è_uódB_u

~

p E (-1, 1) Xt = X

t=u=T

Where p is the instantaneous constant correlation coefficient between W and B

4.1.1 The job termination risk and exercise window

Leung & Sircar incorporate job termination risk in the general model. Indeed, in the general model it is assumed that the Executive can exercise her option without time constraints. In reality the holder cannot exercise her option during the vesting period which is contractually defined and she is subject to a risk induced by her job termination. The problem can be described has follows:

1. if she is still in the firm after the vesting period then she can exercise her option when she wants until the option maturity. She get a gain resulting of her exercise equal to the company stock price level at the time where her ESO is exercised minus the strike price of her option;

2. if she leaves the firm before the vesting period then her option is lost and thus she has no gain from her option

Remark : In this paper the job termination risk is incorporated with a stopping time which is exponentially distributed. This assumption can be relaxed by introducing a random variable estimated by an historical data for each company. But in our case, this distribution function allow us to simplify computations.

4.2 Optimization method

The optimization problem differs from the general problem seen previously in the fact that there exist time constraints. On one hand there is an exercise window under which the ESO can be exercised and on the other hand there is a job termination time which have an impact on the Executive's behaviour and which have to be included in the optimization problem.

By this way we have to define the utility rewarded of immediate exercise at any time t which reflects the executive's gain when she leaves her firm at this time or the gain when t is the optimal stopping time: Proposition 4.1.

The gain Ë(t, x, s) coming from the immediate exercise of an ESO is defined as:

Ë(t,x,s) =G(t,x+(S-K)⁺)

(46)

= e_ã(x+(S_K)+)er(T - t)e_( u-r

ó )2 T -t

2

Where the function G(.) is the value function defined in equation (17)

Hence we have clarified the gain coming from the ESO exercise when the executive leaves her firm at time t.

This gain is obviously at most equal to the gain coming from the optimal exercise where this latest is driven firstly by the optimal trading strategy and secondly by the optimal company's stock price level. Moreover in the region where the early exercise constraint is inactive, the value function G satisfies the following PDE:

GG = 0 (47)

Where G is the differential operator of (X,S) under the historical probability P define in the equation (20).

By combining the immediate utility rewarded constraint with the previous PDE a time dependent Linear Complementary Problem (LCP) is stated for the price of the ESO (Dempster-Hutton (1999)). According to these arguments we can provide the executive's optimization problem:

the Executive which is endowed by 1 unit of ESO can trade dynamically in the risk-free bond and the market index and then the Executive's problem is to choose an admissible strategy and an optimal stopping time such that:

G(t, X, S) = sup sup

ôETt,T 8t,ô

= sup sup

ôETt,T 8t,ô

E^P[I(ô, Xàô + _(Sàô - K)⁺) | Xt = X, St = s

E [_-e-ã(X_ôà+(Sà_ô-K)⁺)e^r(T-ôà) _e^-(T;'^àô)(^u-ró)²|X -XS

t - t =s(48)

Where ôà = ôAô^ë. By linking the EIP with this problem and according to the optimal stopping argument we can define ô^* as:

ô* = inf ft <u< T: G(u, X_u, S_u) = I(u, (S_u - K)⁺, M_u)} (49)

Now we can write the Linear Complementarity Problem of the Executive's Value Function:

Proposition 4.2.

Suppose that the value function J be the solution to the EIP. Then for (t,x,y)E [0, T] x R x (0, oo) the Executive's is face to the following complementary problem:

À
· (G - A) + GG + sup G < 0

0E8

G > A

(À
· (G - A) + GG + sup

0E8

G)
· (A - G) = 0

{

(50)

Where the boundary conditions are:

f

GT (x, s) = -e-ã(x+(s-K)+) Gt(x, 0) = -e-ã(xer(T-t))e-(T2t)(u-ró)2 (51)
Proof. By dynamic principle, the value function G is supposed to maximize the Executive's objective

function.

Now it is interesting to focus us on the assumption made on the utility function.

By its exponentiality form and the constant absolute risk aversion argument we can separate the Executive's initial cash endowment and the trading gain process. Which involves that we can reduce the dimension of the optimization problem.

By this way we can write the value function G(t,x,s) as:

G(t, x, s) = e-ãxer(T-t) x G(t, 0, s)

With G(t,0,s):=V(t,s) and G(t,x,0):=I(t,x). By 1.11 argument and the separation of variables we are allowed to rewrite the value function as:

1

G(t, x, s) = I (t, x)
· p(t, s) 1-ñ²

Remark :

· By the exponential property of the utility function p(t, s) = V (t, s)^(1-P²⁾ is a function of only t and s,

· the function p(t,s) turn out to be related to the ESO indifference price of the executive. According to 1.10 argument we can define also the minimal entropy martingale measure (MEMM) P0 relative to the historical probability P such that the wealth process (X)T is a P0-martingale.

P0(A) = E [te(- (m7)2T1

IAi , A? FT (52)

Thus by this formulation the Executive's optimal exercise time is independant of her wealth and the Market Index price and the free-boundary problem for p is defined as follow:

? ??????? ? ????????

Lp - (1 - ñ2)ëp+ (1 - ñ2)ëe-7(y-K)+er(T -t)p- 1fP2 = 0 p -7(1-P2)(8-K)+er(T-t) = ö(t, s) (53) 1 (pt + Lp - (1 - ñ^2)ë/3 + (1 - ñ^2)ë hö(t, s)p^-12] 1-P2 ) · (ö(t, s) -p) = 0

Where the boundary conditions are:

~

p(t, s) = e-7(1-,2)(8-K)+ p(t, 0) = 1 4.2.1 The Executive's Exercise Boundary

The Executive's optimal exercise boundary s^* is interpreted as the critical company's stock price such that:

s^*(t) = inf ns = 0 :p(ts) = e-7(1_-

8--K)+e 1

p²)(^r(T--t).1.

Then by the optimal stopping time argument we get:

ô* = inf {t = u = T : S_u = s^*}

By Feynman-Kac argument the function p has the following probabilistic representation under the MEMM:

p(t, s) = inf TETt,T

^--'^)PT0 -(1-pMr-t)e-7(1-p^2 Er^(T) )(8-Kre^-' [ + / T t e-- (1--p2)À(U--t) (1 - ñ2)ëe--^y(y--KrEer(T '''8 e^2 .73(u, Su)^-2d ^ul ^P

By definition of the Private Price state previously we can express it via the following proposition: Proposition 4.3. Executive Indifference Price

The Executive's indifference price for her ESO according to 1.5.1 as the following form:

_e-r(T-t)

p(t, s) ^=-ã(1 - ñ2) log(p(t, s)) (54)

Or equivalently

G(t, x, s) = I(t, x)e-7P(t,8)e-r(T-t) (55)

Proof. The same as (28) with I(t,x) instead of V(t,x)

4.2.2 A Partial Differential Equation for the Private Price We have found just above the form of the ESO Private Price.

Recall: We have define earlier the infinitesimal generator of the company stock price process under the MEMM:

Lu - r 8 18²

= + (í-q ñ)s + (çs)²

8t ó 8S 2 8S²

(56)

Thus p solves the following free boundary problem:

? ??

??

Lp -rp - ¹₂y(1 - ñ2)(çs)2er(T-t) ^a2p + À(1 e-ã((s-K)+-p)er(T-t))

as²ã = 0

p = (s - K)⁺

Lp -rp - ¹₂y(1 - ñ²) (çs)2er(T-t) ^a2p + ^À (1 - _eã((sK)⁺ -p)e^r(T-t))

as²ã

· ((s - K)+ - p) = 0

Proof. We are going to begin the proof by the case of the first member of the free-boundary problem. By proposition (3.1) we have:

p(t, s) = e-ã(1-ñ2)er(T-t)p(t,s)

.

MoreoverLpis defined as:

8p +(í - q - u - r çñ)s813 +¹ (çs)2 82 ii 8t

ó8S28S²

With:

^ee_t = (rp-ÿp)y(1 -ñ2)er(T-t)e-ã(1-ñ2)er(T-t)p

(1)

(2)

(3)

af)

-y(1 - _ñ2)_er(T_t) ap _e-ã(1_ñ²)e^r(T--op

aS =

aS

a²r

= L(y(1 _ñ2)_er(T-t) ap2 2 r(

aS) ñ )e ,T-t) a2pi e -ã(1- ñ2)erp-
· -t)

aS²

-(1 - ñ²)ëp = -(1 - ñ2)ëe-ã(1-ñ2)er(T-t)p (4)

(1 - ñ2)ëe-ã(s-K)+er(T -t)p-

^(1(::²⁾ = (1 - ñ2)ëe-ã(s-K)+er(T-t)eãñ2er(T-t)p (5)

By divided each member of the equation by: y(1 - _ñ2)_er(T-t) _e-ã(1-ñ²)e^r(T-t -

) which is strictly posi-

tive we get:

(1)

? (rp - ÿp)

(2) ? - as= -p_s

(3) ? y(1 -ñ2)er(T-t)( _aaD 2 _aa S2p = y(1 -ñ²)e^r(T-t)((p_s)² - p _{s s}) -

(4) ? -Àãe-r(T-t)

(5) ? _ãÀ_e-r(T-t)_e-ã[(s-K)⁺-ñ²p-(1-ñ²)p]e^r(T-t) = _ãÀ_e-r(T-t)_e-ã[(s-K)⁺-p]e^r(T-t)

Then the first inequality of the original free-boundary problem in term of p can be rewritten in term of p:

Lp - (1 - ñ²)Àp + (1 - ñ2)Àe?ã(y-K)+er(T_t) p- ^ñ2

1_ñ2 = 0 ?

(rp - ÿp) - (í - q - u-r

ó çñ)sp_s + 1 ã _e-r(T -t) (1 - e?ã[(s-K)+-p]er(T _t)) = 0 ?

2(çs) ² [ã(1 - ñ² )er(T ^-t) (p_s)² - pss ] - ^ë

Lp - rp - ¹ ₂(çs)²ã(1 - ñ2)er(T ^-t) (p_s)² + ã ë e-r(T ^-t)(1 - e?ã[(s-K)+-p]er(T _t)) <0

With the following boundary condition:

~

p(T, s) = (ST - K)⁺ p(t,0)=0 And thus:

(Lp- rp- ¹ ₂(çs)²ã(1 - ñ² )er(T ^-t) (p_s)² + ã ë e-r(T ^-t)(1 - e?ã[(s-K)+-p]er(T _^t)))· ((s - k)⁺ -p) = 0

(57)

Now we can isolate each region where the Executive can choose the optimal stopping time ô^*:

ô^*= inf{t<u<T:G(u,X_u,S_u)=Ë(u,X_u,S_u)}

= inf{t<u<T:I(u,X_u +p(u,S_u))=I(u,X_u +(S_u -K)⁺)} (58)

= inf{t<u<T:p(u,S_u)=(S_u -K)⁺}

Remark : First of all, by assuming that the first inequality of the free-boundary problem is binding then we can isolate 3 parts:

1. if À = 0 and ñ = 1 then this is just the standard Black-Scholes PDE for an option with the company's stock as underlying.

2. if À = 0 and ñ =6 1 there exists a quadratic pertubation of the standard Black-Scholes PDE which is more important if the correlation coefficient between the company's stock and the Market Index is weak.

This part highlights the unperfect replication driven by the Market Index hedging.

3. if À =6 0 and ñ =6 1 this is the part of the price driven by the job termination risk whose one part highlights the case where the ESO is forfeited (during the vesting period) and the second part shows the case where the Executive have to exercise her option after her departure and when the option is unvested.

4.2.3 The optimal trading strategy

According to the general form of the optimal trading strategy defined by (40) we can find the one that have to be used by the executive int our case.

Recall: The optimal trading strategy have the following general form:

( u - r) ?G

?X + (ñçóS) ?2G

_è** ?S?X

LS = ^-_ó2 ?²G

?X²

Then by replacing each member of the formula by their value in our case we get: ?X = IX = -ãer(T -u)I

?G

?X2 = (ãer(T -u))2I ?²G

?S?X = GXS = -ãe^r(T -u)GS = -ãe^r(T -u)IX ?^2G ?p ?s = (ãe

r(T -u) )2Ips

So:

(u - r) [-ãer(T ^-u)I] + (ñço-S)(ãer(T -u))2I

_è**

LS = - o-²(ãe^r(^T -^u))²Ip_s

(59)

(u - r)e-r(T -u)	ñço-S o-2 ps
(o-ã)^2

4.3 The effects of parameters

During this section we are going to retranscribe propositions suggested by Leung & Sircar. The Reader can found proofs of these propositions in the Appendix.

Let begun this part by the Job Termination Risk effect.

4.3.1 The effect of Job Termination risk

Proposition 4.4. Suppose À2 = À1. Then the utility-maximizing boundary associated with À1 dominates that with À2

In fact this assertion is very intuitive. Indeed, if the agent 1 is less risk-averse than the agent 2, on the like-for-like basis, she is willing to wait more time in order to maximize her gain coming from her ESO than the agent 2. Thus agent 1 has more opportunities to exercise her option in a better condition than the other one.

4.3.2 The effect of risk-aversion

Proposition 4.5. The indifference price is non-increasing with risk aversion. The utility-maximizing boundary of a less risk-averse ESO holder dominates that of a more risk-averse ESO holder.

This proposition confirms also our expectation. More the Executive is risk-averse less she is willing to wait 1 unit more time to exercise her option and thus she doesn't fully exploit the potential gains coming from waiting more.

4.3.3 The correlation effect

Proposition 4.6. Assume á := u - r > 0. Fix any number ñ E (0,1). Denote by p+ and p- the o-

indifference prices corresponding to ñ and -ñrespectively. Then, we have p- = p+. Moreover, the utility-

maximizing boundary corresponding to -ñ dominates that corresponding to ñ

Proposition 4.7. If á := u - r <0. Then the opposite happens. If á = 0 then p+ = p-, and the two o-

exercise boundaries coincide.