Valuation Methods of Executive Stock Options

Conclusion

During this dissertation we have described how can be valuated an ESO. Risk-aversion, vesting period and job termination risk have been incorporated in the models in order to describe accurately the behaviour of the ESO holder.

From the Fundamental Investment Problem to the Leung & Sircar model we have defined what can be the optimal portfolio choice of an Executive endowed by one unit of ESO. But by the tractability argument we have restricted this analysis to the case of exponential utility. A general case can outperform our approach however the case of power utility could been seen in the literature.

The case of multiple ESO has not been seen during this dissertation while it is a huge intensive research area. The paper of Leung & Sircar dedicates a part on this case. It can be said that contrary to the standard American option theory, the ESO's exercising is not made simultaneously but through multiple blocks depending on different critical price level.

Also, discrete models such that Hull & White (2004) which is one of the most popular models have not been discussed here. The Hull & White model is a compliance valuation method according to the US Financial Accounting Standards Board (FASB) 123 standard. They propose a modified binomial tree method to estimate the value of ESOs and assumed that a vested option is exercised whenever the stock price hits a certain constant barrier, or when the option reaches maturity. As the Cvitanic, Wiener and Zapatero (2004) model seen here the exact value of the barrier is left as a free parameter but in continuous time.

This dissertation have described the main aspects of the ESO valuation methods and showed that B & S model is not fit to price a such exotic option.

In reality, hedging and trading constraints are imposed to the ESO holder. We have seen that with an undiversified portfolio and risk-aversion, the rational Executive have to exercise her option early. That explain what can be seen empirically.

But other extensions can be exposed such that the case of stochastic volatility. We can thought that the Executive can influence the stock price only on the stock price volatility. This problem can be linked to the standard Principal-Agent problem where for the Executive making an effort is costly. Thus solving the standard Principal-Agent allows to find an optimal choice for her where the effort can be focused on the stock price volatility. This approach could reveal the trade-off between minimizing costly effort and maximizing the ESO value.

So we can assess the huge field of the ESO valuation methods which cannot definitely be summarized in this dissertation and which will be most probably an extensive research area for the next years.

A Proofs

A.1 Proof (1):

A utiliy function belong in the Hara class if T(x) =-Uxx(x)

Ux(x) = 1

á+âx,?x E D(x) and 3 E R⁺.

There are 2 cases:

1. If 3 = 0 then > 0;

2. If3>0then ER. Where U _x(x) = dU(x) dx and U_xx(x) = d2U(x)

d_x2and D(x) is the interval on which the utility function U is

'y² e^-ã^x.

defined.

For U(x) = -e^-ã^x with 'y> 0, we get U_x(x) = 'ye^-ã^x and U_xx(x) = -

Then T(x) =?(?ã2e-ãx)

ãe-ãx = 'y

therefore T(x) = 1

ã ₊₀with = 1 ã >0 and 3 = 0.

1

Since 3 = 0 we get D(x) = R and U is finite valued for all x.

Thanks to Arrow and Pratt we know the economics interpretation of T(x) as the coefficient of absolute risk aversion which in our case is constant and positive.

A.2 Proof General Investment Problem(17): Proof: Given (P):

? ??

??

I(t,X) = sup

Èt,T

s/t

dXè u = {9_u(u - r) + rX} du + 9uódW u 1

E [U(XT = -e?ãXT | Xt = x]

(75)

To solve this problem we are using the Hamilton-Jacobi-Bellman method. 1st step: Write the Hamiltonian of the system (P):

H(u,p,q) = p{9u(u-r)+rX}+ ¹ ₂q(9_uó)² (76)

Then we maximize H under Èt,T:

H* = max H

Èt,T

?H ?è |è=è^* = 0 if q < 0 by concavity assumption

p(u - r) + qó²9^* = 0 if q < 0 by concavity assumption

9* = -p q

u-r ó2 if q < 0 by concavity assumption

Then we deduce H^*:

H* = prX - p2 q ((u-r

ó )2 + q 2ó2(-p q )2(u-r

ó )2 if q < 0 by concavity assumption

H* = prX - 1 p2 q (u-r

ó )2 if q < 0 by concavity assumption

2

2st step: Then we can write now the HJB of the system (P):

????? ?

????? Where ?I

I(T, X) = U(XT) = -e?ãXT (Terminal Condition)

?u + H*( ?I

?I ?X , ?2I

?u = I_u and ?I

?X2 , u) = 0

?X = IX = p, ?2I

?X2 = IXX = q (77)

3rd step: We assume that I(u,X) has the following form:

I(u, X) = --e-a(u)7X+â(u) (78)

with the following boundary constraints:

f á(T) = 1

1 0(T) = 0

4th step: We solve the HJB system:

{ Iu = --(--ÿáãX + ÿ0)e-a(u)7X+â(u)

IX = --(--á(u)ã)e

IXX = --(--á(u)ã)2 e-a(u)7X+â(u)

-a(u)7X+â(u)

And the HJB is written as:

,_a_.,2_e-2(árã X +â)

(ÿárãX -- ,à)e^-ae7X+â árãe^-a7X+â)rX 7) (p-n2

2(a7re-áãX-Ef3 0

?ÿáã X -- 0^ÿ+ áãr X + ²¹ (^u;^r )² = 0

By divided all members of the equation by ^e?a7X+â which is non negative on IR. So we have the following system:

{
{

{

So

ÿáãX + áãrX = 0

-- 0^ÿ= 0

+ ₂ ( ó

á(T) = 1 boundary condition

0(T) = 0 boundary condition

áÿ = --ár

0ÿ₌ 1₂(u-r

_ó)²

á(T) = 1 boundary condition

0(T) = 0 boundary condition

á(u) = er(T-u)

= (T--u) ( u-r

u

2 ó )2

I(u, X) = --e-7xer(T-.)e- (T2.) ( m-ró)2 (79)

5th step: We deduce è^* the optimal trading strategy.

è* u	IX = -- IXX	u-- r ó2	=	á(u)ãe^-a(u)7X+â^(u)	u-- r ó2	= á(u1)ã	u-- r ó2	(u-- r) = e ãó^2	-r(T-u) (80)
				(á(u)ã)^2e-^a(^u)^7X+â(^u)

I(u, X, S) = sup sup

ô ?T.,T et,.

= sup sup

ô?T.,T et,.

1Y [I(ô, X_ô + (S_ô -- K)⁺) | X_u = X, S_u = s

1P[--e-7(Xô+(Sô-K)+)er(--)em-ró,2 I ) I X_u = X,S_u = s

(81)

A.3 Proof proposition (4.4)

The Private Price satisfies the following variational inequality:

{ }

-pt - Lp + rp + 1

min ₂'y(1 - ñ² )ç2s2er(T -t)_p 2 s + 'y À _(e?ã(s-K)⁺e^r(T -t)eãper(T -t) - 1), p - (s - K)⁺= 0

(82)

Let p1(t, s) and p2(t, s) be the indifference prices associated with À1 and À2 respectively.

Since the coefficient of À is non-negative, the left-hand side is non-decreasing with À. Then, substituting p2 (t, s) into the variational inequality for p1 (t, s) will render the left-hand side less than or equal to zero. Therefore, p2 (t, s) is a subsolution to the variational inequality for p1 (t, s), so p2 (t, s) < p1 (t, s). We conclude from (58) that the optimal exercise time corresponding to À1 is longer than or equal to that corresponding to À2, which implies that the utility-maximizing boundary corresponding to À1 dominates that corresponding to À2.

A.4 Proof proposition (4.5)

We consider the variational inequality in the previous proposition (82). The p2 s term is non-decreasing with 'y. Differentiating the nonlinear term with respect to 'y, we get:

'y²

nÀ 1 + ö(t,s)e^ö(t,s) - eö(t,s)} = 0

With ö(t, s) := 'y(p(t, s) - (s - K⁺))e^r(T-^t) = 0.Hence, the nonlinear term is also non-decreasing with 'y. By comparison principle, this implies the indifference price p is non-increasing with 'y. The second assertion follows from the characterization of the optimal exercise time see equation (reftime).

A.5 Proof proposition (4.6)

We consider the variational inequality 82. Since á > 0 and p_s = 0, the p_s term is non-decreasing in ñ. Therefore, p+ is a subsolution to the variational inequality for p-, so p+ <p-. The last statement in the proposition follows from (58) and that p- = p+

A.6 Proof proposition (4.7)

When the hedging instrument has a positive Sharpe ratio, the employee would prefer a negative correlation
than a positive one. As the correlation becomes even more negative, the employee can hedge more risk
away. Consequently, the employees indifference price increases and he tends to wait longer before exercise.

A.7 Proof proposition (5.5)

This proof have been stated by Leung & Sircar. We first consider the value of a vested ESO. Define the operator L1 such that:

L1C1 = Ct + (r - q)sC_s + ç2 2 sC_ss - (r + À)C + À(s - K)⁺

Let _À1,À2 be the intensity levels such that 0 < À1 < À2. Let Ci(t, s) and ô* i be the cost of a vested ESO and optimal exercise time corresponding to Ài?i E {1, 2}.

By the Partial Differential Equation of the vested ESO equation(70), we have:

L1C1 = 0. Due to the Q-submartingale property of {e^rs(S_s - K)+}s=0 we have Ci(t, s) = (s - K)⁺. Consequently, direct substitution shows that L2C2 = 0.

Next, we apply Itos formula to the function:

Z t

V (t, St) = e^(r+ë1)tC2(t, St) + e^-(r+ë1)sÀ1(S_s - K)⁺ds

0

Then due to L2C2 = 0 and from the proposition (1.2) , the following holds for any ô = t:

E _t,s [V(ô, S_ô)] = V(t, St)

In particular, we take r = r* 2 < r^* 1then we get:

" _{Z ô}*

2

C2(t, s) < _EQ e(r+ë1)(ô* 2 _t)C2(r* ₂ , Sô* ₂ ) + e_^(r+ë1)(s_t)ë1(S_s - K)⁺ds

t,s

t

h = EQ e_r(ô* 2 ?ôÀ1 _t)(Sô* 2 ?ôÀ1 - K)^+] t,s h < EQ e_r(ô* 1 ?ôÀ1_t)(S ô* 1 ?ôÀ1 - K)^+] t,s < C1(t,s)

(83)

A.8 Proof proposition (5.6)

Let 0 < a < b < T. Denote by r* a , r* b the employees exercise time when the vesting periods are a and b years respectively. Then, we have r * a <r * b < T. Since the discounted payoff process {e^rs(S_s - K)+}s=0 is a Q-submartingale it follows from proposition (1.2) that:

_h h _e_r(T _t)(ST _K)⁺]

_EQ e_r(ô* _a _t)(Sô* _a _K)+] b _K)+] h

< _EQ e_r(ô* _b _t)(Sô* < EQ

t,s t,s t,s