Valuation Methods of Executive Stock Options

The state of the art

The huge use of ESO since the last two decades and the issue on their valuation methods have led to growing literature on this topic. The natural way to understand the problem is to take the similarities between standard options and the ESO. The risk-neutral approach studied by Black, Scholes and Merton in 1973 has been the first one to give formally a price to plain vanilla options. One result is that the fully diversified and rational option holder have to wait until maturity in the case of European option but according to empirical studies such that Huddart & Lang (1996) or Bettis et al. (2005) ESO holders tend to exercise their option early.

But the ESO case is different since its holders are not allowed to fully diversified their risk. Rubinstein in 1995 stated the dissimilarities between standard option and ESO.

An other way to find the value of this contract is to see it as a lump-sum payment such that the ESO holder is indifferent between receiving this payment or receiving the ESO payoff. The most representative of the beginning of the certainty equivalence framework's theory for an ESO was written by Richard A. Lambert. David F, Larker and Robert E. Verrecchia in 1991. They proposed a model of certainty equivalent price for valuing ESO from the Executive's perspective and had pointed out that the valuation model have to incorporate the level of the: risk aversion, diversification and Executive's wealth. This model belong to continuous time models.

But an other class of models: binomial-tree have been developped by Huddart et al. (1994). They examined the non-tradability effects and hedging restrictions and computed the certainty equivalent price for an ESO.

But all of these models are restrictive since that they assume that the Executive can only invest in riskfree bond.

Carpenter with " Exercise and Valuation of Executive Stock Options" in 1998 allows for outside investment and Henderson in 2004 had introduced the indifference valuation methodology for pricing ESO. In "The impact of the market portfolio on the valuation, incentives and optimality of executive stock options " Henderson allows investment in a Market Index which is partially correlated with the stock option underlying stock.

She highlights the relation between risk and incentives and separates market risk from idiosyncratic risk.

Leung & Sircar have defined in 2006 a model with Job termination risk, vesting and risk aversion. The difference between this model with those of Ctivanic, Wiener and Zapatero (2004) and Hull White (2004) is that optimal exercise boundary is endogenously stated in Leung & Sircar while in the second ones this frontier and other parameters such that exit rate are completely ad hoc.

In this dissertation we will treat of ESO valuation models through their continuous time component and present the approach of Leung & Sircar (2006) as well as at the same time the one of Ctivanic, Wiener and Zapatero (2004).

1 Definit ions and Theorems

1.1 Introduction

Trough this section, we introduce some key concepts which will use all along this dissertation. Because the purpose of this dissertation is to show how can a contingent claim as an ESO can be priced in the incomplete market some mathematical and economic specific concepts need to be presented. Let this part begun by the definition of an ESO and some basic definitions:

1.2 Executive or Employee Stock Option: ESO

For the sake of clarity we introduce what we might be termed ESO. These have the following properties:

1. ESO are American call option issued by the executive's company on its own securities;

2. there exist a period of time during which options cannot be exercised: the Vesting Period;

3. holders are not allowed to sell their ESO. They could only exercise options and realize a cash benefit by selling the underlying shares after the Vesting Period;

4. holders are not allowed to hedge their position by short selling the company stock;

5. if the holders leave their job during the Vesting Period then they forfeit unvested options. In the case of the Vesting Period is finished then they have to exercise immediatly vested options that are in the money whereas they forfeit options that are out the money;

6. regarding the company side, a new Treasury stock is issued when options are exercised. 1.3 Stochastic calculus

1.3.1 Fundamental definitions

Definition 1.3.1. Filtration

A filtration F = {F(t) : t E R } is a collection of a-algebra satisfying:

0 = u < t F(u) c F(t)

Then a stochastic process {X(t)}t>0 is said to be adapted with respect to F or (F)-adapted if: ?t E R : X(t) is F(t)-measurable.

Remark During this dissertation F(i) will be denoted by Fi.

Definition 1.3.2. Lévy process

A stochastic process {X(t)}t>0 is said to be a Lévy process if the following properties hold:

1. continuity and limit: X has a right continuous paths and left limits,

2. independent increments: X(0) = 0 and given 0 < t1 < t2 < ··· < t_n, the following random variables are independants:

X(t1),X(t2) -X(t1),. .. ,X(t_n) -X(tn_1)

3. time homogeneity: The distribution of the increments X(u) - X(s) is time homogeneous (depends only on u - s)

Remark A Lévy process {W(t)}t>0 which has stationary and normaly distributed increments W(u) - W(s) with 0 mean and u - s variance is called a Brownian motion or Wiener Process.

Definition 1.3.3. Stopping time

A random variable T : Ù ? N U {8} is called a stopping time if ?n E N, {T = n} E F_n

Definition 1.3.4. Martingale

A martingale is couple of a stochastic process and a filtration {{Mt}t>0, {Ft}t>0} such that {Mt} is {Ft}-adapted and ?t ER the following properties hold:

1. E [|Mt|] <oc

2. E[M_s|Tt]=Mt?s=t

By considering respectively a submartingale and a supermartingale, the equation above is replaced by = and <

Theorem 1.1. Optimal stopping time (Doob)

Let (X)_n a martingale (respectively a supermartingale) and T a stopping time. Then:

1. the process (Xmin(n,T))n?N(denoted (Xn?T)n?N) is a martingale (respectively a supermartingale)

2. When T is bounded almost surely (?N E N such that P [T < N] = 1) E [XT] = E [X0] (respectively <)

3. IfP[T<oc]=1andif ? Ysuchthat|X_n?T|<Y?n ENwith=E[Y]<octhen:

E [XT] = E [X0] (respectively <)

Theorem 1.2. The Optional Sampling Theorem

If {Mt}t=0 is a continuous martingale with respect to the filtration {Tt}t=0 and if ô1 and ô2 are two stopping times such that ô1 < ô2 <K where K is a finite real number, then _Mô2 is integrable (that is has finite expectation) and following equation holds:

E [Mô2 | Tô1] = Mô1, P - almost surely (1)

1.3.2 Itô and Feynman-Kac

Definition 1.3.5. Itô process

Given z and ó, 2 respectively n and nxm dimensional Tt-adapted process and W a m-dimensional Brownian motion.

An n-dimensional Itô process, St is a process that can be represented by:

Z t Z t

St = S0 + z_udu + óudWu

0 0

{dS u = z(S _u, u)du + ó(S_u, u)dWu (2) S0 = s

And have the following Stochastic Differential Form:

Lemma 1.3. Itô's Lemma

Assume St a 1-dimensional Itô process satisfying the following Stochastic Differential Equation (SDE):

{

dSu = z_udu + óudWu S0 = s If ö(t, S) : [0, oc) x R ? R is a C^1,2 function and X(t, S) := ö(t, St) then:

?²ö

X(u, S) = ?ö

?u(u, S_u)du + ?ö

?s (u, S_u)dS_u + 1 ?s2 (u, S_u)(dS_u)²

2

(3)

{?ö }

?²ö

= ?u(u, S_u) + zu ?ö

?s (u, S_u) + ¹ ₂ó² du + óu ?ö

?s (u, S_u)dW_u

u ?s2 (u, S_u)

Theorem 1.4. Feynman-Kac

Let St a Ito process defined by equation(2) and assume that a bounded, continuous and twice differentiable function f(.) is the solution of the following Partial Differential Equation (PDE):

? ??

??

(4)

?f

?u(s, u) + z(s, u) ?f

?s (s, u) + ó(s, u) ?2f

?s2 (s, u) - rf(s, u) = 0

f(s,T) = ø(s)

Then f(.) has the following probabilistic representation:

f(s, t) = _er(T -t)E [ø(ST) | St = s] (5)

1.3.3 Radon-Nikodym

Definition 1.3.6. Absolute continuity

Let P, P0 2 measures on the same probability space Ù.

Then VA E F with zero P-measure if P(A) = 0 P0 (A) = 0 P0 is said absolute continuous with respect to P. All along this dissertation, this property will be denote by <<.

Definition 1.3.7. Radon-Nikodym

R

Let (Ù, F, P) be a probability space and M a non-negative F-measurable random variable such that _ÙM(ù)dP(ù) = 1. We can define a new probability measure P0 on Ù such that:

dP0(ù) = M(ù)dP(ù) (6)

Then for all F-measurable functions f such that the integral exists we have the following equality:

ZÙ

_Z

f(ù)dP0(ù) = f(ù)M(ù)dP(ù) (7)

Ù

Theorem 1.5. Radon-Nikodym Theorem

Let (Ù, F, P) be a a-finite measure space and P0 <<P defined on the filtration F.

Then there exists a unique nonnegative finite measurable function f which is called the Radon-Nikodym derivatives of P0 w.r.t P such that V A E F we have:

_Z

P0(A) = fdP

A

dP0

All along this dissertation we denote Radon-Nikodym derivatives by: f = dP .

The following definition allow us to state the distance between 2 probability distributions. 1.3.4 Cameron-Martin and Girsanov

Lemma 1.6. Exponential Martingale

Suppose a standard brownian motion {W(t) }t>0 defined on the probability space (Ù, F, P) with its associated filtration {F(t)}t>0 . Vë E R, define a stochastic process {Më(t)}t>0 as follow:

Më(t) = eëW(t)_ ë2 2 t

Then {Më(t)}t>0 is a positive martingale relative to {F(t)}t>0

Proof. According to the definition of a martingale we need to show that Vu = t > 0

E [Më(t + u) |F(u)] = Më(u)

_{h i}

E [Më(t + u) |F(u)] = E eëWt+u_ ë2 2 (t+u) | F_u

= E_{h i}
eëWu_ ë2 2 ueë(Wt+u_Wu)_ ë2 2 t | F_{u h i}
= eëWu_ ë2 2 uE eë(Wt+u_Wu)_ ë2 2 t | F_u

_{h i}

= Më(u)E eë(Wt+u_Wu)_ë2 2 t | F_u

_{h i}

= Më(u)E eë(Wt+u_Wu)e_ ë2 2 t | F_u

By the fact that _Wt+u- W_u is independant of F_u and factoring we get

= Më(u)e_ ë2 2 tE [_eë(Wt+u_Wu)]

By normality distribution argument we get

= Më(u)e_ ë2 2 t e+ ë2 2 t

= Më(u)

Theorem 1.7. Cameron-Martin Formula

Under the probability measure _P0ë, the standard brownian motion process {W(t)}0<t<T has the same law as the process {W(t) + Àt}0<t<T has under the probability measure P = P0

Theorem 1.8. Novikov Condition

Let À be a real predictable process and Vt E [0,T] Wt be a standard brownian motion w.r.t to the probability measure P and the filtration F. Then if the following condition hold:

E he12 fô iiëti²dt] < 8

Then Vu E [0, T] the process

M_u = efo ëtdWt-z fo ëdt

is a martingale under P and the filtration F.

Theorem 1.9. Girsanov's change of measure Theorem

Suppose a real process À such that e 2 fô iëti2.

Let Mt(ÀW) be the stochastic exponential of ÀW:

Mt(ÀW) = ef0 fôpt.|2du

dP0

According to Novikov condition then the Radon-Nikodym derivatives is equal to M_oo(ÀW):

dP

dP0 dP

= Moo(ÀW) = ef0^7 ë.dW.^-Z e|ë.|2 dP0 dP 1Ft = Mt(ÀW) = ef0^t ë.dW.^-z foPt.|^2

defines a equivalent probability measure P0 = P. And W^P0(t) such that:

WP0 (t) = W(t) - J t Àudu

is a P0-brownian motion

1.3.5 Minimale Entropy Martingale Measure

Definition 1.3.8. Relative entropy

The relative entropy H(P0 |P) of a probability measure P0 with respect to a probability measure P is defined as follow:

H(P0 |P) = {E^P[^P_P⁰ log( p0 )i if P0 << P

8 otherwise

Remark :

1. The function log(.) used in the previous equation have to be understand as the natural logarithme which is sometimes written ln(.).

2. According to Csiszar (1975), we know that: H(P0 |P) = 0 ? P0 = P otherwise H(P0 |P) = 0

Definition 1.3.9. The Minimal Entropy Martingale Measure (MEMM)

Given a Ft-adapted stochastic process {Xt}_t>0 defined on the probability space stated above. Define also, M^xequiv the set of all Equivalent X-Martingale Measures.

If an Equivalent Martingale Measure (EMM) Q^à (cf. 1.9) satisfies:

VP0 E M^x equiv, H( Q^à| P) = H(P0 | P) (8)

Then ^Qà is called the MEMM of X(t).

Theorem 1.10. Yoshio Miyahara

Let Wt = (W1(t), W2 (t), . . . , Wd(t))⁰ be a d-dimensional ((F), P)-brownian process.

Suppose that Ft =FW t = ó {W(s), s =t}.

Suppose also that a diffusion price process is given by Xt = (X1(t), X2(t),. . . , X_n(t))⁰:

Z t d Z t

Xi(t) = Xi(0) + âi(s, X(s))ds + ái,j(s, X(s))dWj(s), ?i E {1, 2, . . . , n} (9)

0 0

j=1

It is assume that âi and _ái,j, ?i E {1, 2,. .. , n} , ?j E {1, 2,.. . , d} satisfy the global Lipschitz condition. If there exist a martingale measure P0 E M (X) such that H(P0 | P) <oc, then there exist the MEMM Q* which is obtained by Girsanov transformation from P.

Definition 1.3.10. Admissible strategy

A 1-dimensional process 9 is said to be an admissible strategy if 9 is F_u-predictable almost surely square

integrable process.

_{(Z T)}

E (9_u)²du <oc (10)

0

1.4 Analytical tools

1.4.1 Distortion

The following proposition is purely technical. It allows to separate variable in the case of exponential utility via a power transformation and then permit to linearize a non linear Partial Differential Equation in one linear.

Proposition 1.11. Distortion by Zariphopoulou (2001)

Suppose the following PDE:

? ??

??

(11)

(12)

Vt + (í - q - u-r

ó çñ)sV_s + ¹ 2ç2 s2 Vss - 1 2(çñs)2 (Vs)2

V - 1 2(u-r

ó )V = 0

With the terminal boundary condition VT(x, s) = -e?ã(x+(s-K)+)

This non-linear PDE can be reduced as a linear on by an appropriate power transformation:

V = p^ä

Where ä = 1

1-p2 .The former PDE is rewritten as:

? ?

?

pt +Ap - 1 2( u-r

ó )²(1 - ñ)² p = 0

With the terminal boundary condition pT (x, s) = -e- ã ä (x+(s-K)+)

Where the differential operator A = (í - q - çñ^u-r

ó )s ?s ?+ 1 2(çs)2 ?2

?s²

Remark : A= L. WhereL we will se later is the infinitesimal generator of the company stock diffusion process S under the probability measure P0.

1.4.2 Pertubation expansion

Suppose a function p(t,s) which solves the following Partial Differential Equation:

pt+Ap-rp+ ₂ãç²(1 - ñ²)s² expr(T -t) p2 = 0

1

With the terminal condition:

p(T,S)= (ST-K)⁺

Where A is the infinitesimal generator defined in (12). Moreover by Feynman-Kac argument p(t,s) as the following probabilistic reprsentation:

p=

_e-r(T -t)_]

)

ã(1 - ñ2) log(E110 [ _e?ã(1-p²)(ST -K)+ | Xt = X, St = s

Let a random variable X have a variance ç² and write ?k E N, uk = E¹¹0 [_Xk] Where P0 is the probability measure defined by equation(21) (see a little farther).

We define the skewness and the kurtosis of X as:

= E^P0 [(X - u1)^3] ~ç^3skew(X)	= u3	- 3u1u2 + 2u3 1
ç^4kurt(X) = E^P0 [(X - u1)^4]	- 3 =	u4 - 3u2 2 + 12u^2 1u2	- 4u1u3	- 6u2 1

Assume f(y)=log(1 + y) with-1 <y < 1 and f(g(x)) with g(.) = e^x. Then the Taylor expansion to 1.5 Economics concepts

Through this dissertation we try to value an asset under a constrained world. Thus we are in incomplete market and according to this we need to identify the ESO value trough the Executive's utility function. This approach is called utility-based Pricing.

In fact the standard pricing theory can identify by replication the unique price of one derivative asset under complete market.

But the issue in incomplete market come from the non-unicity price of such derivative. By the UtilityBased Pricing approach, the ESO price is define as the Private Valuation or Utility Indifference Price which is own for each holder. The Executive is assumed to be rational and according to her own riskaversion she hedges optimaly her risk by trading into the stock market under the constraint inherent to the ESO contract.

Definition 1.5.1. Private Valuation

The Private Valuaton bid price p is the price at which the ESO holder is indifferent between paying nothing and not having her ESO or paying p and having it. In fact this indifference have to take into the sense that given her optimal expected utility the latter remain unchanged between paying and having or not.

Formaly we have:

Let J(x,s) the optimal expected utility of the Executive with initial endowment x and 1 unit of ESO.

J(x,s) = sup E [U(XT + (ST - K)⁺ |Xt = x]

èt?È

Then p is the Private Valuation of the Executive if:

J(x -p,s) = J(x,0)

Definition 1.5.2. Marginal Price The Marginal Price is the price which left the Executive's maximum utility unchanged for an infinitesimal diversion of funds into the purchase or sale of a claim.

Formaly:

p	=

E [Ux(X* _T )(ST - K)⁺]

J_x

Where {X* }0<t<T is the optimal wealth process generated by the optimal trading strategy and all function whichare associated to a subscript denote the derivative of this function w.r.t to variable which defines the subscript.

2 Mode! for Executive's Stock Option va!uation

This section introduces the executive's investment problem and the stochastic setting of the Economy.

2.1 The Economy

To start this dissertation, we introduce in this section a general framework for the Economy in the presence of uncertainty and in which the ESO's holder lives.

Consider the following probability space (Ù, A, P) which represents the uncertainty of the Economy and on which is defined a n-dimensional Brownian motion W= (W',... , W n) 0 over a finite continuous timehorizon [0, T]. The superscript denotes the transposition operator, since W is a column vector as every vector in this dissertation.

We consider a financial market M allowing instantaneous default-free borrowing and lending at continuouslycompounded rate given by the process r. The rest of M is composed by n risky assets which can be traded.

Suppose one risk-neutral firm and its risk-averse Executive in this Economy.

Call options on this firm stock are granted to the Executive as part of her compensation package and to avoid the issue of insider trading, the executive cannot trade in the firm stock.

Moreover these Call options have a vesting period inside which the holder cannot exercises it.

Suppose also that the risk-averse executive's preferences can be modelled by an exponential utility function: U(x) =-e^-ã^x, where ã> 0 define the executive's constant absolute risk aversion and x her wealth. We can see that U(x) is a twice continuously-differentiable function, strictly increasing and stricly concave in x. These properties respectively reflects that the executive preferes more wealth to less and that executive is risk-averse.

Moreover U(x) belong to the Hara utilities class (the proof of this assertion can be found in the appendix).

2.2 Assets Price

Suppose n=2 risky assets: the Firm stock and the Market Index and 1 default-free bond. The ESO's holder is allowed to trade only in the Market Index and the risk-free bond but not on the company stock. Each price of risky asset is modelled as a diffusion process. The first one is the Market index which is partially correlated with the company stock:

dM u = zM_udu+óM_udW' _u, t=u=T (13)

Mt =M

The last one is the company stock price:

dS_u = (í-q)Sudu+iSudW ² _u, t=u=T (14)

St =S

And the price of the default-free bond B:

dR_u = rRudu, t=u=T(15) Rt = 1

Where:

· r is the constant risk-free rate of the Economy.

· u = t is a time index which live in [t, T],

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

· Mt = M is the Market Index's stock price at time t,

· z and í are respectively the constant Market Index and company stock's expected return under the historical measure P,

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

· ó, i are respectively the constant Market Index and company stock's volatility under the historical measure P,

· W ⁱ, ?i E {1, 2} is a Brownian motion defined on the probability space (Ù, F, (F_u), P). The information set is captured by the augmented filtration {F_u : u E [0, t] } where F_u is the augmented ó-algebra generated by {W¹, W², t = u = 0} and their instantaneous correlation p E (--1,1),

· FT C A and F0 is trivial.

Remark : We assume in the equation (14) that any dilution of the Company's stock price is excluded during the lifetime of the option. We could say that the price has been adjusted before grant date.

2.3 The Executive's Investment Problem: EIP

By assumption, the ESO's holder is not allowed to sell her option or to trade her company stock. Therefore, it is central to consider her risk aversion.

The following subsection shows the Executive's Investment Problem in its general form while the next section describes a modified model where the Executive is endowed by 1 unit of ESO.

2.3.1 General results for the EIP

Previously, we have defined her risk preferences as the exponential utility function of her wealth U(x) = --e^-ã^x, where 'y> 0.

We suppose also, throughout the entire period [t, T], that the Executive trades dynamically between the risk-free asset (bond) and the Market Index.

According to definition (1.3.10) let an admissible trading strategy {9_u, T = u = t}. Denote _Èt,T the set of 1-dimensional admissible strategies over the time period [t, T].

Consider now that the ESO's holder uses a admissible strategy 9 in a self-financing way (i.e she invests at time u 9_u in the risky asset (Market Index) and X_u -- 9_u in the bond).

Then for all s = t the executive's trading wealth process evolves according to:

~

dXè u = {9 _u(sa -- r) + rX} du + 9uódW u 1(16)

Xt =X

The Executive's objective is to maximize her expected utility of wealth at time T subject to the Executive's trading wealth process until T (which can be viewed as the budget constraint).

Then the Executive's Investment Problem can be formulate as:

? ??

??

I(u,X) = sup

Èu,T E [U(XT = --e^-ã^XT) |X_u = x]

s/t

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

It follow that:

= (a-r)

ãó2 e

(17)

-r(T -u)

I(u, X) = --e?ãXer(T -u)e - (T -u)

2 (M-r

ó )2

9^*_u

This is the well-known solution of Merton Problem with exponential utility form. The proof of this results can be shown by the reader on the appendix.

Remarks: The optimal expected utility of wealth is defined by two parts:

1. the fist one: --e ?ãxer(T -u) i s the utili ty which come from the investment in the risk-free asset;

(T -u)

2 ( M-r

2. the last one: e- ó )2 is the utility which come from the trading stratagy under the
investment on the Market Index.

Property 2.3.1. Given that 'y, r and X are positive and that G(u,X) is strictly negative then:
= ('yrX + 2( sa -- r

DI
Du

1 ó )2)I(u,X)<0

Then the optimal expected utility is decreasing with time. The aversion depend in this case only on the time increment.