Valuation Methods of Executive Stock Options

ValUatIOn methOdS Of

ExeCUtIVe StOCK OPtIOnS

DisseRtation
MasteR II
Financial maRkets and inteRmediaRies

Pomiès Ismaïl

SupeRvisoR: PR. Villeneuve Stéphane

Dedication

To my wife

ValUatIOn methOdS Of
ExeCUtIVe StOCK OPtIOnS

Abstract

This dissertation develops the main things in continuous time utility-based models for valuing ESO. The first part will be devoted to exposing some useful technical tools from the basics of stochastic calculus to the Minimal Entropy Martingale Measure concepts including some economic key concepts. The second part will deal with the general investment model developed by Merton (1969). The result coming from this model will allow us to give a general framework for valuing an ESO which will be the subject of the third part. By using some statistical tools and a polynomial approximation we will show that the Black & Scholes valuation is an upper bound to the ESO fair-price when its holder is subject to risk-aversion and according to these results we will discuss about the effects of parameters included in the model. The fifths part part will exposed the Leung & Sircar approach (2006). This sophisticated model will allow to value an ESO by taking into account the vesting period and the job termination risk. And finally the firm's perspective will be exposed by treating the firm's cost of issuing an ESO with several models from a naive approach to a more sophisticated model while the parameters effects will conclude dissertation.

Contents

1	Definitions and Theorems		9
	1.1	Introduction	9
	1.2	Executive or Employee Stock Option: ESO	9
	1.3	Stochastic calculus	9
		1.3.1 Fundamental definitions	9
		1.3.2 Itô and Feynman-Kac	10
		1.3.3 Radon-Nikodym	11
		1.3.4 Cameron-Martin and Girsanov	11
		1.3.5 Minimale Entropy Martingale Measure	12
	1.4	Analytical tools	13
		1.4.1 Distortion	13
		1.4.2 Pertubation expansion	13
	1.5	Economics concepts	14
2	Model for Executive's Stock Option valuation		15
	2.1	The Economy	15
	2.2	Assets Price	15
	2.3	The Executive's Investment Problem: EIP	16
		2.3.1 General results for the EIP	16
3	The Executive's Optimal Exercise Policy: the general approach		17
	3.1	utility-based pricing	17
		3.1.1 Introduction	17
		3.1.2 The general form of the EIP with 1 unit of ESO	17
		3.1.3 Private Price of 1 unit of ESO	17
		3.1.4 The Partial Differential Equation of the Private Price	19
	3.2	The Private Price and its Black & Scholes counterpart	20
		3.2.1 Skewness and Kurtosis	20
		3.2.2 The perturbative expansion	21
		3.2.3 Comments	22
	3.3	The optimal trading strategy	22
	3.4	The effects of the parameters	23
		3.4.1 The Private Price	23
		3.4.2 The Optimal Trading Strategy	23
		3.4.3 Incentives effect or ESO delta	23
		3.4.4 The effect of risk-aversion	25
		3.4.5 The effect of correlation	25
4	The Executive's Optimal Exercise Policy: Leung & Sircar Approach (2006)		27
	4.1	Settings	27
		4.1.1 The job termination risk and exercise window	27
	4.2	Optimization method	27
		4.2.1 The Executive's Exercise Boundary	29
		4.2.2 A Partial Differential Equation for the Private Price	30
		4.2.3 The optimal trading strategy	31
	4.3	The effects of parameters	32
		4.3.1 The effect of Job Termination risk	32
		4.3.2 The effect of risk-aversion	32
		4.3.3 The correlation effect	32
5	ESO cost to the firm		33
	5.1	General model for the ESO cost to the firm	33
	5.2	The naive approach	33
	5.3	The ESO cost to the firm with no vesting period and no job termination risk - Ctivanic, Wiener and Zapatero (2004)	34
	5.4	An Intensity based model for the firm's cost - Ctivanic, Wiener and Zapatero (2004) . . .	35

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

and Zapatero (2004) 35

5.6 ESO cost to the firm: Leung & Sircar (2006) 36

5.7 The effects of parameters 37

5.7.1 The job termination risk intensity 38

5.7.2 The vesting period 38

A Proofs 40

A.1 Proof (1): 40

A.2 Proof General Investment Problem(17): 40

A.3 Proof proposition (4.4) 42

A.4 Proof proposition (4.5) 42

A.5 Proof proposition (4.6) 42

A.6 Proof proposition (4.7) 42

A.7 Proof proposition (5.5) 42

A.8 Proof proposition (5.6) 43

Introduction

This dissertation deals with the evaluation methods of Executive Stock Option (ESO) in continuous time. Executive or Employee Stock Option are call options granted by firm's shareholders to their Executives or Employees as compensation in addition to salary. The ESO give the right but not the obligation to buy a number of shares of the underlying company's stock at a predetermined price (strike) and period of time (from the end of vesting period to maturity).

From agency problem point of view, this compensation program allows to add and align incentives to their holders with those of the shareholders. Indeed, there are a lot of situations in which Executive has to take a risk in firm's projects and could have a more conservative or more agressive choice than the one choose by the shareholders. Thus via the ESO program, their holders have an incentive to act as a shareholder: the implied assumption is that the ESO holder has an influence to the stock price.

The main issue is that the ESO cannot be priced by the standard option pricing theory.

Indeed Black, Scholes and Merton in 1973 were the first ones having defined a mathematical understanding of the options pricing but some of main assumptions such that short selling the underlying stock and market completeness do not work in the ESO framework. In the standard theory the call option payoff can be replicated by a portfolio made up by risky and risk-free assets.

But in the case of ESO, the holder is not allowed to trade her company stock leading to an undiversified portfolio for the holder and thus to be exposed to an unhedgeable risk. It result that an infinty of prices could be derive for one derivative.

Empirically, it has been shown that B & S valuation failed to price an ESO.

According to Huddart & Lang (1996), Marquardt (2002) and others empirical studies, the majority of holders tend to exercise their options early which is in contradiction with the prediction made by the B & S model. These studies underline the suboptimal behaviour according to the B & S theory. This suboptimal behaviour arises in fact with risk-aversion and others constraints such that trading constraints and job termination risk.

By this assessment and the risk-aversion principle, we have to develop a continuous time valuation theory based on indifference preferences and to distinguish the ESO from plain vanilla options.

A utility-based valuation allow to find a unique fair-price by taking into account the risk-aversion parameter. The indifference or private price resulting from the model is not the same depending on the level of risk-aversion parameter. That is why we can empirically found that two Employees or Executives granted with the same ESO and whose the exercise time is not the same.

The valuation method can be thought from the Executive's or the firm's perspective.

When a company issues some ESO no trading constraints are imposed and thus there is no unhedgeable risk. Intuitively, B & S valuation method can be used since the shareholders of the company can be assumed as risk-neutral and subsequently the cost of issuing an ESO is easily demonstrable.

Regarding the company side, this naive approach is not an accurate way to valuate ESO, since it is does not care about the suboptimal behaviour of the counterparty.

By adjusting the B & S-model and replace the option expiration date by the expected time to exercise the Regulator wants to improve the company pricing method into their report.

To summarize we have to find a cost model which integrates all constraints imposed to the ESO holders in order to get the better price possible. By deducing the critical level, which is the value-maximizing exercise policy of the holder, and combining Vesting period, job termination risk, trading and hedging constraints we have finaly the option cost.

This dissertation aspires to solve these main issues stated above and will be presented as follows:

After having introduced the state of the art in the ESO valuation method, the first part will be dedicated to some mathematical, statistical and economics concepts which will intervene during the report.

The second part will treated about the fundamental investment model derived from Merton (1969). This model will be useful since the ESO pricing model are no more no less an enhanced Merton's problem. The third part will give the general approach to deal with ESO.

Firstly we will focus on the optimization process and the Private Price, then we will use a polynomial
approximation in order to reveal the B & S price as a component of the Private Price formulation and

discuss about the difference between them. It will be presented also, the optimal trading strategy in the case of one ESO and we will conclude this part by a discussion of the effects of the model parameters. The fourth part will expose the Leung & Sircar's model for valuing an ESO from the Executive's side. While the settings will be presented in the first sub-section, the optimization method will be the subject of the second one. Through the last one we will see how can be defined the Executive's Exercise boundary and then how can we get the Private Price and its associated PDE. We will conclude this sub-section by the optimal trading strategy statement.

Finally, the ESO cost from the firm's side will be tackled.

The first step will show the naive approach which is in fact the B & S model. The second, third and fourth step will present the ESO cost with respectively the following assumptions: the Executive's optimal exercise boundary without vesting period and no job termination risk, the job termination risk without the Executive's optimal exercise boundary and vesting period: a risk-intensity model, and the Executive's optimal exercise boundary with job termination risk and no vesting period. These models coming from the paper written by Ctivanic, Wiener and Zapatero (2004).

We will lead to the Leung & Sircar's model for valuing an ESO from the firm's side. This one integrates all parameters such that job termination risk, vesting period and Executive's optimal exercise boundary and we will conclude this part by the parameters effects.

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.

3 The Executive's Optimal Exercise Policy: the general approach

3.1 utility-based pricing

3.1.1 Introduction

By using the complete market argument, the standard financial theory can valuate a contingent claim by replicating its future payoff via the use of risky and riskfree stocks in the market. The derivative security price found is unique.

However in the incomplete market case, the future payoff cannot be replicated since the agent's environ- ment is constrained and thus there are not enough assets in the market that allow the fully replication of the terminal payoff.

This issue can be solved by considering the utility-based Pricing method.

Suppose that the agent has a utility function dependant on her risk aversion parameter and her initial wealth. Then by founding an optimal trading rule which involves the investment of her wealth between risky and riskfree assets we can find a price p which makes the agent indifferent between having a stock option and paying p or paying nothing and not having the derivatives. In the economic literature, p is called indifference price or private price.

Given the general formulation of the Excutive Investment Problem (17) we can formulate the indifference price idea via the definition (1.5.1).

The main problem in this approach come from the technical difficulty to find an explicit solution. Thus a set of assumptions is imposed to the utility function form. Because the optimization program , technically speaking is hard, it is supposed an exponential form of the utility function in order to allow an easy variable separation.

By this way we can formulate the Executive's Optimal Policy in its general form through a general method.

3.1.2 The general form of the EIP with 1 unit of ESO

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 general result found on the Merton Problem allow us to generalize it in the case where the Executive is endowed by 1 unit of ESO.

Assume all constraints imposed in the previous section hold (recall: the Executive is allowed to trade only in the risk-less asset and the Market Index) and Mt = M (The Market Index price at t is M).

The aim of the Executive is to maximize her expected utility among all trading strategy before the Terminal time T.

Then at time u E [0, T], the EIP associated to the value function G(u,X,M) is defined by:

G(u,X,S) = sup è?È0,T = sup è?È0,T

EP [U(T,XT + (ST - K)^+)| Xt = X,St = s] EP [ ] -e-ã(XT +(ST -K)+)er(T _ u)e - (T _u) 2 ( u_r ó ) 2 | Xt = X, St = s

(18)

Here we have reformulate the EIP general form by introducing only 1 unit of ESO.

Remark : It can be easily formulated this EIP with n identical ESO by putting n as factor before the derivative's payoff.

3.1.3 Private Price of 1 unit of ESO

The second step of this methodology consists of finding the Private Price p. To achieve this objective we are going to use the Private Price definition.

( G(u, X, S) = sup E^P ~U(T, XT + (ST - K)⁺) | Xt = X, St = s

è?È0,T (19)

G(u, X - p, S) = G(u, X, 0)

Using the Bellman dynamic programming principle, G(u,X,S) have to satisfy the following Partial Differential Equation:

(

sup LG = 0
è?È0,T G(u, X - p, S) = G(u, X, 0)

Where L define the inifinisetimal generator of (X,S) under the historical probability measure P: D

L = + [9(sa - r) + rX] D DX + (í - q)S D DS + ¹ ₂(9 ó)2 D2

DX2 + ¹ 2(çS)2 D2

DS2 + (ñçó9S) D2

DSDX (20)

Du

Remark :

1. the differential operator L is not linear in 9. Then if we focus us on the optimization problem we are face on a non linear Hamilton-Jacobi-Bellman equation. By 1.11 argument we are allowed to linearize it by introducing a power transformation;

2. given that G(u,X,S) could be written as G(u, X, S) = e_ã^r(T_^u)XG(u, 0, S) we can reduce the dimension of the original problem (18)

Remark : By using the Girsanov's change of measure argument we know that G(u,X,S) is a martingale under the optimal strategy 9^* define by equation (17). And moreover this martingale is MEMM by 1.10 argument.

According to the last argument we can define the MEMM P0 relative to the historical probability P such that the Utility process is a P0-martingale.

[{ ó )2T )} ]

e(_ u-r

ó WT _ 1 2 ( u-r

P0(A) = E IA, A?FT (21)

The last argument point out that all other strategy are not optimal and define a supermartingale.

1

Given G(u, 0,S) = p 1-ñ2 and using the Bellman dynamic programming principle and the 1.11 argument,

p have to satisfy the programming system:

sup è?È0,T

Lp=0 (22)

With the following boundary conditions:

f

p(T,X,S) = e_ã(1_p2)(ST_K)+ p(T,X,0) = 0 The simplest form of the PDE is:

Lp(t, s) = 0 (23)

Where L is the infinitesimal generator of the process (St) under P0:

L = D + (í - q - sa - r D 2(çs)2 D2

1

DS +

çñ)s (24)

Dt ó DS²

(25)

Proof. We can rewrite the stock price diffusion process (St) under P0 as:

dS_u = (í - q)S_udu + çSudW u 2

= (í - q)S_udu + çñS_udW u By Girsanov argument (cf.1.9)

= (í - q)S_udu + çñSu(dW P0

u - u_r

ó du)

= (í - q - u_r

ó çñ)S_udu + çñSudW 1,P0

u



Now we can write an explicit form for the intermediate function p(t, S).

By Feynman & Kac argument the PDE (23) has the following form under the measure P0:

1

1-ñ2 (27)

]p(t, S) = EP0 [ _e_ã(1_p²)(ST _K)+ | Xt = X, St = s(26) And with this expression we deduce the form of the value function G(t,X,S):

]

G(t, X, S) = -e_ãXer(T -t)_ (T -t)

2 ( u-r

ó )2EP0 [ _e_ã(1_p²)(ST _K)+ | Xt = X, St = s

Now by using the system (19) we can deduce the Private Price of 1 unit of ESO:

Proposition 3.1. Executive Indifference Price

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

2)(S-K)+ | St =S,Xt = x])

_e-r(T-t)

(28)

p(t, s) = P° [-ã(1

ã(1 - ñ2) log(Ee

Or equivalently

G(t, x, s) = V (t, x)e-ãp(t,s)e-r(T-t) (29)

Proof. By definition the Executive's Indifference Price is such that:

G(t, x - p, s) = G(t, x, 0) = V (t, x)

Then by separation variables argument we get:

1

G (t , x - p, 0)p 1- P2 = G(t, x, 0) = V (t, x)

t] -e-ã(x-p)e^r(T-t) e- 2 (T-t)(ur) 2 } EP° [e-ã(1-ñ2)(ST-K)+ |Xt = X, St = s

1 1-P^2

= V(t, x)



1

teãp^er(T-t)V(t, x)} EP° [e-ã(1-ñ2)(ST-K)+| Xt = X, St = s] 1-P² = V(t, x)

1

teãper(T-t)} _EP° [e-ã(1-ñ2)(ST-K)+ | Xt = X, St = s] 1-P² = 1 (Since V(t, x) =6 0)

1

eãper(T-t) = _EP ° [e -ã(1-ñ2)(ST-K)+ | Xt = X, St = s] 1-P²

ãper(T-t) _=-1-1ñ2 log(EP° [e-ã(1-ñ2)(ST-K)+ Xt = X St = s])|

log(EP° [e-ã(1-ñ2)(ST-K)+|Xt = X, St = s])

p =-ã(1-- ñ²)

Now we can deduce the PDE of the private price.

3.1.4 The Partial Differential Equation of the Private Price

From the Private Price's expression we know that p(t,s) have to satisfied the following PDE (which has been defined in the equation (23)):

_e-r(T -t)

p(t, s) ^=-ã(1 - ñ2) log(p) (30)

By Feynman-Kac argument p solves:

Lp= 0

With the boundary conditions:

f /3(T, s) = eã(1-ñ2)(ST-K)+

1 p(T, 0) = 0

And where L defined by (24) is the inifinitesimal generator of the company's stock price process under the MEMM P0.

Thus the Private Price p(t,s) satisfies the following Partial Differential Equation:

Lp(t, s) - rp(t, s) -

1 T --t 819(4 s) )2

(31)

ã(çs)²(1 - ñ2)er( ) = 0

?s

With boundary condition:

p(t, s) = (ST - K)⁺

Proof. By equation(23) we have:

-- r1

pt(t, s) + (í--q ñ)sp_s(t, s) + ₂ (çs)2

.73ss(t, s) = 0

ó

With:

· pt =

ap
at

· p_s = a p

a_s

_a2

· fIC^.1

s s = p

a_s2

But:

_e-r(T-t)

p(t, s) = --

y(1 -- ñ2) log(p(t, s)) ? p(t, s) = e-ã(1-ñ2)p(t,s)er(T-t)

Then:


·

pt(t, s) = --y(1 -- ñ2)e-r(T-t)e-ã(1-ñ2)p(t,s)e-r(T-t) (pt(t, s) -- rp(t, s))

· p_s(t, s) = --y(1 -- ñ2)e-r(T-t)e-ã(1-ñ2)p(t,s)e-r(T-t)

_p2)_e--r(T--t)_e--^y(1--P^2-r(T-t) e_y(i _p 2)_e--r(T--t)_ps /
·,

· /3_ss(t, s) = y(1-- --p_ss(t, s))
p _s (t,s)

Then we get the following PDE in term of p after divided each part of the equation by:

--y(1 -- _ñ2)_e-r(T-t)_e-ã(1-ñ²)p(t,s)e^-r(T-t)

for ñ =6 1 and y =6 0

pt(t, s) -- rp(t, s) + (í -- q -- u-r _óçñ)sp_s(t, s) + 12 (çs)² _(pss (t, s) + y(1-- ñ2)e-r(T-t)p2s(t,s)) = 0

By grouping we get: (32)

Lp(t, s) -- rp(t, s) -- ã2 (ç s)² (1 -- ñ²)e^-r(T-t)p²_s(t, s) = 0



3.2 The Private Price and its Black & Scholes counterpart

During the previous subsection we have derived an explicit form to the Executive Private Price. We have built an incomplete market framework in order to understand how could be the behaviour of an executive with an ESO. We know that in Black&Scholes (B & S) framework failed to fair-valued a such option since assumptions such that unconstrained portfolio and riskless agent are unrealistic. Intuitively, we could say that B & S valuation overstate the fair value of an ESO: the B & S price is an upper bound of the fair price. The idea here is to define an approximate expression of the Private Price derived previously and compare it by the B & S value. First of all we are going to deal with some key statistical concepts and subsequently use an analytical tool (perturbation expansion) in order to approximate the Executive Private Price. This will allow us to derive the Executive Private Price as the B & S price plus a negative pertubation. And finally we are going to conclude that B & S price overstate the fair-value of an ESO.

3.2.1 Skewness and Kurtosis

A random variable could be defined with its moment. Mean and variance which are the most wellknown moment of a random variable are respectively the first moment and second central moment. But some higher moment are interesting such that skewness and kurtosis which are respectively the third and fourth central moment.

This moment are interesting since it measure respectively the lopsidedness and the degree to which a statistical frequency curve is peaked. But in our problem, this moments will serve us to give an polynomial expression to the Private price.

Considere first the expression of the skewness and secondly the one of kurtosis.

By the definition of skew(X) we get:

~u³skew(X) = u3 -- 3u1u2 + 2u2 1 Where uk = E [(X)^k] , ?k E N (34)
Definition 3.2.2. Kurtosis

The kurtosis is the relative peakness or flatness of a distribution compared with the Gaussian distribution. Let X the same random variable as previously. Thus the fourth standardized central moment of X is written by kurt(X) and is defined by:

E[(X -- u1)⁴]

kurt(X) :=

3 (35)

_u4

By definition of kurt(X) we get:

u⁴kurt(X) = u4 -- 3u²2 + 12u²1u2 -- 4u1u3 -- 60. (36)

In the next part we will derive a polynomial form for the Executive Private Price.

3.2.2 The perturbative expansion

By Taylor argument we can approximate each n-differentiable function by its n orders differentials. The idea in this part is to derive a tractable polynomial expression of the Executive Private Price in order to reveal the B & S valuation and thus to be able to compare this two valuation. By equation (3.1) we have an explicit form of the private price.

?ñ2)(S,--K)+ | St = S, Xt = x] )

_e--r(T--t)

p(t, s) =-- log(EP0 [e?ã(1

ã(1 -- ñ²)

Let E := ã(1-- ñ²), z := (St -- K)⁺, f (Ez) = e-^€z - 1 and y := E^P0 [f (Ez) | St = S, Xt = x] = Ellt°,08,x [Ez]. Thus the polynomial expansion of the function e-^€z and log(1 + y) at 0 to the order 4 for the first one and to order 1 for the other one as the following expression:

₈

<>> >

>>>:

(Ez)²

e^-€z = 1 -- Ez

2!

(Ez)3 + ^(Ez)4 O(E4)

3! 4!

log(1 + y) = y + O(y),?|y|= 1

Suppose that uk := EP0 ((ST --K)⁺)^k | St = S, Xt = x = E^llt^',⁰8,x

e

p(t, s) --

log ₍1 -- Eu1 + E2 u2 --Eu3 +E4u41

2! 3! E4!

-r(T-t)



~_e-r(T-t)₂ E2 2 E3 3

Eu1 + _2! (u2 -- + _2! u₁ -- 3! (u3 -- 3u1u2 + 2u²₁) +



3!3 -- 3! 1 2E3 u2 + E4 4! u4 )


·
·
·

~ ~

_e-r(T-t) E2 E

EEPt ,x 2! ,8 [z] + V _tP0,x [z] -- E3 skew^P0 (z) + _4! kurtP ⁰ (z) +

3! é

E

e-r(T-t) (4,08,x [z] vir0

L 2! .,8,x !

[z] + E 32 skew^P0 (z) -- 4E! kurt^P0 (z) -- é)

E

(37)

Where é = ^€2₂u²1 + €33! (3u1u2 -- 2u21) + €43! (3u2 2 -- 12u²1u2 + _4u1u3 +

6u⁴1)

3.2.3 Comments

We have found in the previous part a nice form of the Executive Private Price. In fact we can write the Indifference Price as a linear function of the n-moments of the option payoff.

Without complex calculus in order to find the explicit Private Price expression with all variables defined in the model we can reduce the previous expression and consider only terms with epsilon-power strictly lower than two. In fact we assume that ~ⁿ 0, {?n E N n n = 2}

Thus the simplified form of the Executive Private Price is:

~ ~

[(ST - K)⁺] - ã(1 - ñ²)

p(t, s) _e-r(T -t) E ⁰ V ⁰ [(ST - K)⁺] + E ⁰ [((ST - K)⁺)²]))

t,s,x t,s,x t,s,x

2

p(t, s) pBS(t, s) - Ø(f, S)

Where Ø(f, S) > 0 and

pBS (t, s) is the price of an european call option in B & S framework.

(38) Finally we have shown by a polynomial approximation that the fair-price of an ESO is lower than the price derived in B & S. This inequality come from the risk-aversion of the Executive which cannot perfectly hedge her risk with the set of constraints which are imposed to her.

3.3 The optimal trading strategy

This section treats about the Executive's optimal strategy where in this case she is endowed by 1 unit of ESO. By using a similar way that in the EIP section we are going to solve this by Hamilton-JacobiBellman principle. The problem is stated as:

max è ?Èt,T

LG(u,x,s)=0 (39)

Where L is the inifinitesimal generator of (X,S) under P which is defined in the equation (20). By Hamilton-Jacobi-Bellman argument we have to solve:

(u - r) DG

DX + èó2D2G

DX2 + (ñçóS) D2G

DSDX = 0

And then we have a general form of the optimal trading strategy è^**attimeu:

è** = -( u - r) aG

ax + ( ñçóS) a2G

aSax (40)

ó 2 a²G

ax²

Now we can express all the partial derivative functions:

1. aG

ax = - ãer(T -u)G

2. a2G

ax2 = ( ãe r(T -u))2G

3. a2G

axaS = aG aS = ( ãer(T -u)) 2G ap

ap

axaS

by separation of variables argument.

Then we obtain the optimal strategy 9^** as a function of the differential of the Private Price:

9** =

-'y(u - r)e^r(T-^u)G + (ñióS)('yer(T -u))2G ap aS ó^2('ye^r(^T-^u))^2G

((u-r) ~

9** = _e-r(T -u)

ãó²

- ñçS ap

ó aS

9** = 9^* +ö(S,p,ñ,i,ó,'y)

9** = 9^* If ñ < 0 and 9^** <9^* otherwise.

Where ö(S, p, ñ, i, ó, 'y) = - ñçS aS <0. If ñ> 0 and the reverse otherwise

ap

ó

Since^ap

aS = 0 (we will see this assertion later)

3.4 The effects of the parameters

In the previous subsection we have found a general form for the Executive's Private Price of 1 unit of ESO. Moreover we have derived the Executive's optimal strategy when she is endowed by 1 unit of ESO. We observe that the Executive's initial endowment does not appear in the price's expression and that the optimal strategy is equal to the optimal strategy when the Executive is without ESO plus a certain function that we are going to define in this section.

3.4.1 The Private Price

Thus the ESO Private Price is independant of the Executive's initial wealth. This property is closely linked to the utility function form. We have seen that by exponentiality argument we can separate each variable allowing us to simplify computations. But in reality the agent's utility can be describe by a huge set of utility function. Suppose for instance that the ESO contract involves holder's liability such that if ST - K < 0 then the executive have a penalty (her salary can be reduce by a certain amount of money) then the payoff of the ESO can be negative. Then in this specific case the Private Price could be not well defined.

So it is important to show that the solution found here is closely linked to the utility function's form and can be generalized by relaxing assumptions made for make computations more easy.

3.4.2 The Optimal Trading Strategy

We have found that the optimal trading strategy in the case where the executive is endowed of 1 unit of ESO is linked with the optimal trading strategy without ESO plus a certain function. This function can be interpreted as the adjustment of the initial trading strategy brought by the introduction of the ESO in the executive's wealth. More precisely we are going to define this function as the part of the optimal trading strategy which allows to the executive to hedge her risk brought by the derivative.

Proposition 3.2. The hedging strategy for the ESO at the private price p(t,s) at time t E [0, T] is to hold ê_ushares of the Market Index M at time u E [t, T] such that:

êuMu = -ñiS ó

?p (41) ?S

Remark :The Executive have to be short on the Market Index if it is positively correlated with the underlying stock of her option and have to be long otherwise. By risk-aversion principle she have to invest only in the risk-free bond if the Market Index is totally non-correlated with her companys stock.

3.4.3 Incentives effect or ESO delta

The main argument for shareholders to expense ESO is to align executive's incentives to their owns. Basically the Executive throught her ESO is exposed to an unhedgeable specific risk while by assumption the unspecific risk can be fully hedgeable by the Market Index. The Executive's incentives is closely linked to the part of her risk which cannot be hedgeable but by risk-aversion argument we know that the

executive's value of her ESO is less than its value in the market.

Following the argument exposed just above we will take as incentives definition the option's effect on
motivation for an executive to increase the company's stock price and thus formally sepaking take the

first derivative of the Executive's Private Price with respect to the company's stock price. In the classical option pricing literature the incentives effect is called option delta. We are going to give an explicit solution to the ESO delta and discuss about the effect of the others parameters.

Proposition 3.3. Incentives effect

The Executive 's incentives effect Ä provoked by the ESO is defined by the first derivative of the Executive 's Private Price according to the underlying stock price. Therefore the incentives effect called

the ESO delta is positive and has the following explicit formulation:

_{h i}

EPp I(ST _=K)e?ã(1?ñ²)(ST -K) + | Xt = X, St = s

D

Ä(t, s) = _Dsp(t, s) = _e-r(T -t) _{h i} = 0 (42)

EPp _e?ã(1?ñ²)(ST -K)+ | Xt = X, St = s

Under the new measure Pp = P0 defined by:

dPp = eçW'0?ç2 t 2 dP0 (43)

Proof. First of all, according to equation (30) we have the following equation:

ã(1 - ñ²)

Dp
Ds

e

= -

-r(T-t)

D log(^-p)

Ds

Dp
Ds

e

= -

ã(1-ñ²) ^-p(t,s)

-r(T-t)

D p^-

Ds (t,s)

(44)

Secondly, according to (1.9) argument _Pp defines an equivalent Martingale Measure for the company's stock price process. And thus the company's stock price is an exponential brownian motion with volatility i and drift ((u - q - u-r

ó iñ) + i²) under the new measure Pp.

Then, let ð(t, s) = ^-p_s then we can rewrite the PDE (23) of ^-p(t, s) in term of ð(t, s):

L^-p = p^-ÿ + (u - q - /1 - r

ó iñ)s^- p_s + ¹ ₂(is)^{2 -}pss = 0

p-ÿ Ds +

(u - q - /1 - riñ)s-ps Ds + 1 ó (is)^2 -pss Ds = 0 2

ðÿ+(u-q-/1 - r

ó iñ)sð_s + (u-q- /1 - r

ó iñ)ð+i2sðs +1 2(is)²ðss =0

ðÿ+ (u - q - /1- r

ó iñ+i²)sð_s + (u - q - /1 - r

ó iñ)ð +1 2(is)²ðss = 0

With the boundary condition defined by:

^-p_s(T, s) = -ã(1 - ñ²)(ST - K)⁺I(_s=K)e?ã⁽¹?ñ^2)(s-^K)+ (45)

where ^-p(t, s) is define by the equation (26) and _Pp is defined according to:

Moreover by using Feynman-Kac argument under the new probability measureP_p we obtain the following probabilistic representat ion of ^-p_s:

[

^-p_s(t, s) = ð(t, s) = e{(í?q? u-r

ó çñ)(T -t)}EPp -ã(1 - ñ²)^I(ST = K)e-ã(1 - ñ²)(ST - K)+ | Xt = X,St = s

= 0

Finally by combining the explicit expression of ^-p_s with equations (44) and (26) (taken in the new measure Pp) we get:

[ ]

?p
?s

-r(T-t)

e

= -

'y(1 - ñ²)

e{(í?q? u-r

ó çñ)(T -t)}EPp -'y(1 - ñ²)^I(ST = K)e-'y(1 - ñ²)(ST - K)+ | Xt = X, St = s

[ ]

e{(í-q- u-r

ó çñ)(T _{-t)}EPp e}?ã(1?ñ²)(ST -K)+ | Xt = X, St = s

= e-r(T-t)

h i EPp ^I(ST = K)e-'y(1 - ñ^2)(ST - K)+ | Xt = X, St = s [ i = 0 EPp e?ã(1?ñ^2)(ST -K)+ | Xt =X, St = s

₂'y(ñ2 - ñ1)e-r(T-t) (V^P0 [(ST - K)⁺] + E^P0 [((ST - K)⁺)²]) < 0

t,s,x t,s,x

1

pñ2 - pñ1 -

3.4.4 The effect of risk-aversion

ESO are awarded to the executive in order to align her incentives to those of shareholders. Also we know from standard option theory that more risky is the underlying of the option more its value is important. But in the case of incomplete market framework this assertion does not hold. In fact the ESO holder is risk-averse. Her objective is to maximize her expected wealth (which depends on the ESO payoff) and thus there is two contrary effects:

1. the first one which is to maximize the expected payoff of the ESO which is positively linked to the underlying risk;

2. the last one which is to minimize the risk due to the risk-aversion, which is obviously negatively linked to the underlying risk.

We are going to show that the last effect described just above beats the first one.

So Formally speaking, let 'y1 <'y2 2 risk-aversion parameters and p_ãi(t, s) be the Executive Private Price under 'y , we have to show that:

p_ã1(t,s) = p_ã2(t,s)

Proposition 3.4. Risk-aversion effect

Let 2 Executives i and j respecting our framework conditions and have respectively 'y and 'yj as riskaversion parameters. Suppose also that 'y <'yj.

Then the Executive i Private Price dominates the Executive j Private Price.

Proof. By equation (38) we have an simplified form for the Private Price.

Let _pã1 ^and _pã2 be respectively the Private Price of the Executive i and the Executive j. Then

pã1 - pã2

2('y2-'y1)(1-ñ2)e-r(T-t) ( 1 V^P0[(ST - K)^+] + E^P0 [((ST - K)^+)^2]) > 0 t,s,x t,s,x



In fact the assertion is very intuitive, since that more the Executive is risk-averse less she is willing to wait 1 unit of time more and then she cannot fully exploit all potential gains coming from earlier exercising.

3.4.5 The effect of correlation

Through the Market Index, the executive can partially hedge her risk. More this parameter is lower less the residual risk is important and more the ESO private price is higher. We are going to demontrate this assertion via the following propostion:

Proposition 3.5. Let ñ1 and ñ2 2 correlation parameters. Suppose also that 0 < ñ1 <ñ2. Then the ESO Private Price under the world defined by ñ1 dominates the ESO Private Price under the world defined by ñ2

Proof. Given the price approximation defined by the equation(38). And let ñ1 and ñ2 defined in the proposition. Then:

Remark : Note that in the case of perfect correlation (|p| = 1) and ignoring the higher moments on the Private Price expansion our valuation method gives no more no less the Black & Scholes formulation of the Private Price.

Proposition 3.6. The opposite happens if p2 <p1 <0

These propositions confirm what we are expecting. Indeed, the Executive seeks the asset which is the most negatively correlated with the underlying of her ESO in order to maximizing her hedging. That is why she is willing to pay more her ESO if she can found a substitute which can better hedge her risk inherent to her ESO.

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.



5 ESO cost to the firm

In the previous sections we have derived the Executive's optimal exercise policy. The valuation of the ESO is taken in the holder's perspective. We have shown that the ESO valuation depend closely to the Executive's parameters and are summarized by her risk aversion.

The issue treated in this section is the ESO valuation from the firm or shareholder's perspective. The cost of issuing an option depend closely to the Executive's behaviour which results from a rational trading strategy under a set of constraints.

While the Executive cannot fully hedge her risk by short selling the company's stock, we assume that shareholders can fully hedge their risk.

Therefore the last assumption allows us to build the ESO cost model under the risk-neutral measure Q. More precisely, the executive choose an optimal exercise policy ô that yields a payoff _(S,- - K)⁺ so that the firm's cost to this option at time ô is equal to _(S,- - K)⁺.

Thus how can the firm do anticipate this cost and estimate it precisely?

5.1 General model for the ESO cost to the firm

In this part we are going to generalize the cost beared by shareholders during the issuing of an ESO. From the firm's perspective the option exercising is exogenous since the exercising policy is fully explained by the Executive's behaviour.

Thus we could modelize the critical price which reflects the price level where the ESO is exercised as a barrier.

From the firm's perspective the ESO payoff is a certain amount of cash outflow at an a priori uncertainty time which is the first time where the company's stock price reaches the critical price. The critical price is completely exogeneous from the shareholders perspective since this price is fully explained by the Executive's behaviour that is why the ESO is a barrier-option from the firm's point of view.

5.2 The naive approach

This part describes how can be derived the cost of issuing an ESO without taking into account the Executive's risk aversion. In fact from the firm's perspective the Executive's is not risk-averse and her behaviour is only driven by the maximization of the ESO expected payoff.

By this way the Executive's private price for the ESO is equal to the ESO risk-neutral price or its cost of issuing under complete market assumption. This approach is called naive approach since it does not care about hedging constraints imposed to the Executive. But in fact the result that we will obtain may be interesting since it can be undertood as the upper bound of the ESO cost.

Proposition 5.1. Black-Scholes

Suppose that shareholders and option holder are risk-neutral. Then the aim of the option holder is to maximize the present value of her stock-option.

Thus C^naive(t, s) the cost or the price at time t of a such option is the B&S price of an European Call option and satisfied the following PDE:

Cnaive

t (t, s) + rsC^naive

s (t, s) + ¹ 2(çs)2Cnaive

ss(t , s) - rC^naive(t, s) = 0

(60)

With the boundary condition: C^naive(T, s) = (ST - K)⁺

C^naive(t, s) have the following probabilistic form according to Feynman-Kac argument:

C^naive(t, s) = e_r(T_t)EQ [(ST - K)⁺ | St = s]

Where Q is the risk-neutral measure under which the risk-free rate and the Company 's stock return are equal by non-arbitrage argument.

Here we have found the cost of issuing an ESO where the holder is assumed to be risk-neutral. Since the dividend is assumed smoothed during the time span the optimal choice for the Executive is to wait until the option maturity. That is why the cost to the firm is no more no less the B & S price of an European Call option.

In the next part we are going to define the ESO as a barrier-option. In fact the Executive through

her optimal exercise boundary defines a stock price level where she optimally exercises her option. This bound is completely exogenous from the firm's perspective, since it is fully explained by the Executive's behaviour. We are going to model this barrier by some exogenous barrier S" and find what is the cost when there is no vesting period and no job termination risk.

5.3 The ESO cost to the firm with no vesting period and no job termination risk - Ctivanic, Wiener and Zapatero (2004)

I?]

This section treates about the firm's cost of issuing an ESO when shareholders take into account the fact that the ESO holder is risk-averse. But our assumption here is that the risk-aversion come only from the unperfect hedging and not from the job termination risk. Moreover there is no vesting period under which the Executive cannot exercised her option.

Thus we have to define two parameters:

1. S": the optimal level of the Company's stock price at which the Executive exercises her option;

2. á: the exogenous rate of decay of that barrier as maturity approaches. This parameter captures the fact that !!the Executive is more likely to exercise the option, (that is, for a lower price of the underlying), the closer the maturity.' see Ctivanic, Wiener and Zapatero (2004).

Here the ESO is treated as an American Call option and thus can be exercised at any time during the time span until its maturity T.

So the Executive has the choice to exercise or not her option and thus bring about different costs for the firm. According to the last remark we have to separate two kinds of cost:

1. the cost brought about by the option exercise before maturity;

2. the cost at maturity.

In order to explicit this two sources of cost we have to define the time when the option is exercised or expires.

Let ô be a random stopping time and min(ô, T) be the time when the option is exercised or expires. Given Ft the distribution of ô contionally to the information available to the Company's stock price up to time t:

Ft=Pt[ô=t] (61)

Then we deduce the general formulation of the expected cost to the firm at maturity:

"Z T #

E [Cô?T (t, S)] = E CudFu + CT (1 - FT ) (62)

0

Suppose now the Executive exercises her option at time t so the critical Company's stock price is hit for the first time. The boundary S" t = S"e^át > K is reached for t = T. Thus

ô" = inf{t > 0,St = S" t } = inf {t> 0,Ste^?át = S"} (63)

Then the expected cost to the firm can be expressed as two parts:

1. the cost brought about by the option exercising when the critical price has been reached;

2. the cost when the option matures at time T.

Proposition 5.2. The expected cost to the firm - Ctivanic, Wiener and Zapatero (2004)

According to the equation (62) and (63) the expected cost to the firm at time of issuing an ESO is written as:

iC(0, s) = E [e-rT (ST - K)⁺I{ô^*>T } | S0 = s] + E h (se^?ráô* - Ke?rô* )I{ô^*=T } | S0 = s(64)

The previous proposition highlighted the fact that the cost of issuing an ESO is a combination of the cost brought about by the Executive's optimal exercising behaviour and the cost when the option matures. In the next part we are going to introduce an other parameter the intensity of the job termination risk, but in order to only show this parameter effect we are going to exclude the case of vesting period. The last part will present the final model with all parameters coming from the Leung& Sircar's paper (2006).

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.

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

References

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[Hen3] Vicky Henderson, Valuing the Option to Invest in an Incomplete Market, 2005

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