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
utilitybased 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
fairprice when its holder is subject to riskaversion 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 FeynmanKac

10



1.3.3 RadonNikodym

11



1.3.4 CameronMartin 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

utilitybased 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 riskaversion

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 riskaversion

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 riskfree
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 riskaversion and
others constraints such that trading constraints and job termination risk.
By this assessment and the riskaversion principle, we have to
develop a continuous time valuation theory based on indifference preferences
and to distinguish the ESO from plain vanilla options.
A utilitybased valuation allow to find a unique fairprice by
taking into account the riskaversion parameter. The indifference or private
price resulting from the model is not the same depending on the level of
riskaversion 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
riskneutral 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 & Smodel 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 valuemaximizing
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
subsection, 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 subsection 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 riskintensity 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.
