Memoire Online - Valuation Methods of Executive Stock Options

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.

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

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.

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.

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.

Valuation Methods of Executive Stock Options

Abstract