Valuation Methods of Executive Stock Options

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 -