Valuation Methods of Executive Stock Options

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)

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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).