Valuation Methods of Executive Stock Options

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.