Evaluation of executive stock options in continuous and discrete time

What follows is a treatise on executive stock options (ESOs), employing both theoretical and empirical methods. The title speaks of evaluation, instead of 'plain' valuation, because subjective ESO values depend on the employee's private data including risk aversion, initial wealth, and labor income. Economic theory assumes that utility is drawn from consumption, and consumers are impatient. It follows that the decision to exercise or sell ESOs deals with consumption smoothing. Assuming that the ESOs have vested, the option grantee asks: Is it better to sell now and consume the proceeds, or to wait and possibly enjoy higher consumption later on? Obviously, the answer depends (among other things) on individual risk preferences. ESOs are options in the sense that they provide the call option payo¤. However, since the option grantee cannot usually hedge his position, usual arguments leading to risk-neutral valuation are not applicable, suggesting that ESO pricing is different from standard contracts. This claim can be tested, since ESOs are traded in the Helsinki stock exchange. Now let us consider a generalized Black-Scholes model with (implied) volatility as free parameter. If ESO pricing differs from standard options, it should be seen in their volatilities. It is shown in Market pricing of ESOs that volatilities (i.e. relative prices) of ESOs are lower compared to standard options. Given the prominence of illiquidity and hedging under uncertainty in mathematical finance, we ask if these factors could explain the discount in ESO prices. In the case of Nokia ESOs, illiquidity is not the story, since the underlying stock is fiercely traded. However, hedging is complicated by ESOs having longer maturities than standard contracts. The average maturity of ESOs is 3.1 years, compared to few months for liquid standard options. Hence, hedging ESOs by taking opposite position in standard options is subject to volatility risk. Obviously, hedging is also possibly by shorting the stock and using cash account, but this involves frequent trading and hedging error. Nonetheless, volatility risk and hedging error can explain the discount only to an extent. Finally, we conclude that the employees benefit from public trading of ESOs. It allows the employees to receive some time value1, which they would lose if ESOs were not traded and had to be exercised