Optimal stock option schemes for managers

Except Asian option scheme, all the other types of options can be found in the manager’s compensation scheme in practice. Some examples will be given in the subsequent sections.

Cvitanić and Zhang (2013) provide an excellent overview of contract theory. The three classical cases for principal-agent problems are: Risk-sharing with symmetric information, hidden action and hidden type. In the first case, the principal, which is represented by the shareholders here, and the agent, the manager, have the same information and can contract directly upon the agent’s actions. In this case a first-best outcome can always be achieved and the principal and the agent have to agree on how to share the risk between themselves. An early paper discussing risk sharing is for example Borch (1962). In the second case, the actions of the agent cannot be directly observed or contracted upon. Here the principal influences the agent indirectly to pick certain actions by giving him incentives through a contract. Usually, only a second-best solution can be achieved. Holmström and Milgrom (1987) discuss this setup. In the third case, the principal does not know some key characteristics of the agent, e.g. his ability. Hence, the principal offers a menu of contracts. Under certain conditions the agent will reveal his true type. Generally, the principal only gets a third-best reward in this case. A hidden type example is Cvitanić and Zhang (2007). Our paper falls into the second category. We do not allow the shareholders to directly contract upon the risk of the company, but only to offer incentives through the different stock option schemes. However, we show that a first-best outcome can be achieved in this case.

Usually in the literature the shareholders are assumed to be risk-neutral and can always sell their shares and walk away when the firm’s assets do not perform well. However, those shareholders who are able to decide on compensation scheme are large investors and probably interested in long-term investment in the firm. When the firm’s assets do not perform well, low asset price combined with high transaction costs might cause huge losses for these investors. Hence, they might choose to hold their shares until the terminal date rather than to sell them.

The price \(\Uppi(x)\) denotes the value that the shareholders assign to a certain compensation package. As the shareholders are free to trade in any derivative and markets are assumed to be complete, the price \(\Uppi(x)\) can be calculated as the expectation of the discounted payoffs under the risk-neutral probability measure. However, as managers are not allowed to hedge their stock options, the complete market assumption does not hold for them and thus the price \(\Uppi(x)\) does in general not coincide with their subjective evaluation.

Nevertheless, it might still be better for the shareholders to prefer a certain scheme, particularly under information asymmetry.

Hereby we do not consider the case of an undiversified manager. According to Kanniainen (2010), in that case the option price becomes where θ is the Sharpe ratio and ρ the correlation coefficient between the firm’s asset and the market portfolio. Some other references on the topic of ESOs and undiversification are e.g. Hall and Murphy (2002) and Jin (2002).

A Brownian excursion {Y _t ,0 ≤ t ≤ 1} has the same distribution as a Brownian motion {W _t ,0 ≤ t ≤ 1 }, conditional on W _t  > 0 for all 0 < t < 1 and W ₁ = 0.