Employee Options And Risk-Taking Incentives

Why Option Compensation Can Affect Risk Choices

An employee who owns company stock participates in both its gains and losses. An employee who holds a stock option has a different exposure: the option can produce a positive payoff when the share price is high enough, but it does not require the employee to pay the exercise price when the option is out of the money. That asymmetry can make an option holder more willing to accept risk than someone whose compensation moves one-for-one with the share price.

This is an incentive effect, not a claim that every employee will prefer risky decisions. The employee’s response also depends on the size and timing of the award, the employee’s other wealth and compensation, the probability distribution of outcomes, and the employee’s attitude toward risk. The comparisons below isolate the payoff incentive by holding other compensation fixed.

Definition: In a one-period model, let $S_T$ be the company’s share price at the option’s expiration date $T$, $K$ the exercise price, and $N$ the number of options. Assume the options are vested, the employee retains them until expiration, and taxes, forfeiture, early exercise, and transaction costs are ignored. The employee’s option payoff at expiration is $$ N\max(S_T-K,0). $$ The payoff is zero when $S_T\le K$, and $N(S_T-K)$ when $S_T>K$.

This formula describes the payoff at expiration, not the option’s value before expiration. In particular, the expected payoff calculated under stated outcome probabilities is useful for comparing incentives, but it is not automatically a market price or a risk-adjusted value. A risk-averse employee may not choose the alternative with the largest expected payoff.

The Source of the Risk Incentive

For a share price above the exercise price, each option gains dollar-for-dollar with the share price. Below the exercise price, the payoff cannot become negative. Consequently, a rise in the share price can increase the option payoff substantially, while a fall below the exercise price does not create a corresponding loss on the option itself. The employee may still lose the option’s potential value or face other employment and wealth consequences, but the option payoff alone is bounded below by zero.

The crucial feature is convexity. The payoff $\max(S_T-K,0)$ has slope zero below $K$ and slope one above $K$. Its slope does not decrease as $S_T$ increases, so the payoff is convex. Convexity means that a wider range of outcomes can increase expected option payoff even when the expected share price stays the same.

Result: If one terminal share-price distribution is a mean-preserving spread of another, then it has the same mean but is more dispersed in the sense that it gives at least as large an expectation for every convex payoff function. Since $\max(S_T-K,0)$ is convex, the more dispersed distribution gives at least as large an expected payoff per option. This conclusion applies to the stated distributions and payoff comparison; it does not by itself establish that an employee will choose the riskier action.

The logic follows from the option payoff’s shape. The payoff is the larger of two straight-line functions, $S_T-K$ and zero. The maximum of affine functions is convex. A mean-preserving increase in risk therefore cannot lower the expectation of this payoff. When some outcomes fall below $K$ and others rise above it, the option’s asymmetry can make the increase strict.

Worked Comparisons

Worked Example: Same Expected Share Price, Different Option Payoff

An employee holds 100 vested options with an exercise price of $40. At expiration, the share price is either certain to be $40 under a stable business plan, or equally likely to be $20 or $60 under a riskier plan. Compare the expected option payoff under the two plans. Treat the probabilities as the scenario probabilities and ignore discounting.

Method. Calculate the option payoff in each possible outcome, multiply by its probability, and sum. First compare the expected share prices: $$ E[S_T\mid\text{stable}]=40, \qquad E[S_T\mid\text{risky}]=0.5(20)+0.5(60)=10+30=40. $$ The risky plan has the same expected terminal share price.

Under the stable plan, the share price equals the exercise price, so the payoff per option is $\max(40-40,0)=0$. The total payoff is $100(0)=\$0$, and its expected value is also $0.

Under the risky plan, the per-option payoff is $\max(20-40,0)=0$ in the low outcome and $\max(60-40,0)=\$20$ in the high outcome. Thus the expected payoff per option is $$ 0.5(0)+0.5(20)=\$10. $$ For 100 options, the expected payoff is $100(10)=\$1{,}000$.

Conclusion and trap avoided. The risky plan raises expected option payoff by $1,000 even though expected share price is unchanged. A common mistake is to calculate the option payoff from the expected share price: $\max(E[S_T]-K,0)=\max(40-40,0)=0$. That is not the expected payoff. The correct calculation applies the option payoff separately to each outcome and then takes the expectation.

Worked Example: Risk Incentives With an In-the-Money Option

An employee holds 20 options with an exercise price of $75. Under a lower-risk plan, the terminal share price will be $70 or $90 with equal probability. Under a higher-risk plan, it will be $40 or $120 with equal probability. Both plans have the same expected terminal share price. Find and compare expected option payoffs.

Method. Confirm the common expected share price before calculating the option payoffs: $$ E[S_T\mid\text{lower risk}]=0.5(70)+0.5(90)=35+45=80, $$ $$ E[S_T\mid\text{higher risk}]=0.5(40)+0.5(120)=20+60=80. $$

For the lower-risk plan, payoff per option is $\max(70-75,0)=0$ or $\max(90-75,0)=\$15$. The expected payoff per option is $0.5(0)+0.5(15)=\$7.50$, so the expected payoff on 20 options is $20(7.50)=\$150$.

For the higher-risk plan, payoff per option is $\max(40-75,0)=0$ or $\max(120-75,0)=\$45$. The expected payoff per option is $0.5(0)+0.5(45)=\$22.50$, so the expected payoff on 20 options is $20(22.50)=\$450$.

Conclusion and trap avoided. The higher-risk plan increases expected option payoff by $\$450-\$150=\$300$, despite the equal expected share prices. The comparison does not show that the higher-risk plan creates more expected firm value: the expected terminal share price was deliberately held constant. A common error is to treat an out-of-the-money outcome as a negative option payoff. The employee’s option payoff in the $40 outcome is zero, not $20(40-75)$.

Worked Example: When More Risk Does Not Change Expected Option Payoff

An employee holds 10 options. Compare a lower-risk plan with terminal share prices of $80 and $100, equally likely, against a higher-risk plan with terminal prices of $50 and $130, equally likely. First use an exercise price of $40. Then repeat the comparison with an exercise price of $75.

Method. Both distributions have expected share price $0.5(80)+0.5(100)=\$90$ and $0.5(50)+0.5(130)=\$90$. With $K=\$40$, every outcome is above the exercise price. The payoff per option is therefore $S_T-40$ in every outcome.

For the lower-risk plan, the payoffs per option are $40 and $60, so the expected payoff is $0.5(40)+0.5(60)=\$50$. For the higher-risk plan, the payoffs are $10 and $90, so the expected payoff is $0.5(10)+0.5(90)=\$50$. In each case, 10 options have an expected payoff of $10(50)=\$500$.

Now set $K=\$75$. Under the lower-risk plan, per-option payoffs are $\max(80-75,0)=\$5$ and $\max(100-75,0)=\$25$, giving an expected payoff of $0.5(5)+0.5(25)=\$15$, or $10(15)=\$150$ for the award. Under the higher-risk plan, per-option payoffs are $\max(50-75,0)=0$ and $\max(130-75,0)=\$55$, giving $0.5(0)+0.5(55)=\$27.50$, or $10(27.50)=\$275$ for the award.

Conclusion and trap avoided. At $K=\$40$, the payoff is linear over all possible outcomes, so equal expected share prices produce equal expected option payoffs. At $K=\$75$, the higher-risk distribution spans the exercise price, and the expected payoff is $125 larger. The trap is to assume that greater risk always increases expected option payoff. Here it does not when all outcomes are above the strike and the payoff is linear throughout.

Conditions That Strengthen or Weaken the Incentive

Options are most likely to reward an increase in risk when the possible outcomes straddle the exercise price. Below the strike, the employee receives no option payoff; above it, the employee participates in further gains. A spread that moves probability toward both low and high outcomes can therefore retain the downside floor while increasing the chance of a large payoff. The relevant comparison is not volatility alone: the strike’s position relative to the possible outcomes matters.

When every plausible outcome is above the strike, the payoff is $S_T-K$, a straight line. If expected share price is held fixed, expected option payoff is then $E[S_T]-K$ under either distribution. When every outcome is below the strike, the payoff is zero under either distribution. In both cases, changing dispersion within that range does not change expected payoff. If a change in risk causes outcomes to cross the strike, that linearity argument no longer applies.

Several practical conditions affect how strongly the payoff translates into behavior:

Award size: Holding all else constant, more options magnify the dollar difference in expected payoff between risk choices.
Exercise price: The strike determines where the payoff changes from zero to increasing with the share price. Its location relative to possible outcomes affects the option’s sensitivity to risk.
Other exposure: An employee who also owns company shares, has substantial company-linked wealth, or faces job-loss risk may bear downside not shown in the option payoff alone.
Risk preferences and constraints: A larger expected payoff does not guarantee a choice when the employee is risk-averse, cannot influence the decision, or faces limits on taking risk.
Timing and retention: The simplified comparison assumes the employee keeps the options until expiration. Actual compensation terms, exercise timing, and the possibility of forfeiture can alter the incentives.

For the same reason, option-based compensation should not be described as automatically aligning employee and shareholder interests. Options can reward gains in share value, but their capped downside can also make some risk increases attractive to an employee even when those choices do not improve expected share value. The comparison must specify whose payoff is being measured and which outcomes are being compared.

A Practical Check on the Analysis

For an exam-style incentive comparison, keep the decision sequence explicit. State the option terms and assumptions, compare expected share prices if the question concerns a risk change with no change in the mean, calculate the option payoff in every outcome, and then take the expectation. Finally, explain what the calculation establishes—and what it does not. In particular, expected option payoff is not the same as a guaranteed employee preference, total firm value, or the option’s value before expiration.

Specify the payoff.
Record $N$, $K$, expiration, and any assumptions about vesting, retention, and ignored costs.

List the outcomes.
For each alternative, state every possible $S_T$ and its probability. Check whether the expected share prices are equal if that is part of the comparison.

Apply the floor outcome by outcome.
Calculate $N\max(S_T-K,0)$ in each state before weighting the payoffs by their probabilities.

Interpret the difference carefully.
Report the expected payoff difference and state whether it indicates an incentive, rather than claiming that it proves a particular employee will take a particular action.

Key takeaway: An employee option combines participation in upside with a zero floor on its expiration payoff. Because that payoff is convex, a mean-preserving increase in share-price risk can raise expected option compensation, especially when outcomes cross the exercise price. The incentive can weaken or disappear when the payoff is linear across all possible outcomes, and it does not alone determine employee behavior.

Check Your Understanding

For each problem, calculate the option payoff in each outcome before taking its expectation.

An employee has 12 options with $K=\$50$. The terminal price is $40 or $80 with equal probability. Find the expected total option payoff. (Answer: $180)
Eight options have $K=\$70$. A lower-risk plan gives terminal prices of $65 and $95 with equal probability; a higher-risk plan gives $25 and $135 with equal probability. Compare expected total option payoffs. (Answer: $100 under the lower-risk plan and $260 under the higher-risk plan; increase of $160)
Six options have $K=\$30$. Compare equally likely prices of $70 and $90 with equally likely prices of $50 and $110. Both distributions have the same mean. Give the expected total option payoff under each. (Answer: $300 under each distribution)
Five options have $K=\$60$. Plan A gives $60 or $100 with equal probability; Plan B gives $20 with probability 0.25 and $100 with probability 0.75. Confirm the common expected share price and compare expected total option payoffs. (Answer: Both expected share prices are $80; Plan A pays $100 in expectation and Plan B pays $150, a $50 increase)
An employee receives a fixed salary of $60,000 and 10 options with $K=\$90$. Under Plan A, $S_T=\$100$ with certainty. Under Plan B, $S_T=\$50$ or $150 with equal probability. Assuming a risk-neutral employee and ignoring discounting, calculate expected total compensation under each plan and state the limitation of the conclusion. (Answer: Plan A, $60,100; Plan B, $60,300; Plan B has the higher expected compensation, but the comparison alone does not predict a risk-averse employee’s choice)

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