Payoff Is Not the Same as Compensation Value
An employee stock option gives its holder the right, but not the obligation, to buy a company share at a specified exercise price. At the option’s expiration, its payoff depends on the share price then. But a compensation award has value before expiration too: it gives the employee a claim on a possible future payoff. That distinction matters to the company and its shareholders even if granting the option requires no immediate cash payment.
The simplest option model treats the claim as a call option on one share. It is useful for understanding the economics of an award, but it does not automatically produce the exact value of every employee option. Employee awards may have restrictions, service conditions, or exercise rules that differ from those of a freely traded option. A valuation must reflect the award’s actual terms and an appropriate model.
The current exercise payoff, also called intrinsic value, is \(\max(S_0-K,0)\), where \(S_0\) is the current share price. An option can have positive value even when this amount is zero: there may still be a chance that the share price will exceed \(K\) before expiration. Conversely, the option’s eventual payoff is uncertain. It is therefore not correct to equate an award’s compensation value with either its current intrinsic value or a single possible future payoff.
Valuing a Call in a One-Period Binomial Model
A one-period binomial model assumes that the share price at the end of a single period can take one of two specified values. Let the current share price be \(S_0\), and write the possible end-of-period prices as \(S_u=uS_0\) and \(S_d=dS_0\), with \(u>d\). Assume the effective risk-free interest rate for the period is \(i\), there are no dividends, and investors can borrow and lend at that rate. The model also assumes no taxes, transaction costs, or arbitrage opportunities, and that the option can be exercised only at the end of the period.
For the standard one-period no-arbitrage result to apply, the risk-free gross return must lie between the share’s two gross returns: \(d<1+i<u\). Under these assumptions, the risk-neutral probability assigned to the up state is
This \(q\) is a pricing weight implied by no-arbitrage, not necessarily the actual probability that the share price rises. If the call’s up- and down-state payoffs are \(C_u\) and \(C_d\), respectively, its value today is the risk-neutral expected payoff discounted at the risk-free rate:
Here, \(C_0\) is a model value at the start of the period. The model does not say that the employee will receive \(C_0\) in cash, or that the employee’s eventual payoff will equal \(C_0\). It gives the current no-arbitrage value of the contingent claim under the stated assumptions.
Why the Risk-Neutral Formula Holds
The formula follows by replicating the option’s two possible payoffs with shares and risk-free borrowing or lending. A portfolio holding \(\Delta\) shares and borrowing \(B\) dollars today has end-of-period value \(\Delta S_u-(1+i)B\) in the up state and \(\Delta S_d-(1+i)B\) in the down state. Choose \(\Delta\) and \(B\) so those values match the call’s two payoffs.
Subtract the down-state portfolio value from the up-state value. This gives \(\Delta(S_u-S_d)=C_u-C_d\), so \(\Delta=(C_u-C_d)/(S_u-S_d)\).
Using the down state, set \(\Delta S_d-(1+i)B=C_d\). Thus \(B=(\Delta S_d-C_d)/(1+i)\).
The replicating portfolio costs \(\Delta S_0-B\) today. If the option had a different price, an investor could buy the cheaper of the two identical state-contingent payoffs and sell the more expensive one. Substitution and simplification give \(C_0=[qC_u+(1-q)C_d]/(1+i)\), where \(q=((1+i)-d)/(u-d)\).
The replication argument also explains the model’s limits. It relies on the ability to trade and finance the replicating portfolio under the stated assumptions. An employee’s award may not be transferable or exercisable at the same times as a traded call. Its model-based compensation value must therefore be based on terms and assumptions appropriate to that award.
Worked Example: Valuing a One-Year Employee Call
A company grants an employee a hypothetical one-year option on one share. The current share price is $25 and the exercise price is $25. In one year, the share price will be either $35 or $20. The effective annual risk-free rate is 5%. Assume no dividends, no transaction costs or taxes, no arbitrage, and end-of-year exercise only. No mortality model is used. Find the model value per option.
Method and state returns. The up and down gross share returns are \(u=35/25=1.40\) and \(d=20/25=0.80\). The risk-free gross return is \(1+i=1.05\), which lies between the two stock returns: \(0.80<1.05<1.40\).
Risk-neutral probability.
State payoffs. In the up state, the call pays \(\max(35-25,0)=\$10\). In the down state, it pays \(\max(20-25,0)=\$0\).
Discount the risk-neutral expected payoff.
Rounded to the nearest cent, the model value is $3.97 per option. As a check, the risk-neutral expected payoff before discounting is about $4.1667, and dividing by 1.05 gives about $3.9683. A common mistake is to use \(q=0.3333\); that does not follow from the stated stock returns and risk-free rate.
Physical Expected Payoffs Are Not Fair Values
A physical probability describes beliefs about how likely each outcome is in the real world. A risk-neutral probability is instead chosen to make the price consistent with the risk-free return and the available trading opportunities in the model. They answer different questions. Discounting a physical expected payoff at the risk-free rate generally does not give the option’s no-arbitrage value, because it ignores the market price of risk.
Worked Example: Physical Expectation Versus Model Value
A hypothetical employee option has one year to run, with current share price $60 and exercise price $60. In one year the share price will be either $78 or $48. The physical probability of the up state is 0.70 and of the down state is 0.30. The effective annual risk-free rate is 5%. Assume no dividends, no transaction costs or taxes, no arbitrage, and end-of-year exercise only; no mortality model is used. Calculate the one-period model value and compare it with the discounted physical expected payoff.
State returns and risk-neutral probability. The gross returns are \(u=78/60=1.30\) and \(d=48/60=0.80\). Since \(0.80<1.05<1.30\), the no-arbitrage condition holds, and
Option payoffs. The up-state payoff is \(\max(78-60,0)=\$18\); the down-state payoff is \(\max(48-60,0)=\$0\). Therefore, the model value is
Physical expected payoff. Using the physical probabilities gives \(0.70(\$18)+0.30(\$0)=\$12.60\). Discounting this at 5% gives \(\$12.60/1.05=\$12.00\). That discounted amount is not the model’s no-arbitrage option value; it uses a physical expected payoff with a risk-free discount rate. The model value is $8.57. A common mistake is to treat the physical expected payoff as if it were risk-neutral, producing $12.00 as the option price.
From One Option to a Compensation Claim
A company’s option grant is a package of contingent claims. In a simplified setting where each option has the same model value, the total grant value is the number of options multiplied by the value per option. This does not mean the employee receives that total amount in cash on the grant date. Rather, it estimates the value of the compensation claim being awarded under the valuation assumptions.
For an employee option, the standard call formula is a starting point, not a universal answer. The award’s term, exercise rights, restrictions, and conditions can affect its value. A model used for compensation purposes should reflect the contractual features being valued. If a simplified example treats employee options as standard calls, that is an explicit modeling assumption, not a claim that all employee options are freely traded or have identical terms.
Worked Example: Total Value of a Hypothetical Option Award
A company awards an employee 600 hypothetical one-year options, each with an exercise price of $45. The current share price is $50. In one year, the share price will be either $65 or $40. The effective annual risk-free rate is 4%. Assume no dividends, no transaction costs or taxes, no arbitrage, identical standard end-of-year exercise rights for all options, and no mortality model. For this illustration, treat the award as 600 standard calls and ignore any additional restrictions. Find the model value of the award and compare it with current intrinsic value.
Risk-neutral probability. The up and down gross returns are \(65/50=1.30\) and \(40/50=0.80\). The risk-free gross return is \(1.04\), so \(0.80<1.04<1.30\). Thus,
Payoffs and value per option. The up-state payoff is \(\max(65-45,0)=\$20\); the down-state payoff is \(\max(40-45,0)=\$0\). The value per option is
Total award value. Multiplying by 600 gives \(600(\$9.2308)\approx \$5,538.46\), with rounding performed at the final step. Current intrinsic value per option is \(\max(50-45,0)=\$5\), so the 600 options’ current intrinsic value is \(600(\$5)=\$3,000\). The model value of $5,538.46 is greater because the claim also has a possible future payoff beyond its current exercise value. The $5,538.46 is not a grant-date cash payment to the employee. It is an estimate of the value of the compensation claim under this simplified model.
Why There Can Be an Economic Cost Without Cash Paid
Cash flow and economic cost are not the same measure. When a company grants an option, it may make no immediate cash payment, but it gives the employee a claim whose value depends on future share prices. That claim is compensation for employee services. From the shareholders’ perspective, the award is not costless simply because no cash leaves the company at grant: the company has committed value to compensation rather than leaving all potential value with existing shareholders.
In accounting, the treatment and timing of compensation expense depend on the applicable accounting framework and the award’s terms. The economic distinction remains: a noncash award can have value, and its value can represent a cost of obtaining employee services. Do not confuse an accounting expense, a cash outflow, and the option’s eventual payoff. They are related concepts, but they are not interchangeable amounts.
- Payoff: the amount the option delivers in a particular state at exercise or expiration.
- Model value: the current value of the uncertain claim under specified valuation assumptions.
- Cash flow: money actually paid or received at a particular date.
- Economic compensation cost: the value transferred in return for services, even when the transfer is not an immediate cash payment.
A frequent exam trap is to say that an option costs the company nothing because the employee pays the exercise price if the option is exercised. That reasoning overlooks the claim granted: the employee receives the right to buy at \(K\) when the share may be worth more. The correct analysis distinguishes the cash paid at grant from the value of that right.
Check Your Understanding
For each problem, use the stated one-period assumptions and distinguish the model value from any physical expected payoff. Round dollar answers to the nearest cent.
- A one-year call has \(S_0=\$30\), \(K=\$30\), and terminal share prices of $39 or $24. The effective annual risk-free rate is 5%. Assume no dividends, no transaction costs or taxes, no arbitrage, and end-of-year exercise only. Find the risk-neutral probability and value per option. (Answer: \(q=0.5000\); payoffs are $9 and $0; value is $4.29.)
- A one-year call has \(S_0=\$40\), \(K=\$35\), and terminal prices of $52 or $32. The effective annual risk-free rate is 5%, and the physical up-state probability is 0.60. Under the same no-dividend, frictionless, end-of-year exercise assumptions, find the model value and the discounted physical expected payoff. (Answer: model value $8.10; discounted physical expected payoff $9.71, which is not the no-arbitrage model value.)
- A company grants 250 hypothetical one-year calls with \(S_0=\$80\), \(K=\$75\), and terminal prices of $104 or $64. The effective annual risk-free rate is 4%. Assume no dividends, no transaction costs or taxes, no arbitrage, identical standard end-of-year exercise rights, and no mortality model. Find \(q\), the value per option, and total grant value. (Answer: \(q=0.4800\); value per option $13.38; total grant value $3,346.15.)
- A hypothetical award consists of 120 one-year calls with \(S_0=\$55\), \(K=\$50\), and terminal prices of $71.50 or $44. The effective annual risk-free rate is 5%. Assume no dividends, no transaction costs or taxes, no arbitrage, identical standard end-of-year exercise rights, and no mortality model. Find the risk-neutral probability, total model value, and total current intrinsic value. (Answer: \(q=0.5000\); total model value $1,228.57; total current intrinsic value $600.00.)