Robert M. Pavlik rp14@swt.edu is a professor in the Department of Finance and Economics, Southwest Texas State University.

Financial theory clearly demonstrates that marketable options on non-dividend paying stocks should never be exercised early. However, empirical evidence shows that early exercise of employee stock options (ESOs) is a pervasive phenomenon. Finance theory and actual practice are at odds because ESOs cannot be sold or transferred by their holders -- a violation of the liquidity assumption in option pricing models. An option holder’s inability to sell his/her ESO can make early exercise a rational choice, when, for example, the holder expects the stock price to decline prior to the ESO’s expiration. Holders of ESOs who wish to maximize personal wealth and financial advisors who assist clients in such matters need to understand the circumstances in which early exercise early is beneficial. This paper defines “share leverage”, a previously under-recognized factor in an ESO holder’s early exercise decision, and demonstrates how share leverage affects wealth-maximizing choices regarding the early exercise of ESOs. |

The purposes of this paper are to define share leverage, a previously unrecognized factor in the decision of whether to exercise employee stock options (ESOs) early, and to delineate how wealth maximizing early exercise decisions regarding ESOs are dependent upon share leverage. This investigation is important, because understanding share leverage is essential to holders of employee stock options who wish to maximize personal wealth and to financial advisors who assist clients in such matters.

ESOs are a type of American call option. They provide an employee-holder the right to buy shares of the his/her employer’s stock at the ESO’s exercise price anytime between the ESO’s vesting or grant date and its expiration date. While they have similarities to plain vanilla listed call options, ESOs have other attributes that make their valuation more complex. ESOs are warrant-like in that their exercise increases the issuing firm’s cash account, equity and shares outstanding. Also, similar to warrants, the typical time-to-maturity of five to ten years from the date of grant is very long when compared to the standard listed option. However, for the purposes of this paper, the key valuation-related feature of ESOs is their lack of marketability. ESOs cannot be sold by their holder, and they cannot be transferred other than by will. In addition, ESOs may only be exercised by the holder, so they are worthless to other parties.

Huddart (1994) shows that nonmarketability creates conditions where early exercise of ESOs will be optimal for a sufficiently risk averse holder. Lambert, Larcker, and Verrecchia (1991) and Hemmer, Matsunaga, and Shevlin (1994), show that restrictions placed on ESOs significantly impair the ability of the holder to hedge an ESO position. The nonmarketability of ESOs coupled with legal restrictions on hedging an ESO via a short sale of the underlying stock, result in early exercise sometimes being the rational choice, even for ESOs on non-dividend paying stocks. This result is in conflict with traditional option theory. Merton (1973) demonstrates theoretically that American call options, like European call options, on non-dividend paying stocks should never be exercised prior to maturity. The intuition behind Merton’s argument is straightforward -- prior to maturity, call options have time value (value in excess of the stock price minus the exercise price), and this time value is destroyed by early exercise. The rational holder of a call would never willingly destroy value he or she could garner.

With regard to ESOs, however, the inability of the ESO holder to sell the ESO precludes capturing the option’s time value. If the holder of a vested ESO believes the underlying stock is about to decline in value, the holder cannot sell the option, but must instead exercise it and sell the underlying stock. While existing ESO literature has shown that the combination of marketability and short sale restrictions is sufficient to make early exercise a viable strategy, the literature has not properly noted the impact of another factor on the early exercise decision. Call options are “leveraged” investments in the sense that the purchase of a call option requires a smaller share-for-share dollar investment than the stock itself. A given dollar investment can purchase call options on more shares than it can shares of the underlying stock itself. If the stock advances sharply in price, the call holder earns a higher return than the stockholder. While this leverage feature of options has been noted in existing options literature, its importance in the early exercise decision of ESO holders has not been. The leverage implicit in option transactions is sufficiently different from the leverage that involves borrowing funds to deserve its own name -- call it “share leverage”. Later I define a complex measure of share leverage.

As will be shown, share leverage plays an important role in the early exercise decision, although it greatest importance is to ESO holders who are deciding whether to exercise early, immediately sell the stock, and then repurchase at a later date. When the holder of an ESO exits his/her position via exercise and stock sale, there will always be an immediate loss in share leverage; since the holder’s net proceeds from paying the exercise price and selling the stock will always be less than the stock price. When share leverage is lost, the holder loses some capacity to benefit from share price increases subsequent to the anticipated repurchase date.

This paper is divided into several sections. The first section discusses some additional characteristics of employee stock options. The second section contains the main results of the paper. Finally, the paper concludes with a brief summary.

**I. Personal tax characteristics of ESOs**

There are two broad classes of ESOs – “nonqualified” and “incentive” stock options -- distinguished primarily by different tax status for both the holder and the issuing firm. The concern here is only with the individual holder, and any firm-related tax effects are ignored.

Nonqualified options, when exercised, are taxed as ordinary income to the holder on the difference between the exercise price and market price. Incentive stock options are usually taxed only when the stock is sold, with the difference between the exercise price and the sales price being taxed as a capital gain. The two classes thus differ in the relevant tax rate (T) and the timing of the tax payment. Unless the holder is leaving the firm, there is little reason to exercise an ESO early without simultaneously selling the stock, so with regard to the early exercise decision we can generally treat the two types as alike. To employ the model developed below, one need only insert the appropriate tax rate to account for the difference in tax treatments. See Mozes 1995 for a more complete discussion of taxes and the costs of ESOs to the firm. See Huddart (1998) for a discussion of how a proposed tax rate increase increases the frequency of early exercise for employees likely to be affected by the increase

**II. Notation
and the Model **First consider the
simple case of a risk neutral ESO holder who does not intend to repurchase the
underlying stock at a later date.

Notation and basic
assumptions:

When t = 0, it is the
expiration date of the ESO;

When t = -n, it is the
decision date, n periods before expiration;

X = the option’s
exercise price, assume X > 0;

P_{-n} = the
stock price at t = –n, assume P_{-n} > X, so the option is
in-the-money;

P_{t} = the holder’s forecast
or expected price at some date in the future;

P_{0} = the holder’s forecast
or expected price on the expiration date;

*r *= a continuously
compounded risk free rate.

The early exercise decision for a risk neutral holder who does not intend to
repurchase the stock at a later date depends on basic time value of money
relationships. If the holder forecasts the price of the stock at expiration to
be less than the exercise price, P_{0} < X, the expected t = 0 value
of the ESO will be zero. The holder, given his forecast of the price of the
stock, would exercise the ESO at t = -n and simultaneously sell the stock. If
the holder invests the proceeds of the sale at the risk free rate, r, early
exercise provides the holder with time 0 wealth of (P_{-n} -
X)e^{rn} > 0 versus expected wealth of zero should the options
not be exercised early. Generally, the risk neutral holder should exercise
whenever

(P_{-n} - X)e^{rn} > P_{0} -
X
(1)

Suppose the holder reaches an exercise decision more than two (n>2) dates prior to expiration, and forms a price forecast for a date q periods, where q<n, into the future (i.e., at -n+q), but no further. Since the holder does not intend to repurchase the stock, he should exercise the ESO at t = -n and sell the underlying security with the proceeds invested at r for the q periods, if:

(P_{-n} - X)e^{rq} > P_{-n+q} -
X
(2)

Unless the holder expects a change in tax rate, taxes are not important to the exercise decision, because both sides of (1) and (2) are multiplied by a common factor, (1-T). Equation (2) should only be applied when the holder does not intend to reinvest the proceeds from early exercise at a later date in the same stock.[1] Its application is appropriate if, for example, the holder were intending to use the exercise proceeds to buy a house or make some other investment in the future. If the holder either intends to reinvest or might consider reinvesting in the same stock at a later date, he should weigh any share leverage changes.

Share leverage must be considered, because the holder of an ESO who exercises early experiences an immediate loss in share leverage. Ignoring taxes, the net proceeds of exercising early and immediately selling the stock would fund the immediate repurchase of only (1 – X/P) as many shares as obtained via the early exercise. Without depositing additional funds, the holder will be able to repurchase X/P fewer shares than the number on which he held options. This is a loss in share leverage. Paying taxes on the stock sale would further magnify this loss and can be considered another, though separable, component of the loss in share leverage.

Note that no one would ordinarily exercise an ESO, sell the stock and then immediately repurchase the stock. Share leverage is most important in cases where the holder intends to repurchase the stock at a later date or would consider doing so. Such an intention might be fostered by an expected near term decline in the value of the underlying stock and expected subsequent increase.

Suppose an ESO holder forms a price forecast for q periods in the future. [2] Momentarily ignoring taxes and the reinvestment of the proceeds from early exercise, share leverage will be lost whenever the per share proceeds of early exercise are less than the anticipated t = -n+q repurchase price of the stock. A loss in share leverage means that the net proceeds from the sale of shares received via early exercise will fund the repurchase on t = -n+q of only (1 – X/P) times the number shares sold after the early exercise. When share leverage is lost, the holder's ability to benefit from price increases after the anticipated repurchase date is diminished.

Suppose the holder of an ESO is trying to reach a decision about early
exercise. On the early exercise decision date, t = -n, the holder of the ESO
forecasts the underlying stock price for both the option’s expiration date, t =
0, and for some date prior to expiration, t = -n+q. Call this type of holder a
“trader”, because she intends to trade out of the ESO and back into the stock.
To decide between early exercise and holding the option to maturity, the
risk-neutral trader need only to determine in which instance her wealth will be
greater at expiration. Without early exercise, the trader’s wealth will be
P_{0} - X. If the option is exercised early at t = -n and the net
proceeds (P_{-n} - X)(1-T) invested at r for q periods until t = -n+q,
the trader will be able to repurchase a proportion of the originally optioned
shares equal to:

(3)

In equation (3), µ is the proportion of the number of shares obtained via
early exercise and sold at t = -n which can be repurchased at t = -n+q without
the trader tapping into other sources of funds. As such, µ is a complex measure
of share leverage that assumes the trader has forecast a price for some
anticipated reinvestment date, P_{-n+q.} It can be equal to, greater
than or less that one. For µ to be greater than one, the trader must generally
be expecting large declines in stock price or very high r’s between t = -n and t
= -n+q. [3] When µ ł
1, the trader should exercise at t = -n, invest at r for q periods and then use
the proceeds to repurchase µ ł
1 times of the number of shares received at t = -n. When µ ł
1 and P_{0} > X, the early exercise and repurchase of the stock will
provide the trader with wealth at the ESO’s expiration of µP_{0}, which
will be greater than the wealth from exercising at expiration, P_{0} –
X. Perceived gains in share leverage make the early exercise decision easy – the
trader should exercise her ESOs early and repurchase the stock at a later
date. [4]

When µ < 1, there is a loss of share leverage, but early exercise may still be optimal. All else equal, the wealth maximizing choice will depend on the expected price path of the underlying stock.

**III. Early Exercise When µ < 1**

In this section I use the specialized concept of share leverage developed above to examine optimal exercise behaviors when there is a loss in share leverage, i.e., µ < 1. To keep the notation tractable, we assume that the trader forecasts stock prices for q dates into the future and for the expiration date. Obviously the model below could be applied to terminal dates prior to the expiration date. Given µ < 1, for the trader to be indifferent between exercise at maturity and early exercise with repurchase at t = -n+q, then it must be the case that:

µP_{0}
= P_{0} -
X
(4)

The LHS of (4) represents the expected per share wealth at time 0 of the
trader given exercise at t = -n, investment of the proceeds at r for q periods
and repurchase at t = -n+q. The RHS represents the expected per share wealth
from waiting until expiration to exercise. Any time the LHS of (4) is less than
the RHS, the trader should delay exercise until expiration. Equation (4)
provides an upper bound on the expected stock price at expiration,
P_{0}, above which the trader should not exercise early. For example,
suppose X = 15, P_{-3} = 49, P_{-1} = 40, P_{0} = 55, T
= 0.20 and r = 0.05 (so µ = 0.752). Substituting into (4) we see that the
expected t = 0 wealth resulting from early exercise and subsequent repurchase is
$41.36 while the corresponding wealth from exercise at maturity is $40. In this
instance, the trader would conclude that the optimal exercise strategy is to
exercise early and repurchase the stock at t = -n+q. A general solution on the
upper bound on P_{0} above which early exercise will be sub optimal can
be obtained by rearranging equation (3) as:

(5)

In the prior numerical example, P_{0}^{u} is
$60.48. If the holder expects P_{0} to be above $60.48, the holder
should not exercise early. For the trader who forms price path expectations such
that P_{0} > P_{0}^{u}, the
wealth maximizing strategy would be to exercise the option at maturity.

To review how share leverage as defined here affects the early exercise
decision: in cases where there are gains in share leverage, µ > 1, the trader
should exercise at t=-n and repurchase at t=-n+q; in the more typical cases
where there are losses in share leverage, µ<1, the appropriate exercise
strategy is largely dependent on the trader's price path expectation regarding
the stock price at expiration. Specifically, if P_{0} > X/(1-µ), the
trader should not exercise until expiration.

**IV. Summary**

Once an ESO is vested, the holder has the right to exercise early, but not the right to sell or transfer the ESO. The inability of the holder to sell the ESO prevents the option from possessing realizable time value in excess of the stock price minus the exercise price. While the value of a typical American call option prior to expiration is greater than P - X, the nonmarketability of an ESO effectively eliminates this value difference. As a result, circumstances arise in which early exercise is optimal for a rational holder. In this paper, I stress the importance of share leverage in the ESO holder’s decision process. Without factoring changes in share leverage into his/her early exercise decision, the ESO holder may make sub optimal decisions. To estimate share leverage loses or gains and thereby make a reasoned decision about early exercise, the holder or his/her advisor needs to form opinions about future prices of the underlying stock, the tax rate and a reinvestment rate – obvious inputs to an informed decision.

**End
Notes**

1.
Recall
we had assumed risk-neutrality. This was done to simplify later notation and
analysis. However, the LHS of (2) represents a certain cash flow, while the RHS
is uncertain. To incorporate risk
aversion into the analysis, the following version of (2) would be more
appropriate: Exercise at t=-n and sell the stock, if (P_{-n} - X) >
(P_{-n+q} – X) e^{-kq}, where *k* is a discount rate
reflecting the holder’s level of risk aversion. Hall and Murphy (2000) develop a
model for determining optimal exercise prices based on the employee’s risk
aversion, non-firm related wealth and stock holdings. They find that value is decreasing in
risk aversion and stock holdings and increasing in non-firm related wealth. Their model does not address early
exercise or share leverage.

2.
Heath,
Huddart and Lang (1999) find that, consistent with psychological models of
belief, employees exercise ESOs early in response to stock price trends. They find that exercise is positively
related to recent returns. The
model in this paper would enable employees to make a more informed decision.

3.
I have assumed for simplicity that *r* is a risk free rate, but *r*
can be any reasonable rate the trader chooses to use. The higher the *r* at which the
proceeds of early exercise can be reinvested, the more likely is early
exercise.

4.
To
incorporate risk aversion into (2), the denominator, P_{-n+q} can be
multiplied by e^{-kq}, where *k* is a discount rate
reflecting the holder’s risk aversion. Discounting P_{-n+q} will
increase μ and will make early exercise more likely.

**
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