1. Topics in the Employee Reload Options
Acknowledgement: Slides are based on "Employee Reload Options: Pricing, Hedging, and Optimal Exercise," Dybvig, Philip H. and Mark Loewenstein, 2003, Review of Financial Studies 16, 2003,  
Topics in the Employee Reload Options

2. What Is the Reload Option?
Reload option is, essentially, an American call option with an additional bonus.
If the option is exercised prior to maturity and the exercise price is paid with previously owned share, the holder is entitled to one new share for each option exercised plus new options which reload or replace some of the original option.
Main Feature of the Reload Option:

3. What Is the Reload Option?
Some Variations:
1. How many times can the reloads take place?
Does the reload option allow unlimited reloads?
Or if it only allows a single reload?
2. Is there a waiting period (i.e. time vesting) between exercises?
For example, the employee is not allowed to excise the option until the end of an initial vesting period, and the reload options received after the initial exercise may also be subject to the same vesting period.
3.How many new options are granted upon the exercise?
Is it K/S per option exercised (replacing the number of shares tendered)?
Or, is it 1 per option exercised (replacing all the options exercised)?
4. What’s the maturity of the newly granted reload options?
Do they have the same expiration date as the original reload option?
Or they have a life extending beyond the life of the one previously exercised?

4. What Is the Reload Option?
The reload option, with strike price K and expiration date T, is an option which, if exercised on or before the expiration date and the exercise price is paid with previously owned shares, entitles the holder to one share per option exercised plus one new reload option per share tendered.
The new reload option has a strike price equal to the current stock price and it has the same expiration date as the original option.
The Definition of Reload Option (The one that we will use forward):

5. Is Reload Option a Money Pump?
NO.
The value of the reload option does have both upper and lower bounds, even if the reload option allows an unlimited number of reloads and has infinite time to maturity.
The lower bound is the value of an American call option.
(Recall that the reload option is an American call with a bonus.)
And we will show in the next slide, the upper bound is the underlying stock price.

6. The Upper Bound of Reload Option Value
Let’s start with one share of reload option,
After the 1st exercise at τ1:
The total net share:
The remaining option:
After the 2nd exercise at τ2:
The net additional share:
The total net share:
The remaining option:
After the 3rd exercise at τ3:
The net additional share:
The total net share:
The remaining option:
After the ith exercise at τi:
The total net share: , which is always less than 1.
The remaining option:

7. Is the Reload Option a Money Pump?
Does company lose control over the number of shares issued?
No and No.

8. What’s the Incentive Effect of the Employee Reload Options?
It’s not possible to asses the incentive effect of reload options without including the context of the rest of the compensation package. (e.g. What are the other pieces in the compensation package? What new pieces will be added ? And in what contingencies those new pieces will be added)
However, understanding the hedge ratios and the overall shape of the valuation function clarifies the employee’s risk exposure and incentives

9. What’s the Incentive Effect of the Employee Reload Options?
Let’s compare the reload option with the European call option with the same underlying, strike price, and maturity:
1. Comparison of value
The upper bound of the reload option value is indeed its underlying stock price

10. What’s the Incentive Effect of the Employee Reload Options?
The hedge ratio equals to one as the reload option moves in the money.
2. Comparison of hedge ratio
The derivative of the hedge ratio for at the money reload options does not exist.
As the reload option moves out of the money, its hedge ratio get closer to that of the European call option.

11. The gamma of the reload option is indeed discontinuous at the money.

12. What’s the Incentive Effect of the Employee Reload Options?
3. Comparison of the delta per unit value
The incentive effects of the reload option are quite similar to those of a European call option with the same market value

13. What’s the Incentive Effect of the Employee Reload Options?
Now let’s consider the dynamic nature of the reload package:
(Recall what we have derived)
After the ith exercise at τi:
The total net share:
The remaining option:
As higher stock prices are attained, the reload option holdings decrease as more exercises occur, then the reload package acts like a share of stock;
If, on the other hand, not many exercises occur, is close to or less than K, then the reload package acts like a ordinary call option.
Act like stock
Act like European call

14. What’s the Optimal Exercise Strategy of the Reload Option?
In short, the optimal strategy is to exercise the reload option whenever it is in the money.

15. What’s the Optimal Exercise Strategy of the Reload Option?
Let’s start with the discrete case:
Assuming that exercise is available only on the set of non-stochastic times
{ t1 , t2 , t3 , … , tn }, where 0 = t1 < t2 < t3 < … < tn = T;
An exercise policy is defined to be an increasing family of stopping times, τi taking values on the grid with t1 ≤ τ1 < … < τi <….
Define the strike or exercise price process as:
X(t)=
K ≤ t < τ1
S(τ1) τ1 ≤ t < τ2
S(τ2) τ2 ≤ t < τ3 .

16. What’s the Optimal Exercise Strategy of the Reload Option?
For the derivation of the optimal strategy, we will assume the following:
Assumption 1: The employee is always free to hold additional shares.
Assumption 2: The employee is always holding enough shares to pay the exercise price ( or at least can borrow the necessary shares).
Assumption 3: The exercise decision itself does not affect the employee’s compensation, the sock price, or dividend payments, for example, through the dependence of future wages on exercise, through a dilution of shares, or through signaling.
Assumption 4: The dividend payments are nonnegative and the stock price is strictly positive. The employee prefers more consumption to less and can eventually convert dividend payments and share receipts into desirable subsequent consumption.
Assumption 5: There are no taxes or transaction costs.

17. What’s the Optimal Exercise Strategy of the Reload Option?
Theorem 1. It is an optimal policy to exercise the reload option whenever it is in the money, and refrain from exercising whenever it is out of the money. This strategy results in the exercise process X*(t), where
is the decreasing process that describes the strike price as a function of time under this optimal strategy on the grid with n Points. This is the only optimal strategy (up to indifference about exercising at dates when the option is at the money) if the stock price can always fall between grid dates (which we think of as the ordinary case).

18. What’s the Optimal Exercise Strategy of the Reload Option?
X*(t) captures the maximal stock price from 0 to t
The strike price process results form the optimal exercise strategy:
In the discrete case:
Proof:
Let θ(t) be the process representing the number of shares of the stock held at time t. Consider switching from X(t) to our candidate optimum X*(t), holding all other decisions fixed outside of the exercise decision.
The process θ*(t) will be given by
Subtract the net additional shares from the arbitrary strategy
Add the net additional shares from the candidate optimal strategy
Notice that X*(t) ≥ X(t), thus,

19. Why Do We Care about the Valuation?
1. To prepare accounting statements and tax returns
Because a reload feature is part of the option initially awarded, the Board believes that the value added to those options by the reload feature ideally should be considered in estimating the fair value of the award on the grant date.
However, the Board understands that no reasonable method currently exists to estimate the value added by the reload feature. Accordingly, the Board concluded that the best way to account for an option with a reload feature is to treat both the initial grant and each subsequent grant of a reload option separately.
In 1995, the FASB recommended:
HOWEVER, Treating each reload as a new grant, as suggested by the FASB, could substantially overstate the value of reload options.
2. To understand what value has been promised to the employees and what residual value remains with the shareholders

20. Valuation of the Reload Options with Discrete Exercise
There are two ways to think of it:
1. At time T, we end up with shares of stock per option
units of remaining option per original option,
Then the market value of the reload option can be described as
(1)
2. If we treat each exercise as a cash event,
Then the market value of the reload option can be described as
(2)
*Equations (1) and (2) should have the same value for any given exercise policy.

21. Valuation of the Reload Options with Discrete Exercise
Theorem 2. For the optimal exercise policy in Theorem 1, we have that the market value
can be written equivalently as

22. Valuation of the Reload Options with Continuous Exercise
The strike price process:
In the discrete case:
In the continuous case:
Any jumps in the process S are downward jumps, i.e. S(t)-S(t-)<0
The market value of the reload option under the optimal exercise policy:
In the discrete case:
In the continuous case:
Alternatively, the market value of the reload option under the optimal exercise policy:
In the discrete case:
In the continuous case:
Where

23. Valuation of the Reload Options under Black-Scholes Assumptions
Assuming a constant positive interest rate r, the bond prices follow
Assuming a constant volatility and continuous proportional dividends, the stock price and cumulative dividend processes follow

24. Valuation of the Reload Options under Black-Scholes Assumptions
Proposition 1. Suppose stock and bond returns are given by the Black-Scholes case with dividends and the current stock price is S(0). Consider a reload option with current strike price K and remaining time to maturity T. Its value is
Where the cumulative distribution function of m(t) is given by
, if y < 0
, if y ≥ 0
Where
and is the unit normal cumulative distribution function.

25. Some Thoughts
Should companies include reload options as part of their executive compensation packages?
Yes, I think so.
2. Is Black-Scholes an appropriate tool, since it’s designed for traded options and the reload option is not traded in the market?
I don’t know.