Option Explicit Dim NotifyErrors As Boolean Public Function ESO_BKB(stockPrice As Double, strike As Double, volatility As Double, interestRate As Double, _ dividendRate As Double, tenor As Double, valueDiscount As Double, attritionRate As Double, _ vestingTimes As Variant, vestingLevels As Variant, numSteps As Integer) ' Version 1.0 '(c) Copyright 2005, by Brian K. Boonstra, Ph.D. ' 'A function to price employee stock options. This is similar to modeling ' work I have done over the years as a quant for my various employers and ' modeling examples I have taught to my classes at IIT Stuart Business School. ' ' This code is licensed under the GPL: ' That means that you can freely use it, change it, and redistribute it so long ' as any version you distribute has the source code freely available. See below for ' details. Other licensing terms are available -- I am in the Chicago phonebook, or ' write to me at 1435 S. Federal St, Chicago, Illinois. ' ' Uses a tree to price employee stock options according to the following assumptions: ' (1) Due to holding periods, portfolio constraints, or other reasons, an ' employee receiving stock upon exercise will value it at less than the ' market rate. This undervaluation thus affects an employee's decision about ' whether or not to exercise his/her options. The valueDiscount represents ' how much. Typical valueDiscount: 3-5% ' (2) Option-owning employees leave the company, forced to exercise or lose vested ' stock options. Stock options therefore go to term at a constant attrition ' rate. Note this is NOT the exactly the same as the employee attrition rate, ' since (for example) senior employees holding 75% of the options might depart ' at a rate of 10% per year, while junior employees might depart at a rate of ' 20% per year, yielding an option attrition rate of 12.5%. Typical value: 5-20% ' (3) When choosing to exercise, option holders do so according to a simple ' probability of exercise formula that depends on the current time, the ' option tenor, and perceived and actual exercise values. A function for this ' is supplied in the code, but is easily changeable. ' (4) Options come with a vesting schedule, ranging from nothing vested before ' the first vesting time to fully vested at or before the option tenor. ' Typical schedule (times in years): t={0,1,3}, level={0.33,0.67,1} ' (5) The company stock price follows the Black-Scholes model. The stock will ' pay a constant dividend proportion per unit of stock, whatever the current ' stock price. The risk-free interest rate is a constant. Stock price ' volatility is also a constant. In principle, volatility would best be ' set by the price of an at-the-money over the counter call with a tenor ' similar to that of the employee stock options in question. However, ' historical calibration (to a period roughly the same as the ESO tenor) may ' be the only available means for illiquid names. ' Typical values: S=50, strike=50, vol=50%, r=5%, dividendRate=2% ' (6) The tree has enough timesteps in it to properly approximate a solution to ' the model. Try successively higher timestep counts to ensure you are ' converging on the correct answer. For example, you could try ' 10, 50, 100, 200, 500, 1000, 2000 timesteps and so on, to the limits ' of your patience. Typical value: 50 * (tenor (in years)). Dim perceived_df, perceived_stock_price, dt, root_half_dt, u, u_inv, _ mult_factor, riskfree_discount, attrition_prob, pu, pm, pd, _ carry_cost, current_time, current_stock_price, _ holder_early_exer_val, eex_value, accumulated_vesting, _ attr_cont_rate, drift_speed, issuer_option_value, _ holder_option_value, no_perception_option_value, european_option_value _ As Double Dim bank_no_exer(), ignore_perception_no_exer(), holder_no_exer(), euro_no_exer(), _ issuer(), holder(), ignore_perception(), european(), _ issuer_new(), holder_new(), ignore_perception_new(), european_new(), _ weights(), stock_prices() _ As Double Dim i, j, vest, num_vestings, step_num _ As Integer Dim vested_one_or_zero() As Integer Dim inputsOK As Boolean Dim tree_probs As Variant Dim prob_of_exer, prob_of_exer_attr, prob_of_exer_noperc, prob_of_exer_noperc_attr As Variant NotifyErrors = False inputsOK = ESO_CheckInputs(stockPrice, strike, volatility, interestRate, _ dividendRate, tenor, valueDiscount, attritionRate, _ vestingTimes, vestingLevels, numSteps) If Not inputsOK Then ESO_BKB = -1 Return End If num_vestings = VestingLength(vestingTimes) ' Discount factor yielding perceived value perceived_df = 1 - valueDiscount 'Step sizes dt = tenor / numSteps root_half_dt = Math.Sqr(dt / 2#) u = Exp(2# * volatility * root_half_dt) u_inv = 1# / u ' Set array sizes. We will have 2-d arrays for holding option values...one dimension captures ' option value along a variety of stock prices (as normal for tree solutions to BS) whike ' the other dimension captures values according to vesting cohorts ReDim issuer(-numSteps - 1 To numSteps + 1, num_vestings) ReDim issuer_new(-numSteps - 1 To numSteps + 1, num_vestings) ReDim holder(-numSteps - 1 To numSteps + 1, num_vestings) ReDim holder_new(-numSteps - 1 To numSteps + 1, num_vestings) ReDim ignore_perception(-numSteps - 1 To numSteps + 1, num_vestings) ReDim ignore_perception_new(-numSteps - 1 To numSteps + 1, num_vestings) ReDim european(-numSteps - 1 To numSteps + 1, num_vestings) ReDim european_new(-numSteps - 1 To numSteps + 1, num_vestings) ReDim issuer_no_exer(num_vestings) ReDim ignore_perception_no_exer(num_vestings) ReDim holder_no_exer(num_vestings) ReDim euro_no_exer(num_vestings) ReDim vested_one_or_zero(num_vestings) ReDim weights(num_vestings) ReDim stock_prices(-numSteps - 1 To numSteps + 1) ' A multiplication factor captures the stock price at the current node mult_factor = u ^ (numSteps + 2) ' Set up terminal values For i = numSteps + 1 To -numSteps - 1 Step -1 ' The multiplication factor for this grid point mult_factor = mult_factor * u_inv ' Actual and perceived price of stock stock_prices(i) = stockPrice * mult_factor perceived_stock_price = perceived_df * stockPrice * mult_factor ' Loop over vestings, making final value assignments where appropriate For vest = 1 To num_vestings ' Allow vesting to be after expiry, however odd If vestingTimes(vest) < tenor Then ' Perceived option value, which affects exercise decisions holder(i, vest) = perceived_stock_price - strike If holder(i, vest) < 0# Then ' Option has no value if the stock is perceived to be underwater issuer(i, vest) = 0# holder(i, vest) = 0# Else ' Holder wants to exercise, so the option has value equal to market value issuer(i, vest) = stock_prices(i) - strike End If european(i, vest) = issuer(i, vest) ' We also track and could report prices that do not allow for perceived discounts ignore_perception(i, vest) = stock_prices(i) - strike If ignore_perception(i, vest) < 0# Then ignore_perception(i, vest) = 0# End If End If Next vest Next i ' Per-period discount riskfree_discount = Math.Exp(-interestRate * dt) ' Attrition probability per period, translated from an annual rate to a per-timestep ' rate by assuming continuous attrition attrition_prob = 1 - Math.Exp(Math.Log(1# - attritionRate) * dt) ' The carry cost is the drift of the stock under the risk-free measure carry_cost = interestRate - dividendRate ' Probabilities on the tree tree_probs = ESO_BKB_TreeProbs(u, dt, carry_cost, volatility) pu = tree_probs(1) pm = tree_probs(0) pd = tree_probs(-1) ' We now approximate the PDE solution with a tree (or, if you like, an ' explicit finite-difference scheme for the initial value problem in backwards time). ' ' There are several missing numerical efficiencies, such as finding (and not calculating ' beyond) the exercise boundary, assignment of local constants to avoid array ' references, extra calculations made for values we don't end up returning, ' unpadding arrays, and multiplications we could cache in local variables For step_num = numSteps To 1 Step -1 current_time = (step_num - 1) * dt ' Record which vesting cohorts have vested. This loop could be made ' more efficient, but is a small part of the algorithm cost For vest = 1 To num_vestings If vestingTimes(vest) <= current_time Then vested_one_or_zero(vest) = 1 Else vested_one_or_zero(vest) = 0 End If Next vest ' Loop over stock prices For i = step_num To -step_num Step -1 current_stock_price = stock_prices(i) ' Option holder's perception of value of the stock he/she would receive on exercise perceived_stock_price = perceived_df * current_stock_price ' True early exercise value eex_value = current_stock_price - strike ' Perceived early exercise value holder_early_exer_val = perceived_stock_price - strike ' Loop over vestings, assigning option values of various types For vest = 1 To num_vestings ' One more period of holding the american option, issuer's perspective issuer_no_exer(vest) = riskfree_discount * (pu * issuer(i + 1, vest) + _ pm * issuer(i, vest) + pd * issuer(i - 1, vest)) ' One more period of the option with no perceptin discount ignore_perception_no_exer(vest) = riskfree_discount * (pu * ignore_perception(i + 1, vest) + _ pm * ignore_perception(i, vest) + pd * ignore_perception(i - 1, vest)) ' One more period of holding the american option, holder's perspective holder_no_exer(vest) = riskfree_discount * (pu * holder(i + 1, vest) + _ pm * holder(i, vest) + pd * holder(i - 1, vest)) ' Get the value of holding the european option one more period euro_no_exer(vest) = riskfree_discount * (pu * european(i + 1, vest) + _ pm * european(i, vest) + pd * european(i - 1, vest)) Next vest ' Exercise probabilities by vesting cohort(which will of course be zero for unvested options) prob_of_exer = ESO_Exercise_Function(holder_early_exer_val, holder_no_exer, vested_one_or_zero, current_time, tenor) ' The same, but now the holder is leaving and must decide immediately prob_of_exer_attr = ESO_Exercise_Function(holder_early_exer_val, 0, vested_one_or_zero, current_time, current_time) ' Other exercise probabilities (no perception discount) prob_of_exer_noperc = ESO_Exercise_Function(eex_value, ignore_perception_no_exer, vested_one_or_zero, current_time, tenor) prob_of_exer_noperc_attr = ESO_Exercise_Function(eex_value, 0, vested_one_or_zero, current_time, current_time) ' Backwarddate on the tree to get new option values For vest = 1 To num_vestings issuer_new(i, vest) = attrition_prob * prob_of_exer_attr(vest) * vested_one_or_zero(vest) * eex_value + _ (1 - attrition_prob) * (vested_one_or_zero(vest) * (prob_of_exer(vest) * eex_value + (1 - prob_of_exer(vest)) * issuer_no_exer(vest)) + _ (1 - vested_one_or_zero(vest)) * issuer_no_exer(vest)) ignore_perception_new(i, vest) = attrition_prob * prob_of_exer_noperc_attr(vest) * vested_one_or_zero(vest) * eex_value + _ (1 - attrition_prob) * (vested_one_or_zero(vest) * (prob_of_exer_noperc(vest) * eex_value + (1 - prob_of_exer_noperc(vest)) * ignore_perception_no_exer(vest)) + _ (1 - vested_one_or_zero(vest)) * ignore_perception_no_exer(vest)) holder_new(i, vest) = attrition_prob * prob_of_exer_attr(vest) * vested_one_or_zero(vest) * holder_early_exer_val + _ (1 - attrition_prob) * (vested_one_or_zero(vest) * (prob_of_exer(vest) * holder_early_exer_val + (1 - prob_of_exer(vest)) * holder_no_exer(vest)) + _ (1 - vested_one_or_zero(vest)) * holder_no_exer(vest)) european_new(i, vest) = attrition_prob * prob_of_exer_attr(vest) * vested_one_or_zero(vest) * eex_value + _ (1 - attrition_prob) * euro_no_exer(vest) Next vest Next i ' End loop over stock prices ' Update the values in our vectors issuer = issuer_new ignore_perception = ignore_perception_new holder = holder_new european = european_new Next step_num ' Now figure out average values by weighting according to vesting cohorts accumulated_vesting = 0 For vest = 1 To num_vestings weights(vest) = vestingLevels(vest) - accumulated_vesting accumulated_vesting = vestingLevels(vest) Next vest issuer_option_value = 0# holder_option_value = 0# no_perception_option_value = 0# european_option_value = 0# For vest = 1 To num_vestings issuer_option_value = issuer_option_value + issuer(0, vest) * weights(vest) holder_option_value = holder(0, vest) * weights(vest) no_perception_option_value = ignore_perception(0, vest) * weights(vest) european_option_value = european(0, vest) * weights(vest) Next vest ESO_BKB = issuer_option_value End Function Public Function ESO_Exercise_Function(ByVal exerciseValue As Double, noExerciseValue As Variant, vested As Variant, ByVal nowTime As Double, ByVal tenor As Double) As Double() Dim vest As Integer Dim prob() As Double Dim noExer As Double ReDim prob(LBound(vested) To UBound(vested)) For vest = LBound(vested) To UBound(vested) prob(vest) = 0# If 1 = vested(vest) Then If IsArray(noExerciseValue) Then noExer = noExerciseValue(vest) Else noExer = noExerciseValue End If If exerciseValue > noExer Then prob(vest) = 1# End If End If Next vest ESO_Exercise_Function = prob End Function Public Function ESO_BKB_TreeProbs(ByVal u As Double, ByVal dt As Double, ByVal carry_cost As Double, ByVal volatility As Double) As Variant Dim p(-1 To 1) As Double Dim i As Integer Dim drift, vardrift As Double drift = Math.Exp(carry_cost * dt) vardrift = Math.Exp(dt * (2# * carry_cost + volatility * volatility)) p(1) = (1 - drift * (u + 1) + u * vardrift) / ((u - 1) ^ 2 * (u + 1)) p(0) = (drift * (u ^ 2 + 1) - u * (vardrift + 1)) / ((u - 1) ^ 2) p(-1) = 1# - p(1) - p(0) If p(1) < 0 Or p(0) < 0 Or p(-1) < 0 Then If NotifyErrors Then MsgBox ("Negative tree probabilities encountered. Try more timesteps.") End If For i = -1 To 1 p(i) = 0 Next i End If ESO_BKB_TreeProbs = p End Function Public Function ESO_CheckInputs(stockPrice As Double, strike As Double, volatility As Double, interestRate As Double, _ dividendRate As Double, tenor As Double, valueDiscount As Double, attritionRate As Double, _ vestingTimes As Variant, vestingLevels As Variant, numSteps As Integer) As Boolean Dim numVestings, numVestingTimes As Integer Dim errorMsg As String Dim inputType As String Dim i As Integer Dim v, vold, t, told As Double If numSteps < 1 Then If NotifyErrors Then MsgBox ("Step count must be at least 1") End If ESO_CheckInputs = False Return End If If stockPrice <= 0 Then If NotifyErrors Then MsgBox ("Stock price must be greater than zero") End If ESO_CheckInputs = False Return End If If strike <= 0 Then If NotifyErrors Then MsgBox ("Strike price must be greater than zero") End If ESO_CheckInputs = False Return End If If volatility <= 0 Then If NotifyErrors Then MsgBox ("Volatility must be greater than zero") End If ESO_CheckInputs = False Return End If If interestRate <= 0 Then If NotifyErrors Then MsgBox ("Risk-free interest rate must be greater than zero") End If ESO_CheckInputs = False Return End If If tenor <= 0 Then If NotifyErrors Then MsgBox ("Option tenor must be greater than zero") End If ESO_CheckInputs = False Return End If If valueDiscount < 0 Then If NotifyErrors Then MsgBox ("The value discount cannot be less than zero...it must be between 0 and 1 (0% and 100%)") End If ESO_CheckInputs = False Return End If If valueDiscount > 1 Then If NotifyErrors Then MsgBox ("The value discount cannot be more than 1.0...it must be between 0 and 1 (0% and 100%)") End If ESO_CheckInputs = False Return End If If attritionRate < 0 Then If NotifyErrors Then MsgBox ("The attrition rate cannot be less than zero...it must be at least 0 and less than 1") End If ESO_CheckInputs = False Return End If If attritionRate >= 1 Then If NotifyErrors Then MsgBox ("The attrition rate cannot be 1.0 or greater...it must be at least 0 and less than 1") End If ESO_CheckInputs = False Return End If numVestingTimes = VestingLength(vestingTimes) numVestings = VestingLength(vestingLevels) If numVestingTimes <= 0 Then If NotifyErrors Then errorMsg = "The number of vesting times (" & numVestingTimes & ") must be more than zero" MsgBox (errorMsg) End If ESO_CheckInputs = False Return End If If numVestings <> numVestingTimes Then If NotifyErrors Then errorMsg = "The number of vesting times (" & numVestingTimes & ") must be equal the number of vesting levels (" & numVestings & ")" MsgBox (errorMsg) End If ESO_CheckInputs = False Return End If vold = 0 told = vestingTimes(1) - 0.1 For i = 1 To numVestings v = vestingLevels(i) t = vestingTimes(i) If v < 0 Then If NotifyErrors Then errorMsg = "The vesting level (" & v & ") at index " & i & " was negative" MsgBox (errorMsg) End If ESO_CheckInputs = False Return ElseIf v > 1 Then If NotifyErrors Then errorMsg = "The vesting level (" & v & ") at index " & i & " was greater than 1" MsgBox (errorMsg) End If ESO_CheckInputs = False Return End If If v < vold Then If NotifyErrors Then errorMsg = "The vesting level (" & v & ") at index " & i & " was less than the previous level of " & vold MsgBox (errorMsg) End If ESO_CheckInputs = False Return End If If t <= told Then If NotifyErrors Then errorMsg = "The vesting time (" & t & ") at index " & i & " was less than or equal to the previous time of " & told MsgBox (errorMsg) End If ESO_CheckInputs = False Return End If vold = v told = t Next i ESO_CheckInputs = True End Function Public Function VestingLength(vestingvar As Variant) As Integer Dim inputType As String If "Range" = TypeName(vestingvar) Then VestingLength = vestingvar.Count ElseIf IsArray(vestingvar) Then VestingLength = 1 + UBound(vestingvar) - LBound(vestingvar) Else VestingLength = -1 End If End Function ' GNU GENERAL PUBLIC LICENSE ' Version 2, June 1991 ' ' Copyright (C) 1989, 1991 Free Software Foundation, Inc. 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If the Program ' specifies a version number of this License which applies to it and "any ' later version", you have the option of following the terms and conditions ' either of that version or of any later version published by the Free ' Software Foundation. If the Program does not specify a version number of ' this License, you may choose any version ever published by the Free Software ' Foundation. ' ' 10. If you wish to incorporate parts of the Program into other free ' programs whose distribution conditions are different, write to the author ' to ask for permission. For software which is copyrighted by the Free ' Software Foundation, write to the Free Software Foundation; we sometimes ' make exceptions for this. Our decision will be guided by the two goals ' of preserving the free status of all derivatives of our free software and ' of promoting the sharing and reuse of software generally. ' ' NO WARRANTY ' ' 11. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY ' FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN ' OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES ' PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED ' OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF ' MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS ' TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE ' PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, ' REPAIR OR CORRECTION. ' ' 12. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING ' WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR ' REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, ' INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING ' OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED ' TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY ' YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER ' PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE ' POSSIBILITY OF SUCH DAMAGES. ' ' END OF TERMS AND CONDITIONS ' ' How to Apply These Terms to Your New Programs ' ' If you develop a new program, and you want it to be of the greatest ' possible use to the public, the best way to achieve this is to make it ' free software which everyone can redistribute and change under these terms. ' ' To do so, attach the following notices to the program. It is safest ' to attach them to the start of each source file to most effectively ' convey the exclusion of warranty; and each file should have at least ' the "copyright" line and a pointer to where the full notice is found. ' ' ' Copyright (C) ' ' This program is free software; you can redistribute it and/or modify ' it under the terms of the GNU General Public License as published by ' the Free Software Foundation; either version 2 of the License, or ' (at your option) any later version. ' ' This program is distributed in the hope that it will be useful, ' but WITHOUT ANY WARRANTY; without even the implied warranty of ' MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ' GNU General Public License for more details. ' ' You should have received a copy of the GNU General Public License ' along with this program; if not, write to the Free Software ' Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA ' ' ' Also add information on how to contact you by electronic and paper mail. ' ' If the program is interactive, make it output a short notice like this ' when it starts in an interactive mode: ' ' Gnomovision version 69, Copyright (C) year name of author ' Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'. ' This is free software, and you are welcome to redistribute it ' under certain conditions; type `show c' for details. ' ' The hypothetical commands `show w' and `show c' should show the appropriate ' parts of the General Public License. Of course, the commands you use may ' be called something other than `show w' and `show c'; they could even be ' mouse-clicks or menu items--whatever suits your program. ' ' You should also get your employer (if you work as a programmer) or your ' school, if any, to sign a "copyright disclaimer" for the program, if ' necessary. Here is a sample; alter the names: ' ' Yoyodyne, Inc., hereby disclaims all copyright interest in the program ' `Gnomovision' (which makes passes at compilers) written by James Hacker. ' ' , 1 April 1989 ' Ty Coon, President of Vice ' ' This General Public License does not permit incorporating your program into ' proprietary programs. If your program is a subroutine library, you may ' consider it more useful to permit linking proprietary applications with the ' library. If this is what you want to do, use the GNU Library General ' Public License instead of this License. '