Valuing Employee Stock Options Part 7 docx

PART
Two
Technical Background
of the Binomial Lattice
and Black-Scholes Models
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CHAPTER
7
Brief Technical
Background
BLACK-SCHOLES MODEL
The basic BSM is summarized as follows:
where Φ is the cumulative standard-normal distribution function
S is the value of the forecast stock price at grant date
X is the option’s contractual strike price
rf is the nominal risk-free rate
σ is the annualized volatility
T is the time to expiration of the option
To illustrate its use, let us assume that an option exists such that both the
stock price (S) and the strike price (X) are $100, the time to expiration (T)
is one year with a 5 percent annualized risk-free rate (rf) for the same dura-
tion, while the annualized volatility (σ) of the underlying asset is 25 per-
cent. The BSM calculation yields $12.3360:
Call S
S X rf T
T
Xe
S X rf T
T
Put Xe

S X rf T
T
S
S X rf T
rf T
rf T
=
++
−
+−
=−
+−
−−
++

























−
−
ΦΦ
ΦΦ
ln( / ) ( / ) ln( / ) ( / )
ln( / ) ( / ) ln( / ) ( / )
()
()
σ
σ
σ
σ
σ
σ
σ
22
22
22
22
σσ
T









77
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The cumulative standard-normal distribution function can be solved in Ex-
cel by using its “NORMSDIST( )” function.
In addition, you can create calculation codes within Excel’s Visual
Basic for Applications (VBA) environment. Following are the VBA codes
for the BSM for estimating call and put options. The equations for the
BSM are simplified to functions in Excel named “BlackScholesCall” and
“BlackScholesPut.”
Public Function BlackScholesCall(Stock As Double, Strike As Double,
Time As Double, Riskfree _
As Double, Volatility As Double) As Double
Dim D1 As Double, D2 As Double
D1 = (Log(Stock / Strike) + (Riskfree + 0.5 * Volatility ^ 2 ) * Time) /
(Volatility * Sqr(Time))
D2 = D1 – Volatility * Sqr(Time)
BlackScholesCall = Stock * Application.NormSDist(D1) – Strike *
Exp(–Time * Riskfree) * _
Application.NormSDist(D2)
End Function
Public Function BlackScholesPut(Stock As Double, Strike As Double,
Time As Double, Riskfree _
As Double, Volatility As Double) As Double

Dim D1 As Double, D2 As Double
D1 = (Log(Stock / Strike) + (Riskfree + 0.5 * Volatility ^ 2 ) * Time) /
(Volatility * Sqr(Time))
D2 = D1 – Volatility * Sqr(Time)
BlackScholesPut = Strike * Exp(–Time * Riskfree) *
Application.NormSDist(–D2) – Stock * _
Application.NormSDist(–D1)
End Function
78 BACKGROUND OF THE BINOMIAL LATTICE AND BLACK-SCHOLES MODELS
Call S
S X rf T
T
Xe
S X rf T
T
Call e
rf T
=
++
−
+−
=
++
−






































−
−
ΦΦ
Φ
ln( / ) ( / ) ln( / ) ( / )
$
ln
$
$
()
.
$
()
σ
σ
σ
σ
22
2
0
22
100
100
100
005
1
2
025 1
025 1
100

()
ln
$
$
()
.
$ (.)$(.)(.)$(.)$.(.)$.
05 1
2
100
100
005
1
2
025 1
025 1
100 0 3250 100 0 9512 0 0750 100 0 6274 95 12 0 5299 12 3360
Φ
ΦΦ

























+−
=− =−=Call
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As an example, entering the function in Excel:
“=BlackScholesCall(100,100,1,5%,25%)”
results in 12.3360 and entering the function in Excel:
“=BlackScholesPut(100,100,1,5%,25%)”
results in 7.4589.
Note that Log is a natural logarithm function in VBA and Sqr is
square root, and make sure there is a space before the underscore in
the code.
MONTE CARLO SIMULATION MODEL
In the simulation approach for estimating European options, a series of
forecast stock prices are created using the Geometric Brownian Motion
stochastic process, and the options maximization calculation is applied to
the end point of the series, and discounted back to time zero, at the risk-
free rate. That is, starting with an initial seed value of the underlying stock

price, simulate out multiple future pathways using a Geometric Brownian
Motion, where . That is, the change in asset value
δS
t
at time t is the value of the asset in the previous period S
t–1
multiplied
by the Brownian Motion . The term rf is the risk-free rate,
δt is the time-steps, σ is the volatility, and ε is the simulated value from a
standard-normal distribution with mean of zero and a variance of one.
Other variations of Brownian Motions exist but for illustration purposes,
this simple version will be used.
The first step in Monte Carlo simulation is to decide on the number of
time-steps to simulate. In the example, 10 steps were chosen for simplicity.
Starting with the initial stock price of $100 (S
0
), the change in value from
this initial value to the first period is seen as
.
Hence, the stock price at the first time-step is equivalent to
. The stock price at the second
time-step is hence , and so forth, all
the way until the terminal tenth time-step. Notice that because ε changes
on each simulation trial, each simulation trial will produce an entirely
SS SSSrft t
21 211
=+ =+ +
()
δδσεδ
()

SS SSSrft t
10 100
=+ =+ +
()
δδσεδ
()
δδσεδ
SSrft t
10
=+
()
()
rf t t()
δσεδ
+
()
δδσεδ
SSrft t
tt
=+
()
−1
()
Brief Technical Background 79
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different asset-evolution pathway. At the end of the tenth time-step, the
maximization process is then applied. That is, for a simple European op-
tion with a $100 implementation cost, the function is simply C
10,i
=

Max[S
10,i
– X, 0]. This is the call value C
10,i
at time 10 for the i
th
simulation
trial. This value is then discounted at the risk-free rate to obtain the call
value at time zero, that is, C
0,i
= C
10,i
e
–rf(T)
. This is a single-point estimate for
a single simulated pathway. A forecast distribution of the thousands of
simulated pathways is collected and the mean of the distribution is the ex-
pected value of the option. On the one hand, it must be stressed that
Monte Carlo simulation can be applied only to calculate European op-
tions, and not American options, making it less suitable for use in valuing
ESOs. On the other hand, Monte Carlo can be used to simulate the uncer-
tain input variables that go into the customized binomial lattice model as
seen in Chapter 5.
BINOMIAL LATTICES
Binomial lattices, in contrast to the other methods, are easy to implement
and easy to explain. They are also highly flexible but require significant com-
puting power and time-steps to obtain good approximations, as will be seen
later. It is important to note, however, that at the limit, results obtained
through the use of binomial lattices tend to approach those derived from
closed-form solutions, and hence, it is always recommended that both ap-

proaches be used to benchmark the results. The results from closed-form so-
lutions may be used in conjunction with the binomial lattice approach when
presenting a complete financial options valuation solution in the most basic
European options analysis. However, when real-life cases are added into the
analysis (forfeitures, vesting, suboptimal exercise behavior multiples, and
blackout dates), the results diverge because the closed-form models such as
the BSM or GBM cannot account for these added variables.
Following is an example to illustrate the point of binomial lattices ap-
proaching the results of a closed-form model. Let us look again at the Eu-
ropean call option presented previously, but this time, calculated using the
GBM:
Using the previous example where both the stock price (S) and the strike
price (X) are $100, the time to expiration (T) is one year with a 5 per-
80 BACKGROUND OF THE BINOMIAL LATTICE AND BLACK-SCHOLES MODELS
Call Se
SX rf q T
T
Xe
SX rf q T
T
qT
rT
=
+−+
−
+−−
−
−

















ΦΦ
ln( / ) ( / ) ln( / ) ( / )
σ
σ
σ
σ
22
22
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cent risk-free rate (rf) for the same duration, while the volatility (σ) of
the underlying asset is 25 percent with no dividends (q). The GBM cal-
culation yields $12.3360, similar to the BSM calculations, while using a
binomial lattice we obtain the following results:
N = 10 steps $12.0923
N = 20 steps $12.2132
N = 50 steps $12.2867

N = 100 steps $12.3113
N = 1,000 steps $12.3335
N = 10,000 steps $12.3358
N = 50,000 steps $12.3360
Notice that even in this simplified example, as the number of time-steps
(N) gets larger, the value calculated using the binomial lattice approaches
the GBM closed-form solution. Suffice it to say, many steps are required
for a good estimate using binomial lattices. It has been shown in past re-
search that 1,000 time-steps are usually sufficient for a good approxima-
tion for up to 2 decimals. Chapter 10 provides a case example of how to
find the optimal number of lattice steps and to test for results convergence
in a binomial lattice.
SUMMARY AND KEY POINTS
■ The three mainstream approaches used to solve simple options are the
GBM and BSM closed-form models, path-dependent simulation, and
binomial lattices.
■ BSM is highly inflexible and can be applied to solve only European
options.
■ Path-dependent simulations are also applicable for solving only Euro-
pean options.
■ Binomial lattices are more flexible and can be used to solve both Amer-
ican and European options and are capable of handling other exotic
input variables that exist in real-life ESOs.
Brief Technical Background 81
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