Goizueta Business School
Emory University
FIN 659: Futures and Options
Dr. Nicholas Valerio III
Fall 2022
Assignment: Binomial Option Pricing
This assignment is to be completed using Microsoft xcel. Best practices must always
be adhered to: calculations should be performed using cell references and functions only, and
should not contain any embedded numbers.
1. Use the binomial valuation method for N = 4 periods to determine the value of an
American-style Put option with a striKe price of $60 that expires in 45 days. Note
that ? = 45/365 years, and h = ? /N is the amount of time (in years) covered by one
period. The other relevant data are:
Current stock price:
Annual riskless interest rate:
Annual volatility:
S
i
?
= $57.00 (no dividends)
= 4.00%
= 30%
The values of the parameters u, d, r?, and ? can be found from the equations:
?
u = e? h ,
d=
1
,
u
r? = rh = (1 + i)h ,
?=
r? ? d
u?d
Prominently list on your spreadsheet the values for these parameters displayed to
four decimal places.*
Using the values for the parameters u and d, build the binomial 2ee3howing all of
the possible realizations for the stock price. Next, for each node in the binomial tree,
calculate the corresponding put option value. Speci?cally, for each of the ?ve possible
stock prices at t = 4 (expiration):
Pt = max {0, K ? St }
(1)
And then for all of the ten possible stock prices at t = 0, 1, 2, 3:
1
[? Pu,t+1 + (1 ? ?) Pd,t+1 ]
r? {
}
= max Ptf , K ? St
Ptf =
Pt
(2)
where the last step is the check for early exercise. Clearly display the price of the stock
and the value of the put at each node, displayed to four decimal places.
* )splayed-eans formatted, not rounded.
All calculations should preserve the internal precision of
Excel and not be rounded.
1
2. Next, calculate the respective values (again, displayed to four decimal places) for the
components of the duplicating portfolio (the %lta!nd the amount of lending) at the
start of each period, for t = 0, 1, 2, 3:
Pu,t+1 ? Pd,t+1
St (u ? d)
d Pu,t+1 ? u Pd,t+1
=
r?(u ? d)
?t =
(3)
Bt
(4)
Show that the value of the portfolio at each node (Vt ) matches the option value that
you calculated, for the ten nodes corresponding to t = 0, 1, 2, 3:
V t = ? t St ? B t
(5)
3. With the relevant data from Part 1, use the Black-Scholes formula to calculate and
report the value of an otherwise-similar European-style Call option. From this value,
use the European Put-Call Parity relationship (provided on Page 3) to ?nd the value
of a European-style Put option with the same terms. Calculate the di?erence between
your put value here and the put value found in Part 1, and provide two explanations
of any di?erence that you ?nd.
The Black-Scholes Model
C(S, ? ) = S N (d1 ) ? P V (K) N (d2 )
where:
d1 =
and:
ln
(
S
P V (K)
)
?
+ 12 ? 2 ?
? ?
,
?
d 2 = d1 ? ? ?
P V (K) = Kr??
when:
S
K
?
r
?
N (J?
?
?
?
?
?
the current stock price
the striKe (exercise) price
the time-to-expiration (in years)
the risk free annual factor, = (1 + i)
the standard deviation of the stockàreturns (%/year)
the cumulative standard normal probability distribution.
2
European-Style Put-Call Parity Relationship
P = C ? S + P V (K)
(For a non-dividend-paying stock.)
Useful Excel Functions
ez
?
z
max{y, z}
ln(z)
N (z)
= EXP (z)
= SQRT (z)
= M AX(y, z)
= LN (z)
= N ORM.S.DIST (z, T RU E)
The symbol z indicates a required argument for the function, and the Excel function includes
the T sign.
Note that the NORM.S.DIST function with the logical value of TRUE is for the cumulative
standard normal distribution (the special case of a mean of zero and standard deviation of
one) and only requires two arguments. The Excel function NORM.DIST is more general and
requires you to specify values for the mean and standard deviation, and the logical value
of TRUE for the cumulative distribution. The NORM.DIST function can be used in your
Black-Scholes calculation instead of NORM.S.DIST but then you will need to input the extra
arguments, i.e. N ORM.S.DIST (z, T RU E) = N ORM.DIST (z, 0, 1, T RU E)
3

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