### 09b - NYU Stern

```Session 9b
Overview
Finance Simulation Models
• Securities Pricing
– Black-Scholes
– Electricity Option
• Miscellaneous
– Monte Carlo vs. Latin Hypercube
– Review of Binomial
Decision Models -- Prof. Juran
2
Finance Example
• A European call option on a stock earns the owner an
amount equal to the price at expiration minus the
exercise price, if the price of the stock on which the
call is written exceeds the exercise price. Otherwise,
the call pays nothing.
• A European put option earns the owner an amount
equal to the exercise price minus the price at
expiration, if the price at expiration is less than the
exercise price. Otherwise the put pays nothing.
Decision Models -- Prof. Juran
3
Finance Example
• The Black-Scholes formula calculates the price
of a European options based on the following
inputs:
–
–
–
–
–
today's stock price
the duration of the option (in years)
the option's exercise price
the risk-free rate of interest (per year)
the annual volatility (standard deviation) in stock
price
Decision Models -- Prof. Juran
4
The Black-Scholes model:
C  SN d 1   Ee  rt N d 2 
where:
S
E
r
σ2
t
d1
d2
N(d)
Decision Models -- Prof. Juran
= current stock price
= exercise price
= risk-free rate of return
= variance of the stock’s return
= time to expiration
2
 S    
ln     r 
t

E
2
  

=
 2t
d1   2 t
=
= probability that z < d
5
A Black-Scholes calculator:
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
B
Inputs
1
35
40
0.5
0.05
0.4
Type of option (1 for call, 2 for put)
Stock price
Exercise price
Duration (years)
Riskfree interest rate
Volatility
C
D
E
F
G
H
=IF(B2=1,NORMSDIST(B10),NORMSDIST(-B10))
=(LN(B3/B4)+(B6+B7^2/2)*B5)/(B7*SQRT(B5))
Quantities for Black-Scholes formula
d1
d2
Option price
-0.242
-0.525
=B10-SQRT(B7^2*B5)
2.456
N(d1)
N(d2)
0.404
0.300
=IF(B2=1,NORMSDIST(B11),NORMSDIST(-B11))
=IF(B2=1,B3*E10-B4*EXP(-B5*B6)*E11,-(B3*E10-B4*EXP(-B5*B6)*E11))
Notice the use of “if” statements in cells E10:E11 and B13, so that the
same model can be used for both puts and calls.
Decision Models -- Prof. Juran
6
Example: Diageo (DEO)
Decision Models -- Prof. Juran
7
Assume today is the first trading day of October
and that DEO is selling for \$57.98 per share.
What is a fair price for a six-month call option
with a strike price of \$60.00?
Assume the risk-free rate is 10%.
Two approaches:
•Black-Scholes formula
•Crystal Ball model
Decision Models -- Prof. Juran
8
Data file: deo-data.xlsx
A
1 Date
2 17-Dec-97
3
2-Jan-98
4
2-Feb-98
5
2-Mar-98
6
1-Apr-98
7
1-May-98
8
1-Jun-98
9
1-Jul-98
10
3-Aug-98
11
1-Sep-98
12
1-Oct-98
B
C
D
E
F
G
Open
High
Low
Close
Volume
36.17
36.29
34.12
34.58
201080
25.53
34.12
35.37
32.81
33.26
254960
24.56
33.38
38.8
33.32
37.94
221678
28.02
38.74
45.42
38.4
44.33
181459
32.74
45.98
47.49
42.43
44.64
179619
33.6
44.87
46.44
41.91
42.03
206615
31.63
42.09
46.73
41.91
44.58
115309
33.55
44.41
48.07
43.19
44.93
136350
33.82
43.19
44.58
34.66
35.12
130133
26.43
37.62
39.47
30.65
33.67
160090
25.34
34.08
41.87
33.67
41.58
143327
31.92
Decision Models -- Prof. Juran
9
Remove unnecessary columns and calculate
monthly returns:
A
1 Date
2 17-Dec-97
3
2-Jan-98
4
2-Feb-98
5
2-Mar-98
6
1-Apr-98
7
1-May-98
8
1-Jun-98
A
1 Date
2 17-Dec-97
3
2-Jan-98
4
2-Feb-98
5
2-Mar-98
6
1-Apr-98
7
1-May-98
8
1-Jun-98
B
C
25.53
24.56
-0.0380
28.02
0.1409
32.74
0.1685
33.6
0.0263
31.63
-0.0586
33.55
0.0607
D
=(B3-B2)/B2
B
C
D
1+Return
25.53
24.56
-0.0380
0.9620
28.02
0.1409
1.1409
32.74
0.1685
1.1685
33.6
0.0263
1.0263
31.63
-0.0586
0.9414
33.55
0.0607
1.0607
Decision Models -- Prof. Juran
E
E
F
=C3+1
10
Black-Scholes assumes that the future price is the
following random function of the current price:

pt  p0 exp   0.5 t  Z t
t
p0
pt
exp(x)
e


Z
2

Some specific time in the future
Current price
Price at time t
ex
Root of the natural logarithms (about 2.718282)
Mean percentage growth rate of the stock
Standard deviation of the growth rate (a.k.a. Volatility)
Normal random variable with mean 0 and standard deviation 1
Decision Models -- Prof. Juran
11
Another way to look at it:
ROI log
Decision Models -- Prof. Juran
 pt
 ln 
p
 0



12
A
1 Date
2 17-Dec-97
3
2-Jan-98
4
2-Feb-98
5
2-Mar-98
6
1-Apr-98
7
1-May-98
8
1-Jun-98
93
94
95
96
97
98
99
100
101
102
A
1-Jul-05
1-Aug-05
1-Sep-05
3-Oct-05
B
C
D
E
25.53
24.56 -0.0380
0.9620
-0.0387
28.02 0.1409
1.1409
0.1318
32.74 0.1685
1.1685
0.1557
33.6 0.0263
1.0263
0.0259
31.63 -0.0586
0.9414
-0.0604
33.55 0.0607
1.0607
0.0589
B
C
54.44 -0.0612
56.43 0.0366
58.01 0.0280
57.98 -0.0005
D
0.9388
1.0366
1.0280
0.9995
E
-0.0632
0.0359
0.0276
-0.0005
Monthly Mean
StDev
0.0087
0.0653
Annual
0.1047
0.2261
Decision Models -- Prof. Juran
Mean
StDev
F
G
=LN(D3)
F
G
H
=AVERAGE(E3:E96)
=STDEV(E3:E96)
=E98*12
=E99*SQRT(12)
13
A
1
2
3
4
5
6
7
8
9
10
11
12
13
Type of option (1 for call, 2 for put)
Stock price
Exercise price
Duration (years)
Riskfree interest rate
Volatility
B
C
Inputs
1
57.98
60
0.5
0.1
0.226148277
Quantities for Black-Scholes formula
d1
d2
0.178
0.019
Option price
4.137
Decision Models -- Prof. Juran
D
E
N(d1) 0.571
N(d2) 0.507
14
Now we create a new sheet that uses estimated
parameters from the data to calculate the future price of
DEO and the resulting cash flow from the option.
The present value of the expected payout is \$0.17, but
Black-Scholes says \$4.14. Why?
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A
Inputs
Current Price
Strike Price
Mean Annual Return
StDev Annual Return
Risk-Free Rate
Duriation (Years)
\$ 57.98
\$ 60.00
0.1047
0.2261
0.1
0.5
Simulation
Z
Stock Price (at Risk-Free Rate)
Option Cash Flow
0.0000
\$ 60.18
\$ 0.18
Output
Discounted Value
\$
Decision Models -- Prof. Juran
B
0.17
C
D
E
F
G
=data!E101
=data!E102
=B2*EXP((B6-0.5*B5^2)*B7+B5*B10*SQRT(B7))
=MAX(B11-B3,0)
=EXP(-B7*B6)*B12
15
Notes on the formulas:
B11:

pt  p0 exp   0.5 t  Z t
2

(random future DEO price)
B12:
C  MAXpt  E,0
(payoff from the option)
B15:
PVC   e
trf 
(present value of the payoff)
Decision Models -- Prof. Juran
16
A green cell: B10 is now a standard normal
random variable
A blue cell: B15 is now the present value of the
random cash flow from the option
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
B
Inputs
Current Price
Strike Price
Mean Annual Return
StDev Annual Return
Risk-Free Rate
Duriation (Years)
\$ 57.98
\$ 60.00
0.1047
0.2261
0.1
0.5
Simulation
Z
Stock Price (at Risk-Free Rate)
Option Cash Flow
0.0000
\$ 60.18
\$ 0.18
Output
Discounted Value
\$
Decision Models -- Prof. Juran
0.17
C
D
E
F
G
=data!E101
=data!E102
=B2*EXP((B6-0.5*B5^2)*B7+B5*B10*SQRT(B7))
=MAX(B11-B3,0)
=EXP(-B7*B6)*B12
17
Simulation Results
Decision Models -- Prof. Juran
18
OK, so simulation can do the same thing as
Black-Scholes.
Maybe Black-Scholes is easier and/or quicker
than running a simulation.
So, why do we need the simulation at all?
Decision Models -- Prof. Juran
19
• Black-Scholes is an analytical result; if specific
assumptions hold true, then we can calculate the
expected value of the payout on an option.
• Analytical solutions do not exist in general for all
types of financial instruments.
• In the absence of analytical results, Monte Carlo
simulation offers an alternative approach.
• Analytical solutions may exist for expected value, but
not for other important parameters.
Decision Models -- Prof. Juran
20
Example: Asian Option
George Brickfield’s business is highly exposed to volatility in
the cost of electricity.
He has asked his investment banker, Lisa Siegel, to propose an
option whereby he can hedge himself against changes in the
cost of a kilowatt hour of electricity over the next twelve
months.
Decision Models -- Prof. Juran
21
Lisa thinks that an Asian option would work nicely for George’s
situation.
An Asian option is based on the average price of a kilowatt
hour (or other underlying commodity) over a specified time
period.
Decision Models -- Prof. Juran
22
In this case, Lisa wants to offer George a one year Asian option
with a target price of \$0.059.
•If the average price per kilowatt hour over the next twelve
months is greater than this target price, then Lisa will pay
George the difference.
•If the average price per kilowatt hour over the next twelve
months is less than this target price, then George loses the price
he paid for the option (but he is happy, because he ends up
Decision Models -- Prof. Juran
23
What is a fair price for Lisa to charge for 1 million kwh worth
of these options?
Use the historical data provided and Monte Carlo simulation to
arrive at a fair price.
Decision Models -- Prof. Juran
24
Analysis of historical data: Our model will be based not
on the actual prices, but on monthly percent changes in
price (a.k.a. returns):
A
B
C
1 Month \$/kwh
Return
2
Jan \$ 0.051
3
Feb \$ 0.056 0.09804
4
Mar \$ 0.054 -0.03571
5
Apr \$ 0.052 -0.03704
6
May \$ 0.052 0.00000
7
Jun \$ 0.057 0.09615
8
Jul \$ 0.067 0.17544
9
Aug \$ 0.064 -0.04478
10
Sep \$ 0.064 0.00000
11
Oct \$ 0.058 -0.09375
12
Nov \$ 0.054 -0.06897
13
Dec \$ 0.057 0.05556
Decision Models -- Prof. Juran
D
E
F
mean
stdev
0.1768% 7.3462%
=(B3-B2)/B2
G
H
I
=STDEV(C2:C121)
=AVERAGE(C2:C121)
25
Histogram of Electricity Returns
30
25
Frequency
20
15
10
5
0
-0.200 -0.175 -0.150 -0.125 -0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300
Monthly Price Change
Decision Models -- Prof. Juran
26
• Returns are approximately normal.
• We’ll use the sample mean and sample standard deviation
from the data (0.001768 and 0.073462, respectively).
Decision Models -- Prof. Juran
27
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
A
Initial Electricity Price
Target Price
Mean monthly return
Std dev monthly return
# kwh per option
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
B
Return
1.13%
-0.21%
7.82%
-5.77%
6.54%
8.31%
-12.05%
9.71%
1.56%
6.84%
9.39%
-0.50%
Decision Models -- Prof. Juran
D
C
\$0.05684
\$0.05900
0.18%
7.35%
1,000,000
Price
0.05748
0.05736
0.06185
0.05828
0.06209
0.06725
0.05915
0.06489
0.06590
0.07041
0.07703
0.07664
E
F
G
Average Price
0.06486
=AVERAGE(C8:C26)
Payout
5,861.89
\$
=C5*MAX(E8-C2,0)
H
=C15*(1+B16)
28
In B8:B19 we have 12 Crystal Ball assumption cells, normally distributed with the
mean and standard deviation from our sample data (C3 and C4).
In C8:C19 we use the random percent returns to calculate monthly prices, which
are averaged in E8 for the whole year.
E11 calculates the payout on the option (a Crystal Ball forecast cell).
The average value of E11 over many trials will be a reasonable estimate of the
fair price for this option.
Decision Models -- Prof. Juran
29
We’ll add a graph, to show the change in electricity prices over the course of each
simulated year:
F
G
H
I
J
K
L
Price
0.12
0.11
0.09
0.08
0.07
0.06
0.05
0.04
ar
Ap
r
M
ay
Ju
n
Ju
l
Au
g
Se
p
O
ct
N
ov
D
ec
Payout
\$
5,861.89
0.10
b
Average Price
0.06486
M
Price
0.05748
0.05736
0.06185
0.05828
0.06209
0.06725
0.05915
0.06489
0.06590
0.07041
0.07703
0.07664
E
n
Return
1.13%
-0.21%
7.82%
-5.77%
6.54%
8.31%
-12.05%
9.71%
1.56%
6.84%
9.39%
-0.50%
C
D
\$0.05684
\$0.05900
0.18%
7.35%
1,000,000
Fe
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
B
Ja
A
Initial Electricity Price
Target Price
Mean monthly return
Std dev monthly return
# kwh per option
Price (\$/kwh)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Month
Decision Models -- Prof. Juran
30
Decision Models -- Prof. Juran
31
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A
B
Statistics
Payout
Trials
1000000
Base Case
\$5,861.89
Mean
\$2,927.16
Median
\$0.00
Mode
\$0.00
Standard Deviation
\$5,249.37
Variance
\$27,555,886.95
Skewness
2.31
Kurtosis
9.16
Coeff. of Variation
1.79
Minimum
\$0.00
Maximum
\$63,307.77
Range Width
\$63,307.77
Mean Std. Error
\$5.25
C
D
E
alpha
z-value
UCL
LCL
X
 1.96
\$2,927.16
 1.96
0.05
1.959964
\$2,937.45
\$2,916.87
F
G
H
=-NORMSINV(E2/2)
=\$B\$4+\$E\$3*\$B\$15
=\$B\$4-\$E\$3*\$B\$15
s
n
\$5,249.67
1,000,000
 1.96\$5.25
 \$10 .29
95% confident that the true fair price is between \$2,916.87 and \$2,937.45.
Could narrow the interval by running a longer simulation.
Decision Models -- Prof. Juran
32
Monte Carlo vs. Latin Hypercube
Decision Models -- Prof. Juran
33
Example: Standard normal distribution
(mean = 0, standard deviation = 1)
Divided into 8 equal-probability ranges
-4
-3
-2
Decision Models -- Prof. Juran
-1
0
1
2
3
4
34
Probability that the first random independent
observation falls into any one range is 0.125.
-4
-3
-2
Decision Models -- Prof. Juran
-1
0
*
1
2
3
4
35
Probability that the first two observations fall into
any one range is 0.1252 = 0.01563.
-4
-3
-2
Decision Models -- Prof. Juran
-1
0
*
*
1
2
3
4
36
Probability that the first three observations fall
into any one range is 0.1253 = 0.001953.
A small (but not zero!) chance of an
unrepresentative sample.
-4
-3
-2
Decision Models -- Prof. Juran
-1
0
*
*
*
1
2
3
4
37
Latin Hypercube ensures that each range gets one
observation before any range gets a second
observation (but with more than 8 ranges).
-4
-3
-2
* * * ** * * *
Decision Models -- Prof. Juran
-1
0
1
2
3
4
38
Monte Carlo vs. Latin Hypercube
A
1
2
3
4
5
Decision Models -- Prof. Juran
B
C
0
0
=B2
39
Decision Models -- Prof. Juran
40
Decision Models -- Prof. Juran
41
Decision Models -- Prof. Juran
42
Decision Models -- Prof. Juran
43
Decision Models -- Prof. Juran
44
Binomial Random Variable
Decision Models -- Prof. Juran
45
The number of successes in a Bernoulli process is a binomial random variable.
The probability that a binomial variable X will take on any particular value x is
given by the binomial formula:
P X  x n , p 
n
   p x 1  p n x
x

 x
n!
 p 1  p n x
 
 x! n  x ! 
If X is a Binomial(n, p) random variable, then
Expected Value:
Variance:
Standard Deviation:
Decision Models -- Prof. Juran
E(X) = np
Var(X) = np(1 - p)
(X) =
np( 1  p )
46
A Bernoulli process is a sequence of n identical trials of a random experiment
such that each trial:
(1) produces one of two possible complimentary outcomes that are
conventionally called success and failure and
(2) is independent of any other trial so that the probability of success or
failure is constant from trial to trial.
Note that the success and failure probabilities are assumed to be constant from trial to trial, but
they are not necessarily equal to each other.
Decision Models -- Prof. Juran
47
1
2
3
4 X
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
B
n
p
0
1
2
3
4
5
6
7
8
9
10
C
D
E
F
G
H
I
J
10
0.44
0.0030
0.0238
0.0843
0.1765
0.2427
0.2289
0.1499
0.0673
0.0198
0.0035
0.0003
=BINOMDIST(A5,\$C\$1,\$C\$2,0)
Binomial Distribution
0.3
0.3
Probability
A
0.2
0.2
0.1
0.1
0.0
Decision Models -- Prof. Juran
0
1
2
3
4
5
6
7
8
9
10
Number of "Successes"
48
Summary
Finance Simulation Models
• Securities Pricing
– Black-Scholes
– Electricity Option
• Miscellaneous
– Monte Carlo vs. Latin Hypercube
– Review of Binomial
Decision Models -- Prof. Juran
49
```