# Playing the Odds with Black-Scholes

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```							Playing the Odds with Black-Scholes

Portfolio Analysis and Simulation
for better Investing.
Marc Elpel
BA225 Derivatives and Risk Management
Abstract
Option analysis adds another dimension to investment management by accounting for
potential value of future conditions and opportunities. With automated numerical
analysis tools we are now able to provide complex portfolio simulations and analysis
in real time enabling us to optimize our portfolio positions.
The Black-Scholes equations provide a closed form solution to valuation of options
based on inputs including time, current and exercise price, risk free rate, and stock
volatility.
Binomial tree analysis uses a probability tree to derive the chances of a discrete set of
outcomes from a given point. An advantage of the binomial method over Black-
Scholes is the ability to account for intermediate transactions which may occur in
American options.
Monte Carlo analysis generates predictive outcomes through averaged random
simulations based on stock characteristics. Like the binomial model Monte Carlo
uses a branching structure, but through automation the number of levels may be
extended, and thousands of simulations may be run to provide mean and statistical
investment analysis.
An automated solution is presented for each analysis including Black-Scholes options
pricing, Binomial Tree analysis, and Monte Carlo simulation. By leveraging these
tools the user will be more informed prior to committing to investment decisions, and
can evaluate risk management alternatives based on the most recent market data.
Options – Risk Reduction
PUT Option
Prob.

1                                       10

CALL Option (Sell)
Prob.
Call – Sell the upside
potential for immediate cash

1                                       10

PUT Option   CALL Option (Sell)
Prob.

1                                       10
Black-Scholes Equations
Normally Distribution for Percent Returns
Log-Normal Distribution for stock price

c = S0 N (d1 ) − K e− rT N (d 2 )
p = K e−rT N (−d 2 ) − S0 N (−d1 )   -3.00   -2.00   -1.00   0.00   1.00   2.00   3.00

ln(S0 / K ) + (r + σ 2 / 2)T
where d1 =                                     1. TIME

σ T                      2. STOCK PRICE
3. OPTION PRICE
4. RISK-FREE RATE
d 2 = d1 − σ T                        5. VOLATILITY
Binomial Tree Analysis
Monte Carlo Simulations
Random walk based on
real stock parameters.
Average of Multiple
simulations provides
expected return and
distribution
Monte Carlo Stock Simulation                                              Monte Carlo Stock Histogram

80                                                                                 300
70                                                                                 250
60                                                                                 200
Stock Price

50                                                                                 150
40                                                                                 100
30                                                                                  50
20                                                                                   0
10
0
.00

4

5

6

0

6

3

6

1

2
1

2

2

0

4

\$1 3
.66
4.4

6.9

2.7

3.9

3.3

8.8

4.8

1.5

6.9
9.7

6.1

9.8

8.4

8.8

5.8
\$0

05
1
19

37

55
73

91
109

127

145
163

181

199
217

235

\$2

\$2

\$3

\$4

\$5

\$5

\$6

\$7

\$8
\$2

\$3

\$3

\$4

\$7

\$9
Stock Hist.
Portfolio Analysis introduction
Sample Portfolios
\$ 3,000.00                                                                                                                                    \$ 5.00           \$ 400.00

\$ 200.00
\$ 0.00
\$ 2,000.00

\$ 0.00
(\$ 5 . 0 0 )                \$ 10.00             \$15.00                 \$ 20.00            \$ 25.00             \$ 30.00              \$ 35.00             \$ 40.00
\$ 1,000.00
(\$ 200.00)

(\$ 10.00)

\$ 0.00                                                                                                                                                    (\$ 400.00)

\$ 0.00   \$ 5.00      \$ 10.00             \$ 15.00       \$ 20.00   \$ 25.00   \$ 30.00       \$ 35.00             \$40.00      \$ 45.00
(\$ 15.00)

-p(20) + -c(25)
(\$ 600.00)

(\$ 1,000.00)

(\$ 20.00)
(\$ 800.00)

(\$ 2,000.00)
(\$ 25.00)      (\$ 1,000.00)

(\$ 3,000.00)
S0 + p(22.5) – c(22.5)                                                                                     (\$ 30.00)      (\$ 1,200.00)

\$ 100.00                                                                                                                                                     \$ 300.00

\$ 50.00                                                                                                                                                    \$ 200.00

\$ 0.00                                                                                                                                                    \$ 100.00

\$ 15.00          \$ 17.00             \$ 19.00             \$21.00        \$ 23.00       \$ 25.00             \$ 27.00            \$ 29.00

(\$ 50.00)                                                                                                                                                      \$ 0.00

\$ 15.00        \$ 17.00      \$ 19.00   \$ 21.00             \$23.00   \$ 25.00   \$ 27.00        \$ 29.00   \$ 31.00         \$ 33.00   \$ 35.00

(\$ 100.00)                                                                                                                                                   (\$ 100.00)

(\$ 150.00)                                                                                                                                                   (\$ 200.00)
c(20) – c(25)
(\$ 200.00)                                                                                                                                                   (\$ 300.00)

(\$ 250.00)
-c(20) + 2*c(22.50) -c(25)                                                                                                   (\$ 400.00)

stock at options

\$10,000 Portfolio
Choose only from listed

Max \$1,000 options Sold
Portfolio Competition

4/2
8/
20                                                                             4/2
05

\$35.00
\$40.00
\$45.00
\$50.00
\$55.00
\$60.00
\$65.00

8/2
5/5                                                                                        00                         4/2
5
\$35.00
\$40.00
\$45.00
\$50.00
\$55.00
\$60.00
\$65.00

/20                                                                                                                  8/2
5/5                                    00
5
\$35.00
\$40.00
\$45.00
\$50.00
\$55.00
\$60.00
\$65.00

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Option
Models
Online

http://www.sunsetmaui.com/finance_models.htm

```
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