A SIMULATION EXAMPLE: RETURN ON A PORTFOLIO WITH A
STOCK AND AN OPTION ON THE STOCK
Objective Hierarchies:
An investor buys one share of AnTech stock at the current price ($42) and a European
call option on this stock for $4.76 (calculated using the Black-Scholes Model). The
option has an expiration date of 6 months with $40 as an exercise price. Also, the stock
price has an annual standard deviation of 20% with anticipated annual growth rate equal
to 15% per year. The risk-free rate is 10% per year.
The problem here is to use simulation techniques to find the return on the investor’s
portfolio as of the exercise date.
Hedge Portfolio
Value
Maximize
Option Value
Maximize
Stock Value
Variables, Attributes and Terms:
Option: A contract that gives the buyer the right to purchase or sell an asset from the
seller at a pre-agreed price by a specified date.
Call option is the right to purchase.
Put option is the right to sell.
Covered Call is the combination of selling a call on a stock and buying the stock.
Protective Put is the purchase of stock combined with a put option that guarantees
minimum proceeds equal to the put’s exercise price.
Exercise (or Strike) Price: The price at which the buyer of a call may purchase (or the
buyer of a put may sell) the commodity.
Expiration Date: The date on which an option contract expires.
European Option: An option that can only be exercised at expiration, unlike the
American option.
Premium: A term for the price of an option paid by the buyer to the option writer (issuer
of an option).
Risk-Free Rate: The rate for a similar maturity treasury instrument.
Standard Deviation: A measure of stock price volatility (in % terms).
Influence Diagram:
Portfolio Value
Stock Price
Growth
Rate
Volatility
Risk-Free
Rate
Options
Duration
Risk-Free
Rate
Strike
Price
Premium
Model and Mathematical Representation:
Mathematical Model
The Black-Scholes model, developed by Fischer Black and Myron Scholes is the premier
and widely used formula for determining option prices. The assumptions employed by
the model are:
1.
that asset prices adjust to prevent arbitrage opportunity.
2.
that both the risk-free rate and the stock price volatility are constant over the
life of the option.
3.
that stock returns follow a log-normal distribution.
4.
that the European call options have no dividends.
C = SN(d1) – Xe-rtN(d2)
Where,
d1=[Ln (S/X) + ( r + 2/2)T]/ T
d2 = d1 - T
and here,
C = current call price
S = current stock price
N(d) = cumulative standard normal density function
X = exercise or strike price
e = 2.71828
r = risk free rate
T = time to maturity
Ln = natural log function
 = standard deviation of the stock’s rate of return
Spreadsheet Model
Simulation is the use of computer models to imitate a real-life situation. techniques in
@Risk are used to simulate the behavior of the portfolio under the current portfolio
objective, that is to hedge our risk exposure in purchasing the AnTech stock and thus
limit our portfolio value exposure.
The Spreadsheet Model Development is as follows:
Inputs
Enter the inputs in the shaded cells to include the known price of the call option.
Future Stock Price
Create the stock price (randomly) in 6 months in cell B13 with the formula. This uses the
equation and the stock’s mean growth rate.
Option Cash Flow
Calculate the cash flow using the formula =MAX(FutPrice-ExerPrice, 0) in cell B14.
Portfolio Value
At expiration, the value of the portfolio will be worth the price of the stock plus cash low
from the option. This is calculated in cell B16 using the formula =SUM(FutPrice,
OptCFlow). In cells B17 and B18, calculate the amount we paid for the portfolio and its
return with the formulas =CurrPrice + OptPrice and =RISKOUTPUT() + (EndPortValPortCost)/PortCost.
Using @Risk and Summary Statistics
First, set the number of iterations to 10,000 and the number of simulations to 1. After
running @Risk, we obtain the values in the range B21 to B27. The mean return for this
portfolio is 9.44%. However, there is a lot of volatility. For instance, there is a 5%
probability that the portfolio will see a 24% decline or a 56% gain. Also the probability
for a positive return is about 59%, not very encouraging. The summary results generated
by @Risk are displayed on the spreadsheet.
The DSS Use and Implementation:
The DSS is displayed below.
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A
B
C
Return on a portfolio with stock and a call option on the stock
Input section
Current stock price
Exercise price
Mean annual return
StDev of annual return
Risk-free rate
Option duration (years)
Price of option (from Black-Scholes)
$42
$40
15%
20%
10%
0.5
$4.76
Simulation section
Stock price in 6 months (growing at stock's rate)
Option cash flow at termination
$44.821
$4.821
Ending value of portfolio
Initial cost
Return from portfolio
$49.641
$46.760
6.162%
D
E
F
Range names used:
CurrPrice - B4
ExerPrice - B5
MeanReturn - B6
Volatility - B7
RFRate - B8
Duration - B9
OptPrice - B10
FutPrice - B13
OptCFlow - B14
EndPortVal - B16
PortCost - B17
PortReturn - B18
Summary measures from @Risk (based on 10,000 iterations)
Mean return
9.439%
Stdev of return
25.546%
Min return
-45.848%
Max return
155.365%
5th percentile of return
-24.048%
95th percentile of return
56.362%
Probability of a positive return
0.591
The results seem to indicate this strategy of buying a stock and a call is risky, in that if
the stock price falls, she loses money on her stock value as well as well as his/her
premium spent on a worthless call option. However, the portfolio gains only with a rise
in the stock price.
The accuracy of part of the model was verified using the Option Software model to
determine the price of the call option. Our models indicate that the call price is $4.76
while the Option software quotes the price as $4.73 (which is close).
What-if, Sensitivity and Scenario Analyses
Various analyses were performed to test the validity of the DSS as well as the effects of
each variable on the price of the call. The results seem to be in line with the intuition
behind the valuation of options. Below are the relationships suggested by our model
which are in line with what we expect. For illustrative purposes, dividend and the
relation with put value (not tested) is also displayed below:
Relationship Between Call Price and Put Price with various Variables
Call
Put
S
+
-
X
+
T
+
+
r
+
-

+
+
Dividend
+
In addition, several variables were altered to see their effects on mean returns, standard
deviations, etc. For instance, with price, exercise price, mean annual return, duration
remaining the same, the change of standard deviation from 20% to 30%, risk-free rate
from 10% to 15%, and consequently, option price from $4.76 to $6.29, results in a 7.79%
mean return compared to 9.44%. Below is the output for our simulation after changing
standard deviation and risk-free rate to 30% and 15%, respectively.
Summary measures from @Risk (based on 10,000 iterations)
Mean return
7.794%
Stdev of return
36.117%
Min return
-46.333%
Max return
130.073%
5th percentile of return
-35.460%
95th percentile of return
74.964%
Probability of a positive return
0.500
Conclusion:
To hedge against losing portfolio value, one should engage in a protective put, that is
purchase a put for each share of stock purchased. In this strategy, the investor is covered
in both directions of stock price movements. This strategy decreases the downside risk at
the expense of the upside gains.