OptionsIQ
Montgomery Investment Technology, Inc.
Financial Modeling Software and Consulting
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Contents
Contents:
BASIC TOPICS:
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ADVANCED TOPICS:
Option Contracts Defined
Intrinsic and Time Value
Type of Options
Theoretical or Fair Value
Volatility
The Black-Scholes Option
Pricing Model
The Binomial Method of Pricing
Options
Risk Sensitivities
Delta Hedging
Trading Strategies Involving
Options
Exotic Options
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The Markov Process and the
Efficient Market
Hurst Exponent to Measure
Trend in the Time Series
Normality of the Return Rates
Log-Normality of the Stock
Prices
Autocorrelation of the Return
Rates
Ito and Stratonovich
Interpretations
The Risk-Free Interest Rate vs.
Return Rate
Contents
Option Contracts Defined

An option is a derivative security which gives the holder the
right, but not the obligation, to buy or sell an underlying
asset by a certain date for a certain price.
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A call option gives the holder the right to buy, while a put
option gives the right to sell the underlying asset. The price
designated in the contract is known as the exercise or
strike price. The time to expiration is calculated based on
the time from the value date to the expiration date.
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American-style options can be exercised at anytime up to
expiration, while European-style options can only be
exercised at expiration.
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Option Contracts Defined
Question 1:
An option gives the holder...
A. the obligation to buy the underlying asset.
B. the right to sell the underlying asset.
C. the obligation to buy or sell the underlying
asset.
D. the right to buy or sell the underlying asset.
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Option Contracts Defined
Question 1: Answer D
An option gives the holder...
A. the obligation to buy the underlying asset.
B. the right to sell the underlying asset.
C. the obligation to buy or sell the underlying
asset.
D. the right to buy or sell the underlying asset.
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Option Contracts Defined
Question 2:
What is the difference between an American
option and a European option?
A. An American option is traded on American exchanges, while
European options are traded on European exchanges.
B. An American option is written on an the assets of an
American company, while a European option is written on the
assets of a European company.
C. An American option can only be exercised at expiration,
while a European option can be exercised at anytime up to
expiration.
D. An American option can be exercised at anytime up to
expiration, while a European option can be exercised only at
expiration.
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Option Contracts Defined
Question 2: Answer D
What is the difference between an American
option and a European option?
A. An American option is traded on American exchanges, while
European options are traded on European exchanges.
B. An American option is written on an the assets of an
American company, while a European option is written on the
assets of a European company.
C. An American option can only be exercised at expiration,
while a European option can be exercised at anytime up to
expiration.
D. An American option can be exercised at anytime up to
expiration, while a European option can be exercised only
at expiration.
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Intrinsic and Time Value

The premium, or price of an option, can be divided into two
components, intrinsic and time value. Intrinsic value is the
payoff if the option were to be exercised immediately.
Intrinsic value is always greater than or equal to zero. For
example, a certain asset is trading for $30. The intrinsic
value of a $25 call is therefore $5.
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Usually the price of an option in the marketplace will be
greater than its intrinsic value. The difference between the
market value of an option and its intrinsic value is called
the time value (or extrinsic value) of an option. An option is
trading at parity when the price of the option is equal to its
intrinsic value (the time value is zero).
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Intrinsic and Time Value
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An option which has a positive intrinsic value is considered
to be in-the-money by the amount of the intrinsic value. If a
stock is trading at $50, a $40 call is $10 in-the-money. A $50
call on the same stock would be considered to be at-themoney, and a $55 call would be considered to be out-of-themoney.
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Intrinsic and Time Value
Question 1:
What is the intrinsic value of a $30 put if the
underlying asset is trading at $23?
A. zero
B. $7
C. -$7
D. There is not enough information to determine the
intrinsic value.
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Intrinsic and Time Value
Question 1: Answer B
What is the intrinsic value of a $30 put if the
underlying asset is trading at $23?
A. zero
B. $7
C. -$7
D. There is not enough information to determine the
intrinsic value.
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Intrinsic and Time Value
Question 2:
The underlying asset of a $55 call is trading at
$48. This call would be considered...
A. In-the-money.
B. At-the-money.
C. Out-of-the-money.
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Intrinsic and Time Value
Question 2: Answer C
The underlying asset of a $55 call is trading at
$48. This call would be considered...
A. In-the-money.
B. At-the-money.
C. Out-of-the-money.
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Intrinsic and Time Value
Question 3:
A certain underlying asset is trading at $50. A
$42 call is trading in the marketplace for $10.
What is the time value of the option?
A. zero
B. $8
C. $2
D. $10
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Intrinsic and Time Value
Question 3: Answer C
A certain underlying asset is trading at $50. A
$42 call is trading in the marketplace for $10.
What is the time value of the option?
A. zero
B. $8
C. $2
D. $10
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Types of Options

A person who has purchased an option is considered to be
long the option (owner). A person who has sold the option
is considered short the option (seller). However, these
terms (long and short) are also used to refer to a position in
the market. A person who thinks that the market will rise (or
is bullish about the market) will make a long market
position (buy a call or sell a put), while a person who thinks
that the market will decline (or is bearish about the market)
will make a short market position (sell a call or buy a put).
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Types of Options
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The owner of a call (put) benefits from price increases
(decreases) in the stock with limited downside risk. The
most that can be lost is the price or premium of the option.
However, the seller of a call (put) benefits from price
decreases (increases) with unlimited loss potential and
limited gains. The most that can be gained is the premium
of the option.
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Types of Options
Question 1:
An investor is bullish about the market for a
particular underlying asset. Which of the
following strategies might this investor
pursue?
A. Long a put.
B. Short a put.
C. Short a call.
D. None of the above.
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Types of Options
Question 1: Answer B
An investor is bullish about the market for a
particular underlying asset. Which of the
following strategies might this investor
pursue?
A. Long a put.
B. Short a put.
C. Short a call.
D. None of the above.
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Types of Options
Question 2:
A trader is bearish about the market for a
certain underlying asset. Which of the
following strategies might this trader pursue?
A. Long a put.
B. Short a put.
C. Long a call.
D. Short a call.
E. A & D.
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Types of Options
Question 2: Answer E
A trader is bearish about the market for a
certain underlying asset. Which of the
following strategies might this trader pursue?
A. Long a put.
B. Short a put.
C. Long a call.
D. Short a call.
E. A & D.
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Theoretical or Fair Value
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The theoretical or fair value of an option is the price one
would expect to pay in order to just break even in the long
run. Theoretical value can be thought of as the "production
cost" of the option.
The cost of purchasing an option in the marketplace is
called the option premium. This amount is often different
from the theoretical value.
There are several characteristics involved in the pricing of
an option. They include: underlying price, exercise price,
amount of time remaining until expiration, the volatility of
the underlying asset, the risk-free rate of interest over the
life of the option, and the dividend yield rate of the asset.
There are several models available to price options using
these characteristics. The two most commonly used
models are the Black-Scholes and the Binomial models.
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Theoretical or Fair Value
Question 1:
What is the theoretical value of an option?
A. An option value generated by a mathematical
model given certain assumptions
B. The price one would expect to pay for an option to
just break even in the long run.
C. The option value determined by inputting values
into the Black-Scholes model.
D. All of the above.
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Theoretical or Fair Value
Question 1: Answer D
What is the theoretical value of an option?
A. An option value generated by a mathematical
model given certain assumptions
B. The price one would expect to pay for an option to
just break even in the long run.
C. The option value determined by inputting values
into the Black-Scholes model.
D. All of the above.
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Volatility

Volatility is one of the key inputs to an option pricing
model. Volatility is the degree to which the price of an
underlying asset tends to fluctuate over time. More
generally, it is a measure of how uncertain we are about
future stock price movements. If an underlying asset has a
small volatility or price variability, then an option on that
asset would not have much value to the holder.
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Volatility
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There are several different types of volatility. Future or
projected volatility is based on the expected future
distribution of prices for a particular underlying asset.
Implied volatility is calculated based on the option price
traded in the marketplace. It is the volatility which would
have to be input into a theoretical pricing model in order to
yield a theoretical value equal to the market value of the
option. Historical volatility is calculated based on a range of
historical prices. At lease 20 observations are usually
desirable to ensure statistical significance. Seasonal
volatility comes into affect with certain commodities, for
example as a consequence of changes in weather
conditions or demand.
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Volatility
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n order to estimate the volatility of an underlying asset
using historical data (e.g. daily, weekly, monthly), the
following formula can be used:
Definitions:
n : number of observations
S(i) : stock price at the end of the ith interval (i=0,1,2,...,n)
T : length of time interval in years
s* : the standard deviation of s (volatility)
u(i)= ln (S(i)/S(i-1) for i= 0,1,2,...,n
n
n
s = [ {1/(n-1)} E [ u(i)]^2 - {1/n(n-1)} {E u(i) }^2 ]^1/2
i=1
i=1
s*= s/(T^1/2)
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Volatility
Question 1:
What is volatility?
A. It is the degree to which an asset price fluctuates
over time.
B. It is a measure of speed of the market.
C. It is a measure of uncertainty of future stock price
distributions.
D. All of the above.
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Volatility
Question 1: Answer D
What is volatility?
A. It is the degree to which an asset price fluctuates
over time.
B. It is a measure of speed of the market.
C. It is a measure of uncertainty of future stock price
distributions.
D. All of the above.
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Volatility
Question 2:
Given E u(i) = 0.09531 and E [u(i)]^2 =0.00333 for a
twenty trading day period, assume that there are
252 trading days per year. What is the approximate
volatility per year?
A. 10%
B. 20%
C. 30%
D. 40%
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Volatility
Question 2: Answer B
Given E u(i) = 0.09531 and E [u(i)]^2 =0.00333 for a
twenty trading day period, assume that there are
252 trading days per year. What is the approximate
volatility per year?
A. 10%
B. 20%
C. 30%
D. 40%
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The Black-Scholes Option Pricing Model
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The Black-Scholes model is one of the most basic pricing
models used. It is designed for use with European options
on non-dividend paying stocks. The following is the
mathematical equation for the Black-Scholes model.
C
=
P
=
U
=
E
=
t
=
v
=
r
=
e
=
ln
=
N'(x) =
N(x) =
theoretical value of a call
theoretical value of a put
price of the underlying asset
exercise price
time to expiration in years
annual volatility expressed as a decimal fraction
risk-free interest expressed as a decimal fraction
base of the natural logarithm
natural logarithm
the normal distribution curve
the cumulative normal density function
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The Black-Scholes Option Pricing Model
C = UN(h) - Ee^(-rt) N (h-v(t^1/2))
N(x) = 1 - N'(x) (a1*k + a2*k^2 + a3*k^3)
1 - N (-x)
x>0
x<0
P = -UN(-h) + Ee(-rt) N (v(t^1/2) -h)
h = [ ln(U/E) + (r + (v^2)/2) t]
[ v(t^1/2)]
N'(x) = [e^(-(x^2)/2)] / (2 Pi)^1/2
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k = 1 / (1 + yx)
y = 0.33267
a1 = 0.4361836
a2 = -0.1201676
a3 = 0.9372980
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The Black-Scholes Option Pricing Model
Question 1:
What is the approximate theoretical value (using the
Black-Scholes pricing model) of a call if U=50,
E=45,t=1, v=30%, and r=5%?
A. $10.
B. $12
C. $15
D. None of the above.
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The Black-Scholes Option Pricing Model
Question 1: Answer A
What is the approximate theoretical value (using the
Black-Scholes pricing model) of a call if U=50,
E=45,t=1, v=30%, and r=5%?
A. $10.
B. $12
C. $15
D. None of the above.
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The Black-Scholes Option Pricing Model
Question 2:
What is the approximate theoretical value (using the
Black-Scholes pricing model) of a call if U=80,
E=65,t=.5, v=25%, and r=6%?
A. $15.
B. $17
C. $20
D. $25
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The Black-Scholes Option Pricing Model
Question 2: Answer B
What is the approximate theoretical value (using the
Black-Scholes pricing model) of a call if U=80,
E=65,t=.5, v=25%, and r=6%?
A. $15.
B. $17
C. $20
D. $25
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The Binomial Method of Pricing Options
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The Binomial method, detailed in a 1979 journal article by
Cox, Ross and Rubinstein, can be used to accurately value
American-style options. The "open architecture" allow
flexibility in relaxing some of the constraints that are
imposed by the Black-Scholes model. The binomial method
is used extensively to price both standard and nonstandard option contracts. The degree of accuracy can be
specified by selecting a desired number of nodes or
"iterations". Discrete cash flows or dividend yields may be
incorporated into the option calculation when using the
binomial or lattice approach.
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The Binomial Method of Pricing Options
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In order to provide the binomial tree which will
approximate the lognormal distribution, the following is
defined:
u = e^{v(t/n)^1/2}
d = 1/u
where:
n = the number of periods to expiration (number of branches
of the binomial tree)
v = the annual volatility of the underlying asset
t = the time to expiration in years
j = the underlying price (from 0 - n)
i = the period (from 0 - n)
rr = the risk free rate of interest over the life of the option
defined as rr = 1 + (rt / n)
p = the probability defined as p = (rr-d) / (u-d)
E = the exercise price
U = the underlying asset price
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The Binomial Method of Pricing Options
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C(i , j) = max [ pC (i + 1, j) + (1 - p) C (i + 1, j + 1) ]
[ rr ' U (i , j) – E ]
P(i , j) = max [ pP (i + 1, j) + (1 - p) P (i + 1, j + 1) ]
[ rr ' E - U (i , j) ]
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The Binomial Method of Pricing Options
Question 1:
The binomial method is often used to calculate option
values when:
A. the exercise style is American.
B. discrete cash flows are generated by the
underlying asset.
C. speed of recalculation is not the overriding factor.
D. all of the above.
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The Binomial Method of Pricing Options
Question 1: Answer D
The binomial method is often used to calculate option
values when:
A. the exercise style is American.
B. discrete cash flows are generated by the
underlying asset.
C. speed of recalculation is not the overriding factor.
D. all of the above.
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The Binomial Method of Pricing Options
Question 2:
What is the approximate theoretical value of a put
option with the following properties: U=85, E=82,
t=1,v=35%, n=6 and ri=7%?
A. $8
B. $12
C. $16
D. $20
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The Binomial Method of Pricing Options
Question 2: Answer A
What is the approximate theoretical value of a put
option with the following properties: U=85, E=82,
t=1,v=35%, n=6 and ri=7%?
A. $8
B. $12
C. $16
D. $20
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Risk Sensitivities (Greeks)
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Delta (also known as the hedge ratio) is the sensitivity of an
option's theoretical value to a change in the price of the
underlying contract. Calls have deltas ranging from zero to
one hundred. Puts have deltas ranging from zero to
negative one hundred. An underlying contract always has a
delta of one hundred
delta = change in the option price
change in the stock price
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Risk Sensitivities (Greeks)
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The Gamma of a portfolio of derivatives on an underlying asset is
the rate of change of the portfolio's delta with respect to the price
of the underlying asset. If gamma is large, delta is highly sensitive
to the price of the underlying asset.
gamma = change in the value of the portfolio * change in time
change in stock price
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The Theta of a portfolio of derivatives is the rate of change of the
value of the portfolio with respect to time with all else remaining
the same. It is sometimes referred to as the time decay of the
portfolio.
theta = change in the value of the portfolio
change in time
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Risk Sensitivities (Greeks)
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The Vega of a portfolio of derivatives is the rate of change of the
value of the portfolio with respect to the volatility of the
underlying asset. Vega may also be known as lambda, kappa, or
sigma.
vega =
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change in the value of the portfolio
change in the volatility of the underlying asset
The Rho of a portfolio of derivatives is the rate of change of the
value of the porfolio to the interest rate. It is a measure of the
sensitivity of the portfolio's value to interest rates.
rho =
change in the value of the porfolio
change in interest rates
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Risk Sensitivities (Greeks)
Question 1:
Delta is a measure of:
A. time decay of the portfolio.
B. the sensitivity of the portfolio's value to interest
rates.
C. the sensitivity of an option's value to a change in
the price of the underlying asset.
D. none of the above..
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Risk Sensitivities (Greeks)
Question 1: Answer C
Delta is a measure of:
A. time decay of the portfolio.
B. the sensitivity of the portfolio's value to interest
rates.
C. the sensitivity of an option's value to a change
in the price of the underlying asset.
D. none of the above..
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Risk Sensitivities (Greeks)
Question 2:
Rho is a measure of:
A. time decay of the portfolio.
B. the sensitivity of the portfolio's value to interest
rates.
C. the sensitivity of an option's value to a change in
the price of the underlying asset.
D. the rate of change of the value of the portfolio with
respect to the volatility.
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Risk Sensitivities (Greeks)
Question 2: Answer B
Rho is a measure of:
A. time decay of the portfolio.
B. the sensitivity of the portfolio's value to
interest rates.
C. the sensitivity of an option's value to a change in
the price of the underlying asset.
D. the rate of change of the value of the portfolio with
respect to the volatility.
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Delta Hedging

Delta hedging is a hedging technique that attempts to make
portfolios immune to small changes in the price of the
underlying asset for short intervals of time. In order to form
a delta neutral hedge, a position with total delta being equal
to zero, one must buy or sell options and shares of stock
such that the delta positions cancel one another out.
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For example, buying 2000 put options, each with a delta of .5 and buying 1000 shares of the stock would create a delta
neutral hedge. In order to maintain this hedge, continual
adjusting of the position must occur. This is called
rebalancing. Hedging schemes that require continual
rebalancing are called dynamic hedging schemes.
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Delta Hedging
Question 1:
If an investor is long 500 shares of stock, which of the
following will create a delta neutral hedge:
A. if the investor goes long on 1000 puts, each with a
delta of -.4.
B. if the investor goes short on 1000 calls, each with
a delta of .5.
C. if the investor goes long on 1000 calls, each with a
delta of .5.
D. if the investor goes short on 1000 puts, each with
a delta of -.4..
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Delta Hedging
Question 1: Answer B
If an investor is long 500 shares of stock, which of the
following will create a delta neutral hedge:
A. if the investor goes long on 1000 puts, each with a
delta of -.4.
B. if the investor goes short on 1000 calls, each
with a delta of .5.
C. if the investor goes long on 1000 calls, each with a
delta of .5.
D. if the investor goes short on 1000 puts, each with
a delta of -.4.
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Delta Hedging
Question 2:
If an investor is short 200 shares of stock, which of the
following with create a delta neutral hedge:
A. if the investor goes short on 400 puts, each with a
delta of -.5.
B. if the investor goes long 200 calls, each with a
delta of .5 and goes short on 300 puts, each with
a delta of -.33.
C. if the investor goes long on 500 calls, each with a
delta of .4.
D. all of the above.
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Delta Hedging
Question 2: Answer D
If an investor is short 200 shares of stock, which of the
following with create a delta neutral hedge:
A. if the investor goes short on 400 puts, each with a
delta of -.5.
B. if the investor goes long 200 calls, each with a
delta of .5 and goes short on 300 puts, each with
a delta of -.33.
C. if the investor goes long on 500 calls, each with a
delta of .4.
D. all of the above.
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Trading Strategies Involving Options

There are several strategies that involve the use of options.
An investor can have a long position in a stock and a short
position in a call. This is known as writing a covered call.
The reverse of this is having a short position in a stock and
having a long position in a call. A protective put involves
buying a put and the stock itself. The reverse of this
involves going short in a put and the stock itself.
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Trading Strategies Involving Options

There are several strategies that involve the use of more
than one kind of option at a time. These strategies are
known as spreads. A bull spread is created by buying a call
option with a certain strike price and selling a call option
with a higher strike price. A bear spread is created by
buying a call option with a certain strike price and selling a
call with a lower strike price. A butterfly spread involves
positions in options with three different strike prices. A call
option with a relatively low strike price is purchased, a call
option with a relatively high strike price is purchased, and
two call options with a strike price halfway between the
other two are sold. This strategy only leads to profits if the
price of the stock doesn't change significantly. A calendar
spread can be created by selling a call option with a certain
strike price and maturity and buying a call option with the
same strike price but with a longer maturity.
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Trading Strategies Involving Options

There are also strategies known as combinations that
involve taking positions in calls and puts. One such
strategy is called a straddle. This involves buying a call and
a put with the same strike price and expiration date. A strip
involves a long position in one call and two puts with the
same strike price and expiration date. A strap involves a
long position in two calls and one put with the same strike
price and expiration date. A strangle, also called a bottom
vertical combination, involves purchasing a call and a put
with the same expiration date and different strike prices.
The call strike price is higher than the put strike price.
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Trading Strategies Involving Options
Question 1:
The trading strategy that involves the purchase of a call
and a put on the same underlying asset at the same
strike price and same expiration date is known as:
A. a strip.
B. a straddle.
C. a strangle.
D. a bottom vertical combination.
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Trading Strategies Involving Options
Question 1: Answer B
The trading strategy that involves the purchase of a call
and a put on the same underlying asset at the same
strike price and same expiration date is known as:
A. a strip.
B. a straddle.
C. a strangle.
D. a bottom vertical combination.
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Trading Strategies Involving Options
Question 2:
The trading strategy that consists of buying a call
option with a certain strike price and selling a call
option with a higher strike price is called:
A. a calendar spread.
B. a bull spread.
C. a protective put.
D. a butterfly spread.
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Trading Strategies Involving Options
Question 2: Answer B
The trading strategy that consists of buying a call
option with a certain strike price and selling a call
option with a higher strike price is called:
A. a calendar spread.
B. a bull spread.
C. a protective put.
D. a butterfly spread.
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Exotic Options

A Bermudan option is a type of non-standard American option in
which early exercise is limited to certain dates during the life of
the option. Also referred to as "hybrid-style" exercise.

A forward start option is an option that is paid for now, but does
not begin until some later date.

A compound option is an option on an option. Compound
options have two strike prices and two expiration dates. For
example, a call on a call is purchased. At some specified date in
the future, a person will have the right but not the obligation of
purchasing a call option.
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Exotic Options

A chooser option, also called an "as you like it" option, allows
the holder to choose after a specified period of time whether the
option is a call or a put.

A barrier option is an option in which the payoff depends on
whether the underlying asset's price reaches a certain level
during the life of the option. An up-and-out option becomes
worthless once the underlying asset price reaches a specified
boundary price. An up-and-in option requires the underlying
asset price to reach the boundary price before the option can be
activated.

A rainbow option is an option involving two or more risky assets.
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Exotic Options




A lookback option is an option whose payoffs depend on the
maximum or minimum the stock price has reached over the life
of the option.
An Asian option, also called an average price option, is an
option whose payoff depends on the average price of the
underlying asset (rather than the stock price itself) over some
specified amount of time during the life of the option.
A spread option is an option whose strike price is the spread
between two underlying assets. For example, there are crack
spreads on the spread between the price of crude oil and its
resulting by-products.
A basket option is an option whose payoff depends upon a
portfolio of assets.
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Exotic Options
Question 1:
An option whose payoff depends upon whether the
underlying asset price has reached a certain
boundary over the life of the option is called:
A. a chooser option.
B. an average price option.
C. a barrier option.
D. a bermudan option.
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Exotic Options
Question 1: Answer C
An option whose payoff depends upon whether the
underlying asset price has reached a certain
boundary over the life of the option is called:
A. a chooser option.
B. an average price option.
C. a barrier option.
D. a bermudan option.
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Exotic Options
Question 2:
An example of a "spread" option is:
A. an Employee Stock "out-performance" option.
B. a NYMEX "crack" spread option.
C. an OTC Motorola vs. Intel option.
D. all of the above.
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Exotic Options
Question 2: Answer D
An example of a "spread" option is:
A. an Employee Stock "out-performance" option.
B. a NYMEX "crack" spread option.
C. an OTC Motorola vs. Intel option.
D. all of the above.
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The Markov Process and the Efficient Market

A Markov process implies that the probability distribution
of the stock price at the next moment is a function of the
current stock price, only; previous stock prices are
irrelevant. The Markov process is consistent with the
efficient market hypothesis. An efficient market means that
the present stock price impounds all the information
contained in a record of past prices. If the market were not
efficient, then investors could make above-average returns
by interpreting the past history of the stock prices. The
competition among investors and the regulations of the
marketplace tend to ensure the efficiency of the market.
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The Markov Process and the Efficient Market
Question 1:
An efficient market is:
A. a market that returns on the average 20% or
more.
B. a market that returns on the average 0%.
C. a market where future evolution of the stock price
is determined by the present value of the stock price,
past price history being irrelevant.
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The Markov Process and the Efficient Market
Question 1: Answer C
An efficient market is:
A. a market that returns on the average 20% or
more.
B. a market that returns on the average 0%.
C. a market where future evolution of the stock
price is determined by the present value of the
stock price, past price history being irrelevant.
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The Markov Process and the Efficient Market
Question 2:
A market in order to be efficient requires:
A. A large influx of capital.
B. Immediate access to information
("Transparency").
C. Competition.
D. Over-regulation.
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The Markov Process and the Efficient Market
Question 2: Answer B & C
A market in order to be efficient requires:
A. A large influx of capital.
B. Immediate access to information
(“Transparency").
C. Competition.
D. Over-regulation.
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The Markov Process and the Efficient Market
Question 3:
A market that is not efficient can be modeled by a
Markov process.
A. True
B. False
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The Markov Process and the Efficient Market
Question 3: Answer B
A market that is not efficient can be modeled by a
Markov process.
A. True
B. False
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The Markov Process and the Efficient Market
Question 4:
The Black-Scholes model can be applied when:
A. the market is efficient.
B. the market is inefficient.
C. the market is either efficient or inefficient.
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The Markov Process and the Efficient Market
Question 4: Answer A
The Black-Scholes model can be applied when:
A. the market is efficient.
B. the market is inefficient.
C. the market is either efficient or inefficient.
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Hurst Exponent

For a random walk process there is no correlation between
past and future increments and the Hurst exponent is 0.5 . A
Hurst exponent greater than 0.5 indicates persistence in the
data: the trend in the time series (either increasing or
decreasing) will likely continue and therefore the next event
is more likely to repeat the present event (i.e., up follows
up, down follows down). A Hurst exponent less than 0.5
indicates antipersistence in the data: the trend will likely
reverse itself and therefore the next event is less likely to
repeat the present event (i.e., up follows down, down
follows up).
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Hurst Exponent

The efficient market hypothesis implies a random walk in
terms of return rates. Therefore, the Hurst exponent can be
used as a check of the efficient market validity. When the
Hurst exponent is significantly different with respect to 0.5,
it may be possible to estimate how long the market memory
is. An efficient market has zero memory.
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Hurst Exponent
Question 1:
The Black-Scholes model can be applied when:
A. the Hurst exponent is greater than 0.5.
B. the Hurst exponent is 0.5.
C. the Hurst exponent is less than 0.5.
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Hurst Exponent
Question 1: Answer B
The Black-Scholes model can be applied when:
A. the Hurst exponent is greater than 0.5.
B. the Hurst exponent is 0.5.
C. the Hurst exponent is less than 0.5.
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Hurst Exponent
Question 2:
The Markov model can be used when:
A. the Hurst exponent is greater than 0.5
B. the Hurst exponent is 0.5.
C. the Hurst exponent is less than 0.5.
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Hurst Exponent
Question 2: Answer B
The Markov model can be used when:
A. the Hurst exponent is greater than 0.5
B. the Hurst exponent is 0.5.
C. the Hurst exponent is less than 0.5.
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Normality of the Return Rates

The return rates, according to the Black-Scholes model,
follow a Gaussian white noise stochastic process, i.e. with
zero expectation and the stationary normalized
autocorrelation function given by the "Dirac delta function".
The return rates should be computed for each date, and the
corresponding time series should be checked for normality.

Among the different statistical tests available, we
recommend the D'Agostino tests for skewness, for
kurtosis, and omnibus test. Other tests seem to be less
powerful. Departure from normality can significantly effect
the predictions of the Black-Scholes model.
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Normality of the Return Rates
Question 1:
If the distribution is skewed to the left:
A. Black-Scholes overprices out-of-the-money calls and in-themoney puts. It underprices out-of-the-money puts and in-themoney calls.
B. Black-Scholes overprices out-of-the-money puts and in-themoney calls. It underprices in-the-money puts and out-of-themoney calls.
C. Black-Scholes underprices out-of-the-money and in-themoney calls and puts.
D. Black-Scholes overprices out-of-the-money and in-the-money
calls and puts.
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Normality of the Return Rates
Question 1: Answer A
If the distribution is skewed to the left:
A. Black-Scholes overprices out-of-the-money calls and inthe-money puts. It underprices out-of-the-money puts and
in-the-money calls.
B. Black-Scholes overprices out-of-the-money puts and in-themoney calls. It underprices in-the-money puts and out-of-themoney calls.
C. Black-Scholes underprices out-of-the-money and in-themoney calls and puts.
D. Black-Scholes overprices out-of-the-money and in-the-money
calls and puts.
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Normality of the Return Rates
Question 2:
If the distribution is skewed to the right:
A. Black-Scholes overprices out-of-the-money puts and in-themoney calls. It underprices in-the-money puts and out-of-themoney calls.
B. Black-Scholes overprices out-of-the-money calls and in-themoney puts. It underprices out-of-the-money puts and in-themoney calls.
C. Black-Scholes underprices out-of-the-money and in-themoney calls and puts.
D. Black-Scholes overprices out-of-the-money and in-the-money
calls and puts.
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Normality of the Return Rates
Question 2: Answer A
If the distribution is skewed to the right:
A. Black-Scholes overprices out-of-the-money puts and inthe-money calls. It underprices in-the-money puts and outof-the-money calls.
B. Black-Scholes overprices out-of-the-money calls and in-themoney puts. It underprices out-of-the-money puts and in-themoney calls.
C. Black-Scholes underprices out-of-the-money and in-themoney calls and puts.
D. Black-Scholes overprices out-of-the-money and in-the-money
calls and puts.
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Normality of the Return Rates
Question 3:
If the distribution is leptokurtic:
A. Black-Scholes underprices out-of-the-money and in-themoney calls and puts.
B. Black-Scholes overprices out-of-the-money puts and in-themoney calls. It underprices in-the-money puts and out-of-themoney calls.
C. Black-Scholes overprices out-of-the-money calls and in-themoney puts. It underprices out-of-the-money puts and in-themoney calls.
D. Black-Scholes overprices out-of-the-money and in-the-money
calls and puts.
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Normality of the Return Rates
Question 3: Answer A
If the distribution is leptokurtic:
A. Black-Scholes underprices out-of-the-money and in-themoney calls and puts.
B. Black-Scholes overprices out-of-the-money puts and in-themoney calls. It underprices in-the-money puts and out-of-themoney calls.
C. Black-Scholes overprices out-of-the-money calls and in-themoney puts. It underprices out-of-the-money puts and in-themoney calls.
D. Black-Scholes overprices out-of-the-money and in-the-money
calls and puts.
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Normality of the Return Rates
Question 4:
If the distribution is platikurtic:
A. Black-Scholes overprices out-of-the-money and in-the-money
calls and puts.
B. Black-Scholes overprices out-of-the-money puts and in-themoney calls. It underprices in-the-money puts and out-of-themoney calls.
C. Black-Scholes overprices out-of-the-money calls and in-themoney puts. It underprices out-of-the-money puts and in-themoney calls.
D. Black-Scholes underprices out-of-the-money and in-themoney calls and puts.
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Normality of the Return Rates
Question 4: Answer A
If the distribution is platikurtic:
A. Black-Scholes overprices out-of-the-money and in-themoney calls and puts.
B. Black-Scholes overprices out-of-the-money puts and in-themoney calls. It underprices in-the-money puts and out-of-themoney calls.
C. Black-Scholes overprices out-of-the-money calls and in-themoney puts. It underprices out-of-the-money puts and in-themoney calls.
D. Black-Scholes underprices out-of-the-money and in-themoney calls and puts.
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Log-Normality of the Stock Prices

As an assumption of the Black-Scholes model, the
distribution of the stock prices, at any moment in the future,
follows the log-normal law. It means that an increase of the
stock price, by a given factor, has the same probability to
occur as the decrease by the same factor. Non-positive
stock values are not expected.

Because at a given time we have a single realization for that
particular stock price, the log-normality of the stock prices
cannot be tested directly.
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Log-Normality of the Stock Prices
Question 1:
The Black-Scholes model can be applied when:
A. The return rates are log-normally distributed.
B. The return rates are normally distributed.
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Log-Normality of the Stock Prices
Question 1: Answer B
The Black-Scholes model can be applied when:
A. The return rates are log-normally distributed.
B. The return rates are normally distributed.
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Log-Normality of the Stock Prices
Question 2:
The Black-Scholes model concludes that:
A. The stock prices are log-normally distributed.
B. The stock prices are normally distributed.
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Log-Normality of the Stock Prices
Question 2: Answer A
The Black-Scholes model concludes that:
A. The stock prices are log-normally distributed.
B. The stock prices are normally distributed.
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Log-Normality of the Stock Prices
Question 3:
The Black-Scholes model can be applied in a market
where the stock prices are not expected to decrease
or increase by the same factor.
A. True.
B. False.
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Log-Normality of the Stock Prices
Question 3: Answer B
The Black-Scholes model can be applied in a market
where the stock prices are not expected to decrease
or increase by the same factor.
A. True.
B. False.
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Log-Normality of the Stock Prices
Question 4:
The log-normal distribution of the stock prices is a basic
assumption of the Black-Scholes model.
A. True.
B. False.
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Log-Normality of the Stock Prices
Question 4: Answer B
The log-normal distribution of the stock prices is a basic
assumption of the Black-Scholes model.
A. True.
B. False.
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Log-Normality of the Stock Prices
Question 5:
The log-normal distribution of the stock prices can be
tested using the D'Agostino tests.
A. True.
B. False.
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Log-Normality of the Stock Prices
Question 5: Answer B
The log-normal distribution of the stock prices can be
tested using the D'Agostino tests.
A. True.
B. False.
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Autocorrelation of the Return Rates

The return rates, according to the Black-Scholes model,
follow a Gaussian white noise stochastic process, i.e. with
zero expectation and the stationary normalized
autocorrelation function given by the "Dirac delta function".

The return rates should be computed for each date, and the
corresponding time series should be checked for
autocorrelation.

The computation of the autocorrelation function requires
evenly spaced data. To by-pass the condition of evenly
spaced data, we may use the Lomb periodogram. Whenever
the return rates are correlated, the Gaussian white noise
stochastic process does not provide an acceptable
description of the market.
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Autocorrelation of the Return Rates
Question 1:
The autocorrelation function can be computed only for
evenly spaced data.
A. True.
B. False.
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Autocorrelation of the Return Rates
Question 1: Answer A
The autocorrelation function can be computed only for
evenly spaced data.
A. True.
B. False.
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Autocorrelation of the Return Rates
Question 2:
The Lomb periodogram can be computed for:
A. Data that are evenly spaced..
B. Data that are unevenly spaced.
C. Any of the above.
D. None of the above.
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Autocorrelation of the Return Rates
Question 2: Answer C
The Lomb periodogram can be computed for:
A. Data that are evenly spaced..
B. Data that are unevenly spaced.
C. Any of the above.
D. None of the above.
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Autocorrelation of the Return Rates
Question 3:
The Black-Scholes model can be applied when the
return rates are correlated.
A. True.
B. False.
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Autocorrelation of the Return Rates
Question 3: Answer B
The Black-Scholes model can be applied when the
return rates are correlated.
A. True.
B. False.
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Ito and Stratonovich Interpretations

The fluctuations of the return rate can be considered as a
Gaussian white noise stochastic process, that is with zero
expectation and the stationary autocorrelation function
given by the "Dirac delta function" multiplied by a constant.
This implies that the return rate can change infinitely fast.
White noise is not physically realizable, because no
process can change infinitely fast. Nevertheless it is often
employed as a model for random physical systems. It is
related to the Wiener process, a continuous parameter
Gaussian process with zero expectation and stationary
independent increments. Although the Wiener process is
not differentiable, it can be shown that formally its
derivative is the white noise process.
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Ito and Stratonovich Interpretations


There exists two alternative interpretations of the
stochastic differential equations, the Ito and Stratonovich
interpretations. These different interpretations generally
yield different solutions and there is no mathematical
reason to prefer one interpretation over the other. The fact
that there are two interpretations of the white noise which
yield two different solutions is due to the pathological
nature of the white noise and Wiener processes. When
more realistic correlated noise models are used, the Ito and
Stratonovich interpretations become identical.
As long as a model based upon the white noise is fitted to
the market values, the two interpretations will provide
different estimates of the parameters, but identical values
concerning the predicted stock prices.
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Ito and Stratonovich Interpretations
Question 1:
The two parameters of the Black-Scholes model (the
expected return rate and the volatility) should be
identified from real market data using the Ito
interpretation.
A. the Ito interpretation.
B. the Stratonovich interpretation.
C. either the Ito or the Stratonovich interpretation.
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Ito and Stratonovich Interpretations
Question 1: Answer C
The two parameters of the Black-Scholes model (the
expected return rate and the volatility) should be
identified from real market data using the Ito
interpretation.
A. the Ito interpretation.
B. the Stratonovich interpretation.
C. either the Ito or the Stratonovich interpretation.
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Ito and Stratonovich Interpretations
Question 2:
When the two parameters of the Black-Scholes model
(the expected return rate and the volatility) have be
identified from real market data using the Ito
interpretation, the predictions for future stock values
should be based upon:
A. the Ito interpretation.
B. the Stratonovich interpretation.
C. either the Ito or the Stratonovich interpretation.
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Ito and Stratonovich Interpretations
Question 2: Answer A
When the two parameters of the Black-Scholes model
(the expected return rate and the volatility) have be
identified from real market data using the Ito
interpretation, the predictions for future stock values
should be based upon:
A. the Ito interpretation.
B. the Stratonovich interpretation.
C. either the Ito or the Stratonovich interpretation.
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Ito and Stratonovich Interpretations
Question 3:
When the two parameters of the Black-Scholes model
(the expected return rate and the volatility) have be
identified from real market data using the
Stratonovich interpretation, the predictions for future
stock values should be based upon:
A. the Ito interpretation.
B. the Stratonovich interpretation.
C. either the Ito or the Stratonovich interpretation.
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Ito and Stratonovich Interpretations
Question 3: Answer B
When the two parameters of the Black-Scholes model
(the expected return rate and the volatility) have be
identified from real market data using the
Stratonovich interpretation, the predictions for future
stock values should be based upon:
A. the Ito interpretation.
B. the Stratonovich interpretation.
C. either the Ito or the Stratonovich interpretation.
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The Risk-Free Interest Rate vs. Return Rate



The Black-Scholes model for stock option valuation uses
the risk-free interest rate, no matter what is the actual value
of the expected return rate.
When the return rate for a given company has been
consistently underperforming with respect to the risk-free
interest rate, we may conclude:
1) It will continue to underperform in the future; in this case
we should not invest in that company.
2) It will perform significantly better in the future; in this
case we should invest and the use of the risk-free interest
rate is reasonable.
Due to the availability of the risk-free interest rate, it does
not make sense to use the expected return rate for stock
option valuation.
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The Risk-Free Interest Rate vs. Return Rate
Question 1:
The expected return rate should be used to predict
stock values.
A. True.
B. False.
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The Risk-Free Interest Rate vs. Return Rate
Question 1: Answer A
The expected return rate should be used to predict
stock values.
A. True.
B. False.
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The Risk-Free Interest Rate vs. Return Rate
Question 2:
The risk-free interest rate should be used to predict
stock values.
A. True.
B. False.
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The Risk-Free Interest Rate vs. Return Rate
Question 2: Answer B
The risk-free interest rate should be used to predict
stock values.
A. True.
B. False.
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The Risk-Free Interest Rate vs. Return Rate
Question 3:
The stock option valuation should be based upon:
A. The expected return rate.
B. The risk-free interest rate.
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The Risk-Free Interest Rate vs. Return Rate
Question 3: Answer B
The stock option valuation should be based upon:
A. The expected return rate.
B. The risk-free interest rate.
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