Www.fintools.comContents OptionsIQ Montgomery Investment Technology, Inc. Financial Modeling Software and Consulting

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OptionsIQ

Montgomery Investment Technology, Inc.Financial Modeling Software and Consulting


Contents:BASIC 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 Op

tions Risk Sensitivities Delta Hedging Trading Strategies Involving Optio

ns Exotic Options

ADVANCED TOPICS: The Markov Process and the

Efficient Market Hurst Exponent to Measure Trend i

n the Time Series Normality of the Return Rates Log-Normality of the Stock Prices Autocorrelation of the Return Rate

s Ito and Stratonovich Interpretations The Risk-Free Interest Rate vs. Re

turn Rate


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.

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.

American-style options can be exercised at anytime up to expiration, while European-style options can only be exercised at expiration.


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.


Option Contracts Defined

Question 1: Answer DAn 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.


Option Contracts DefinedQuestion 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.


Option Contracts DefinedQuestion 2: Answer DWhat 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.


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.

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).


Intrinsic and Time Value

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-the-money, and a $55 call would be considered to be out-of-the-money.



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.



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.



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.



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.



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



Question 3: Answer CA 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


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).


Types of Options 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.


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.


Types of Options

Question 1: Answer BAn 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.


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.


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.


Theoretical or Fair Value 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.


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.


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.



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.


Volatility 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.


Volatility 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)


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.



Volatility

Question 1: Answer DWhat 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.



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%


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%


The Black-Scholes Option Pricing Model 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 = theoretical value of a call P = theoretical value of a put U = price of the underlying asset E = exercise price t = time to expiration in years v = annual volatility expressed as a decimal fraction r = risk-free interest expressed as a decimal fraction e = base of the natural logarithm ln = natural logarithm N'(x) = the normal distribution curve N(x) = the cumulative normal density function


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) x>0 1 - N (-x) x<0

P = -UN(-h) + Ee(-rt) N (v(t^1/2) -h) k = 1 / (1 + yx) h = [ ln(U/E) + (r + (v^2)/2) t] y = 0.33267 [ v(t^1/2)] a1 = 0.4361836 a2 = -0.1201676

N'(x) = [e^(-(x^2)/2)] / (2 Pi)^1/2 a3 = 0.9372980



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.



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




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



Question 2: Answer BWhat 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


The Binomial Method of Pricing Options

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 non-standard 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.


The Binomial Method of Pricing Options 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



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) ]



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.



Question 1: Answer DThe 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.




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



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


Risk Sensitivities (Greeks) 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


Risk Sensitivities (Greeks) 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

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


Risk Sensitivities (Greeks) 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 = 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


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..



Question 1: Answer CDelta is a measure of:




D. none of the above..



Question 2:Rho is a measure of:




D. the rate of change of the value of the portfolio with respect to the volatility.



Question 2: Answer BRho is a measure of:




D. the rate of change of the value of the portfolio with respect to the volatility.


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.

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.


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..


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.


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.



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.



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.


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.


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.


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.



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.



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.



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.


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.


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.


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.


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.


Exotic Options

Question 1: Answer CAn 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.


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.



Exotic Options

Question 2: Answer DAn 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.



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.



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.



Question 1: Answer CAn 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.



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.



Question 2: Answer B & CA market in order to be efficient requires:

A. A large influx of capital.

B. Immediate access to information

(“Transparency").

C. Competition.

D. Over-regulation.



Question 3:A market that is not efficient can be modeled by a

Markov process.

A. True

B. False



Question 3: Answer BA market that is not efficient can be modeled by a

Markov process.

A. True

B. False



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.



Question 4: Answer AThe 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.


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).


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.


Hurst Exponent


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.


Hurst Exponent

Question 1: Answer BThe Black-Scholes model can be applied when:

A. the Hurst exponent is greater than 0.5.




Hurst Exponent

Question 2:The Markov model can be used when:

A. the Hurst exponent is greater than 0.5




Hurst Exponent

Question 2: Answer BThe Markov model can be used when:

A. the Hurst exponent is greater than 0.5




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.


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-the-

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-the-money calls. It underprices in-the-money puts and out-of-the-money calls.

C. Black-Scholes underprices out-of-the-money and in-the-money calls and puts.

D. Black-Scholes overprices out-of-the-money and in-the-money calls and puts.



Question 1: Answer AIf the distribution is skewed to the left: A. Black-Scholes overprices out-of-the-money calls and in-

the-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-the-money calls. It underprices in-the-money puts and out-of-the-money calls.





Question 2:If the distribution is skewed to the right: A. Black-Scholes overprices out-of-the-money puts and in-the-

money calls. It underprices in-the-money puts and out-of-the-money calls.

B. Black-Scholes overprices out-of-the-money calls and in-the-money puts. It underprices out-of-the-money puts and in-the-money calls.





Question 2: Answer AIf the distribution is skewed to the right: A. Black-Scholes overprices out-of-the-money puts and in-

the-money calls. It underprices in-the-money puts and out-of-the-money calls.

B. Black-Scholes overprices out-of-the-money calls and in-the-money puts. It underprices out-of-the-money puts and in-the-money calls.





Question 3:If the distribution is leptokurtic: A. Black-Scholes underprices out-of-the-money and in-the-

money calls and puts. B. Black-Scholes overprices out-of-the-money puts and in-the-


C. Black-Scholes overprices out-of-the-money calls and in-the-money puts. It underprices out-of-the-money puts and in-the-money calls.




Question 3: Answer AIf the distribution is leptokurtic: A. Black-Scholes underprices out-of-the-money and in-the-







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-the-



D. Black-Scholes underprices out-of-the-money and in-the-money calls and puts.



Question 4: Answer AIf the distribution is platikurtic: A. Black-Scholes overprices out-of-the-money and in-the-




D. Black-Scholes underprices out-of-the-money and in-the-money calls and puts.


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.


Log-Normality of the Stock Prices


A. The return rates are log-normally distributed.

B. The return rates are normally distributed.



Question 1: Answer BThe Black-Scholes model can be applied when:

A. The return rates are log-normally distributed.

B. The return rates are normally distributed.



Question 2:The Black-Scholes model concludes that:

A. The stock prices are log-normally distributed.

B. The stock prices are normally distributed.



Question 2: Answer AThe Black-Scholes model concludes that:

A. The stock prices are log-normally distributed.

B. The stock prices are normally distributed.



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.



Question 3: Answer BThe 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.



Question 4:The log-normal distribution of the stock prices is a basic

assumption of the Black-Scholes model.

A. True.

B. False.



Question 4: Answer BThe log-normal distribution of the stock prices is a basic

assumption of the Black-Scholes model.

A. True.

B. False.



Question 5:The log-normal distribution of the stock prices can be

tested using the D'Agostino tests.

A. True.

B. False.



Question 5: Answer BThe log-normal distribution of the stock prices can be

tested using the D'Agostino tests.

A. True.

B. False.


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.


Autocorrelation of the Return Rates

Question 1:The autocorrelation function can be computed only for

evenly spaced data.

A. True.

B. False.



Question 1: Answer AThe autocorrelation function can be computed only for

evenly spaced data.

A. True.

B. False.



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.




Question 2: Answer CThe Lomb periodogram can be computed for:

A. Data that are evenly spaced..

B. Data that are unevenly spaced.

C. Any of the above.




Question 3:The Black-Scholes model can be applied when the

return rates are correlated.

A. True.

B. False.



Question 3: Answer BThe Black-Scholes model can be applied when the

return rates are correlated.

A. True.

B. False.


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.


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.


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.



Question 1: Answer CThe 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.






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:






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:






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:






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:





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.



Question 1:The expected return rate should be used to predict

stock values.

A. True.

B. False.



Question 1: Answer AThe expected return rate should be used to predict

stock values.

A. True.

B. False.



Question 2:The risk-free interest rate should be used to predict

stock values.

A. True.

B. False.



Question 2: Answer BThe risk-free interest rate should be used to predict

stock values.

A. True.

B. False.



Question 3:The stock option valuation should be based upon:

A. The expected return rate.

B. The risk-free interest rate.



Question 3: Answer BThe stock option valuation should be based upon:

A. The expected return rate.

B. The risk-free interest rate.

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