Question 1

One of the variables that influences the price of the option is the expected return on the stock.

Accepted Answer

The price of an option is influenced by factors such as the stock price, strike price, time to expiration, volatility, interest rates, and dividends, but not directly by the expected return on the stock.

Question 2

What happens when the volatility is zero in the Black-Scholes-Merton model?
A)the option price converges to either zero or the lower bound
B)the option price converges to the intrinsic value
C)the option automatically expires out of the money
D)the gamma and delta converge
E)none of the above

Accepted Answer

The answer of What happens when the volatility is zero...

Question 3

The Black-Scholes-Merton model is the discrete time limit to the binomial model.

Accepted Answer

The binomial model is actually the discrete time approximation to the Black-Scholes-Merton model, which is a continuous time model.

Question 4

Which of the following statements is incorrect about the historical volatility?
A)if used in the Black-Scholes-Merton model,it gives the current market price
B)it is based on the volatility of the log return on the stock
C)it requires a sample of recent returns
D)it should be converted to an annualized volatility
E)none of the above

Accepted Answer

Historical volatility, when used in the Black-Scholes-Merton model, gives the theoretical option price, not the current market price. The current market price is determined by supply and demand in the market.

Question 5

A hedge portfolio is established and maintained by constantly adjusting the relative proportions of stock and options,a process referred to as
A)actively managing
B)continuous reconciliation
C)marking to market
D)dynamic trading
E)none of the above

Accepted Answer

The process of constantly adjusting the relative proportions of stock and options in a hedge portfolio is referred to as dynamic trading. This involves actively managing the portfolio to ensure it remains effective in hedging against market risks. Continuous reconciliation and marking to market are related processes but do not accurately describe the specific action of adjusting the portfolio's proportions.

Question 6

The relationship between the option price and the exercise price is called
A)the gamma
B)the vega
C)the omega
D)the zeta
E)none of the above

Accepted Answer

The relationship between the option price and the exercise price is not given a specific name or symbol. Therefore, none of the choices provided are correct.

Question 7

The implied volatility is obtained by finding the standard deviation that,when used in the Black-Scholes-Merton model,makes the
A)model price expire at zero
B)model price equal the market price of the option
C)model price such that it exceeds currently traded market option values
D)model price equal the intrinsic value of the underlying stock
E)none of the above

Accepted Answer

The implied volatility is obtained by finding the standard deviation that, when used in the Black-Scholes-Merton model, makes the model price equal the market price of the option. This is because the Black-Scholes-Merton model is used to price options based on certain assumptions, including the volatility of the underlying asset. If the market price of the option is known, the implied volatility can be found by inputting the market price and other relevant information into the model and solving for the volatility.

Question 8

The option's rate of time value decay is represented by its theta.

Accepted Answer

Theta measures how much an option's value decreases as time passes, indicating the rate of time value decay. Therefore, the statement is true.

Question 9

The option's delta is approximately the change in the option price for a change in the stock price.

Accepted Answer

This statement is true. Delta measures the sensitivity of an option price to changes in the underlying stock price. A delta of 1 means that for every $1 change in the stock price, the option price will change by approximately $1. A delta of 0.5 means that for every $1 change in the stock price, the option price will change by approximately $0.50.

Question 10

The standard normal random variable used in the calculation of cumulative normal probabilities within the Black-Scholes-Merton option pricing model is
A)the lognormal distribution
B)the d1 and d2 statistic
C)the z statistic
D)the f distribution
E)none of the above

Accepted Answer

The standard normal random variable used in the Black-Scholes-Merton option pricing model is the z statistic, which is obtained by standardizing the log return. The d1 and d2 statistics are derived from the z statistic and are used in the calculation of Black-Scholes option prices, but they are not the standard normal random variable themselves. The lognormal distribution and the f distribution are not used in the calculation of cumulative normal probabilities within the Black-Scholes-Merton option pricing model.

Question 11

Which of the following is not correct about a call's gamma?
A)it is the same as a put's gamma
B)it is large when the call is at-the-money
C)it can be viewed as a measure of the risk of the delta
D)it is a source of risk that can be hedged only by using another option
E)none of the above

Accepted Answer

Gamma is a measure of the rate of change of delta, and it is indeed the same for both calls and puts due to the properties of options. Gamma is largest when the option is at-the-money, reflecting higher sensitivity of delta to changes in the underlying's price at this point. It can be considered a measure of the risk of the delta, as it indicates how much the delta could change with movements in the underlying asset's price. Gamma risk can be hedged using various strategies, not only by using another option, but also through dynamic hedging with the underlying asset, making statement D incorrect. However, since the question asks for what is not correct and all statements except D are correct descriptions of gamma, the answer that none of the provided statements are incorrect about gamma is the most fitting.

Question 12

Which of the following statements about the volatility is not true?
A)the implied volatility often differs across options with different exercise prices
B)the implied volatility equals the historical volatility if the option is correctly priced
C)the implied volatility is determined by trial and error
D)the implied volatility is nearly linearly related to the option price
E)none of the above

Accepted Answer

Implied volatility is the market's expectation of how volatile the underlying asset will be in the future. It is determined by trial and error, meaning that it is found by inputting different values into an options pricing model until the model's output matches the market price of the option. Therefore, there is no guarantee that implied volatility will equal historical volatility, and option pricing models often use a range of inputs to determine implied volatility. Choice A is true because the implied volatility can differ across options with different exercise prices due to varying market expectations for volatility. Choice C is true because implied volatility is determined by trial and error in options pricing models. Choice D is true because there is often a nearly linear relationship between implied volatility and option price, with higher implied volatility leading to higher option prices.

Question 13

The binomial model always gives the same option price as the Black-Scholes-Merton model.

Accepted Answer

The binomial model and Black-Scholes-Merton model are two different mathematical models used to price options. While they can both provide similar results, they are not guaranteed to always give the same option price. Factors such as the underlying asset's volatility and interest rates can cause differences in the calculated option price.

Question 14

The values of N(d₁)and N(d₂)are called risk neutral probabilities.

Accepted Answer

The statement is true. The values of N(d₁) and N(d₂) are risk neutral probabilities.

Question 15

The pattern of volatility across exercise prices is often called
A)the price-fluctuation graph
B)the volatility smile
C)the term structure of implied volatility
D)the skew
E)none of the above

Accepted Answer

The pattern of volatility across exercise prices is commonly referred to as the "volatility smile." It is called a smile because the shape of the graph often appears to curve upward on both sides, resembling a smiling mouth. The other options listed do not accurately describe the pattern of volatility across exercise prices.

Question 16

The Black-Scholes-Merton model for European puts,obtained by applying put-call parity to the Black-Scholes-Merton model for European calls,is customarily expressed by which of the following:
A) $\mathrm { P } = \mathrm { Xe } ^ { - \mathrm { I } _ { \mathrm { e } } \mathrm { T } } \mathrm { N } \left( - \mathrm { d } _ { 2 } \right) - \mathrm { S } _ { 0 } \mathrm {~N} \left( - \mathrm { d } _ { 1 } \right)$
B) $\mathrm { P } = \mathrm { X } ( 1 + \mathrm { r } ) ^ { - \mathrm { T } } \mathrm { N } \left( - \mathrm { d } _ { 2 } \right) - \mathrm { S } _ { 0 } \mathrm {~N} \left( - \mathrm { d } _ { 1 } \right)$
C) $\mathrm { P } = \mathrm { X } ( 1 + \mathrm { r } ) ^ { - \mathrm { T } } \mathrm { N } \left( - \mathrm { d } _ { 1 } \right) - \mathrm { S } _ { 0 } \mathrm {~N} \left( - \mathrm { d } _ { 2 } \right)$
D) $\mathrm { P } = \mathrm { Xe } ^ { - \mathrm { I } _ { \mathrm { e } } \mathrm { T } } \mathrm { N } \left( - \mathrm { d } _ { 1 } \right) - \mathrm { S } _ { 0 } \mathrm {~N} \left( - \mathrm { d } _ { 2 } \right)$
E)none of the above

Accepted Answer

The correct formula for a European put option in the Black-Scholes-Merton model is $ P = Xe^{-rT}N(-d_2) - S_0N(-d_1) $, where $X$ is the strike price, $e^{-rT}$ is the discount factor for the risk-free rate $r$ over time $T$, $S_0$ is the current stock price, and $N(-d_1)$ and $N(-d_2)$ are values from the cumulative distribution function of the standard normal distribution. Choice A is the closest representation of this formula, with $I_e$ presumably representing the risk-free interest rate $r$, making it the correct expression for a European put option derived from the Black-Scholes-Merton model.

Question 17

An option's gamma represents the risk of the delta changing.

Accepted Answer

Gamma measures the rate of change in an option's delta in response to changes in the underlying asset's price. It represents the risk of the delta changing, since the delta is a measure of the option's sensitivity to price changes in the underlying asset.

Question 18

Since dividends could trigger an early exercise of an American call,the Black-Scholes-Merton dividend adjustment will provide the correct price of an American call.

Accepted Answer

The Black-Scholes-Merton model is designed for pricing European options and does not account for early exercise features, which are relevant for American options. For American options, especially calls on dividend-paying stocks, adjustments or alternative models are needed to accurately account for the early exercise possibility due to dividends.

Question 19

The Black-Scholes-Merton model assumes that the volatility does not change throughout the option's life.

Accepted Answer

The Black-Scholes-Merton model assumes that the volatility of the underlying asset is constant throughout the option's life. This is known as constant volatility or static volatility. However, in reality, the volatility of the underlying asset can change over time, which can affect the price of the option. This is known as dynamic volatility. Various models have been developed to account for dynamic volatility, such as the stochastic volatility model.

Question 20

In the Black-Scholes-Merton model,stock prices are assumed to behave randomly.

Accepted Answer

In the Black-Scholes-Merton model, stock prices are assumed to follow a random walk process.

Quiz 5: Option Pricing Models: the Black-Scholes-Merton Model

One of the variables that influences the price of the option is the expected return on the stock.

What happens when the volatility is zero in the Black-Scholes-Merton model?

The Black-Scholes-Merton model is the discrete time limit to the binomial model.

Which of the following statements is incorrect about the historical volatility?

A hedge portfolio is established and maintained by constantly adjusting the relative proportions of stock and options,a process referred to as

The relationship between the option price and the exercise price is called

The implied volatility is obtained by finding the standard deviation that,when used in the Black-Scholes-Merton model,makes the

The option's rate of time value decay is represented by its theta.

The option's delta is approximately the change in the option price for a change in the stock price.

The standard normal random variable used in the calculation of cumulative normal probabilities within the Black-Scholes-Merton option pricing model is

Which of the following is not correct about a call's gamma?

Which of the following statements about the volatility is not true?

The binomial model always gives the same option price as the Black-Scholes-Merton model.

The values of N(d1)and N(d2)are called risk neutral probabilities.

The pattern of volatility across exercise prices is often called

The Black-Scholes-Merton model for European puts,obtained by applying put-call parity to the Black-Scholes-Merton model for European calls,is customarily expressed by which of the following:

An option's gamma represents the risk of the delta changing.

Since dividends could trigger an early exercise of an American call,the Black-Scholes-Merton dividend adjustment will provide the correct price of an American call.

The Black-Scholes-Merton model assumes that the volatility does not change throughout the option's life.

In the Black-Scholes-Merton model,stock prices are assumed to behave randomly.

The values of N(d₁)and N(d₂)are called risk neutral probabilities.