Question 1

In the preceding question, the state price in the lower node after one month is equal to

Accepted Answer

A)  0.4996 
B)  0.5000 
C)  0.5368 
D)  0.5372 
A)  0.4996 
B)  0.5000 
C)  0.5368 
D)  0.5372 C

Question 2

Which of the following assumptions made in deriving the Black-Scholes formula are commonly violated in the real world?

Accepted Answer

A)  Log-returns are normally distributed. 
B)  The volatility of the stock is constant. 
C)  Prices evolve continuously, i.e., there are no market "gaps." 
D)  All of the above. 
A)  Log-returns are normally distributed. 
B)  The volatility of the stock is constant. 
C)  Prices evolve continuously, i.e., there are no market "gaps." 
D)  All of the above. D

Question 3

An option-trading firm is using the Black-Scholes (1973) model to price their options using the same level of volatility for all strikes. The market anticipation is that sharp negative gapping behavior is likely given that sudden recessionary information is being released in spurts. By using the Black-Scholes model with a constant volatility the firm is

Accepted Answer

A)  Underpricing out-of-the-money calls relative to in-the-money puts. 
B)  Underpricing out-of-the-money puts relative to in-the-money calls. 
C)  Underpricing out-of-the-money puts and in-the-money puts relative to those at-the-money. 
D)  Underpricing out-of-the-money puts relative to those at-the-money. 
A)  Underpricing out-of-the-money calls relative to in-the-money puts. 
B)  Underpricing out-of-the-money puts relative to in-the-money calls. 
C)  Underpricing out-of-the-money puts and in-the-money puts relative to those at-the-money. 
D)  Underpricing out-of-the-money puts relative to those at-the-money. D

Question 4

For the same problem in the preceding two questions, find the state price at the middle node after two periods, given that the 117.3008--strike call is priced at $4 and the 85.2509--strike put is priced at $4, both options of two-months maturity. Assume that the middle node after two periods has the same stock price as the initial stock price.

Accepted Answer

A)  0.5016 
B)  0.5023 
C)  0.5044 
D)  0.5117 
A)  0.5016 
B)  0.5023 
C)  0.5044 
D)  0.5117

Question 5

A Wall Street trading firm is using the Merton (1976) jump-diffusion model to price their index options. They are pricing European calls and then using put-call parity to compute the prices of puts. The problem with this is

Accepted Answer

A)  Put-call parity is not valid for models with jumps. 
B)  Put-call parity works only if jumps are symmetric. 
C)  Put-call parity works with jumps only if there are no dividends. 
D)  Nothing---there is no problem with using put-call parity even if there are jumps in the stock price. 
A)  Put-call parity is not valid for models with jumps. 
B)  Put-call parity works only if jumps are symmetric. 
C)  Put-call parity works with jumps only if there are no dividends. 
D)  Nothing---there is no problem with using put-call parity even if there are jumps in the stock price.

Question 6

In comparing the ARCH $( q )$ model with GARCH $( p , q )$ for $p \geq 1$ , which of the following statements is most likely to be valid?

Accepted Answer

A)  The GARCH model always delivers a higher volatility than the ARCH model because of the extra terms in the variance equation. 
B)  The GARCH model is aimed at delivering more persistent volatility. 
C)  The volatility of volatility is lower in the GARCH model than in ARCH. 
D)  The volatility of volatility is higher in the GARCH model than in ARCH. 
A)  The GARCH model always delivers a higher volatility than the ARCH model because of the extra terms in the variance equation. 
B)  The GARCH model is aimed at delivering more persistent volatility. 
C)  The volatility of volatility is lower in the GARCH model than in ARCH. 
D)  The volatility of volatility is higher in the GARCH model than in ARCH.

Question 7

The GARCH process for stock prices has been used to better fit option prices. Which of the following best describes why GARCH may be used to fit the options smile?

Accepted Answer

A)  GARCH is just one form of a stochastic volatility process and therefore random volatility causes the smile in the same way as does stochastic volatility. 
B)  GARCH implies persistence in changes in volatility, which in turn usually makes the return distribution have non-zero skewness and kurtosis. 
C)  GARCH volatility is deterministic and therefore causes the smile with certainty. 
D)  None of the above. 
A)  GARCH is just one form of a stochastic volatility process and therefore random volatility causes the smile in the same way as does stochastic volatility. 
B)  GARCH implies persistence in changes in volatility, which in turn usually makes the return distribution have non-zero skewness and kurtosis. 
C)  GARCH volatility is deterministic and therefore causes the smile with certainty. 
D)  None of the above.

Question 8

The current stock price is $100. A $101--strike call option is priced at $7.55. The risk-free one-month rate of interest is 1% in continuously-compounded and annualized terms. Using the Derman-Kani (1994) tree technology for a single period, what is the price of the one-month at-the-money put option?

Accepted Answer

A)  $6.12 
B)  $6.34 
C)  $7.55 
D)  $7.92 
A)  $6.12 
B)  $6.34 
C)  $7.55 
D)  $7.92

Question 9

An option-trading firm is using the Black-Scholes (1973) model (with the same constant volatility for all strikes) to price index options. Market sentiment is that the stock return volatility is stochastic and changes in volatility are negatively correlated with stock returns. By using the Black-Scholes model with a constant volatility the firm is

Accepted Answer

A)  Underpricing out-of-the-money calls relative to in-the-money puts. 
B)  Underpricing out-of-the-money puts relative to in-the-money calls. 
C)  Underpricing out-of-the-money puts and in-the-money puts relative to those at-the-money. 
D)  Underpricing out-of-the-money puts relative to those at-the-money. 
A)  Underpricing out-of-the-money calls relative to in-the-money puts. 
B)  Underpricing out-of-the-money puts relative to in-the-money calls. 
C)  Underpricing out-of-the-money puts and in-the-money puts relative to those at-the-money. 
D)  Underpricing out-of-the-money puts relative to those at-the-money.

Question 10

The Heston (1993) model generalizes the Black-Scholes setting to one in which

Accepted Answer

A)  Volatility itself evolves according to a geometric Brownian motion. 
B)  Volatility is a mean-reverting stochastic process. 
C)  Volatility is a mean-averting stochastic process. 
D)  Volatility evolves according to an arithmetic Brownian motion. 
A)  Volatility itself evolves according to a geometric Brownian motion. 
B)  Volatility is a mean-reverting stochastic process. 
C)  Volatility is a mean-averting stochastic process. 
D)  Volatility evolves according to an arithmetic Brownian motion.

Question 11

A stochastic volatility model generates negative skewness when

Accepted Answer

A)  The correlation between the stock return and changes in volatility is zero. 
B)  The correlation between the stock return and changes in volatility is negative. 
C)  The correlation between the stock return and changes in volatility is positive. 
D)  The volatility process exhibits mean-reversion. 
A)  The correlation between the stock return and changes in volatility is zero. 
B)  The correlation between the stock return and changes in volatility is negative. 
C)  The correlation between the stock return and changes in volatility is positive. 
D)  The volatility process exhibits mean-reversion.

Question 12

If the volatility of a stock is not constant and changes randomly over time, it generates the implied volatility "smile" or "skew". Which of the following statements about the smile is most valid?

Accepted Answer

A)  The smile arises because changing volatility increases the overall variance of the stock return relative to that assumed by the Black-Scholes formula. 
B)  The return distribution with stochastic volatility is thinner-tailed than assumed in the Black-Scholes model. 
C)  The smile arises because changing volatility decreases the overall skewness of the stock return relative to that assumed by the Black-Scholes formula. 
D)  None of the above. 
A)  The smile arises because changing volatility increases the overall variance of the stock return relative to that assumed by the Black-Scholes formula. 
B)  The return distribution with stochastic volatility is thinner-tailed than assumed in the Black-Scholes model. 
C)  The smile arises because changing volatility decreases the overall skewness of the stock return relative to that assumed by the Black-Scholes formula. 
D)  None of the above.

Question 13

By augmenting the geometric Brownian motion process with a Poisson-driven jump process, jump-diffusion models

Accepted Answer

A)  Introduce kurtosis into the returns distribution. 
B)  Introduce skewness but not kurtosis into the returns distribution. 
C)  Introduce skewness and kurtosis into the returns distribution. 
D)  Create a lognormal distribution that is fat-tailed. 
A)  Introduce kurtosis into the returns distribution. 
B)  Introduce skewness but not kurtosis into the returns distribution. 
C)  Introduce skewness and kurtosis into the returns distribution. 
D)  Create a lognormal distribution that is fat-tailed.

Question 14

If the implied volatility surface is flat (i.e., all options have the same implied volatility), then

Accepted Answer

A)  Derman-Kani implied binomial trees cannot be constructed because of the lack of variability in the data. 
B)  The Derman-Kani tree may price away-from-the-money options inaccurately. 
C)  The Derman-Kani implied binomial tree coincides with the Cox-Ross-Rubinstein binomial tree. 
D)  The Derman-Kani implied binomial tree may still differ from the Cox-Ross-Rubinstein (CRR) tree because implied binomial trees are a form of stochastic volatility models whereas CRR trees assume constant volatility. 
A)  Derman-Kani implied binomial trees cannot be constructed because of the lack of variability in the data. 
B)  The Derman-Kani tree may price away-from-the-money options inaccurately. 
C)  The Derman-Kani implied binomial tree coincides with the Cox-Ross-Rubinstein binomial tree. 
D)  The Derman-Kani implied binomial tree may still differ from the Cox-Ross-Rubinstein (CRR) tree because implied binomial trees are a form of stochastic volatility models whereas CRR trees assume constant volatility.

Question 15

A stock has a current price of $100. Assume a CRR-style jump-to-default model in which the volatility is 30%. Let the risk-neutral probability of default in three months be 10%. The 3-month risk-free rate is 2% in continuously-componded and annualized terms. What is the price of a three-month at-the-money put option on this stock in a one-period jump-to-default tree model?

Accepted Answer

A)  $7.22 
B)  7.72 
C)  $11.82 
D)  $13.42 
A)  $7.22 
B)  7.72 
C)  $11.82 
D)  $13.42

Question 16

Stochastic volatility models are said to incorporate the "leverage" effect. The presence of the leverage effect

Accepted Answer

A)  Results in a left skew in the options smile because stock price drops are associated with higher firm leverage and so with increased volatility. 
B)  Results in a right skew in the options smile because stock price increases are associated with higher firm leverage and so with increased volatility. 
C)  Results in a symmetric distribution of stock returns, resulting in higher option prices both in- and out-of-the-money. 
D)  All option models have a leverage effect because the option is a leveraged position in stock. The stochastic volatility model exacerbates this effect. 
A)  Results in a left skew in the options smile because stock price drops are associated with higher firm leverage and so with increased volatility. 
B)  Results in a right skew in the options smile because stock price increases are associated with higher firm leverage and so with increased volatility. 
C)  Results in a symmetric distribution of stock returns, resulting in higher option prices both in- and out-of-the-money. 
D)  All option models have a leverage effect because the option is a leveraged position in stock. The stochastic volatility model exacerbates this effect.

Question 17

If the stock "gaps" or "jumps," an implied volatility smile results. Which of the following reasons explains why this happens?

Accepted Answer

A)  Holding the variance constant, jumps always increase the kurtosis of the distribution. 
B)  Holding the variance constant, jumps always increase the skewness of the distribution. 
C)  Holding the variance constant, jumps always increase the risk aversion of investors who buy options. 
D)  All of the above. 
A)  Holding the variance constant, jumps always increase the kurtosis of the distribution. 
B)  Holding the variance constant, jumps always increase the skewness of the distribution. 
C)  Holding the variance constant, jumps always increase the risk aversion of investors who buy options. 
D)  All of the above.

Question 18

A stock has a probability of jumping to default based on the first arrival of a Poisson process with $\lambda = 0.05$ . What is the probability of a jump-to-default in any month?

Accepted Answer

A)  0.4158% 
B)  0.4167% 
C)  0.4195% 
D)  0.5000% 
A)  0.4158% 
B)  0.4167% 
C)  0.4195% 
D)  0.5000%

Question 19

Two stocks A and B both have a current price of $100 and are identical in every way except that the risk-neutral probability of default of A in three months is 10%, and that of B is zero. Assume a CRR-style jump-to-default model in which the volatility of both stocks is 30%. The risk-free rate is 2%. Consider the price of three-month at-the-money call options on these two stocks in a one-period jump-to-default tree model. Which of the following statements is valid?

Accepted Answer

A)  The call option on A is worth less than the call option on B. 
B)  The call option on A is worth the same as the call option on B. 
C)  The call option on A is worth more than the call option on B. 
D)  There is insufficient information to determine which option is worth more. 
A)  The call option on A is worth less than the call option on B. 
B)  The call option on A is worth the same as the call option on B. 
C)  The call option on A is worth more than the call option on B. 
D)  There is insufficient information to determine which option is worth more.

Question 20

The constant elasticity of variance (CEV) Ito process is as follows: $d S = \mu S d t + \sigma S ^ { \prime \prime } d W$ In order to mimic the leverage effect it is required that

Accepted Answer

A) $\gamma > 1$ . B) $\gamma = 1$ . C) $0 < \gamma < 1$ . D) $\gamma \leq 0$ . A) $\gamma > 1$ . B) $\gamma = 1$ . C) $0 < \gamma < 1$ . D) $\gamma \leq 0$ .

Exam 16: Beyond Black-Scholes

In the preceding question, the state price in the lower node after one month is equal to

Which of the following assumptions made in deriving the Black-Scholes formula are commonly violated in the real world?

A Wall Street trading firm is using the Merton (1976) jump-diffusion model to price their index options. They are pricing European calls and then using put-call parity to compute the prices of puts. The problem with this is

In comparing the ARCH (q)( q )(q) model with GARCH (p,q)( p , q )(p,q) for p≥1p \geq 1p≥1 , which of the following statements is most likely to be valid?

The GARCH process for stock prices has been used to better fit option prices. Which of the following best describes why GARCH may be used to fit the options smile?

The Heston (1993) model generalizes the Black-Scholes setting to one in which

A stochastic volatility model generates negative skewness when

If the volatility of a stock is not constant and changes randomly over time, it generates the implied volatility "smile" or "skew". Which of the following statements about the smile is most valid?

By augmenting the geometric Brownian motion process with a Poisson-driven jump process, jump-diffusion models

If the implied volatility surface is flat (i.e., all options have the same implied volatility), then

Stochastic volatility models are said to incorporate the "leverage" effect. The presence of the leverage effect

If the stock "gaps" or "jumps," an implied volatility smile results. Which of the following reasons explains why this happens?

A stock has a probability of jumping to default based on the first arrival of a Poisson process with λ=0.05\lambda = 0.05λ=0.05 . What is the probability of a jump-to-default in any month?

The constant elasticity of variance (CEV) Ito process is as follows: dS=μSdt+σS′′dWd S = \mu S d t + \sigma S ^ { \prime \prime } d WdS=μSdt+σS′′dW In order to mimic the leverage effect it is required that

Overview

Futures Markets

Pricing Forwards and Futures I

Pricing Forwards Futures II

Hedging With Futures Forwards

Interest-Rate Forwards Futures

Options Markets

Options: Payoffs Trading Strategies

No-Arbitrage Restrictions

Early-Exercise Put-Call Parity

Option Pricing: an Introduction

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Implementing the Binomial Model

The Black-Scholes Model

Mathematics of Black-Scholes

The Option Greeks

Path-Independent Exotic Options

Exotic Options II: Path-Dependent Options

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Term Structure of Interest Rates: Concepts

Estimating the Yield Curve

Modeling Term Structure Movements

Factor Models of the Term Structure

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Structural Models of Default Risk

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In comparing the ARCH $( q )$ model with GARCH $( p , q )$ for $p \geq 1$ , which of the following statements is most likely to be valid?

A stock has a probability of jumping to default based on the first arrival of a Poisson process with $\lambda = 0.05$ . What is the probability of a jump-to-default in any month?

The constant elasticity of variance (CEV) Ito process is as follows: $d S = \mu S d t + \sigma S ^ { \prime \prime } d W$ In order to mimic the leverage effect it is required that