Question 1

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. A

Question 2

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. B

Question 3

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. D

Question 4

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 5

Local volatility models

Accepted Answer

A) Look to describe volatility accurately locally in time (i.e. ,around the current time)but not necessarily in the longer run. 
B) Look to describe volatility accurately locally in space (i.e. ,around the current stock price $S$
)but not necessarily when stock prices move away from current levels. 
C) Look to identify the form $\sigma \left( S _ { t } , t \right)$
For evolution of volatility that will give rise to the currently observed set of option prices. 
D) And stochastic volatility models are exactly the same thing. 
A) Look to describe volatility accurately locally in time (i.e. ,around the current time)but not necessarily in the longer run. 
B) Look to describe volatility accurately locally in space (i.e. ,around the current stock price $S$
)but not necessarily when stock prices move away from current levels. 
C) Look to identify the form $\sigma \left( S _ { t } , t \right)$
For evolution of volatility that will give rise to the currently observed set of option prices. 
D) And stochastic volatility models are exactly the same thing.

Question 6

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 7

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

Accepted Answer

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

Question 8

Stochastic volatility models commonly assume

Accepted Answer

A) There are jumps in the evolution of the volatility process. 
B) Volatility follows a geometric Brownian motion process. 
C) Volatility is normally distributed. 
D) Volatility follows a mean-reverting process. 
A) There are jumps in the evolution of the volatility process. 
B) Volatility follows a geometric Brownian motion process. 
C) Volatility is normally distributed. 
D) Volatility follows a mean-reverting process.

Question 9

GARCH models

Accepted Answer

A) Are discrete-time expressions of stochastic volatility models. 
B) Are designed to capture the empirically-observed leverage effect in equity returns. 
C) Are models in which volatility is not separately stochastic but evolves in a manner dependent on the stock return process. 
D) Are useful for describing stock returns empirically but not for pricing options on equity. 
A) Are discrete-time expressions of stochastic volatility models. 
B) Are designed to capture the empirically-observed leverage effect in equity returns. 
C) Are models in which volatility is not separately stochastic but evolves in a manner dependent on the stock return process. 
D) Are useful for describing stock returns empirically but not for pricing options on equity.

Question 10

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.

Question 11

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 12

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 13

The Black-Scholes model is time-inconsistent in the way it is applied in practice.This is because the model assumes that volatility is constant,even though the trader changes it every day to obtain a new price.Using this definition of time-inconsistency,which of the following statements is valid?

Accepted Answer

A) The stochastic volatility model is time-consistent because the volatility of volatility is allowed to change in the model. 
B) The GARCH model is time-consistent because traders who use GARCH do not change their inputs to the model frequently. 
C) Implied binomial trees are time-consistent even though the implied volatility smile used to calibrate it changes over time. 
D) The stochastic volatility model is time-inconsistent. 
A) The stochastic volatility model is time-consistent because the volatility of volatility is allowed to change in the model. 
B) The GARCH model is time-consistent because traders who use GARCH do not change their inputs to the model frequently. 
C) Implied binomial trees are time-consistent even though the implied volatility smile used to calibrate it changes over time. 
D) The stochastic volatility model is time-inconsistent.

Question 14

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

Question 15

The Merton (1976)model

Accepted Answer

A) Modifies the Black-Scholes model by replacing geometric Brownian motion with a Poisson-driven jump process. 
B) Modifies the Black-Scholes model by adding a Poisson-driven jump process as a second source of noise in addition to geometric Brownian motion. 
C) Modifies the Black-Scholes model by allowing for jumps at specified points in time to account for dividend payments. 
D) Replaces the Black-Scholes model's geometric Brownian motion assumption (i.e. ,lognormal returns)with a Poisson-augmented arithmetic Brownian motion process (i.e. ,normal returns). 
A) Modifies the Black-Scholes model by replacing geometric Brownian motion with a Poisson-driven jump process. 
B) Modifies the Black-Scholes model by adding a Poisson-driven jump process as a second source of noise in addition to geometric Brownian motion. 
C) Modifies the Black-Scholes model by allowing for jumps at specified points in time to account for dividend payments. 
D) Replaces the Black-Scholes model's geometric Brownian motion assumption (i.e. ,lognormal returns)with a Poisson-augmented arithmetic Brownian motion process (i.e. ,normal returns).

Question 16

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 17

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 18

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 19

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 20

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.

Exam 16: Beyond Black-Scholes

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

A stochastic volatility model generates negative skewness when

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

Local volatility models

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

Stochastic volatility models commonly assume

GARCH models

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

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

The Black-Scholes model is time-inconsistent in the way it is applied in practice.This is because the model assumes that volatility is constant,even though the trader changes it every day to obtain a new price.Using this definition of time-inconsistency,which of the following statements is valid?

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

The Merton (1976)model

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

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

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?

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?

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

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

Binomial Option Pricing

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

Value at Risk

Swaps and Floating Rate Products

Equity Swaps

Currency and Commodity Swaps

Term Structure of Interest Rates: Concepts

Estimating the Yield Curve

Modeling Term Structure Movements

Factor Models of the Term Structure

The Heath-Jarrow-Morton Hjmand Libor Market Model LMM

Credit Derivative Products

Structural Models of Default Risk

Reduced-Form Models of Default Risk

Modeling Correlated Default

Filters

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

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?