Deck 16: Beyond Black-Scholes

Question 1

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) 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.

Answer 1

The GARCH model is aimed at delivering more persistent volatility.

Question 2

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?
(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.

Answer 2

C.

Question 3

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

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

Answer 3

$0 < \gamma < 1$ .

Question 4

A stochastic volatility model generates negative skewness when
(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 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
(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

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
(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 7

Which of the following assumptions made in deriving the Black-Scholes formula are commonly violated in the real world?
(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 8

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?
(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 9

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) $6.12
(b) $6.34
(c) $7.55
(d) $7.92

Question 10

A Wall Street trading firm is using a jump-diffusion model to price their index options. They determine that the arrival rate of jumps in the market is 4 times a year, and that the jumps have a mean size of $- 2 \%$ and standard deviation of 10%. If the implied volatility of the stock index is 40%, what is the diffusion parameter ( $\sigma$ ) that they should use in their model?

A) 0.35
B) 0.40
C) 0.45
D) 0.50

Question 11

The asymmetric GARCH model was developed to mimic the following feature of other models that is not captured in the standard GARCH formulation:
(a) Mean reversion in returns.
(b) Persistence in returns.
(c) Skewness in returns.
(d) Kurtosis in returns.

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?
(a) $7.22
(b) 7.72
(c) $11.82
(d) $13.42

Question 13

If the stock "gaps" or "jumps," an implied volatility smile results. Which of the following reasons explains why this happens?
(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 14

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?
(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 15

In the preceding question, the state price in the lower node after one month is equal to
(a) 0.4996
(b) 0.5000
(c) 0.5368
(d) 0.5372

Question 16

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?

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

Question 17

Stochastic volatility models are said to incorporate the "leverage" effect. The presence of the leverage 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 18

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.
(a) 0.5016
(b) 0.5023
(c) 0.5044
(d) 0.5117

Question 19

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?
(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 20

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
(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 21

If the implied volatility surface is flat (i.e., all options have the same implied volatility), then
(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 22

The Heston (1993) model generalizes the Black-Scholes setting to one in which
(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 23

By augmenting the geometric Brownian motion process with a Poisson-driven jump process, jump-diffusion models
(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 24

Stochastic volatility models commonly assume
(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 25

GARCH models
(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 26

The Merton (1976) model
(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 27

Local volatility models

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.