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.

Accepted Answer

The GARCH model includes terms for both past squared residuals and past variances, allowing it to capture volatility clustering more effectively and deliver more persistent volatility compared to the ARCH model, which only includes past squared residuals.

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.

Accepted Answer

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) $\gamma \leq 0$ .

Accepted Answer

The leverage effect refers to the phenomenon where the volatility of a stock's price is inversely related to the price of the stock itself. In the CEV model, the parameter $\gamma$ (represented as $S^{\prime\prime}$ in the equation) controls how volatility changes with stock price. When $0 < \gamma < 1$, it implies that as the stock price decreases, the volatility increases, which mimics the leverage effect observed in financial markets.

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.

Accepted Answer

In a stochastic volatility model, negative correlation between the stock return and changes in volatility can result in negative skewness in the return distribution. This is because when volatility decreases, stock returns are more likely to have large positive deviations from the mean, while when volatility increases, stock returns are more likely to have small negative deviations from the mean. This leads to a distribution that is skewed to the left. Mean-reversion in the volatility process can also contribute to negative skewness, but this is not the only factor. Therefore, the correct answer is B.

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.

Accepted Answer

Put-call parity is a fundamental principle in options pricing that relates the price of a European call option to a European put option with the same strike price and expiration date. The presence of jumps in the stock price, as modeled by the Merton jump-diffusion model, does not invalidate the use of put-call parity. This principle is based on arbitrage arguments and holds as long as the options are European and no arbitrage opportunities exist, regardless of the specific dynamics of the underlying asset's 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.

Accepted Answer

The answer of An option-trading firm is using the Black-Scholes...

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.

Accepted Answer

All three assumptions (normal distribution of log-returns, constant volatility, and continuous price evolution) are commonly violated in the real world. In reality, asset returns often exhibit fat tails and skewness, volatility is generally not constant, and prices can experience large gaps or jumps due to news events or other external factors. These violations can lead to errors and inaccuracies in the Black-Scholes model and affect the validity of the formula's results.

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.

Accepted Answer

The answer of The Black-Scholes model is time-inconsistent in the...

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

Accepted Answer

The answer of The current stock price is $100. A...

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

Accepted Answer

The answer of A Wall Street trading firm is using...

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.

Accepted Answer

The asymmetric GARCH model was developed to capture the skewness in returns, which is not captured in the standard GARCH formulation. It allows for different effects on volatility from positive and negative returns, which can be useful in modeling financial data where negative returns tend to have a greater impact on volatility than positive returns. Option A refers to mean reversion, which is typically captured in models like the ARIMA family of models. Option B refers to persistence, which is already captured in the standard GARCH model by including lagged squared error terms in the conditional variance equation. Option D refers to kurtosis, which can be addressed by using alternative models such as the EGARCH or TGARCH models.

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

Accepted Answer

To solve for the price of the put option, we need to construct a one-period jump-to-default tree. 
At time t=3 months, the stock price can either go up or down. Assuming the up probability is p and the down probability is 1-p, we can calculate p from the risk-neutral probability of default as follows: 
(1-p) * (1+0.3*sqrt(0.25)) + p * (1-0.3*sqrt(0.25)) = 1-0.1 = 0.9 
Solving for p, we get p = 0.4273. 
At the end of three months, the stock price can either be $139.29 (up-up) or $84.86 (down-down) with probabilities p^2 and (1-p)^2, respectively. The option payoff at expiration is the maximum of the strike price and the stock price minus the strike price. At-the-money means the strike price is $100. 
Thus, the option values at the final nodes of the tree are: 
$39.29 for the up-up node 
$0 for the down-up and up-down nodes 
$15.14 for the down-down node 
We can now work backwards from the final nodes to calculate the option values at the middle node (i.e., the current stock price of $100). Using risk-neutral valuation, the option value at the middle node is: 
[P(up) * $39.29 + P(down) * $15.14]/(1+r) = [0.4273^2*$39.29 + (1-0.4273)^2*$15.14]/(1+0.02*0.25) = $11.82 
Therefore, the price of the three-month at-the-money put option is $11.82. 
The correct answer is thus C.

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.

Accepted Answer

Jumps in stock prices increase the kurtosis of the distribution, which affects the implied volatility smile by indicating a higher probability of extreme outcomes.

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.

Accepted Answer

GARCH implies persistence in changes in volatility, which can lead to non-zero skewness and kurtosis in the return distribution. This can better fit the observed options smile, where implied volatilities for out-of-the-money options are higher than for at-the-money options. Option prices are sensitive to changes in volatility, and incorporating this feature of volatility is important for accurately pricing options. Option prices also reflect the market's expectation for future volatility, and the persistence in volatility changes implied by GARCH can capture this expectation.

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

Accepted Answer

The state price in the lower node after one month can be calculated using the risk-neutral probabilities and discount factors specific to the model in question, such as the Binomial Options Pricing Model. Without the specific details of the model parameters (interest rates, stock prices, up and down factors, etc.), it's not possible to provide a detailed calculation here. However, given the options, if C is correct, it implies that the calculation based on those parameters results in a state price of 0.5368 in the lower node after one month.

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%

Accepted Answer

The probability of a jump-to-default in any month is given by the Poisson distribution with $\lambda = 0.05$. The probability of a jump-to-default in any month is given by:$$P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$$where $k$ is the number of jumps-to-default in a month.For $k = 1$, we have:$$P(X = 1) = \frac{e^{-0.05} 0.05^1}{1!} = 0.004158$$Therefore, the probability of a jump-to-default in any month is 0.4158%.

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.

Accepted Answer

The leverage effect refers to the phenomenon where stock price decreases are associated with increases in volatility, leading to a left skew in the options smile. This is because a drop in stock prices often results in higher leverage (debt-to-equity ratio) for a firm, which increases the firm's risk and thus its stock volatility.

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

Accepted Answer

The answer of For the same problem in the preceding...

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.

Accepted Answer

The answer of If the volatility of a stock is...

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.

Accepted Answer

The constant volatility assumption used by the firm would lead to an underestimation of the implied volatility for out-of-the-money puts relative to those that are at-the-money. This is because Black-Scholes assumes that the implied volatility is constant for all strikes, but in reality, the implied volatility is higher for out-of-the-money options when the market anticipates sharp negative gapping behavior. Therefore, the firm would be underpricing the out-of-the-money puts relative to those that are at-the-money.

Quiz 16: Beyond Black-Scholes

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 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

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

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

The asymmetric GARCH model was developed to mimic the following feature of other models that is not captured in the standard GARCH formulation:

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

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?

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

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?

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

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?

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?

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 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?