Question 1

The Black-Scholes model is based on a posited stochastic process for stock prices, where the movements in the stock are represented mathematically by a stochastic differential equation (SDE). Which of the following statements is most valid?
A) The SDE is a differential equation that changes over time.
B) The solution to the SDE is a random function of time.
C) The solution to the SDE is a deterministic function of time.
D) The solution to the SDE is the Black-Scholes formula.

Accepted Answer

The solution to the SDE is a random function of time. The SDE is used to model the stochastic and unpredictable movements of stock prices, and the solution to the equation will also be stochastic and unpredictable.

Question 2

Option pricing in continuous time makes use of Wiener processes. Which of the following is not a property of a Wiener process $W _ { t }$ , given $W _ { 0 } = 0$ ?
A) The process has independent increments $W _ { t } - W _ { s }$ , for $t > s$ .
B) Increments are normally distributed.
C) For each $t$ , $W _ { t }$ is normally distributed with mean zero and variance $t ^ { 2 }$ .
D) The process $\left( W _ { t } \right)$ is a symmetric random walk around zero.

Accepted Answer

The correct variance of a Wiener process $W_t$ at time $t$ is $t$, not $t^2$, making option C incorrect. Wiener processes have increments that are normally distributed with mean 0 and variance proportional to the time increment, specifically variance $t$ for $W_t$.

Question 3

Given that $d Y _ { t } = b d W _ { t }$ and $X _ { t } = e ^ { Y t }$ , what is $d X _ { t }$ ?
A) $e ^ { b d W _ { t } }$
B) $X _ { t } e ^ { b ^ { 2 } dt + Y _ { t } d W _ { t } }$
C) $\ln \left( X _ { t } \right) \left[ \frac { 1 } { 2 } b ^ { 2 } d t + b d W _ { t } \right]$
D) $b X _ { t } \left[ \frac { 1 } { 2 } b d t + d W _ { t } \right]$

Accepted Answer

Using Itô's lemma, $dX_t = X_t \left( b dW_t + \frac{1}{2} b^2 dt \right)$.

Question 4

A call option in the Black-Scholes model is a function of the stock price and time, i.e., $C ( S , t )$ . Which of the following statements is valid with regards to the change in the option price over time, i.e., $d C ( S , t )$ ?
A) The expected change $E ( d C )$ is not a function of stock volatility-taking expectations eliminates the Wiener process term.
B) The expected change over time is not a function of the remaining maturity of the option, only of the amount of time over which the change is examined.
C) The expected change in the call price is not a function of the risk-free interest rate but only the growth rate of the stock at the specific point in time.
D) None of the above.

Accepted Answer

The Black-Scholes model shows that the expected change in the option price over time is indeed affected by stock volatility, the remaining maturity of the option, and the risk-free interest rate, contrary to the statements made in options A, B, and C.

Question 5

Given the following Ito process for a stock: $d S _ { t } = 0.2 S _ { t } d t + 0.4 S _ { t } d W$ , what is the expected value of the stock after 3 years if the current price of the stock is $50?
A) $71.67
B) $86.40
C) $91.11
D) $115.82

Accepted Answer

The answer of Given the following Ito process for a...

Question 6

Consider a stock that is trading at $50, has a volatility of 0.5, and pays no dividends. The risk-free rate is 4%. If the beta of the stock is 1.1, what is the beta of a 52-strike, one-year call option on this stock?
A) 0.55
B) 1.1
C) 3.3
D) 4.4

Accepted Answer

The beta of a call option can be significantly higher than the beta of the underlying stock due to the leverage effect of options. Given the parameters, the Black-Scholes model (or similar option pricing models) would show that the delta of the option (sensitivity of the option price to changes in the underlying stock price) is less than 1 but the leverage effect (given by the option's elasticity, which is the percentage change in option price divided by the percentage change in stock price) amplifies the beta of the option compared to the beta of the stock. Without doing the exact calculation, which requires the Black-Scholes formula or numerical methods, we know the beta of the option will be higher than the beta of the stock due to this leverage effect. Among the given choices, 3.3 is a plausible amplified beta for the call option, reflecting the increased sensitivity of the option's value to market movements.

Question 7

Which of the following is not a characteristic of a price process $Y _ { t }$ that follows a geometric Brownian motion (GBM)?
A) $Y _ { t }$ is an exponential function of a linear Ito process $a t + b W _ { t }$ , where $\alpha$ and $b$ are constants.
B) $Y _ { t }$ is normally distributed.
C) $Y _ { t }$ is a continuous process, i.e., there are no market "gaps."
D) $Y _ { t }$ is non-negative.

Accepted Answer

Geometric Brownian Motion (GBM) implies that the logarithm of $Y_t$ is normally distributed, not $Y_t$ itself. $Y_t$ follows a log-normal distribution, meaning that while the logarithm of the process follows a normal distribution, the process itself does not.

Question 8

Given that $d Y _ { t } = b d W _ { t }$ and $X _ { t } = \ln \left( Y _ { t } \right)$ , what is $d X _ { t }$ ?
A) $X _ { t } e ^ { b ^ { 2 } dt + Y _ { t } d W _ { t } }$
B) $\ln \left( X _ { t } \right) \left[ \frac { 1 } { 2 } b ^ { 2 } d t + b d W _ { t } \right]$
C) $- \frac { 1 } { 2 } b ^ { 2 } e ^ { 2 X _ { t } } d t + b e ^ { - X _ { t } } d W _ { t }$ .
D) $b X _ { t } \left[ \frac { 1 } { 2 } b d t + d W _ { t } \right]$

Accepted Answer

To find $dX_t$, we use Itô's Lemma, which for a function $f(Y_t) = \ln(Y_t)$, gives $df(Y_t) = \frac{\partial f}{\partial Y_t} dY_t + \frac{1}{2} \frac{\partial^2 f}{\partial Y_t^2} (dY_t)^2$. Given $dY_t = b dW_t$, we substitute and simplify to find the correct expression for $dX_t$. None of the other options correctly apply Itô's Lemma to the given functions.

Question 9

The fundamental asset pricing partial differential equation (PDE) is used to derive the Black-Scholes formula. Which of the following statements is not true about the fundamental PDE?
A) The PDE depends on the growth rate of the stock.
B) The PDE is independent of the the utility function of the investor buying the option.
C) The PDE is the same for both calls and puts, the only difference being in the boundary conditions.
D) The solution to the PDE is the Black-Scholes option pricing formula.

Accepted Answer

The fundamental asset pricing PDE, used in deriving the Black-Scholes formula, is independent of the stock's growth rate. It instead relies on the risk-free rate, the stock's volatility, and other factors, but not directly on the stock's growth rate.

Question 10

Given that $d Y _ { t } = b d W _ { t }$ and $X _ { t } = Y _ { t } ^ { 3 }$ , what is $d X _ { t }$ ?
A) $3 b X _ { t } ^ { 1 / 3 } \left[ 2 b d t + X _ { t } ^ { 1 / 3 } d W _ { t } \right]$
B) $3 b Y _ { t } \left[ Y _ { t } d t + b d W _ { t } \right]$
C) $b ^ { 3 } d W _ { t } ^ { 3 }$
D) $b ^ { 3 } d W d t$

Accepted Answer

Using Ito's Lemma, for $X_t = f(Y_t) = Y_t^3$, we have $dX_t = f'(Y_t) dY_t + \frac{1}{2} f''(Y_t) (dY_t)^2$. Given $dY_t = b dW_t$, substituting and simplifying yields the expression in option A.

Question 11

Consider a stock that is trading at $50. A six-month at-the-money put option on the stock has a price of 2.21 and a delta of $- 0.40$ . The stock volatility is 20%, the risk-free rate is 4%, and the beta of the stock is 1.1. What is the beta of the put?
A) $- 0.44$
B) $- 1.1$
C) $+ 1.1$
D) $- 9.95$

Accepted Answer

The answer of Consider a stock that is trading at...

Question 12

Option pricing models are based on Ito processes. Which of the following statements best describes Ito processes? Ito processes $Y _ { t }$ are
A) A special case of Wiener processes $W _ { t }$ .
B) Are functions of Wiener processes: in SDE form, $d Y ( t ) = a d t + b d W ( t )$ .
C) Are functions of Wiener processes: in SDE form, $d Y ( t ) = a d t + b d W ( t )$ , with the restriction that $\alpha$ and $b$ have to be constants.
D) Are functions of Wiener processes: in SDE form, $d Y ( t ) = a d t + b d W ( t )$ , with the restriction that $\alpha$ and $b$ have to be constants or functions of time $t$ alone.

Accepted Answer

Ito processes are indeed functions of Wiener processes, described by stochastic differential equations (SDEs) of the form $dY(t) = a dt + b dW(t)$, where $a$ and $b$ can be functions of both the process $Y$ and time $t$, not necessarily constants or functions of time alone.

Question 13

Consider a stock that is trading at $50. A six-month at-the-money call option on the stock has a price of 3.45 and a delta of 0.60. The stock volatility is 20%, the risk-free rate is 4%, and the beta of the stock is 1.1. What is the beta of the call?
A) 0.66
B) 1.1
C) 9.56
D) 15.94

Accepted Answer

The beta of the call is given by the formula:Beta_call = Delta * Beta_stockSubstituting the given values, we get:Beta_call = 0.60 * 1.1 = 0.66Therefore, the correct answer is C) 0.66.

Question 14

Which of the following properties of a put option's beta is most valid?
A) The beta of a put increases as the stock price increases.
B) The beta of a put decreases if the beta of the stock increases.
C) The beta of a put is bounded between $( - 1 , + 1 )$ .
D) The beta is always positive.

Accepted Answer

The beta of a put option is related to the beta of the underlying stock, but in an inverse manner. If the beta of the stock increases, indicating that the stock's returns are more volatile compared to the market, the put option, which gains in value when the stock price decreases, would have its beta decrease as it becomes a more attractive hedge or protective instrument against the stock's volatility.

Question 15

Which of the following is necessary in order to solve the fundamental PDE to obtain the price of a derivative security?
A) The utility function of the representative investor trading in that derivative security.
B) The growth rate of the price of the derivative security.
C) The boundary conditions for the payoffs of the security.
D) A benchmark price of a related derivative security that is correlated with the security being priced.

Accepted Answer

The boundary conditions for the payoffs of the security are necessary in order to solve the fundamental PDE to obtain the price of a derivative security. The other options (A, B, and D) may be relevant factors in determining the price of a derivative security, but they do not directly pertain to the fundamental PDE itself.

Quiz 15: Mathematics of Black-Scholes

The Black-Scholes model is based on a posited stochastic process for stock prices, where the movements in the stock are represented mathematically by a stochastic differential equation (SDE) . Which of the following statements is most valid?

Option pricing in continuous time makes use of Wiener processes. Which of the following is not a property of a Wiener process $W _ { t }$ , given $W _ { 0 } = 0$ ?

Given that $d Y _ { t } = b d W _ { t }$ and $X _ { t } = e ^ { Y t }$ , what is $d X _ { t }$ ?

A call option in the Black-Scholes model is a function of the stock price and time, i.e., $C ( S , t )$ . Which of the following statements is valid with regards to the change in the option price over time, i.e., $d C ( S , t )$ ?

Given the following Ito process for a stock: $d S _ { t } = 0.2 S _ { t } d t + 0.4 S _ { t } d W$ , what is the expected value of the stock after 3 years if the current price of the stock is $50?

Consider a stock that is trading at $50, has a volatility of 0.5, and pays no dividends. The risk-free rate is 4%. If the beta of the stock is 1.1, what is the beta of a 52-strike, one-year call option on this stock?

Which of the following is not a characteristic of a price process $Y _ { t }$ that follows a geometric Brownian motion (GBM) ?

Given that $d Y _ { t } = b d W _ { t }$ and $X _ { t } = \ln \left( Y _ { t } \right)$ , what is $d X _ { t }$ ?

The fundamental asset pricing partial differential equation (PDE) is used to derive the Black-Scholes formula. Which of the following statements is not true about the fundamental PDE?

Given that $d Y _ { t } = b d W _ { t }$ and $X _ { t } = Y _ { t } ^ { 3 }$ , what is $d X _ { t }$ ?

Consider a stock that is trading at $50. A six-month at-the-money put option on the stock has a price of 2.21 and a delta of $- 0.40$ . The stock volatility is 20%, the risk-free rate is 4%, and the beta of the stock is 1.1. What is the beta of the put?

Option pricing models are based on Ito processes. Which of the following statements best describes Ito processes? Ito processes $Y _ { t }$ are

Consider a stock that is trading at $50. A six-month at-the-money call option on the stock has a price of 3.45 and a delta of 0.60. The stock volatility is 20%, the risk-free rate is 4%, and the beta of the stock is 1.1. What is the beta of the call?

Which of the following properties of a put option's beta is most valid?

Which of the following is necessary in order to solve the fundamental PDE to obtain the price of a derivative security?

Quiz 15: Mathematics of Black-Scholes

The Black-Scholes model is based on a posited stochastic process for stock prices, where the movements in the stock are represented mathematically by a stochastic differential equation (SDE) . Which of the following statements is most valid?

Option pricing in continuous time makes use of Wiener processes. Which of the following is not a property of a Wiener process WtW _ { t }Wt​ , given W0=0W _ { 0 } = 0W0​=0 ?

Given that dYt=bdWtd Y _ { t } = b d W _ { t }dYt​=bdWt​ and Xt=eYtX _ { t } = e ^ { Y t }Xt​=eYt , what is dXtd X _ { t }dXt​ ?

A call option in the Black-Scholes model is a function of the stock price and time, i.e., C(S,t) C ( S , t ) C(S,t) . Which of the following statements is valid with regards to the change in the option price over time, i.e., dC(S,t) d C ( S , t ) dC(S,t) ?

Given the following Ito process for a stock: dSt=0.2Stdt+0.4StdWd S _ { t } = 0.2 S _ { t } d t + 0.4 S _ { t } d WdSt​=0.2St​dt+0.4St​dW , what is the expected value of the stock after 3 years if the current price of the stock is $50?