Deck 15: Mathematics of Black-Scholes

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.

Answer 1

B.

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.

Answer 2

For each $t$ , $W _ { t }$ is normally distributed with mean zero and variance $t ^ { 2 }$ .

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) $b X _ { t } \left[ \frac { 1 } { 2 } b d t + d W _ { t } \right]$

Answer 3

$b X _ { t } \left[ \frac { 1 } { 2 } b d t + d W _ { t } \right]$

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.

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

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

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.

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) $b X _ { t } \left[ \frac { 1 } { 2 } b d t + d W _ { t } \right]$

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.

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) $b ^ { 3 } d W d t$

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) $- 9.95$

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.

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

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.

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.