Deck 14: The Black-Scholes Model

Question 1

The current price of a stock is $100. What is the Black-Scholes model price of a six-month put option at strike $98, given an interest rate of 2% and a dividend rate of 1%? The volatility is 45%.
(a) $11.02
(b) $11.22
(c) $11.68
(d) $11.73

Answer 1

B.

Question 2

Which of the following is not an assumption underlying the Black-Scholes model?
(a) The rate of interest is constant.
(b) The dividend rate must be less than the interest rate.
(c) Stock volatility is constant.
(d) There are no taxes and transactions costs.

Answer 2

B.

Question 3

A stock is currently trading at $S _ { 0 } = 25.85$ . It is not expected to pay dividends over the next year. You price a six-month call option on the stock with a strike of $K = 15$ using the Black-Scholes model and find the following numbers: $$\begin{array} { r l r l } d _ { 1 } & = 2.115 & d _ { 2 } & = 1.832 \\N \left( d _ { 1 } \right) & = 0.983 & N \left( d _ { 2 } \right) & = 0.967\end{array}$$ Given this information, the delta of the call is

A) 0.967
B) 0.983
C) 1.832
D) 2.115

Answer 3

0.983

Question 4

A put option can be replicated by holding a position in stock and bonds, i.e., $P = B + \Delta S$ where $\Delta$ is the delta of the put option. Comparing the replication formula to the Black-Scholes formula, and assuming no dividends, what can you say about the delta of the option?

A) The delta is equal to the probability that the option will end up in the money.
B) The delta is equal to $N \left( d _ { 1 } \right)$ .
C) The delta is equal to $N \left( - d _ { 1 } \right)$ .
D) The delta is the long position in stock needed to replicate the option.

Question 5

If the Black-Scholes call delta (assume a non-dividend-paying stock) is equal to 1/2, then which of the following statements is most valid?

A) The call is at-the-money, i.e., $S = K$ .
B) The call is at-the-money forward, i.e., $S = P V ( K )$ .
C) $P V ( K ) > S$ .
D) $P V ( K ) < S$ .

Question 6

A stock is currently trading at $S _ { 0 } = 26.15$ . It is not expected to pay dividends over the next year. You price a one-month put option on the stock with a strike of $K = 25$ using the Black-Scholes model and find the following numbers: $$\begin{array} { r l r l } d _ { 1 } & = 0.717 & d _ { 2 } & = 0.645 \\N \left( d _ { 1 } \right) & = 0.763 & N \left( d _ { 2 } \right) & = 0.740\end{array}$$ Given this information, the delta of the put is

A) $- 0.355$ .
B) $- 0.283$ .
C) $- 0.260$ .
D) $- 0.237$ .

Question 7

Let $E ^ { * } ( . )$ denote risk-neutral expectations in the Black-Scholes setting. Then, the Black-Scholes formula may calculated by taking the following expectation:

A) $e ^ { - r T } E ^ { * } \left[ S _ { T } - K \right] N \left( d _ { 2 } \right)$
B) $e ^ { - r T } E ^ { *} \left[ S _ { T } - K \right] N \left( d _ { 1 } \right)$
C) $e ^ { - r T } E ^ { * } \left[ \max \left( 0 , S _ { T } \right) \right] - K$ .
D) $e ^ { - r T } E ^ { * } \left[ \max \left( 0 , S _ { T } - K \right) \right]$ .

Question 8

A stock is currently trading at $S _ { 0 } = 21.30$ . It is not expected to pay dividends over the next year. You price a one-month put option on the stock with a strike of $K = 22.50$ using the Black-Scholes model and find the following numbers: $$\begin{array} { r l r l } d _ { 1 } & = - 0.666 & d _ { 2 } & = - 0.738 \\N \left( d _ { 1 } \right) & = 0.253 & N \left( d _ { 2 } \right) & = 0.230\end{array}$$ Given this information, the probability of the put finishing out-of-the-money is

A) 25.3%
B) 23.0%
C) 74.7%
D) 77.0%

Question 9

The implied volatility of an option
(a) Is the volatility that would have to be plugged into a given option-pricing model to obtain the observed market price.
(b) Can only be calculated using the Black-Scholes model.
(c) Is the volatility implied by the underlying stock price over a given historical period.
(d) Is equal to the volatility that will actually be realized over the life of the option.

Question 10

Which of the following quantities associated with equity option pricing is model dependent?
(a) Stock price.
(b) Implied volatility.
(c) Interest rate.
(d) Dividend rate.

Question 11

The Black-Scholes model differs from the binomial in that
(a) The mathematics it requires is much simpler.
(b) It provides closed-form solutions for option prices, so can price European options faster than the binomial model.
(c) It can handle stochastic interest rates more efficiently than the binomial.
(d) It was developed after the binomial model and is therefore more current.

Question 12

A stock is currently trading at $S _ { 0 } = 25.85$ . It is not expected to pay dividends over the next year. You price a six-month put option on the stock with a strike of $K = 15$ using the Black-Scholes model and find the following numbers: $$\begin{array} { r l r l } d _ { 1 } & = 2.115 & d _ { 2 } & = 1.832 \\N \left( d _ { 1 } \right) & = 0.983 & N \left( d _ { 2 } \right) & = 0.967\end{array}$$ Given this information, the risk-neutral probability of the put finishing in the money is

A) 1.7%
B) 1.832%
C) 2.115%
D) 3.3%

Question 13

The current price of a stock is $100. What is the Black-Scholes model price of a six-month call option at strike $101, given an interest rate of 2% and a dividend rate of 1%? The volatility is 25%.
(a) $6.30
(b) $6.52
(c) $6.56
(d) $6.78

Question 14

The Black-Scholes formula is based on
(a) A field of mathematics known as Brownian geometry.
(b) An assumption that stock prices are distributed normally.
(c) An assumption that continuously-compounded stock returns are distributed normally.
(d) A geometric process that allows for both positive and negative stock prices.

Question 15

A stock is currently trading at $S _ { 0 } = 21.30$ . It is not expected to pay dividends over the next year. You price a one-month put option on the stock with a strike of $K = 22.50$ using the Black-Scholes model and find the following numbers: $$\begin{array} { r l r l } d _ { 1 } & = - 0.666 & d _ { 2 } & = - 0.738 \\N \left( d _ { 1 } \right) & = 0.253 & N \left( d _ { 2 } \right) & = 0.230\end{array}$$ Given this information, the delta of the put is

A) $- 0.770$ .
B) $- 0.334$ .
C) $- 0.262$ .
D) $- 0.747$ .

Question 16

In the Black-Scholes setting, the prices of American options
(a) Cannot be determined because the model assumes away the human impatience that leads to early exercise of options.
(b) Cannot be determined because the model does not describe the intermediate stock prices that are required to decide on whether to exercise an option early.
(c) Cannot be determined because the model assumes a continuous-time process, whereas to determine early-exercise you need a discrete-time model.
(d) Cannot be determined in closed-form.

Question 17

A call option can be replicated by holding a position in stock and shorting bonds, i.e., $C = \Delta S - \bar { B }$ where $\Delta$ is the delta of the call option. Comparing the replication formula to the Black-Scholes formula (assume a non-dividend-paying stock), what can you say about the delta of the option?

A) The delta is equal to the probability that the option will end up in the money.
B) The delta is equal to $N \left( d _ { 1 } \right)$ .
C) The delta is the short position in stock needed to replicate the option.
D) There is insufficient information to say anything about the delta.

Question 18

The current price of a stock is $100. Consider the Black-Scholes model price of a six-month call option at strike $101, given an interest rate of 2% and a dividend rate of 1%? The volatility is 25%. What is the risk-neutral probability of the option ending up in the money?
(a) 0.45
(b) 0.48
(c) 0.49
(d) 0.50

Question 19

The current price of a stock is $100. Consider the Black-Scholes model price of a six-month call option at strike $101, given an interest rate of 2% and a dividend rate of 1%? The volatility is 25%. What is the real-world (physical) probability of the option ending up in the money if the growth rate of the stock is expected to be 5% per year?
(a) 0.45
(b) 0.48
(c) 0.49
(d) 0.50

Question 20

A stock is currently trading at $S _ { 0 } = 26.15$ . It is not expected to pay dividends over the next year. You price a one-month call option on the stock with a strike of $K = 25$ using the Black-Scholes model and find the following numbers: $$\begin{array} { r l r l } d _ { 1 } & = 0.717 & d _ { 2 } & = 0.645 \\N \left( d _ { 1 } \right) & = 0.763 & N \left( d _ { 2 } \right) & = 0.740\end{array}$$ Given this information, the risk-neutral probability of the call finishing in the money is

A) 71.7%
B) 64.5%
C) 76.3%
D) 74.0%

Question 21

A variance swap is an option on the realized variance of a stock's return over a defined period of time. A variance swap may be replicated using
(a) A static position in forwards and options on the stock.
(b) A static position in stock, forwards and options.
(c) A dynamic position in forwards and options on the stock.
(d) None of the above

Question 22

The three-month S&P 500 futures contract is trading at a level of 1250. The rate of interest is 2%. The average rate of dividends for stocks in the index is 3%. Index volatility is 20%. What is the Black-Scholes price of a one-year at-the-money put option on the futures?
(a) $97.34
(b) $97.60
(c) $98.33
(d) $99.12

Question 23

The implied volatility skew observed in stock indices cannot be attributed to which of the following reasons?
(a) The distribution of index returns is non-normal.
(b) The skewness of index returns is much greater than zero.
(c) The excess kurtosis of index returns is much greater than zero.
(d) There is crash risk in the index.

Question 24

The dollar-euro exchange rate is $1.30/€. The dollar interest rate is 2% and the euro interest rate is 3%. What is the price of a six-month call option to buy euros at a strike price of $1.25/€? The volatility of the $/€ exchange rate is 40%.
(a) $0.82
(b) $0.86
(c) $0.96
(d) $1.06

Question 25

Consider a Black-Scholes setting. When a call option is deep in-the-money, a decrease in interest rates results in, ceteris paribus,
(a) A decrease in the delta of the option.
(b) A decrease in the insurance value of the option.
(c) An increase in the intrinsic value of the option.
(d) A decrease in the time value of the option.

Question 26

Most major stock indices, like the S&P 500, exhibit an implied volatility skew. This means if we consider three put options, $P \left( K _ { 1 } \right) , P \left( K _ { 2 } \right) , P \left( K _ { 3 } \right)$ at strikes $K _ { 1 } < K _ { 2 } < K _ { 3 } \leq S$ (where $S$ is the current index level) and the options have implied volatilities $\sigma _ { 1 } , \sigma _ { 2 } , \sigma _ { 3 }$ , respectively, then the most likely pattern is

A) $\sigma _ { 1 } < \sigma _ { 2 } < \sigma _ { 3 }$ .
B) $\sigma _ { 1 } = \sigma _ { 2 } < \sigma _ { 3 }$ .
C) $\sigma _ { 1 } = \sigma _ { 2 } = \sigma _ { 3 }$ .
D) $\sigma _ { 1 } > \sigma _ { 2 } > \sigma _ { 3 }$ .

Question 27

The VIX is an implied volatility index for roughly what maturity?
(a) One month.
(b) Three months.
(c) Six months.
(d) Twelve months.

Question 28

The Black-Scholes price of a three-month 50-strike put option is $0.75. The stock is trading at $49. Given an interest rate of 2%, and no dividends, what is the implied volatility of the stock extracted from this option?
(a) 0.55
(b) 0.66
(c) 0.77
(d) 0.88

Question 29

A volatility swap is an option on the realized standard deviation of a stock's return over a defined period of time. A volatility swap may be replicated using
(a) A static position in forwards and options on the stock.
(b) A static position in stock, forwards and options.
(c) A dynamic position in forwards and options on the stock.
(d) None of the above

Question 30

Consider a Black-Scholes setting. When a call option is deep in-the-money, an increase in volatility results in, ceteris paribus,
(a) A decrease in the delta of the option.
(b) A decrease in the insurance value of the option.
(c) An increase in the intrinsic value of the option.
(d) An increase in the time value of the option.

Question 31

Consider a call option on a stock that pays dividends at the rate $q > 0$ . Which of the following statements is most valid for the Black-Scholes model?

A) The call is worth more than a call on an equivalent stock that pays no dividends.
B) The probably of the option finishing up in the money is greater than that of a call on an equivalent stock that pays no dividends.
C) The time value of the call is greater than that of a call on an equivalent stock that pays no dividends.
D) None of the above.

Question 32

The S&P 500 index is trading at a level of 1200. The rate of interest is 2%. The average rate of dividends for stocks in the index is 3%. Index volatility is 20%. What is the Black-Scholes price of a one-year at-the-money put option on the index?
(a) $98.90
(b) $99.10
(c) $99.20
(d) $99.25