Deck 11: Option Pricing: an Introduction

Question 1

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If an investment of a dollar at the risk-free rate returns $1.001668 after one month, what is the price of an Arrow security in the state where the stock price moves up?
(a) 0.4974
(b) 0.4983
(c) 0.5000
(d) 0.5075

Answer 1

D.

Question 2

Which of the following statements best describes the delta of vanilla options?
(a) Call and put deltas are both positive.
(b) The call delta is positive and the put delta is negative.
(c) The call delta is negative and the put delta is positive.
(d) Call and put deltas are both negative.

Answer 2

B.

Question 3

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If the risk-neutral probability of the stock going up is equal to 0.52, what is the price of a one-month call option at a strike price of $102?
(a) $2.84
(b) $3.14
(c) $3.84
(d) $4.14

Answer 3

D.

Question 4

Pricing options in the risk-neutral world implies that
(a) Trading of options is without risk.
(b) All assets are risk-free in this world.
(c) All assets earn the risk-free rate of return.
(d) All investors are risk-neutral.

Question 5

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If an investment of a dollar at the risk-free rate returns $1.001668 after one month, what is the expected gross return of a 100-strike one-month call option under the risk-neutral probabilities?
(a) 1.0010
(b) 1.0017
(c) 1.0100
(d) 1.0167

Question 6

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. The risk-free gross return per time step is 1.001668. If a 98-strike call option is trading at $2, how much arbitrage profit can you make in present value terms?
(a) $2.09
(b) $4.09
(c) $6.09
(d) $12.09

Question 7

In the binomial model, if the stock moves up by a factor $u$ and down by a factor $d$ , and a $1 investment in a risk-free bond returns an amount $R$ per time step, which of the following statements is true in a market that is free from arbitrage?

A) $d < u < R$
B) $\mathcal { U } < R < d$
C) $d < R < u$
D) $R < d < u$

Question 8

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If an investment of a dollar at the risk-free rate returns $1.001668 after one month, and the 98-strike put option is trading at $2, how much arbitrage profit can you make in present value terms?
(a) $1.93
(b) $2.92
(c) $3.93
(d) $8.00

Question 9

In a portfolio insurance strategy, when stock prices drop, the portfolio is rebalanced by
(a) Buying stock and investing at the risk-free rate.
(b) Buying stock and borrowing at the risk-free rate.
(c) Selling stock and investing at the risk-free rate.
(d) Selling stock and borrowing at the risk-free rate.

Question 10

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If the risk-neutral probability of the stock going up is equal to 0.52, what is the one-month forward price of the stock?
(a) $99.48
(b) $100.40
(c) $100.48
(d) $100.52

Question 11

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If an investment of a dollar at the risk-free rate returns $1.001668 after one month, what is the standard deviation of monthly stock return under the risk-neutral probabilities?
(a) 9.93%
(b) 9.97%
(c) 10.00%
(d) 20.00%

Question 12

Which of the following statements best describes the range of possible values of the delta of a call option?

A) Lies between $- 1$ and $+ 1$ .
B) Lies between $0$ and $+ 1$ .
C) Lies between $- 1$ and $0$ .
D) Lies between $- \infty$ and $+ \infty$ .

Question 13

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If the risk-free rate is 0.1668% per month in simple terms, which of the following choices best describes the replicating portfolio for one hundred 99-strike one-month call options?
(a) Long 50 shares of stock and long 50 units of $1 bonds.
(b) Long 55 shares of stock and long 49 units of $1 bonds.
(c) Long 55 shares of stock and short 49 units of $1 bonds.
(d) Long 55 shares of stock and short 50 units of $1 bonds.

Question 14

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If the risk-free rate is 0.1668% per month in simple terms, what is the price of a 99-strike one-month put option?
(a) $4.02
(b) $4.42
(c) $4.49
(d) $4.57

Question 15

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If an investment of a dollar at the risk-free rate returns $1.001668 after one month, how many 100-strike one-month call options and how many units of the stock will be needed to be combined to obtain a riskless $1 portfolio?
(a) Long 0.02 calls and long 0.01 shares.
(b) Long 0.01 calls and long 0.02 shares.
(c) Short 0.01 calls and short 0.02 shares.
(d) Short 0.02 calls and long 0.01 shares.

Question 16

Assuming all else is constant, which of the following statements best describes the delta of a put option?

A) The delta of a put at a stock price $S _ { 1 }$ is greater than the delta of the same put at stock price $S _ { 2 } > S _ { 1 }$ .
B) The delta of a put at a stock price $S _ { 1 }$ is smaller than the delta of the same put at stock price $S _ { 2 } > S _ { 1 }$ .
C) The delta of a put at a stock price $S _ { 1 }$ is equal to the delta of the same put at stock price $S _ { 2 } > S _ { 1 }$ .
D) None of the above is always true for all puts.

Question 17

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $115 or fall to $80 after three months. If risk-free investment has a gross return of 1.005 per three month time-step, what is the best replicating portfolio of one hundred 101-strike three-month put options from the following options?
(a) Short 60 shares of stock and long 69 units of $1 bonds.
(b) Short 65 shares of stock and long 70 units of $1 bonds.
(c) Short 65 shares of stock and short 69 units of $1 bonds.
(d) Short 65 shares of stock and short 70 units of $1 bonds.

Question 18

Suppose that in a binomial model, the stock moves up by a factor $\mathcal { U } = e ^ { \Sigma \sqrt { h } }$ and down by a factor $d = e ^ { - \sigma - \sqrt { h } }$ , where $h _ { c }$ is time in years. Letting $h =$ one month and $u = 1.05$ , what is the annualized volatility $\sigma$ of the stock?

A) 6.93%
B) 10.39%
C) 13.86%
D) 16.90%

Question 19

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. What is the delta of a 99-strike one-month put option?

A) $- 0.50$
B) $- 0.45$
C) $+ 0.45$
D) $+ 0.50$

Question 20

In a one-period binomial model, assume that the current stock price is $100, and that it will rise to $110 or fall to $90 after one month. If the risk-neutral probability of the stock going up or down is equal, what is the one-month risk-free interest rate in continuously-compounded and annualized terms?
(a) 0%
(b) 1%
(c) 2%
(d) 3%

Question 21

You hold a portfolio consisting of 300 calls (short) and 200 puts (long) on a given stock. The delta of the calls is $+ 0.62$ and the delta of the puts is $- 0.57$ . To delta hedge this portfolio, you should

A) Short 74 shares of the stock.
B) Buy 300 shares of the stock.
C) Buy 500 shares of the stock.
D) Buy 74 shares of the stock.

Question 22

The current price of a stock is 40. After one period, it will be worth either 41.80 or 38.20 with probabilities 0.60 and 0.40, respectively. The risk-free rate of interest over this period, in gross terms, is 1.05 (i.e., a dollar invested at the beginning of the period returns 1.05 at the end of the period). The no-arbitrage value of a one-period call option on this stock with a strike of 40
(a) is approximately 1.03.
(b) is approximately 1.81.
(c) can be anything between 0 and 1.80.
(d) does not exist in the given setting.

Question 23

"Portfolio insurance" refers to a trading strategy in which
(a) You insure a portfolio of bonds against default at the aggregate level rather than buying insurance on each separate bond.
(b) You insure a portfolio of bonds against default by buying an insurance policy that pays a fixed sum of money if a trigger number of defaults occurs.
(c) You look to create a synthetic put on a given portfolio by dynamically trading in the assets in the portfolio.
(d) You insure a portfolio of stocks against a fall in their value by buying an insurance policy that pays a fixed sum of money if your portfolio value falls below a trigger level.

Question 24

You are long 300 at-the-money calls on Stock ABC, each with a delta of $+ 0.54$ , and short 200 in-the-money calls on Stock XYZ, each with a delta of $+ 0.66$ . Which of the following statements is most accurate in this context?

A) The aggregate delta of your portfolio is $+ 294$ .
B) The aggregate delta of your portfolio is $+ 30$ .
C) The aggregate delta of your portfolio is $+ 60$ .
D) The notion of an "aggregate portfolio delta" is not meaningful in this setting.

Question 25

You hold a portfolio of European options on a stock that is (i) long 200 at-the-money calls, each with a delta of $0.52$ , (ii) short 200 at-the-money puts, and (iii) long 100 shares of stock. The aggregate delta of your portfolio is

A) 100.
B) 108.
C) 300.
D) Cannot be calculated from the given information.

Question 26

The risk-neutral pricing of options
(a) Assumes investors are neutral to risk.
(b) Assumes some investors are risk-neutral and others risk-loving, but on average they are neutral to risk.
(c) Is a no-arbitrage pricing approach.
(d) Is valid only in a world with risk-neutral investors and not in the real world.