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Options, Futures, and Other Derivatives

John C. Hull

Chapter 15

The Black–Scholes–Merton model - all with Video Answers

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Chapter Questions

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Problem 1

What does the Black-Scholes-Merton stock option pricing model assume about the probability distribution of the stock price in one year? What does it assume about the probability distribution of the continuously compounded rate of return on the stock during the year?

James Kiss
James Kiss
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Problem 2

The volatility of a stock price is $30 \%$ per annum. What is the standard deviation of the percentage price change in one trading day?

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Problem 3

Explain the principle of risk-neutral valuation.

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Problem 4

Calculate the price of a 3-month European put option on a non-dividend-paying stock with a strike price of $$\$ 50$$ when the current stock price is $$\$ 50$$, the risk-free interest rate is $10 \%$ per annum, and the volatility is $30 \%$ per annum.

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01:56

Problem 5

What difference does it make to your calculations in Problem 15.4 if a dividend of $$\$ 1.50$$ is expected in 2 months?

Narayan Hari
Narayan Hari
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07:13

Problem 6

What is implied volatility? How can it be calculated?

Jai Chadha
Jai Chadha
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04:23

Problem 7

A stock price is currently $$\$ 40$$. Assume that the expected return from the stock is $15 \%$ and that its volatility is $25 \%$. What is the probability distribution for the rate of return (with continuous compounding) earned over a 2-year period?

Manasvee Singh
Manasvee Singh
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Problem 8

A stock price follows geometric Brownian motion with an expected return of $16 \%$ and a volatility of $35 \%$. The current price is $$\$ 38$$.
(a) What is the probability that a European call option on the stock with an exercise price of $$\$ 40$$ and a maturity date in 6 months will be exercised?
(b) What is the probability that a European put option on the stock with the same exercise price and maturity will be exercised?

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13:39

Problem 9

Using the notation in this chapter, prove that a $95 \%$ confidence interval for $S_T$ is between $S_0 e^{\left(\mu-\sigma^2 / 2\right) T-1.96 \sigma \sqrt{T}}$ and $S_0 e^{\left(\mu-\sigma^2 / 2\right) T+1.96 \sigma \sqrt{T}}$.

Abhirup Pal
Abhirup Pal
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00:51

Problem 10

A portfolio manager announces that the average of the returns realized in each year of the last 10 years is $20 \%$ per annum. In what respect is this statement misleading?

Tanishq Gupta
Tanishq Gupta
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Problem 11

Assume that a non-dividend-paying stock has an expected return of $\mu$ and a volatility of $\sigma$. An innovative financial institution has just announced that it will trade a security that pays off a dollar amount equal to $\ln S_T$ at time $T$, where $S_T$ denotes the value of the stock price at time $T$.
(a) Use risk-neutral valuation to calculate the price of the security at time $t$ in terms of the stock price, $S$, at time $t$. The risk-free rate is $r$.
(b) Confirm that your price satisfies the differential equation (15.16).

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Problem 12

Consider a derivative that pays off $S_T^n$ at time $T$, where $S_T$ is the stock price at that time. When the stock pays no dividends and its price follows geometric Brownian motion, it can be shown that its price at time $t(t \leqslant T)$ has the form $h(t, T) S^n$, where $S$ is the stock price at time $t$ and $h$ is a function only of $t$ and $T$.
(a) By substituting into the Black-Scholes-Merton partial differential equation, derive an ordinary differential equation satisfied by $h(t, T)$.
(b) What is the boundary condition for the differential equation for $h(t, T)$ ?
(c) Show that $h(t, T)=e^{\left[0.5 \sigma^2 \pi(n-1)+r(n-1)\right](T-t)}$, where $r$ is the risk-free interest rate and $\sigma$ is the stock price volatility.v

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Problem 13

What is the price of a European call option on a non-dividend-paying stock when the stock price is $$\$ 52$$, the strike price is $$\$ 50$$, the risk-free interest rate is $12 \%$ per annum, the volatility is $30 \%$ per annum, and the time to maturity is 3 months?

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Problem 14

What is the price of a European put option on a non-dividend-paying stock when the stock price is $$\$ 69$$, the strike price is $$\$ 70$$, the risk-free interest rate is $5 \%$ per annum, the volatility is $35 \%$ per annum, and the time to maturity is 6 months?

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Problem 15

Consider an American call option on a stock. The stock price is $$\$ 70$$, the time to maturity is 8 months, the risk-free rate of interest is $10 \%$ per annum, the exercise price is $$\$ 65$$, and the volatility is $32 \%$. A dividend of $$\$ 1$$ is expected after 3 months and again after 6 months. Show that it can never be optimal to exercise the option on either of the two dividend dates. Use DerivaGem to calculate the price of the option.

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Problem 16

A call option on a non-dividend-paying stock has a market price of $$\$ 2 \frac{1}{2}$$. The stock price is $$\$ 15$$, the exercise price is $$\$ 13$$, the time to maturity is 3 months, and the risk-free interest rate is $5 \%$ per annum. What is the implied volatility?

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Problem 17

With the notation used in this chapter:
(a) What is $N^{\prime}(x)$ ?
(b) Show that $S N^{\prime}\left(d_1\right)=K e^{-r(T-t)} N^{\prime}\left(d_2\right)$, where $S$ is the stock price at time $t$ and
$$
d_1=\frac{\ln (S / K)+\left(r+\sigma^2 / 2\right)(T-t)}{\sigma \sqrt{ } T-t}, \quad d_2=\frac{\ln (S / K)+\left(r-\sigma^2 / 2\right)(T-t)}{\sigma \sqrt{ } T-t}
$$
(c) Calculate $\partial d_1 / \partial S$ and $\partial d_2 / \partial S$.
(d) Show that when $c=S N\left(d_1\right)-K e^{-r(T-t)} N\left(d_2\right)$, it follows that
$$
\frac{\partial c}{\partial t}=-r K e^{-r(T-t)} N\left(d_2\right)-S N^{\prime}\left(d_1\right) \frac{\sigma}{2 \sqrt{T-t}}
$$
where $c$ is the price of a call option on a non-dividend-paying stock.
(e) Show that $\partial c / \partial S=N\left(d_1\right)$.
(f) Show that $c$ satisfies the Black-Scholes-Merton differential equation.
(g) Show that $c$ satisfies the boundary condition for a European call option, i.c., that $c=\max (S-K, 0)$ as $t \rightarrow T$.

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Problem 18

Show that the Black-Scholes-Merton formulas for call and put options satisfy put-call parity.

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Problem 19

A stock price is currently $$\$ 50$$ and the risk-free interest rate is $5 \%$. Use the DerivaGem software to translate the following table of European call options on the stock into a table of implied volatilities, assuming no dividends. Are the option prices consistent with the assumptions underlying Black-Scholes-Merton?
$$
\begin{array}{cccc}
\hline & {3}{c}{\text { Maturity (months) }} \\
{ 2 - 4 } \text { Strike price }(\$) & 3 & 6 & 12 \\
\hline 45 & 7.0 & 8.3 & 10.5 \\
50 & 3.7 & 5.2 & 7.5 \\
55 & 1.6 & 2.9 & 5.1 \\
\hline
\end{array}
$$

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Problem 20

Explain carefully why Black's approach to evaluating an American call option on a dividend-paying stock may give an approximate answer even when only one dividend is anticipated. Does the answer given by Black's approach understate or overstate the true option value? Explain your answer.

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Problem 21

Consider an American call option on a stock. The stock price is $$\$ 50$$, the time to maturity is 15 months, the risk-free rate of interest is $8 \%$ per annum, the exercise price is $$\$ 55$$, and the volatility is $25 \%$. Dividends of $$\$ 1.50$$ are expected in 4 months and 10 months. Show that it can never be optimal to exercise the option on either of the two dividend dates. Calculate the price of the option.

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Problem 22

Show that the probability that a European call option will be exercised in a risk-neutral world is, with the notation introduced in this chapter, $N\left(d_2\right)$. What is an expression for the value of a derivative that pays off $$\$ 100$$ if the price of a stock at time $T$ is greater than $K$ ?

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Problem 23

Use the result in equation (15.17) to determine the value of a perpetual American put option on a non-dividend-paying stock with strike price $K$ if it is exercised when the stock price equals $H$ where $H<K$. Assume that the current stock price $S$ is greater than $H$. What is the value of $H$ that maximizes the option value? Deduce the value of a perpetual American put with strike price $K$.

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Problem 24

A company has an issue of executive stock options outstanding. Should dilution be taken into account when the options are valued? Explain your answer.

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Problem 25

A company's stock price is $$\$ 50$$ and 10 million shares are outstanding. The company is considering giving its employees 3 million at-the-money 5-year call options. Option exercises will be handled by issuing more shares. The stock price volatility is $25 \%$, the 5-year risk-free rate is $5 \%$, and the company does not pay dividends. Estimate the cost to the company of the employee stock option issue.

Rashmi Sinha
Rashmi Sinha
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Problem 26

If the volatility of a stock is $18 \%$ per annum, estimate the standard deviation of the pereentage price change in (a) 1 day, (b) 1 week, and (c) 1 month.

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Problem 27

A stock price is currently $$\$50$$. Assume that the expected return from the stock is 18%
and its volatility is 30%. What is the probability distribution for the stock price in 2 years? Calculate the mean and standard deviation of the distribution. Determine the 95% confidence interval. Determine the
95% confidence interval.

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Problem 28

Suppose that observations on a stock price (in dollars) at the end of each of 15 consecutive
weeks are as follows:
30:2; 32:0; 31:1; 30:1; 30:2; 30:3; 30:6; 33:0; 32:9; 33:0; 33:5; 33:5; 33:7; 33:5; 33:2
Estimate the stock price volatility. What is the standard error of your estimate?

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Problem 28

Suppose that observations on a stock price (in dollars) at the end of each of 15 consecutive weeks are as follows: 30.2, 32.0, 31.1, 30,1. 30.2, 30.3, 30.6, 33.0, 32.9, 33.0, 33.5, 33.5, 33.7, 33.5, 33.2 Estimate the stock price volatility. What is the standard error of your estimate?

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Problem 29

A financial institution plans to offer a security that pays off a dollar amount equal to $S_T^2$ at time $T$, where $S_T$ is the price at time $T$ of a stock that pays no dividends.
(a) Use risk-neutral valuation to calculate the price of the security at time $t$ in terms of the stock price $S$ at time $t$ and other variables.

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Problem 30

Consider an option on a non-dividend-paying stock when the stock price is $$\$ 30$$, the exercise price is $$\$ 29$$, the risk-free interest rate is $5 \%$, the volatility is $25 \%$ per annum, and the time to maturity is 4 months.
(a) What is the price of the option if it is a European call?
(b) What is the price of the option if it is an American call?

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Problem 31

Assume that the stock in Problem 15.30 is due to go ex-dividend in $1 \frac{1}{2}$ months. The expected dividend is 50 cents.
(a) What is the price of the option if it is a European call?
(b) What is the price of the option if it is a European put?
(c) If the option is an American call, are there any circumstances under which it will be exercised early?

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Problem 32

Consider an American call option when the stock price is $$\$ 18$$, the exercise price is $$\$ 20$$, the time to maturity is 6 months, the volatility is $30 \%$ per annum, and the risk-free interest rate is $10 \%$ per annum. Two equal dividends are expected during the life of the option with ex-dividend dates at the end of 2 months and 5 months. Assume the dividends are 40 cents. Use Black's approximation and the DerivaGem software to value the option. How high can the dividends be without the American option being worth more than the corresponding European option?

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Problem 33

The appendix derives the key result
Show that
$$
\begin{gathered}
E[\max (V-K, 0)]=E(V) N\left(d_1\right)-K N\left(d_2\right) \\
E[\max (K-V, 0)]=K N\left(-d_1\right)-E(V) N\left(-d_2\right)
\end{gathered}
$$
and use this to derive the Black-Scholes-Merton formula for the price of a European put option on a non-dividend-paying stock.

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