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Options, Futures, and Other Derivatives

John C. Hull

Chapter 14

Wiener processes and Ito’s lemma - all with Video Answers

Educators

Chapter Questions

Problem 1

What would it mean to assert that the temperature at a certain place follows a Markov process? Do you think that temperatures do, in fact, follow a Markov process?

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00:36

Problem 2

Can a trading rule based on the past history of a stock's price ever produce returns that are consistently above average? Discuss.

Anjali Kurse
Anjali Kurse
Numerade Educator
06:52

Problem 3

A company's cash position, measured in millions of dollars, follows a generalized Wiener process with a drift rate of 0.5 per quarter and a variance rate of 4.0 per quarter. How high does the company's initial cash position have to be for the company to have a less than $5 \%$ chance of a negative cash position by the end of 1 year?

Stanley Enemuo
Stanley Enemuo
Numerade Educator
01:15

Problem 4

Variables $X_1$ and $X_2$ follow generalized Wiener processes, with drift rates $\mu_1$ and $\mu_2$ and variances $\sigma_1^2$ and $\sigma_2^2$. What process does $X_1+X_2$ follow if:
(a) The changes in $X_1$ and $X_2$ in any short interval of time are uncorrelated?
(b) There is a correlation $\rho$ between the changes in $X_1$ and $X_2$ in any short time interval?

Ameer Said
Ameer Said
Numerade Educator
01:52

Problem 5

Consider a variable $S$ that follows the process
$$
d S=\mu d t+\sigma d z
$$
For the first three years, $\mu=2$ and $\sigma=3$; for the next three years, $\mu=3$ and $\sigma=4$. If the initial value of the variable is 5 , what is the probability distribution of the value of the variable at the end of year 6 ?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:39

Problem 6

Suppose that $G$ is a function of a stock price $S$ and time. Suppose that $\sigma$ and $\sigma$ are the volatilities of $S$ and $G$. Show that, when the expected return of $S$ increases by $\lambda \sigma$, the growth rate of $G$ increases by $\lambda \sigma$, where $\lambda$ is a constant.

James Kiss
James Kiss
Numerade Educator

Problem 7

Stock $\mathrm{A}$ and stock $\mathrm{B}$ both follow geometric Brownian motion. Changes in any short interval of time are uncorrelated with each other. Does the value of a portfolio consisting of one of stock A and one of stock B follow geometric Brownian motion? Explain your answer.

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Problem 8

The process for the stock price in equation (14.8) is
$$
\Delta S=\mu S \Delta t+\sigma S \epsilon \sqrt{\Delta t}
$$
where $\mu$ and $\sigma$ are constant. Explain carefully the difference between this model and each of the following:
$$
\begin{aligned}
& \Delta S=\mu \Delta t+\sigma \epsilon \sqrt{\Delta t} \\
& \Delta S=\mu S \Delta t+\sigma \epsilon \sqrt{\Delta t} \\
& \Delta S=\mu \Delta t+\sigma S \epsilon \sqrt{\Delta t}
\end{aligned}
$$
Why is the model in equation (14.8) a more appropriate model of stock price behavior than any of these three alternatives?

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Problem 9

It has been suggested that the short-term interest rate $r$ follows the stochastic process
$$
d r=a(b-r) d t+r c d z
$$
where $a, b, c$ are positive constants and $d z$ is a Wiener process. Describe the nature of this process.

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Problem 10

Suppose that a stock price $S$ follows geometric Brownian motion with expected return $\mu$ and volatility $\sigma$ :
$$
d S=\mu S d t+\sigma S d z
$$
What is the process followed by the variable $S^n$ ? Show that $S^n$ also follows geometric Brownian motion.

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Problem 11

Suppose that $x$ is the yield to maturity with continuous compounding on a zero-coupon bond that pays off $$\$ 1$$ at time $T$. Assume that $x$ follows the process
$$
d x=a\left(x_0-x\right) d t+s x d z
$$
where $a, x_0$, and $s$ are positive constants and $d z$ is a Wiener process. What is the process followed by the bond price?

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Problem 12

A stock whose price is $$\$ 30$$ has an expected return of $9 \%$ and a volatility of $20 \%$. In Excel, simulate the stock price path over 5 years using monthly time steps and random samples from a normal distribution. Chart the simulated stock price path. By hitting F9, observe how the path changes as the random samples change.

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Problem 13

Suppose that a stock price has an expected return of $16 \%$ per annum and a volatility of $30 \%$ per annum. When the stock price at the end of a certain day is $$\$ 50$$, calculate the following:
(a) The expected stock price at the end of the next day
(b) The standard deviation of the stock price at the end of the next day
(c) The $95 \%$ confidence limits for the stock price at the end of the next day.

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01:19

Problem 14

A company's cash position, measured in millions of dollars, follows a generalized Wiener process with a drift rate of 0.1 per month and a variance rate of 0.16 per month. The initial cash position is 2.0.
(a) What are the probability distributions of the cash position after 1 month, 6 months, and 1 year?
(b) What are the probabilities of a negative cash position at the end of 6 months and 1 year?
(c) At what time in the future is the probability of a negative cash position greatest?

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator

Problem 15

Suppose that $x$ is the yield on a perpetual government bond that pays interest at the rate of $$\$ 1$$ per annum. Assume that $x$ is expressed with continuous compounding, that interest is paid continuously on the bond, and that $x$ follows the process.
$$
d x=a\left(x_0-x\right) d t+s x d z
$$
where $a, x_0$, and $s$ are positive constants, and $d z$ is a Wiener process. What is the process followed by the bond price? What is the expected instantaneous return (including interest and capital gains) to the holder of the bond?

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Problem 16

If $S$ follows the geometric Brownian motion process in equation (14.6), what is the process followed by
(a) $y=2 S$
(b) $y=S^2$
(c) $y=e^s$
(d) $y=e^{r(T-t)} / S$.
In each case express the coefficients of $d t$ and $d z$ in terms of $y$ rather than $S$.

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Problem 17

A stock price is currently 50. Its expected return and volatility are $12 \%$ and $30 \%$, respectively. What is the probability that the stock price will be greater than 80 in 2 years?

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Problem 18

Stock $A$, whose price is $$\$ 30$$, has an expected return of $11 \%$ and a volatility of $25 \%$. Stock B, whose price is $$\$ 40$$, has an expected return of $15 \%$ and a volatility of $30 \%$. The processes driving the returns are correlated with correlation parameter $\rho$. In Excel, simulate the two stock price paths over 3 months using daily time steps and random samples from normal distributions. Chart the results and by hitting F9 observe how the paths change as the random samples change. Consider values for $\rho$ equal to $0.25,0.75$, and 0.95 .

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