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Options, Futures, and Other Derivatives

John C. Hull

Chapter 28

Martingales and measures - all with Video Answers

Educators


Chapter Questions

Problem 1

How is the market price of risk defined for a variable that is not the price of an investment asset?

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01:44

Problem 2

Suppose that the market price of risk for gold is zero. If the storage costs are $1 \%$ per annum and the risk-free rate of interest is $6 \%$ per annum, what is the expected growth rate in the price of gold? Assume that gold provides no income.

Achintya Suden
Achintya Suden
Numerade Educator

Problem 3

Consider two securities both of which are dependent on the same market variable. The expected returns from the securities are $8 \%$ and $12 \%$. The volatility of the first security is $15 \%$. The instantaneous risk-free rate is $4 \%$. What is the volatility of the second security?

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01:33

Problem 4

An oil company is set up solely for the purpose of exploring for oil in a certain small area of Texas. Its value depends primarily on two stochastic variables: the price of oil and the quantity of proven oil reserves. Discuss whether the market price of risk for the second of these two variables is likely to be positive, negative, or zero.

Edward Adams
Edward Adams
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Problem 5

Deduce the differential equation for a derivative dependent on the prices of two nondividend-paying traded securities by forming a riskless portfolio consisting of the derivative and the two traded securities.

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02:54

Problem 6

Suppose that an interest rate $x$ follows the process $d x=a\left(x_0-x\right) d t+c \sqrt{x} d z$ where $a, x_0$, and $c$ are positive constants. Suppose further that the market price of risk for $x$ is $\lambda$. What is the process for $x$ in the traditional risk-neutral world?

Amany Waheeb
Amany Waheeb
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Problem 7

Prove that, when the security $f$ provides income at rate $q$, equation (28.9) becomes $\mu+q-r=\lambda \sigma$. (Hint: Form a new security $f^*$ that provides no income by assuming that all the income from $f$ is reinvested in $f$.)

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Problem 8

Show that when $f$ and $g$ provide income at rates $q_f$ and $q_g$, respectively, equation (28.15) becomes
$f_0=g_0 e^{\left(q_f-q_g\right) T} E_g\left(\frac{f_T}{g_T}\right)$
(Hint: Form new securities $f^*$ and $g^*$ that provide no income by assuming that all the income from $f$ is reinvested in $f$ and all the income in $g$ is reinvested in $g$.)

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Problem 9

"The expected future value of an interest rate in a risk-neutral world is greater than it is in the real world." What does this statement imply about the market price of risk for (a) an interest rate and (b) a bond price. Do you think the statement is likely to be true? Give reasons.

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Problem 10

The variable $S$ is an investment asset providing income at rate $q$ measured in currency A. It follows the process
$d S=\mu_S S d t+\sigma_S S d z$
in the real world. Defining new variables as necessary, give the process followed by $S$, and the corresponding market price of risk, in:
(a) A world that is the traditional risk-neutral world for currency A
(b) A world that is the traditional risk-neutral world for currency $\mathrm{B}$
(c) A world defined by a numeraire equal to a zero-coupon currency A bond maturing at time $T$
(d) A world defined by a numeraire equal to a zero-coupon currency B bond maturing at time $T$.

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00:40

Problem 11

Explain the difference between the way a forward interest rate is defined and the way the forward values of other variables such as stock prices, commodity prices, and exchange rates are defined.

Rae Xin
Rae Xin
Numerade Educator

Problem 12

Prove the result in Section 28.5 that when
and $\begin{aligned} d f & =\left[r+\sum_{i=1}^n \lambda_i \sigma_{f, i}\right] f d t+\sum_{i=1}^n \sigma_{f, i} f d z_i \\ d g & =\left[r+\sum_{i=1}^n \lambda_i \sigma_{g, i}\right] g d t+\sum_{i=1}^n \sigma_{g, i} g d z_i\end{aligned}$
with the $d z_i$ uncorrelated, $f / g$ is a martingale for $\lambda_i=\sigma_{g, i}$. (Hint: Start by using equation (14A.11) to get the processes for $\ln f$ and $\ln g$.)

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Problem 13

Show that when $w=h / g$ and $h$ and $g$ are each dependent on $n$ Wiener processes, the $i$ th component of the volatility of $w$ is the $i$ th component of the volatility of $h$ minus the $i$ th component of the volatility of $g$. (Hint: Start by using equation (14A.11) to get the processes for $\ln g$ and $\ln h$.)

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00:49

Problem 14

"If $X$ is the expected value of a variable, $X$ follows a martingale." Explain this statement.

Hossam Mohamed
Hossam Mohamed
Numerade Educator

Problem 15

A security's price is positively dependent on two variables: the price of copper and the yen/dollar exchange rate. Suppose that the market price of risk for these variables is 0.5 and 0.1 , respectively. If the price of copper were held fixed, the volatility of the security would be $8 \%$ per annum; if the yen/dollar exchange rate were held fixed, the volatility of the security would be $12 \%$ per annum. The risk-free interest rate is $7 \%$ per annum. What is the expected rate of return from the security? If the two variables are uncorrelated with each other, what is the volatility of the security?

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Problem 16

Suppose that the price of a zero-coupon bond maturing at time $T$ follows the process
$$
d P(t, T)=\mu_P P(t, T) d t+\sigma_P P(t, T) d z
$$
and the price of a derivative dependent on the bond follows the process
$$
d f=\mu_f f d t+\sigma_f f d z
$$
Assume only one source of uncertainty and that $f$ provides no income.
(a) What is the forward price $F$ of $f$ for a contract maturing at time $T$ ?
(b) What is the process followed by $F$ in a world defined by the numeraire $P(t, T)$ ?
(c) What is the process followed by $F$ in the traditional risk-neutral world?
(d) What is the process followed by $f$ in a world defined by a numeraire equal to a bond maturing at time $T^*$, where $T^* \neq T$ ? Assume that $\sigma_P^*$ is the volatility of this bond.

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Problem 17

Consider a variable that is not an interest rate:
(a) In what world is the futures price of the variable a martingale?
(b) In what world is the forward price of the variable a martingale?
(c) Defining variables as necessary, derive an expression for the difference between the drift of the futures price and the drift of the forward price in the traditional riskneutral world.
(d) Show that your result is consistent with the points made in Section 5.8 about the circumstances when the futures price is above the forward price.

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