Questions and Answers of Investment Risk Management

A young woman uses the first procedure to deduce her utility function $U(x)$ over the range $A \leq x \leq B$. She uses the normalization $U(A)=A, U(B)=B$. To check her result, she repeats the whole
One advantage of the annual worth method is that it simplifies the comparison of investment projects that are repetitive but have different cycle times. Consider the automobile purchase problem of
Consider a 3 -year $10 \%$ coupon bond. The underlying short rate of interest follows a lattice with initial value of $R=1.15$ and then has an factor of 1.02 , a down factor of .99 , and
Using the density function of the stopping time probability for a fixed $\lambda$, find the average time to the first event over the entire interval $[0, \infty)$.
For the HiTech example (17.1), find the risk-neutral probability of default. Example 17.1 (A leveraged firm) The HiTech firm currently has a value of $1 mil- lion and is financed in part by a 5-year
Construct a short rate lattice for periods (years) 0 through 9 with an initial rate of $6 \%$ and with successive rates determined by a multiplicative factor of either $u=1.2$ or $d=9$. Assign
Let $V_{k s}$ be the value of an adjustable-rate loan initiated at period $k$ and state $s$ with initial principal of 100 . The loan is to be fully paid at period $n$. The interest rate
Suppose that the adjustable-rate auto loan of Example 16.5 is modified by the provision of CAP that guarantees the borrower that the interest rate to be applied will never exceed $11 \%$. What is
Explain how you would find the value of a bond futures option.
Use the forward equation to find the spot rate curve for the lattice constructed in Exercise 1.Data from Exercises 1Construct a short rate lattice for periods (years) 0 through 9 with an initial rate
Use the Black-Derman-Toy model with $b=.01$ to match the term structure of Example 16.7.Data from Example 16.7The 12-year term structure has been extended here to 14 years. We will assume that this
Show that for the Ho-Lee model the (risk-neutral) standard deviation of the one-period rate is exactly $b_{k} / 2$.
Consider a plain vanilla interest rate swap where party A agrees to make six yearly payments to party $B$ of a fixed rate of interest on a notional principal of $\$ 10$ million and in exchange
A swaption is an option to enter a swap arrangement in the future. Suppose that company B has a debt of $\$ 10$ million financed over 6 years at a fixed rate of interest of $8.64 \%$. Company A
Suppose a short rate process in a risk-neutral world is defined by\[\mathrm{d} r=\mu(r, t) \mathrm{d} t+\sigma(r, t) \mathrm{d} \hat{z},\]where $\hat{z}(t)$ is a standardized Wiener process. A
Refer to Example 16.11. Let $F(t)$ be the forward rate from 0 to $t$. By the basic definition of the forward rate, we have the identity\[e^{-F(t) t}=P(r, 0, t)\]Find an explicit formula for
Gavin wants to dig deep into pricing theory, so he decides to work out an application of Eq. (16.11). He suggests to himself that a simple model of interest rates in the risk-neutral world might
Consider a bond with a face value of $\$ 1,000$ and coupon payment at the end of each period $k$ given by a rate $c_{k}=\max \left[6 \%-r_{k}, 0\right]$, where $r_{k}$ is the short rate for
Consider a payoff $C$ that will occur in 2 years, taking one of the three possible values $C_{0}, C_{1}, C_{2}$. The short rate lattice for these 2 years is shown in Figure 16.16, with \(d_{i
For the HiTech bond of Example 17.4, suppose that default is recognized only at maturity and that no restitution is made, that is, the default is ignored. What is the value of that bond? - Example
Suppose that the firm of Example 17.6 has a value of $\$ 200,000$ instead of $\$ 100,000$. Perhaps this makes the bond more secure. What, in fact, is the value of the bond in this case? Example
Suppose there are only 3 rating categories: A, B, D. Their one-year transition probabilities are given by the following matrix.Construct the transition probability matrix for two years. Ending
Consider a situation where $r$ and $\lambda$ are constant. A zero-coupon bond has face value $F$ and maturity $T$. In the case of default at $t$, there is partial recovery equal to
Let $p(t)$ be the probability of survival from 0 to $t$. The probability of surviving to $s$ given survival to $t$ is then\[p(s \mid t)=\frac{p(s)}{p(t)}\]Let
Let $q(t)$ be the survival probability and let $q^{-1}$ be its inverse function. Also, let $U$ be a uniform random variable on $[0,1]$. For each realization $u$, let $\tau$ be chosen such
Consider a 10-year zero-coupon bond with face value $\$ 100$. The interest rate is fixed at 5\%. The credit spread for the bond is estimated to be $1 \%$ (except in part (a)). Calculate:(a) The
Suppose we wish to estimate the probability of a rare event (such as a default probability). Let the random variable $X$ be equal to 1 if the event occurs and to zero otherwise. Then
Consider a total return swap of a coupon bond versus a fixed-rate payment. Discuss whether the payoff is path dependent.
In Example 17.11, assume that the promised amounts to $A$ and $B$ are $\$ 110$ and $\$ 90$, respectively. Develop the new table of results. What are the prices of $A$ and $\mathrm{B}$ ?
Consider a strategy of the form $(\gamma, 0,0)$ for the investment wheel. Show that the overall factor multiplying your money after $n$ steps is likely to be \((1+2 \gamma)^{n / 2}(1-\gamma)^{n /
In a certain state lottery, people select eight numbers in advance of a random drawing of six numbers. If someone's selections include the six drawn, they receive a large prize, but this prize is
Show that $\left(\frac{1}{2}, \frac{1}{2}\right)$ is the optimal policy for Example 18.2 . Example 18.2 (Volatility pumping) Suppose there are two assets available for investment. One is a stock
Consider a wheel with $n$ sectors. If the wheel pointer lands on sector $i$, the payoff obtained is $r_{i}$ for every unit bet on that sector. The chance of landing on sector $i$ is \(p_{i},
Using the notation of Exercise 4, assume that $\sum_{i=1}^{n} 1 / r_{i}=1$, but try to find a solution where one of the $\alpha_{k}$ 's is zero. In particular, suppose the segments are ordered in
Suppose there are $n$ stocks. Each of them has a price that is governed by geometric Brownian motion. Each has $v_{i}=15 \%$ and $\sigma_{i}=40 \%$. However, these stocks are correlated, and
The Dow Jones Industrial Average is an average of the prices of 30 industrial stocks with equal weights applied to all 30 stocks (but the sum of the weights is greater than 1). Occasionally (about
You are managing a pension fund with a goal of maximizing the long-term growth rate. There are three assets available. Asset 1 has a risk-free return of 5%. Assets 2 and 3 each are driven by
This exercise explores the sensitivity of $\log$-optimality to the rebalancing frequency. Consider a market consisting of a risk-free asset with zero rate of interest and a stock that over 1 year
Consider two binomial assets, each with price equal to 1.The first is a stock that at the end of a period pays either 3 or 0 with probabilities $p$ and $1-p$, respectively. The second asset is
A game of chance based on a spinning wheel is available that pays $n$ times money bet in the case of a win and nothing in the case of a loss. A gambler has developed a device by which he may
Consider the power utility defined by the function $F(S)=\frac{1}{\gamma} S^{\gamma}$, for $\gamma \leq 1$. There are available $n$ assets, each of which follows geometric Brownian motion,
Suppose there is a stock and a bond governed by the equationsIt is desired to construct a portfolio of these two securities that gives the maximum expected $\log$ of return. However, although
Suppose there are $n$ assets. Asset $i, i=$ $1,2, \ldots, n$, has rate of return $r_{i}$ over a single period. There is also a risk-free asset with rate of return $r_{f}$. The log-optimal
A certain underlying state graph is a tree where each node has three successor nodes, indexed $a,b, c$. There are two assets defined on this tree which pay no dividends except at the terminal time
Consider a short rate binomial lattice where the risk-free rate at $t=0$ is $10 \%$. At $t=1$ the rate is either $10 \%$ (for the upper node) or $0 \%$ (for the lower node). Trace out the
Assuming the short rate process of Exercise 2 and risk-neutral probabilities of .5 , consider a zero-coupon bond that pays $\$ 1$ at time $t=2$. Find the value at time $t=0$ of this bond in two
Find the values of the 5 -month call option of Example 19.1 using the same trinomial lattice used in that example but employing the utility function $U(x)=\sqrt{x}$. What is $\alpha$ ? Example
Use the information about the Complexico mine of Example 14.8. Chapter 14, but assume that gold prices and interest rates are governed by the models of Example 19.3. Find the value of the Complexico
Suppose that in the double stochastic Simplico gold mine example the real probability of an up move in gold is 6 and the real probability of an up move in the short rate is .7. Suppose also that gold
Calculate the volatility and the current price of oil futures implied by the call 1600 August and the call 1700 August of Figure 19.8 by using the Black-Scholes formula with $T=.25$. OIL CRUDE OIL
A company issues a $10 \%$ coupon bond that matures in 5 years. However, this company is in trouble, and it is estimated that each year there is a probability of .1 that it will default that year.
Mr. Smith wants to buy a car and is deciding between brands A and B. Car A costs $\$ 20,000$, and Mr. Smith estimates that at the rate he drives he will sell it after 2 years and buy another of the
Consider a continuous-time environment, with $e$ as a variable outside of the market.(a) Suppose the final payoff is $V\left(x_{e}, T\right)=x_{e}(T)$. Find $V\left(x_{e}, t\right)$.(b) Find a
Mr. Jones was considering a new grapefruit venture that would generate a random sequence of yearly cash flows. He asked his son. Gavin, "People tell me I should use a cost of capital figure to
An option based on a variable that is not traded is called a real option or sometimes a soft option. Find the projection price of the soft option with the following parameters and compare with the
A stock price is governed by \[\frac{\mathrm{d} S}{S}=\mu \mathrm{d} t+\sigma \mathrm{d} z\] where $z$ is a standardized Wiener process. Interest is constant at rate $r$. An investor wishes to
The current price of gold is $\$ 412$ per ounce. The storage cost is $\$ 2$ per ounce per year, payable quarterly in advance. Assuming a constant interest rate of $9 \%$ compounded quarterly,
Suppose that a forward contract on an asset is written at time zero and there are $M$ periods until delivery. Suppose that the proportional carrying charge in period $k$ is $q S(k)$, where
At the beginning of April one year, the silver forward prices (in cents per troy ounce) were as follows:The carrying cost of silver is about 20 cents per ounce per year, paid at the beginning of each
Suppose that a continuous-time compounding framework is used with a fixed interest rate $r$. Suppose that the carrying charge per unit of time is proportional to the spot price; that is, the charge
Complete the second half of the proof of the "forward price formula with carrying cost". To construct the arbitrage, go long one unit of a forward and short one unit spot. To execute the short, it is
Consider the situation of Example 12.10. Rather than shorting a futures contract, the U.S. firm could borrow $500 /\left(1+r_{G}\right)$ euros (where $r_{G}$ is the 90-day interest rate in
A certain 10-year bond is currently selling for $\$ 920$. A friend of yours owns a forward contract on this bond that has a delivery date in 1 year and a delivery price of $\$ 940$. The bond pays
Derive formula (12.6) by converting a cash flow of a bond to that of the fixed portion of the swap.Formula 12.6 X V=MSo IB(M,C) - 100d (0,M)]}N, (12.6)
Suppose at time 0 you have arranged to be paid at time $T$ the amount $\int_{0}^{T} S(t) \mathrm{d} t$, where $S(t)$ is the spot price at $t$ of a commodity that can be shorted and has zero
The interest rates in the UK and the United States are, respectively, $4 \%$ and $6 \%$ per annum compounded continuously. The spot price of the UK pound is $\$ 1.6$. The forward price for a
Mr. A. Gaylord manages a pension fund and believes that his stock selection ability is excellent. However, he is worried because the market could go down. He considers entering an equity swap where
The floating rate portion of a plain vanilla interest rate swap with yearly payments and a notional principal of one unit has cash flows at the end of each year defining a stream starting at time 1
Suppose the current term structure of interest rates is $(.070, .073, .077$, $.081, .084, .088)$. A plain vanilla interest rate swap will make payments at the end of each year equal to the
Derive the mean-variance hedge formula given by (12.12). h FT-Fo 2r var(FT) cov(x, FT) var(FT) (12.12)
Farmer D. Jones has a crop of grapefruit that will be ready for harvest and sale as 150,000 pounds of grapefruit juice in 3 months. Jones is worried about possible price changes, so he is considering
Assume that cash flow is given by $y=S_{T} W+\left(F_{T}-F_{0}\right) h$. Let $\sigma_{S}^{2}=\operatorname{var}\left(S_{T}\right), \sigma_{F}^{2}=\operatorname{var}\left(F_{T}\right)$, and
A pension fund has just paid some of its liabilities, and as a result of this transaction the fund is no longer fully immunized. The fund manager decides that instead of changing the portfolio, the
Suppose the wealth that is to be received at a time $T$ in the future has the formwhere $a$ is a constant and $x$ is a random variable. The value of the variable $h$ can be selected by the
Suppose that revenue has the formwhere $h$ can be chosen and $x$ and $y$ are random variables. The distribution of $x$ and $y$ is symmetric about $(0,0)$; that is, $-x,-y$ has the same
Suppose that, as in the corn farm example, the farm has random production and the final spot price is governed by the same demand function. However, the crop of the farm is not perfectly correlated
A stock with current value $S(0)=100$ has an expected growth rate of its logarithm of $u=12 \%$ and a volatility of that growth rate of $\sigma=20 \%$. Find suitable parameters of a binomial
A stock price $S$ is governed by the modelwhere the period length is 1 month. Let $u=\mathrm{E}[w(k)]$ and $\sigma^{2}=\operatorname{var}[w(k)]$ for all $k$. Now suppose the basic period
Suppose that $v_{1}, v_{2}, \ldots, v_{n}$ are positive numbers. The arithmetic mean and the geometric mean of these numbers are, respectively,(a) It is always true that $v_{A} \geq v_{G}$. Prove
Suppose that $u=e^{w}$, where $w$ is normal with expected value $\bar{w}$ and variance $\sigma^{2}$. ThenShow thatUse the fact thatto evaluate $\bar{u}$. = 1 22 00- ewe-(w-w)/202 dw.
Use the method of Exercise 4 to find the variance of a lognormal variable in terms of the parameters of the underlying normal variableData from Exercises 4Suppose that $u=e^{w}$, where $w$ is
A stock price is governed by geometric Brownian motion with $\mu=.20$ and $\sigma=.40$. The initial price is $S(0)=1$. Evaluate the four quantities E[In S(1)], E[S(1)], stdev[In S(1)]
A stock price $S$ is governed bywhere $z$ is a standardized Wiener process. Find the process that governs dS as dt +bs dz.
Gavin Jones was mystified by Ito's lemma when he first studied it, so he tested it. He started with $S$ governed byand found that $Q=\ln S$ satisfiesHe then applied Ito's lemma to this last
Suppose $F(S, t)$ is the forward price of a commodity with no storage cost and governed by $\mathrm{d} S(t)=\mu \mathrm{d} t+\sigma \mathrm{d} z$ and terminating at time $T$. What is the
Let $w=e^{\sigma z-\frac{1}{2} t^{2}}$, where $z$ is a stardard Wiener process. Find the equation governing $w$.
Consider an asset whose price follows the geometric Brownian motion processwhere $z$ is a standard Wiener process.(a) At time $t$ (when $S(t)$ is known), what is the expected value of the
An alternative to using $d=1 / u$ in a binomial model is to use the available degree of freedom by setting $p=1 / 2$.(a) Let $p=1 / 2$, and find the values of $u$ and $d$ that satisfy the
A useful expansion isUse this to express the exponential in equation (13.20) in linear terms of powers of $\Delta t$ up to first order. Note that this differs from the expression in (13.19), so
Consider a stock price $S$ governed by the geometric Brownian motion process(a) Using $\Delta t=1 / 12$ and $S(0)=1$, simulate several (i.e., many) years of this process using either method,
An investor who is bullish about a stock (believing that it will rise) may wish to construct a bull spread for that stock. One way to construct such a spread is to buy a call with strike price
Suppose over the period $[0, T]$ a certain stock pays a dividend whose present value at interest rate $r$ is $D$. Show that the put-call parity relation for European options at $t=0$,
One year ago, Bioette, a biotech incubator, entered into a forward contract to sell one of its patents to Pharm, a major drug company, in 2 years for $\$ 10$ million.Currently, with only 1 year
Consider a family of call options on a non-dividend-paying stock, each option being identical except for its strike price. The value of the call with strike price $K$ is denoted by $C(K)$. Prove
Suppose that a stock will pay a dividend of amount $D$ at time $\tau$. We wish to determine the price of a European call option on this stock using the lattice method. Accordingly, the time
Consider a European call option on a non-dividend-paying stock. The strike price is $K$, the time to expiration is $T$, and the price of one unit of a zero-coupon bond maturing at $T$ is
A perpetual option is one that never expires. (Such an option must be of American style.) Use Exercise 6 to show that the value of a perpetual call on a non-dividend-paying stock is $C=S$.Data from
Consider a deterministic cash flow stream $\left(x_{0}, x_{1}, x_{2}, \ldots, x_{n}\right)$ with all positive flows. Let $\mathrm{PV}(r)$ denote the present value of this stream at an interest
There are two propositions:(a) I flip a coin. If it is heads, you are paid \$3; if it is tails, you are paid $\$ 0$. It costs you $\$ 1$ to participate in this proposition. You may do so at any
A New York firm is offering a new financial instrument called a "happy call." It has a payoff function at time $T$ equal to $\max (.5 S, S-K)$, where $S$ is the price of a stock and $K$ is a

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