- Rework Problem 8 assuming that Genatron Manufacturing expects its sales to increase by 20 percent in 2015. What is the amount of external financing needed?Rework Problem 8Genatron Manufacturing
- In Problem 10, we assumed the current asset and liability accounts decrease proportionately with Genatron’s sales. This is probably unrealistic following a decline in sales. What will be the impact
- Financial researchers at Smith Sharon, an investment bank, estimate the current security market line as E(Ri) = 4.5 + 6.8(????i).a. Explain what happens to expected return as beta increases from 1.0
- Get stock price data from http://finance.yahoo.com/ for ten stocks in the Dow Jones Industrial Average (DJIA) for the prior ten days and use these prices to compute a price-weighted index for each of
- Suppose the estimated security market line is E(Ri) = 4.0 + 7(????i).a. What is the current Treasury bill rate?b. What is the current market risk premium?c. What is the current expected market
- Genatron Manufacturing expects its sales to increase by 10 percent in 2016. Estimate the firm’s external financing needs by using the percent-of-sales method for the 2017 data. Assume that no
- Using the information in Tables 14.1 and 14.2, compute the financial ratios we discussed in this chapter for Walgreens using the 2013 and 2012 data.
- Given Robinson’s 2016 and 2017 financial information presented in Problems 2 and 4,a. Compute its operating and cash conversion cycle in each year.b. What was Robinson’s net investment in working
- Robinson expects its 2018 sales and cost of goods sold to grow by 5 percent over their 2017 levels.a. What will be the effect on its levels of receivables, inventories, and payments if the components
- Robinson expects its 2018 sales and cost of goods sold to grow by 20 percent over their 2017 levels.a. What will be the effect on its levels of receivables, inventories, and payments if the
- Genatron Manufacturing expects its sales to increase by 10 percent in 2018. Estimate the firm’s investment in accounts receivable, inventory, and accounts payable in 2018.
- With concerns of increased competition, Genatron is planning in case its 2018 sales fall by 5 percent from their 2017 levels. If cost of goods sold and the current asset and liability accounts
- Redo Problem 14, using the following monthly sales
- Robinson Company (recall their data from Problems 2, 3, and 4) has a 2017 profit margin of 5 percent. It is examining the possibility of loosening its credit policy. Analysis shows that sales may
- Genatron Manufacturing (from Problem 8) is considering changing its credit standards. Analysis shows that sales may fall 5 percent from 2017 levels, with no bad debts from the change in sales. The
- Visit a firm’s website and obtain historical quarterly balance sheet information from it or from its SEC EDGAR fi lings http://www.walmart.com and http://www.walgreens.com may be two good sites to
- Use various Internet resources and information contained in this text to estimate the cost of debt, cost of retained earnings, the cost of new equity, and the WACC for the following firms: Walgreens,
- Through library or Internet resources, find information regarding the sources of long-term financing for AT&T. What are the current market prices for its outstanding bonds and stock? Estimate its
- Go to the Federal Reserve Bank of St. Louis website at http://www.stlouisfed.org, and fi nd interest rates on U.S. Treasury securities and on corporate bonds with diff erent bond ratings.a. Prepare a
- We consider a bond with maturity \(T\), priced \(P(t, T)=\) \(\mathbb{E}^{*}\left[\mathrm{e}^{-\int_{t}^{T} r_{s} d s} \mid \mathcal{F}_{t}\right]\) at time \(t \in[0, T]\).a) Using the forward
- Consider a tenor structure \(\left\{T_{1}, T_{2}\right\}\) and a bond with maturity \(T_{2}\) and price given at time \(t \in\left[0, T_{2}\right]\) by\[P\left(t, T_{2}\right)=\exp
- Assume that the price \(P(t, T)\) of a zero-coupon bond with maturity \(T>0\) is modeled as\[P(t, T)=\mathrm{e}^{-\mu(T-t)+X_{t}^{T}}, \quad t \in[0, T],\]where \(\left(X_{t}^{T}\right)_{t \in[0,
- Consider a short rate process \(\left(r_{t}\right)_{t \in \mathbb{R}_{+}}\)of the form \(r_{t}=h(t)+X_{t}\), where \(h(t)\) is a deterministic function of time and
- a) Given two LIBOR spot rates \(L(t, t, T)\) and \(L(t, t, S)\), compute the corresponding LIBOR forward rate \(L(t, T, S)\).b) Assuming that \(L(t, t, T)=2 \%, L(t, t, S)=2.5 \%\) and \(t=0, T=1,
- a) Compute the forward rate \(f(t, T, S)\) in the Ho-Lee model (17.48) with constant deterministic volatility.In the next questions we take \(a=0\).b) Compute the instantaneous forward rate \(f(t,
- Consider a floorlet on a three-month LIBOR rate in nine month's time, with a notional principal amount of \(\$ 10,000\) per interest rate percentage point. The term structure is flat at \(3.95 \%\)
- Consider a payer swaption giving its holder the right, but not the obligation, to enter into a 3-year annual pay swap in four years, where a fixed rate of \(5 \%\) will be paid and the LIBOR rate
- Consider a receiver swaption which is giving its holder the right, but not the obligation, to enter into a 2-year annual pay swap in three years, where a fixed rate of \(5 \%\) will be received and
- Jamshidian's trick (Jamshidian (1989)). Consider a family \(\left(P\left(t, T_{l}\right)\right)_{l=i, \ldots, j}\) of bond prices defined from a short rate process \(\left(r_{t}\right)_{t \in
- We work in the short rate model\[d r_{t}=\sigma d B_{t}\]where \(\left(B_{t}\right)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion under \(\mathbb{P}^{*}\), and \(\widehat{\mathbb{P}}_{2}\) is
- Analysis of user login activity to the DBX digibank app showed that the times elapsed between two logons are independent and exponentially distributed with mean\(1 / \lambda\). Find the CDF of the
- Consider a standard Poisson process \(\left(N_{t}\right)_{t \in \mathbb{R}_{+}}\)with intensity \(\lambda>0\), started at \(N_{0}=0\).a) Solve the stochastic differential equation\[d S_{t}=\eta
- Consider a standard Poisson process \(\left(N_{t}\right)_{t \in \mathbb{R}_{+}}\)with intensity \(\lambda>0\).a) Solve the stochastic differential equation \(d X_{t}=\alpha X_{t} d t+\sigma d N_{t}\)
- Consider a standard Poisson process \(\left(N_{t}\right)_{t \in \mathbb{R}_{+}}\)with intensity \(\lambda>0\).a) Solve the stochastic differential equation \(d X_{t}=\sigma X_{t^{-}} d N_{t}\) for
- Let \(\left(N_{t}\right)_{t \in \mathbb{R}_{+}}\)be a standard Poisson process with intensity \(\lambda>0\), started at \(N_{0}=0\).a) Is the process \(t \mapsto N_{t}-2 \lambda t\) a submartingale,
- Affine stochastic differential equation with jumps. Consider a standard Poisson process \(\left(N_{t}\right)_{t \in \mathbb{R}_{+}}\)with intensity \(\lambda>0\).a) Solve the stochastic differential
- Consider the compound Poisson process \(Y_{t}:=\sum_{k=1}^{N_{t}} Z_{k}\), where \(\left(N_{t}\right)_{t \in \mathbb{R}_{+}}\)is a standard Poisson process with intensity \(\lambda>0\), and
- Show, by direct computation or using the moment generating function (20.10), that the variance of the compound Poisson process \(Y_{t}\) with intensity \(\lambda>0\)
- Consider an exponential compound Poisson process of the form\[S_{t}=S_{0} \mathrm{e}^{\mu t+\sigma B_{t}+Y_{t}}, \quad t \geqslant 0\]where \(\left(Y_{t}\right)_{t \in \mathbb{R}_{+}}\)is a compound
- Let \(\left(N_{t}\right)_{t \in[0, T]}\) be a standard Poisson process started at \(N_{0}=0\), with intensity \(\lambda>0\) under the probability measure \(\mathbb{P}_{\lambda}\), and consider the
- Consider a standard Poisson process \(\left(N_{t}\right)_{t \in[0, T]}\) with intensity \(\lambda>0\) and a standard Brownian motion \(\left(B_{t}\right)_{t \in[0, T]}\) independent of
- Consider a standard Poisson process \(\left(N_{t}\right)_{t \in \mathbb{R}_{+}}\)with intensity \(\lambda>0\) under a probability measure \(\mathbb{P}\). Let \(\left(S_{t}\right)_{t \in
- Consider a long forward contract with payoff \(S_{T}-K\) on a jump diffusion risky asset \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\)given by\[d S_{t}=\mu S_{t} d t+\sigma S_{t} d B_{t}+S_{t^{-}} d
- Consider \(\left(N_{t}\right)_{t \in \mathbb{R}_{+}}\)a standard Poisson process with intensity \(\lambda>0\) under a probability measure \(\mathbb{P}\). Let \(\left(S_{t}\right)_{t \in
- Consider a standard Poisson process \(\left(N_{t}\right)_{t \in \mathbb{R}_{+}}\)with intensity \(\lambda>0\) under a probability measure \(\mathbb{P}\). Let \(\left(S_{t}\right)_{t \in
- Pricing by the Esscher transform (Gerber and Shiu (1994)). Consider a compound Poisson process \(\left(Y_{t}\right)_{t \in[0, T]}\) with
- Show that when the terminal condition is a constant \(\phi(T, x)=c>0\) the implicit scheme (22.8) recovers the known solution \(\phi(s, x)=c \mathrm{e}^{-r(T-s)}, s \in[0, T]\).
- Let \(X_{t}\) be the geometric Brownian motion given by the stochastic differential equation\[d X_{t}=r X_{t} d t+\sigma X_{t} d W_{t}\]a) Compute the Euler discretization
- Consider a two-step binomial model \(\left(S_{k}\right)_{k=0,1,2}\) with interest rate \(r=0 \%\) and risk-neutral probabilities \(\left(p^{*}, q^{*}\right)\) :a) At time \(t=1\), would you exercise
- Consider an American butterfly option with the following payoff function.Price the perpetual American butterfly option with \(r>0\) in the following cases.a) \(\widehat{K} \leqslant L^{*}
- (Barone-Adesi and Whaley (1987)) We approximate the finite expiration American put option price with strike price \(K\) as\[f(x, T) \simeq \begin{cases}\operatorname{BS}_{\mathrm{p}}(x,
- Consider the process \(\left(X_{t}\right)_{t \in \mathbb{R}_{+}}\)given by \(X_{t}:=t Z, t \in \mathbb{R}_{+}\), where \(Z \in\{0,1\}\) is a Bernoulli random variable with
- Consider a dividend-paying asset priced as\[S_{t}=S_{0} \mathrm{e}^{(r-\delta) t+\sigma \widehat{B}_{t}-\sigma^{2} t / 2}, \quad t \geqslant 0\]where \(r>0\) is the risk-free interest rate,
- American call options with dividends, see § 9.3 of Wilmott (2006). Consider a dividend-paying asset priced as \(S_{t}=S_{0} \mathrm{e}^{(r-\delta) t+\sigma \widehat{B}_{t}-\sigma^{2} t / 2}, t
- Consider an underlying asset whose price is written as\[S_{t}=S_{0} \mathrm{e}^{r t+\sigma B_{t}-\sigma^{2} t / 2}, \quad t \geqslant 0\]where \(\left(B_{t}\right)_{t \in \mathbb{R}_{+}}\)is a
- Perpetual American binary options.a) Compute the price\[C_{b}^{\mathrm{Am}}\left(t, S_{t}\right)=\operatorname{Sup}_{\substack{\tau \geqslant t \\ \tau \text { Stopping time }}}
- An American binary (or digital) call (resp. put) option with maturity \(T>0\) on an underlying asset process \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}=\left(\mathrm{e}^{r t+\sigma B_{t}-\sigma^{2}
- American forward contracts. Consider \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\)an asset price process given by\[\frac{d S_{t}}{S_{t}}=r d t+\sigma d B_{t}\]where \(r>0\) and
- Consider an underlying asset price process written as\[S_{t}=S_{0} \mathrm{e}^{r t+\sigma \widehat{B}_{t}-\sigma^{2} t / 2}, \quad t \geqslant 0\]where \(\left(\widehat{B}_{t}\right)_{t \in
- Let \(p \geqslant 1\) and consider a power put option with payoff\[\left(\left(\kappa-S_{\tau}\right)^{+}\right)^{p}= \begin{cases}\left(\kappa-S_{\tau}\right)^{p} & \text { if } S_{\tau} \leqslant
- Same questions as in Exercise 15.14 , this time for the option with payoff \(\kappa-\left(S_{\tau}\right)^{p}\) exercised at time \(\tau\), with \(p>0\).Data from Exercise 15.14Let \(p \geqslant 1\)
- Let \(\left(B_{t}\right)_{t \in \mathbb{R}_{+}}\)be a standard Brownian motion started at 0 under the riskneutral probability measure \(\mathbb{P}^{*}\). Consider a numéraire \(\left(N_{t}\right)_{t
- Consider two zero-coupon bond prices of the form \(P(t, T)=F\left(t, r_{t}\right)\) and \(P(t, S)=G\left(t, r_{t}\right)\), where \(\left(r_{t}\right)_{t \in \mathbb{R}_{+}}\)is a short-term interest
- Using a change of numéraire argument for the numéraire \(N_{t}:=P(t, T), t \in[0, T]\), compute the price at time \(t \in[0, T]\) of a forward (or future) contract with payoff \(P(T, S)-K\) in a
- Consider a price process \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\)given by \(d S_{t}=r S_{t} d t+\sigma S_{t} d B_{t}\) under the risk-neutral probability measure \(\mathbb{P}^{*}\), where \(r
- Consider two bonds with maturities \(T\) and \(S\), with prices \(P(t, T)\) and \(P(t, S)\) given by\[\frac{d P(t, T)}{P(t, T)}=r_{t} d t+\zeta_{t}^{T} d W_{t}\]and\[\frac{d P(t, S)}{P(t, S)}=r_{t} d
- Compute the price \(\mathrm{e}^{-(T-t) r} \mathbb{E}^{*}\left[\mathbb{1}_{\left\{R_{T} \geqslant \kappa\right\}} \mid \mathcal{F}_{t}\right]\) at time \(t \in[0, T]\) of a cashor-nothing "binary"
- Assume that \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\)and \(\left(N_{t}\right)_{t \in \mathbb{R}_{+}}\)satisfy the stochastic differential equations\[d S_{t}=r_{t} S_{t} d t+\sigma_{t}^{S} S_{t}
- Consider an asset priced \(S_{t}\) at time \(t\), with\[d S_{t}=r S_{t} d t+\sigma^{S} S_{t} d W_{t}^{S}\]and an exchange rate \(\left(R_{t}\right)_{t \in \mathbb{R}_{+}}\)given by\[d
- In the mean-reverting Vasicek model (17.47) with \(b>0\), compute:i) The asymptotic bond yield, or exponential long rate of interest\[r_{\infty}:=-\lim _{T \rightarrow \infty} \frac{\log P(t,
- Consider the Marsh and Rosenfeld (1983) short-term interest rate model\[d r_{t}=\left(\beta r_{t}^{\gamma-1}+\alpha r_{t}\right) d t+\sigma r_{t}^{\gamma / 2} d B_{t}\]where \(\alpha \in \mathbb{R}\)
- Consider the Cox et al. (1985) (CIR) process \(\left(r_{t}\right)_{t \in \mathbb{R}_{+}}\)solution of\[d r_{t}=-a r_{t} d t+\sigma \sqrt{r_{t}} d B_{t},\]where \(a, \sigma>0\) are constants
- Convertible bonds. Consider an underlying asset price process \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\)given by\[d S_{t}=r S_{t} d t+\sigma S_{t} d B_{t}^{(1)}\]and a short-term interest rate
- Write down the bond pricing PDE for the function\[F(t, x)=\mathbb{E}^{*}\left[\mathrm{e}^{-\int_{t}^{T} r_{s} d s} \mid r_{t}=x\right]\]and show that in case \(\alpha=0\) the corresponding bond price
- Consider a zero-coupon bond with prices \(P(1,2)=91.74 \%\) and \(P(0,2)=\) \(83.40 \%\) at times \(t=0\) and \(t=1\).a) Compute the corresponding yields \(y_{0,1}, y_{0,2}\) and \(y_{1,2}\) at times
- Consider a one-step interest rate model, in which the short-term interest rate \(r_{0}\) on \([0,1]\) can turn into two possible values \(r_{1}^{u}=r_{0} \mathrm{e}^{\mu \Delta t+\sigma \sqrt{\Delta
- Compute the price\[\mathrm{e}^{-(T-t) r} \mathbb{E}^{*}\left[\left.\left(\exp \left(\frac{1}{T} \int_{0}^{T} \log S_{u} d u\right)-K\right)^{+} \rightvert\, \mathcal{F}_{t}\right], \quad 0 \leqslant
- Pricing Asian options by PDEs. Show that the functions \(g(t, z)\) and \(h(t, y)\) are linked by the relation\[g(t, z)=h\left(t, \frac{1-\mathrm{e}^{-(T-t) r}}{r T}+\mathrm{e}^{-(T-t) r} z\right),
- Given \(S_{t}:=S_{0} \mathrm{e}^{\sigma B_{t}+r t-\sigma^{2} t / 2}\) a geometric Brownian motion and letting\[\widetilde{Z}_{t}:=\frac{\mathrm{e}^{-(T-t) r}}{S_{t}}\left(\frac{1}{T} \int_{0}^{t}
- Consider an underlying asset price process \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\)modeled as \(d S_{t}=(\mu-\delta) S_{t} d t+\sigma S_{t} d B_{t}\), where \(\left(B_{t}\right)_{t \in
- Compute the first and second moments of the time integral \(\int_{\tau}^{T} S_{t} d t\) for \(\tau \in[0, T)\), where \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\)is the geometric Brownian motion
- Consider the short rate process \(r_{t}=\sigma B_{t}\), where \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion.a) Find the probability distribution of the time integral
- Asian call option with a negative strike price. Consider the asset price process\[S_{t}=S_{0} \mathrm{e}^{r t+\sigma B_{t}-\sigma^{2} t / 2}, \quad t \geqslant 0\]where \(\left(B_{t}ight)_{t \in
- Consider the Asian forward contract with payoff\[\begin{equation*}\frac{1}{T} \int_{0}^{T} S_{u} d u-K \tag{13.36}\end{equation*}\]where \(S_{u}=S_{0} \mathrm{e}^{\sigma B_{u}+r u-\sigma^{2} u / 2},
- Hedging Asian options (Yang et al. (2011)).a) Compute the Asian option price f(t,St,Λt)f(t,St,Λt) when Λt/T⩾KΛt/T⩾K.b) Compute the hedging portfolio allocation (ξt,ηt)(ξt,ηt) when
- Let \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)be a standard Brownian motion started at 0 , i.e. \(B_{0}=0\).a) Is the process \(t \longmapsto\left(2-B_{t}ight)^{+}\)a submartingale, a martingale or
- Stopping times. Let \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)be a standard Brownian motion started at 0 .a) Consider the random time \(u\) defined by\[u:=\inf \left\{t \in \mathbb{R}_{+}:
- Consider a standard Brownian motion \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)started at \(B_{0}=0\), and let\[\tau_{L}=\inf \left\{t \in \mathbb{R}_{+}: B_{t}=Light\}\]denote the first hitting time
- Consider \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)a Brownian motion started at \(B_{0}=x \in[a, b]\) with \(a
- Consider a standard Brownian motion \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)started at \(B_{0}=0\), and let\[\tau:=\inf \left\{t \in \mathbb{R}_{+}: B_{t}=\alpha+\beta tight\}\]denote the first
- (Doob-Meyer decomposition in discrete time). Let \(\left(M_{n}ight)_{n \in \mathbb{N}}\) be a discretetime submartingale with respect to a filtration \(\left(\mathcal{F}_{n}ight)_{n \in
- a) Give the probability density function of the maximum of drifted Brownian motion \(\operatorname{Max}_{t \in[0,1]}\left(B_{t}+\sigma t / 2ight)\).b) Taking \(S_{t}:=\mathrm{e}^{\sigma
- Let \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)denote a standard Brownian motion.a) Compute the expected value\[\mathbb{E}\left[\operatorname{Max}_{t \in[0,1]}
- Consider a risky asset whose price \(S_{t}\) is given by\[d S_{t}=\sigma S_{t} d B_{t}+\sigma^{2} S_{t} d t / 2\]where \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion.a)
- The digital drawdown call option with qualifying period pays a unit amount when the drawdown period reaches one unit of time, if this happens before fixed maturity \(T\), but only if the size of
- a) Check explicitly that the boundary conditions (12.3a)-(12.3c) are satisfied.b) Check explicitly that the boundary conditions (12.14a)-(12.14b) are satisfied.
- Compute the expected realized variance on the time interval [0, T] in the Heston model, with$$d v_{t}=-\lambda\left(v_{t}-might) d t+\eta \sqrt{v_{t}} d B_{t}, \quad 0 \leqslant t \leqslant T$$
- Compute the variance swap rate$$\mathrm{VS}_{T}:=\frac{1}{T} \mathbb{E}\left[\lim _{n ightarrow \infty} \sum_{k=1}^{n}\left(\frac{S_{k T / n}-S_{(k-1) T / n}}{S_{(k-1) T /
- Convexity adjustment (§ 2.3 of Broadie and Jain (2008)).a) Using Taylor's formula$$\sqrt{x}=\sqrt{x_{0}}+\frac{x-x_{0}}{2 \sqrt{x_{0}}}-\frac{\left(x-x_{0}ight)^{2}}{8 x_{0}^{3 /
- Consider an asset price $\left(S_{t}ight)_{t \in \mathbb{R}_{+}}$with log-return dynamics$$d \log S_{t}=\mu d t+Z_{N_{t^{-}}} d N_{t}, \quad t \geqslant 0$$i.e. $S_{t}:=S_{0} \mathrm{e}^{\mu