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Questions and Answers of Marketing Strategy Planning

John Lawrence Bailey is employed in Country T by American Conglomerate Corporation. Bailey has resided with his wife and three children in Country T for seven years. He made one five-day business
Mark Meadows funded a trust in 2014 with Merchants Bank named as trustee. He paid no gift tax on the transfer. The trustee in its discretion is to pay out income, but not principal, to Mark's
Stephen R. and Rachel K. Bates, both U.S . citizens, resided in Country K for the entire current year except when Stephen was temporarily assigned to his employer's home office in the United States.
In 2014, Leon Lopez funded Lopez Trust #3, an irrevocable trust, at First Bank, 125 Seaview, Northwest City, WA 98112, for the benefit of his twin children, Loretta and Jorge. The trust's tax ID
Prepare an estate tax return (Form 706) for Adam Zugg of 45 Cornfield Place, Midwest City, IL 60000. Adam died October 31, 2017. He was survived by his wife, Callie, and their son, Zebulon. At the
Prepare an estate tax return (Form 706) for Marcia Miller, who died July 23, 2017. Marcia (born April 2, 1930) resided at 117 Brandywine Way, Eastern City, PA 19000 and was a lifelong Pennsylvania
Healthwise Medical Supplies Company is located at 2400 Second Street, City, ST 12345. The company is a general partnership that uses the calendar year and accrual basis for both book and tax
The Dapper-Dons Partnership was formed ten years ago as a general partnership to custom tailor men's clothing. Dapper-Dons is located at 123 Flamingo Drive in City, ST, 54321. Bob Dapper manages the
Flying Gator Corporation and its 100%-owned subsidiary, T Corporation, have filed consolidated tax returns for many years. Both corporations use the hybrid method of accounting and the calendar year
Permtemp Corporation formed in 2016 and, for that year, reported the following book income statement and balance sheet, excluding the federal income tax expense, deferred tax assets, and deferred tax
Use the change-of-num?raire toolkit and the DC-operator to derive the drift of the n-th forward LIBOR under the spot LIBOR measure QB, for an arbitrary time 0 ? t n?1. Keep in mind that the process
Show that Bd(t) as defined in Problem 21.1 is almost surely a continuous process.Problem 21.1Suppose that the tenor structure T−1, T0,..., TN is equispaced, withΔt = Tn − Tn−1 and T−1 = 0.
Suppose that the tenor structure T?1, T0, . . . , TN is equispaced, with ?t = Tn ? Tn?1 and T?1 = 0. For 0 ? t ? TN, the discretely rebalanced bank account Bd(t) has present value. with where
Verify this statement by reviewing (20.107) on page 476. (20.107) т аr., D 3D Zoи, )| ,9dsd + Zo,0, Taw,0) q=1 q=1
In fitting the historical volatility structure of the HJM-factors, quadratic splines were used. Suppose there are N observations ?(t, ?n), for n = 1, . . . , N. Show that all spline coefficients can
Check that A(T, T) = B(T, T) = 0 holds.
Compute the convexity adjustment in the Hull–White-extended Vasicek-model.
Show that the GARCH(1,1)-process xt, defined by? with independent and identically distributed innovations zt ? N(0, 1), is not a Markov-process. X, = Vh,z, h; = w + ax_, +Bh;-1, + Bh1-1
Consider the short rate model Show that this model is a member of the affine term structure class and derive the ATS-functions A(t, T) and B(t, T). dr(1) = e "dt + o,dW(t)
Suppose pricing is conducted with a one-factor short rate model. Use the Jamshidian-decomposition to compute the generic price of a receiver swaption, expiring at T0, with tenor structure t < T0,
Derive the discount bond volatility σP(t, T) in the Vasicek-model and verify that σP(T, T) = 0 holds.
Prove that the forward swap rate can be understood as a weighted average of all forward LIBORs within the tenor of the swap. P(t, To) – P(t, TN) rs(t) = Δι ΣΚ Ρ(, Τ)
Use the definition of the forward LIBOR, (19.31) on page 447, to show that the present value of the floating leg of a plain vanilla interest rate swap is indeed (19.31) Floating leg = At> P(1,
Let today be time t. Suppose you want to price a plain vanilla option with expiry date T on a discount bond, maturing at S, within the Black-76-model, with t Compute the quantity d+/?, required in
Use an arbitrage argument, analogous to the one in Table 11.1 on page 212, to prove that in a stochastic interest rate world, the strike price of a forward contract Ft(K, T) that can be entered
Suppose you have an expectation with respect to the T-forward measure, based on the information available at time t T, Derive the Radon?Nikodym-derivative dQT/dQS|Ft and write the solution for Vt
Consider the financing conditions in Table 18.2 on page 436 for companies A and B. Assume that a financial intermediary C offers both companies a payer and a receiver swap to realize comparative
Show that the bootstrap formula (18.49) on page 435 is equivalent to a finite difference approximation of (18.44), if the respective bonds are expressed in terms of their yield to maturity y(t, Tn)
Show that Macaulay-duration is a weighted average of times to maturity where the weights wn are the fractions of the present value of the bond, represented by the n-th cashflow. Macaulay-duration=
Use the banking account as a num?raire to show that has to hold. le-S" ro»d|/
Relate this formula to Figure 18.1 right. Figure 18.1 т
A consol is a perpetual bond that pays coupons forever, without repaying the principal. Consider a standardized consol, issued today at t = 0, with annual coupon payments c. Show that the
Show that in the Black–Scholes-framework, the Esscher-transform of the Wiener-process Wt is equivalent to the Girsanov-transformation.
The normal inverse Gaussian process is a subordinated Brownian motion, where the subordinator is a process with drift η = 0, and Levy-measure Prove that the Laplace-exponent is e- -dx p(dx) =
Suppose xu is an upper barrier. Use the reflection principle to prove that with t ? 0, is the probability density function of the stopping time ?, when the Wiener-process Wt hits the barrier xu for
Analyze the characteristic exponent of the Kou-model of Problem 17.2. What is the proper drift adjustment to obtain the exponential Q-martingale Yt and is the necessary strip condition
The double exponential jump-diffusion of Kou (2002) replaces the log-normal distributed jumps of the Merton-model, by Laplace-distributed ones, with probability density function and 0 f(x) = 27
Consider the random variable N ∼ Poi(λ), which has expectation value E[N] = λ. Show that the variance is also Var[N] = λ.
Consider the SABR-model with β = 1. Show that the at-the-money smile for F0 = K ± δ, with small δ, is symmetric. Use that δ is so small that O(δ2) terms can be neglected.
Polynomials, orthogonal with respect to the weighting function w(x) = 1 in the interval [?1, 1], are called the Legendre-polynomials. They are generated by? Compute the Legendre-polynomials up to N
Prove that the payoff function of a protective put position W(x) = max (ex, K) has no transformed payoff function ŵ(z), regular in a connected strip Sw = {z = u + iv: v ∈(α, β)}.
Shifting the integration contours to v = 0 and v = 1, respectively, the complex line integrals in Problem 16.2 become principal value integrals and pick up one half of the associated residue. The
Show that the fair price of a European plain vanilla call option can be represented as where I1(v) and I2(v) are the complex line integrals evaluated along the contour v ? (0, 1). Co(K, T) =
Derive the extended Black–Scholes-equation for the Heston-model.
The Heston?Nandi-model for option pricing (Heston and Nandi, 1997, 2000) is specified as where zt?is again independent and identically standard normally distributed. In going from probability
The Duan-model is a modified version of the GARCH-in-mean specification where again ?t = ?htzt, and zt ? N(0, 1). Show that under the asymmetric variance dynamics Ax, = µ + Ah, + E %3D
Show that the kurtosis in an ordinary GARCH(1,1)-model is K=3- 1- (a + B)? 1- 2a2 – (a + B)?
Assume that actual variance has the mean reverting term structure where ?2?is the stationary variance and ?2(0)?=??20. What is the implied volatility of an option at time t = 0? (t) do²(1) = 1(o -
Consider a floating strike version of an exponential arithmetic Asian call (Problem 14.6) with European exercise right, contingent on a forward contract with TF > T. Assume that S0, T, r, σ, and
A continuously exponential averaged arithmetic Asian option is a contract, where the average price of the underlying is computed as The parameter ? ? 0 is an arbitrary weight factor. Rewrite the
An important variance reduction technique in Monte Carlo simulation is importance sampling. The idea behind this concept is to approximate an expectation by a weighted average value where the
Monte Carlo simulation approximates an unknown expectation value by an arithmetic mean, computed from a random sample. This mean can be understood as the expectation value with respect to the
Suppose the underlying S follows the standard geometric Brownian motion Use the associated Fokker?Planck-equation, together with the definition of the probability current to prove that the
A power option is a contract, whose value depends on the payoff V(S, T), with S = Sa. Show that the generalized Black–Scholes-equation can be expressed in terms of , with adjusted cost-of-carry
Verify that the mixed boundary conditions x(0) = c1 and x?(?) = c2 require the solution x(t) = c1 cos t ? c2 sin t of (14.25). (14.25) dx(1) = -x(t) dt?
Derive an explicit formula for the value of the liabilities in the Merton-model of the firm value.
There is another Greek called vanna, defined by the mixed partial derivative Derive the vanna of a European plain vanilla contract and show that it is identical for both put and call options. vanna
Prove that the so-called European put-call symmetry holds in the generalized Black?Scholes-framework. ( e2b(T=1) S; K |C,(K, T) = eb(T-1) S, K
Consider the fair price of a European plain vanilla binary contract in the Black– Scholes-framework. What is the delta of a binary call and put option?
The evolution of the bank account is governed by which can be understood as geometric Brownian motion with ? = 0. Manipulate the resulting Black?Scholes-equation to show that it is equivalent to
In demonstrating the transition from the binomial model to the Black?Scholesmodel, the parametrization u/d = e????t chosen in step 3, with ?t = ?/T.?Show that the martingale principle results in the
Imagine the stopping time τ is a fixed deterministic time τ = s, with 0 < s < T. What are the stopping events?
Look at the binomial tree in Figure 12.14. Assume that the risk-free interest rate vanishes, r = 0%, and price the European plain vanilla put option by computing the hedge-portfolios ?(1)1, ?(1)1 and
Assume an investor holds long positions in a covered call and a protective put. Both options have identical expiries and exercise prices. What is the payoff of this combined position at expiry?
Describe the position an investor holds, if she is long in a covered call and short in a protective put, with both options having the same exercise price and time to expiry.
It is possible to generalize the butterfly position even more to the form with K12 3, and a, b > 0 How are the coefficients a and b to be chosen, to generate a vanishing payoff for ST ? K1 and
Consider a modified butterfly position with K12 3, and a > 0. How is a to be chosen to guarantee a vanishing payoff for ST = K1 and ST = K3? IlButterfly = C(K1, T) – a · C(K2, T) + C(K3, T)
Establish a parity relation for binary calls and puts by adding their payoff functions and derive a formula for 0 ≤ t ≤ T.
Sketch the arbitrage portfolio for the contract initiated at t = τ.
An option that pays off one unit of currency at expiry, in case of ST ≥ K, and zero else, is called a binary or digital call option. Likewise a binary put option pays one unit of currency, if ST
Suppose a stock pays a fixed percentage dividend stream q that is continuously reinvested, such that an initial investment S0 has time T value eqTST. Construct an arbitrage portfolio for a forward
A random variable X is said to dominate another random variable Y statewise, if X(ω) ≥ Y(ω) holds, for all ω ∈ Ω, with strict inequality for at least one ω. Statewise dominance implies first
Birnbaum and Navarrete (1998) analyzed choices between two gambles. One of their decision problems was the following: 73% of the test subjects chose gamble B. Provide a fundamental argument why
Sketch the distribution functions of L1 and L2.
Reconsider Problem 10.2 and its solution to show that w(p), like the original weighting function of Kahneman and Tversky, is S-shaped, with concave curvature for small p and convex curvature for
Prelec (1998) suggested the alternative probability weighting function with 0 w(p) = e-(-logp)
Consider the following problem presented by Tversky and Kahneman (1983): Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned
Show that the interest rate in the Campbell-Cochrane-model can be made time dependent of the form? for some fixed r0, if the steady state surplus consumption ratio is modified to be What is the
The generalized Epstein?Zin-utility functional is based on the idea of a discounted certainty equivalent. One possible form is Is this functional time separable if the utility function is u(c) = ?c
Consider an economy at t = 0 and t = T. Show that under CARA-utility, the stochastic discount factor is if the usual time separable von Neumann-Morgenstern-utility functional is assumed.
The rational valuation formula for stochastic discount factor models is How does this formula simplify, if the SDF can be assumed conditionally uncorrelated, E Man Fi ELD+k\Fi] + >Cov|| | Mirn,
In adding up an infinite sum, one usually assumes the following two properties to hold Show that using these properties, the surprising result can be established. Σ Sk and ask =a >: 2 Sk = So +
Show that for ? = 0 and invertible P, the Black?Litterman posterior distribution collapses to What does invertibility of P mean? μBL) PΙ) and ΣpL = 0
Assume that you can invest a fraction π of your wealth in a portfolio P that yields a risky return of RP, and put the other fraction 1 − π in a bank account, paying the risk-free interest rate
The theoretical information ratio of a portfolio P is defined as where RP is the portfolio return, and RMP is the return on the market portfolio. Show that this ratio can never be larger than the
Confirm the functional derivative.
Show that the modified factor model with E[Fq] = ?Fq, Cov[Fp, Fq] = ?pq, Cov [Fq, ?n] = 0, E [?n] = 0, AND COV [?m , ?n]= ?mn?2n, generates the correct APT-equation (7.57). (7.57) |R) = R [1)
The original three-factor portfolio model of Fama and French (1993, 1996) is formulated in the form where SMB indicates the market capitalization spread (?small minus big?), and HML is the spread in
For an arbitrary random variable Y and a ?-algebra FA, generated by observing some event A, the variance decomposition holds. Show that the unconditional variance of the least-squares estimator|???
Check that the expectation of (7.15) is the CAPM equation. (7.15) Rs = Ro + Bs(RMP – Ro) + €s %|
Look at the general mean reversion structure where 0 ? ? ? 1 is an arbitrary coefficient. How is ? to be chosen if the intrinsic period is one month, to maintain the term structure of the yearly
Assume there are two security funds, one based on corporate bonds B, and the other on stocks S, with μB < μS, σB < σS, and ρBS = 0. Construct a market portfolio out of these two funds,
For a random variable X ∼ N(μ, σ2), the probability of realizing a value x outside the range μ ± 2σ is roughly 5%. What proportion of the market portfolio should an agent hold, if she is
Assume there are different risk-free rates for borrowing and lending, and Rb0 > RI0. Sketch the capital market curve graphically and explain what happens to the tangential portfolio.
Which expected return would an agent in the world of Problem 6.1 require to hold the market portfolio, if her risk aversion is ? = 1? Problem 6.1 Think of a world without the opportunity of
Think of a world without the opportunity of risk-free borrowing or lending. You can only hold money, which means saving is permitted, or invest in a portfolio of risky assets. Show that in this world
Convince yourself that the set K in Figure 5.2 left is not convex. Figure 5.2. K M
Imagine a financial market with one zero-coupon bond and one stock. The risk-free rate of interest over one period of time is r = 25% and the initial value of the stock is S0 = $10. There are two
Consider the payoff matrix Check if the financial market is complete, and whether or not any cyclical permutation of the security price form results in an arbitrage free market. (1 2 D= 3 (s] =
Show that the risk premium of a security with state contingent payoff |dn is given by Cov[M, dn]. Use the relation in the process. Cov[X, Y]= E[XY] – E[X\E[ Y]

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