Financial Engineering And Computation Principles Mathematics Algorithms 1st Edition Yuh-Dauh Lyuu - Solutions

Exercise 24.2.1 (1) Supply the arbitrage argument for Eq. (24.1). (2) Generalize(1) by describing a strategy that replicates the forward contract on a coupon bond that may make payments before the delivery date.
Programming Assignment 23.4.2 Add the annualized term structure of yield volatilities to the output of the program of Programming Assignment 23.2.10.
Exercise 23.4.1 Suppose we add a binomial process for the stock price to our binomial interest rate model. In other words, stock price S can in one period move to Su or Sd. What are the constraints on u and d?
Programming Assignment 23.3.7 Write a program to price European options on the Treasuries.
Exercise 23.3.6 Derive the put–call parity for options on coupon bonds.
Exercise 23.3.5 Prove that an American option on a zero-coupon bond will not be exercised early.
Exercise 23.3.4 (1) How do we compute the forward price for a forward contract on a bond? (2) Calculate the forward price for a 2-year forward contract on a 1-year T-bill.
Programming Assignment 23.3.3 Implement a differential tree method for the implied volatility of American options under the BOPM [172, 625].
Programming Assignment 23.3.2 Implement the algorithm in Fig. 23.13.
Exercise 23.3.1 Does the idea of spread assume parallel shifts in the term structure?
Programming Assignment 23.2.11 Calibrate the tree with the secant method.
Programming Assignment 23.2.10 Program the algorithm in Fig. 23.9 with the Newton–Raphson method.
Exercise 23.2.9 Start with a binomial interest rate tree but without the branching probabilities, such as Fig. 23.2. (1) Suppose the state price tree is also given.(2) Suppose only the state prices at the terminal nodes are given and assume that the path probabilities for all paths reaching the
Exercise 23.2.8 Compute the n discount factors implied by the tree in O(n2)time.
Exercise 23.2.7 (1) Suppose we are given a binomial state price tree and wish to price a security with the payoff function c at time j by using the risk-neutral pricing formula d( j ) E[ c ]. What is the probability of each state’s occurring at time j?(2) Take the binomial state price tree in
Exercise 23.2.6 (1) Based on the sample term structure and its associated binomial interest rate tree in Fig. 23.8, what is the next baseline rate if the four-period spot rate is 4.4%? (2) Confirm Theorem 23.2.2 by demonstrating that the baseline rate produced by Eq. (23.4) is smaller than the one
Exercise 23.2.5 Fix a period. (1) Show that the forward rate for that period equals the expected future spot rate under some risk-neutral probability measure. (2) Show further that the said forward rate is a martingale. (Hint: Exercise 13.2.13.)
Exercise 23.2.4 (1) Prove Theorem 23.2.2 for two-period zero-coupon bonds.(2) Prove Theorem 23.2.2 in its full generality.
Exercise 23.2.3 Suppose the probability of moving from r to r is 1−q and that of moving to rh is q. Also assume that a period has length t. (1) Show that the variance of ln r after a period is q(1−q)(ln rh −ln r)2. (2) Hence, if we define σ2 to be the above divided by t, then rh r=
Exercise 23.2.2 Consider a short rate model such that the two equally probable short rates from the current rate r are reμ+σ√t and reμ−σ√t , where μ may depend on time. Verify that this model can result from the binomial interest rate tree when the volatilities σj are all equal to some
Exercise 23.2.1 Verify that the variance of ln r in period k equals σ2 k (k−1)t.(Consistent with Eq. (23.3), the variance of ln r (t) equals σ(t)2t in the continuoustime limit.)
Exercise 23.1.2 Let it denote the period interest rate for the period from time t −1 to t. Assume that 1+it follows a lognormal distribution, ln(1+it ) ∼ N(μ, σ2).(1)What is the value of $1 after n periods? (2)What are its distribution, mean, and variance?
Exercise 23.1.1 Ariskless security with cash flow C1,C2, . . . ,Cn has a market price ofn i=1 Cid(i ). The discount factor d(i) denotes the PV of $1 at time i from now.Is the formula still valid if the cash flow depends on interest rates?
Exercise 22.4.2 Assume continuous compounding. Justify the following claims.(1) If the forward rate curve should be a continuous function, a quadratic spline is the lowest-order spline that can fit the discount function. (2) If the forward rate curve should be continuously differentiable, a spline
Exercise 22.4.1 Verify Eq. (22.4).Pi −1−Cini = k=1 ak fk(ni )+Ci ni j=1 fk( j ), 1 ≤ i ≤ m. (22.4)
Exercise 22.3.2 Suppose we want to fit an exponential curve y = aebx to the data, but we have only a linear-regression solver. How do we proceed?
Exercise 22.3.1 Show how to fit a quadraticfunc tion d(t) = a0 +a1t +a2t2 to the discount factors by using multiple regression.
Exercise 22.2.1 Show that exponential interpolation scheme (22.1) for the discount function is equivalent to the linear interpolation scheme for the spot rate curve when spot rates are continuously compounded.
Exercise 22.1.1 Verify the relation between s(t) and d(t) given above.
Exercise 21.4.8 Prove that a cap is more valuable than an otherwise identical swaption.
Exercise 21.4.7 Afirm holds long-duration corporate bonds. It uses swaps to create syntheticfloating-rate assets at attractive spreads to LIBOR and to shorten the duration much as A does in Fig. 21.9. Why should the fact that most corporate bonds are callable trouble the firm? How may callable
Exercise 21.4.6 Replicate swaps with interest rate caps and floors.
Exercise 21.4.5 Consider a swap with zero value. How much up-front premium should the fixed-rate payer pay in order to lower the fixed-rate payment from C to$C?
Exercise 21.4.4 Verify the equivalence of the two views on interest rate swaps.
Exercise 21.4.3 Use Example 21.4.3’s data to calculate the fixed rate that makes the swap value zero.
Exercise 21.4.2 A firm buys a $100 million par of a 3-year floating-rate bond that pays the 6-month LIBOR plus 0.5% every 6 months. It is financed by borrowing $100 million for 3 years on terms requiring 10% annual interest rate paid every 6 months.Show that these transactions create a synthetic
Exercise 21.4.1 Party A wants to take out a floating-rate loan, and B wants to take out a fixed-rate loan. They face the borrowing rates below:Fixed Floating A F A% LIBOR +SA%B F B% LIBOR +SB%Party A agrees to pay the bank a floating rate of (LIBOR −SA)% in exchange for a fixed rate of (FA +
Exercise 21.2.3 Prove the equivalency for the floorlet.
Exercise 21.2.2 Verify that an FRA to borrow at a rate of r can be replicated as a portfolio of one long caplet and one short floorlet with identical strike rate x and expiration date equal to the settlement date of the FRA.
Exercise 21.2.1 An investor owns T-bills and expects rising short-term rates and falling intermediate-term rates. To profit from this reshaping of the yield curve, he sells T-bills, deposits the cash in the bank, and purchases puts on the l0-year Treasury yield. Analyze this strategy.
Exercise 21.1.6 (1) Prove Exercise 5.6.3 with an arbitrage argument. (2) Verify Eq. (21.3).
Exercise 21.1.5 A pension fund manager wants to take advantage of the high yield offered on T-notes, but unusually high policy payouts prohibit such investments. It is, however, expected that cash flow will return to normal in September. What can be done?
Exercise 21.1.4 Calculate the conversion factor for a 13% coupon bond with 19 years and 11 months to maturity at the delivery date.
Exercise 21.1.3 A deliverable 8% coupon bond must have a conversion factor of one regardless of its maturity as long as the accrued interest is zero. Why?
Exercise 21.1.2 Calculate the annualized implied 9-month LIBOR rate between June 1995 and March 1996 from the data in Fig. 21.3.
Exercise 21.1.1 Derive the formula for the change in the T-bill futures’ invoice price per tick with 91 days to maturity.
EXAMPLE 20.2.1 Here is a lognormal model. Let the processes {Ut } and { Vt } be independent of each other, {Ut } is Gaussian white noise, and ln Vt ∼ N(a, b2). One simple way to achieve this as well as to make both { |Xt −μ| }’s and { (Xt −μ)2 }’s autocorrelations positive is to posit
Exercise 20.2.2 For the lognormal model, show that (1) the kurtosis of Xt is 3e4b2,(2) Var[ Vt ] = e2a+b2 (eb2 −1), (3) Var[ |Xt −μ| ] = e2a+b2 (eb2 −2/π), (4) Var[ V2 t ] =e4a+4b2 (e4b2 −1), and (5) Var[ (Xt −μ)2 ] = e4a+4b2(3e4b2 −1).
Exercise 20.2.1 Assume that the processes { Vt } and {Ut } are stationary and independent of each other. Show that the kurtosis of Xt exceeds that of Ut provided that both are finite.
Exercise 20.1.15 Write the equivalent LS problem for function (20.9).
Exercise 20.1.14 GivenWold’s decomposition (20.8), show that λτ , the autocovariance at lag τ , equals σ2∞j=0 c j c j+τ .
Exercise 20.1.13 Let { Xt } be a sequence of independent, identically distributed random variables with zero mean and unit variance. Prove that the process{Yt ≡l k=0 akXt−k } with constant ak is stationary.
Exercise 20.1.12 Why are near-zero autocorrelations important for returns to be unpredictable?
Exercise 20.1.11 Verify that if { Xt } is strict white noise, then so are { |Xt| } and{ X2 t}.
Exercise 20.1.10 Show that if price changes are uncorrelated, then the variance of prices must increase with time.
Exercise 20.1.9 Consider a stationary process { Xt } with known mean and autocovariances.Derive the optimal linear prediction a0 +a1Xt +a2Xt−1 + · · · + atX1 in the mean-square-error sense for Xt+1. (Hint: Exercise 6.4.1.)
Exercise 20.1.8 Show that, for a stationary process with known mean, the optimal predictor in the mean-square-error sense is the mean μ.
Exercise 20.1.7 Prove that λτ = λ−τ for stationary processes.
Programming Assignment 20.1.6 Write the simulator in Fig. 18.5 to generate stock prices. Then experiment with the ML estimators for goodness of fit.
Exercise 20.1.5 The constant elasticity variance (CEV) process follows dS/S =μdt +λSθ dW, where λ > 0. Derive the ML estimators with μ = 0.
Exercise 20.1.4 Use the process for ln r to obtain theMLestimators for the Ogden model.
Exercise 20.1.3 Derive the formulas for β and μ.
Exercise 20.1.2 Assume that the stock price follows Eq. (20.1). The simple rate of return is defined as [ S(t)− S(0) ]/S(0). Suppose that the volatility of the stock is that of simple rates of return, σs, instead of the instantaneous rates of return, σ.Express σ in terms of σs, the horizon t,
Exercise 20.1.1 Derive the ML estimators for μ and σ2 based on simple rates of returns, Si ≡ Si+1 − Si , i = 1, 2, . . . , n.
Exercise 19.3.6 Write the matrix A in Eq. (19.19).
Exercise 19.3.5 Write the matrices for A, B,b, and d.
Exercise 19.3.4 Construct the spline in Example 19.3.1 when y2 = 1.3.
Exercise 19.3.3 Verify Eq. (19.18).hi f i−1 +2(hi +hi+1) f i +hi+1 f i+1 = 6 yi+1 − yi hi+1 − yi − yi−1 hi , (19.18)
Exercise 19.3.2 Write the tridiagonal system for the natural spline.
Exercise 19.3.1 Verify Eq. (19.16).f (xi+)=− 2 hi+1 (2 f i + f i+1)+6 yi+1 − yi h2 i+1 . (19.16)
Exercise 19.2.16 How would you define slope duration and curvature duration?
Exercise 19.2.15 Prove Cov[ yt , ft ] = L.
Exercise 19.2.14 Once the appropriate estimated factor loadings have been obtained, verify that the factors themselves can be estimated by ft = (LT!−1L)−1 LT!−1(yt −μ).
Exercise 19.2.13 (Lawson–Hanson Algorithm) It is known that any A∈ Rm×n can be decomposed as A= QTR such that Q∈ Rm×m is orthogonal and R∈ Rm×n is zero below the main diagonal. This is called the QR decomposition. Solve the LS problem with linear constraints by using the QR decomposition
Exercise 19.2.12 Design an algorithm for the constrained LS problem by using the SVD.
Exercise 19.2.11 Prove Eq. (19.14). (Hint: Exercise 19.2.6.)ATC−1 Ax = ATC−1b. (19.14)
Exercise 19.2.10 (Underdetermined Linear Equations) Suppose as before that A∈ Rm×n and b ∈ Rm, but m≤ n. Assume further that m= rank(A). Let U VT be the SVD of A. Argue that all solutions to Ax = b are of the form$x = V+UTb+V0 y}m} n−m= A+b+V2y (19.10)for arbitrary y ∈ Rn−m, where V
Exercise 19.2.9 Prove that the pseudoinverse of the pseudoinverse is itself, i.e.,(A+)+ = A.
Exercise 19.2.8 Prove that if the A in Theorem 19.2.3 has full column rank, then A+ = (ATA)−1AT. (This implies, in particular, that A+ = A−1 when A is square.)
Exercise 19.2.7 Verify Eq. (19.8).Cov[ xL S ] = σ2(ATA)−1 . (19.8)
Exercise 19.2.6 Suppose that σ2C is the covariance matrix of for a positive definite C in linear regression model (19.7). How do we solve the LS problem Ax = b by using the Gauss–Markov theorem?
Exercise 19.2.5 Show that the sample residuals of the OLS estimate, AxL S −b, are orthogonal to the columns of A.
Exercise 19.2.4 Define the bandwidth of a banded matrix A as l +u+1, where l is the lower bandwidth and u is the upper bandwidth. Prove that ATA’s bandwidth is at most ω−1 if A has bandwidth ω.
Exercise 19.2.3 Let (x) ≡ (1/2) $Ax −b$2. Prove that its gradient vector,%(x) ≡∂(x)∂x1,∂(x)∂x2, . . . ,∂(x)∂xnT, equals AT(Ax −b), where x ≡ [ x1, x2, . . . , xn ]T. (Normal equations are %(x) = 0).
Exercise 19.2.2 (1) Phrase multiple regression as an LS problem. (2) Write the normal equations.
Exercise 19.2.1 What are the normal equations for linear regression (6.13)?
Exercise 19.1.5 (1) Prove Eq. (19.4). (2) Show that A and AT share the same singular values.
Exercise 19.1.4 Prove that WX ∼ N(Wμ,WCWT) if X ∼ N(μ,C).
Exercise 19.1.3 Verify the correctness of the procedure for generating the bivariate normal distribution in Subsection 6.1.2.
Exercise 19.1.2 Prove that the jth principal component pj has the maximum variance among all the normalized linear combinations of the xi s that are uncorrelated with p1, p2, . . . , pj−1.
Exercise 19.1.1 Let C ≡ [ ci j ] denote the covariance matrix of x ≡ [ x1, x2, . . . , xn ]T. Show that C and x’s correlation matrix P are related by C = P , where≡ diag[√c11,√c22, . . . ,√cnn ]. (Hence, C is positive definite if and only if P is.)
Programming Assignment 18.3.6 Price European options with quasi-random sequences.The computational framework is identical to the Monte Carlo method except that random numbers are replaced with quasi-random numbers with ndimensional sequences for problems with n time periods.
Programming Assignment 18.3.5 Implement the Faure sequence. Pay attention to evaluatingi jmod p efficiently.
Exercise 18.3.4 Verify the sequence for the third component in Example 18.3.2.
Programming Assignment 18.3.3 Implement the Sobol’ sequence.
Programming Assignment 18.3.2 Implement the two-dimensional Halton sequence,apply it to numerically evaluating 1 0 x2 dx, and compare it with Monte Carlo integration.
Exercise 18.3.1 Compute the first 10 one-dimensional Halton points in base 3.
Exercise 18.2.13 Suppose you are searching in set A for any element from set B⊆ A. The Monte Carlo approach selects N elements randomly from A and checks if any one belongs to B. An alternative partitions the set A into m disjoint subsets A1, A2, . . . , Am of equal size, picks N/m elements from
Programming Assignment 18.2.12 Implement the control-variates method for arithmeticaverage-rate calls and puts.
Exercise 18.2.11 Why is it a mistake to use independent random numbers in generating X and Y?

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Financial Engineering And Computation Principles Mathematics Algorithms 1st Edition Yuh-Dauh Lyuu - Solutions

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