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

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2. In which of the following situations would we use tuples instead of structs or classes?a) When we wish to create lists of logically related data in client code.b) When we are not concerned with nested data structures.c) When the tuple data is just a repository.d) A tuple is similar to a class
1. What are the top two advantages of using tuples?a) As return types of functions.b) As input arguments to functions.c) To align data on word boundaries.d) To avoid copying data.
11. (Factorials)Write a recursive metafunction to compute factorials:n! = 1 if n = 0 n! = n(n − 1)! if n ≥ 1.
10. The Reynolds number (Re) is an important dimensionless quantity in fluid mechanics that is used to help predict flow patterns in different fluid flow situations. It is widely used in applications ranging from liquid flow in a pipe to the passage of air over an aircraft wing. It is also related
9. (Ackermann Function)Consider the Ackermann function:A(m, n) =⎧⎪⎨⎪⎩n + 1 if m = 0 A(m − 1, 1) if m > 0 and n = 0 A(m − 1, A(m, n − 1)) if m > 0 and n > 0 Implement this definition in C++ and experiment with the following examples:A(1, 2) = 4 A(4, 3) = 2265536 − 3 A(1, 3) =
8. (Overlapping Enabler Conditions)Function templates with enabling conditions that are not mutually exclusive can lead to ambiguities. Once the compiler has examined the enabling conditions and included the function into the overload resolution set, normal C++ overload resolution rules are then
7. (Advantages of std::enable_if)Which of the following statements can be considered useful features of std::enable_if?:a) More user-friendly error messages than when using ‘raw’ (unrestricted) template parameters.b) Its use can lead to more robust code.c) It restricts templates to types that
6. (C++11 and STL Algorithms)In Section 4.4.1 we discussed how to extend STL algorithms using a number of features in C++. The objective of this exercise is to analyse and review the code to compute the inner product of two containers containing arithmetic data.Answer the following questions:a)
5. (References)Let A be an empty class. Determine which of the following types are references, lvalue references or rvalue references: A, A&. What about A&&, int, int&, int&&?
4. (Code Inspection)The objective of this exercise is to inspect the code in Section 4.3.4 in order to understand it. The questions are:a) Inspect each line and determine if the result is true or false; for example:std::cout
3. (Value Categories)Determine by inspection if the following expressions are xvalue, lvalue or prvalue:a) a ? b : c(ternary conditional expression for somea, b and c).b) a+b, a%b, &a.c) “Hello world”.d) nullptr.e) ++a, ––a.f) a++, a––.Can you use type traits to answer this question as
2. (Variadic Functions)The goal of this exercise is to compute the standard deviation of the arguments of a variadic function using the functionality in Section 4.2.2. We propose two phases; first create a va_list called args1 and use it to compute the mean, then copy args1 using va_copy to compute
1. (Implicit Conversions ‘101’)Review and run the code that implements the cases A1, A2, A3 and A4 in Section 4.2.1.Answer the following questions:a) Determine which compiler errors and/or warnings are generated by this code. Determine the seriousness of these errors.b) Determine how to modify
6. (Becoming Acquainted with Callable Objects)In Section 3.5 we introduced five kinds of callable objects that can be used as target methods of std::function. The objective here is to write code to compute the inner product of two vectors using each callable object type and then to generalise the
5. (Re-engineering Legacy Code)We discuss a specific example that is based on the Bridge pattern that we used to create a flexible framework for numerical quadrature in one dimension (see Duffy, 2004, pp. 290–292). The design is based on a number of assumptions and techniques: Traditional OOP
4. (Exception Handling and Functions)Exceptions can be generated when working with functions and they can be caused by not having defined a function target, for example. In these cases an exception of type std::bad_function_call will be thrown. We take an example of a function that takes a double
3. (Numerical Quadrature Scheme)In Section 3.9 we developed some numerical quadrature schemes. In this exercise we extend the functionality by implementing the basic trapezoidal rule:∫b af (x)dx ≈ (b − a)2[f (a) + f (b)].Answer the following questions:a) Implement the code for this scheme by
2. (Using Libraries and Code)Let us assume that we have a library of modules for three-dimensional geometry. One module computes the distance between two points in 3D space:// Library adapter double Taximan3D(double x1, double y1, double z1, double x2, double y2, double z2){ // Distance between two
1. (General Brainstorming Questions)We wish to compare the relative advantages and disadvantages of using the objectoriented approach (classes, inheritance and subtype polymorphism) and the more general approach taken in this chapter. To keep the discussion concrete you can refer to the example on
9. (Move Semantics ‘101’)The objective of this exercise is to get you used to move semantics and rvalue references.Answer the following questions:a) Create a string and move it to another string. Check the contents of the source and target strings before and after the move.b) Create a vector
8. (Smart Pointers Review)We include some easy exercises on the use of smart pointers.Answer the following questions:a) Use move semantics to create a shared pointer from another shared pointer and from a unique pointer.b) Swap two shared pointers; swap two unique pointers.c) Create a weak pointer
7. (Alias Template and its Advantages Compared to typedef)The keyword typedef does not work with templates (at least not directly) and we need to do a lot of contortions when working with template classes. To this end, we create a class that is composed of a container:template class Client{ // An
6. (Weak Pointers)A weak pointer is an observer of a shared pointer. It is useful as a way to avoid dangling pointers and also when we wish to use shared resources without assuming ownership.Answer the following questions:a) Create a shared pointer, print its use count and then create a weak
5. (Custom Deleter)Shared and unique pointers support deleters. A deleter is a callable object that executes some code before an object goes out of scope. A deleter can be seen as a kind of callback function. We first give a simple example to show what we mean: we create twodimensional points as
4. (Smart Pointers and STL Algorithms)In this exercise we create a simple example of STL containers whose members are smart pointers.To this end, consider the following class hierarchy:class Base{ // Base class private:public:Base() { }virtual void print() const = 0;protected:virtual ~Base() { cout
3. (The Smart Pointer std::auto_ptr)This pointer type is deprecated in C++11. Consider the following code:using std::auto_ptr;// Define auto_ptr pointers instead of raw pointers std::auto_ptr d(new double (1.0));.// Dereference *d = 2.0;// Change ownership of auto_ptr // (after assignment, d is
2. (Shared Pointers)The objective of this exercise is to show shared ownership using smart pointers in C++.Create two classes C1 and C2 that share a common heap-based object d as data member:std::shared_ptr d;Answer the following questions:a) Create the code for the classes C1 and C2 each of which
1. (First Encounters with Smart Pointers: Unique Pointers)Consider the following code that uses raw pointers:{ // Block with raw pointer lifecycle double* d = new double (1.0);Point* pt = new Point(1.0, 2.0); // Two-d Point class// Dereference and call member functions*d = 2.0;(*pt).X(3.0); //
Exercise 31.4.3 Prove that (1) C = PPT, where P’s ith column is the eigenvector ui , (2) P−1 = diag[ λ−1 1 , λ−1 2 , . . . , λ−1 n ] PT, (3) P [dZ1,dZ2, . . . ,dZn ]T =[dW1,dW2, . . . ,dWn ]T, and (4) PTP = diag[ λ1, λ2, . . . , λn ].
Exercise 31.4.2 If the stock price follows dS = Sμdt + Sσ dW, what is its VaR τyears from now at c confidence?
Exercise 31.4.1 What is the VaR of a futures contract on a stock?
Exercise 31.3.5 Complete the proof of the APT under the general factor models.
Exercise 31.3.4 Prove Theorem 31.3.1.
Exercise 31.3.3 For the one-factor APT, what will become of λ1 if the CAPM holds?
Exercise 31.3.2 Describe a procedure to convert a set of correlated factors into a set of uncorrelated factors, which are easier to handle.
Exercise 31.3.1 Assume that the single factor f is the market rate of return, rM.Write the return processes as ri −rf = αi +bi (rM −rf)+i . As usual, E[ i ] = 0 andi is uncorrelated with the market return. Show that bi = Cov[ ri , rM ]/Var[ rM ] , as in the CAPM.
Exercise 31.2.12 Prove that using any efficient portfolio for the risky assets as the proxy for the market portfolio results in linear relations between the expected rates of return and the betas, just as in the CAPM.
Exercise 31.2.11 Why are security analysts’ 1-year forecasts worse than 5-year ones?
Exercise 31.2.10 A bank offers the following financial product to a mutual fund manager planning to buy a certain stock in the near future. If the stock price is over$50, the manager buys it at $50. If the stock price is below $40, the manager buys it at $40. If the stock price is between the two,
Exercise 31.2.9 A mutual fund manager believes that the market is going to be relatively calm in the near future and writes a covered index call. Analyze it by following the same logicas that of the protective put.
Exercise 31.2.8 Consider a portfolio worth $1,000 times the S&P 500 Index and with a beta of 1.0 against the index. Argue that buying 10 put index options with a strike price of 1,000 insures against the portfolio value’s dropping below $1,000,000.
Exercise 31.2.7 Redo Example 31.2.2 with S = 1000, β = 2, q = 0.01, and r = 0.05.
Exercise 31.2.6 (1) Verify that pricing formula (31.4) is linear (the price of the sum of two assets is the sum of their prices, and the price of a multiple of an asset is the same multiple of the price). (2) Derive the same results from the no-arbitrage principle.
Exercise 31.2.5 Why must all portfolios with the same expected rate of return but different total risks fall on the same point on the security market line?
Exercise 31.2.4 If an asset has a negative beta, the CAPM says its expected rate of return should be less than the riskless rate even if this asset is very risky with a large standard deviation. Why? (For example, we saw in Chap. 24 that IO strips earn less than the riskless rate despite their high
Exercise 31.2.3 For an asset uncorrelated with the market (that is, with zero beta), the CAPM says its expected rate of return is the riskless rate even if this asset is very risky with a large standard deviation.Why?
Exercise 31.2.2 Prove the security market line formula in Theorem 31.2.1.
Exercise 31.2.1 Verify that the market portfolio is efficient.
Exercise 31.1.7 Two portfolio selection models are strictly equivalent if they have the same set of obtainable mean–standard deviation combinations. Prove that any model that does not impose the nonnegative constraint on ω is strictly equivalent to some general portfolio selection model, which
Exercise 31.1.6 What would the one-fund theorem imply about trading volumes?
Exercise 31.1.5 Consider a portfolio P of n assets each following an independent geometric Brownian motion process with identical mean and variance, dSi /Si = μdt +σ dWi . Each asset has the same weight of 1/n in the portfolio.Show that this portfolio’s expected rate of return, E[ ln(P(t)/P(0))
Exercise 31.1.4 Let P(t) denote the asset price at time t. Define r (T) ≡[ P(T)/P(0) ]−1 as the holding period rate of return for a period of length T and rc(T) ≡ ln(P(T)/P(0)) as the continuous holding period rate of return for the same period. Under the assumption that asset prices are
Exercise 31.1.3 Let C ≡ [ σi j ] be a positive definite matrix. (1) Prove that maxi σii is the maximum value ofij ωiωjσi j under the constraintsi ωi = 1 and ωi ≥ 0.(2) How about the minimum value under the same constraints? (You may assume that the row sums of C−1 are all
Exercise 31.1.2 Construct a portfolio with zero risk from two perfectly negatively correlated assets without short sales.
Exercise 31.1.1 Express the efficient portfolio in matrix form.
Programming Assignment 30.3.2 Implement the cash flow generator for sequential PAC bonds.
Programming Assignment 30.3.1 Implement the cash flow generator in Fig. 30.2.
Exercise 30.2.2 Argue that the maximum coupon rate that could be paid to a floater is higher than would be possible without the inclusion of an inverse floater.
Exercise 30.2.1 Repeat the calculations in the text by using the following formula:(initial LIBOR−50 basis points)+1.5×(change in LIBOR).
Programming Assignment 29.4.2 Implement the OAS computation for the fourtranche sequential CMO under the BDT model. Assume a constant SMM.
Exercise 29.4.1 Argue that the OAS with zero interest rate volatility, called the zero-volatility OAS, corresponds to the static spread.
Exercise 29.3.1 Suppose that MBSs are priced based on the premise that there are no prepayments until the 12th year, at which time the pool is repaid completely. This is called the FHA 12-year prepaid-life concept. Argue that premium-priced MBSs are overvalued and discount MBSs are undervalued if
Exercise 29.2.3 Firms that derive income from servicing mortgages can be viewed as taking a long position in IOs. Why?
Exercise 29.2.2 From Exercise 29.1.12, show that the prices of PO and IO strips are extremely sensitive to prepayment speeds.
Exercise 29.2.1 Divide the borrowers into slow and fast refinancers. (More refined classification is possible.) The slow refinancers are assumed to respond to refinancing incentive at a higher rate than fast refinancers. Describe how this setup models burnout.
Programming Assignment 29.1.16 Implement the algorithm in Fig. 29.12 for the cash flows of a four-tranche sequential CMO with a Z tranche. Assume that each tranche carries the same coupon rate as the underlying pool’s mortgage rate. Figures 29.13 and 29.14 plot the cash flows and remaining
Exercise 29.1.15 Calculate the monthly prepayment amounts for Fig. 29.11.
Programming Assignment 29.1.14 Implement the algorithm in Fig. 29.10.
Exercise 29.1.13 Show that a pass-through backed by traditional mortgages with a mortgage rate equal to the market yield is priced at par regardless of prepayments.Assume either zero servicing spread or a pass-through rate equal to the market yield. (Prices of par-priced pass-throughs are hence
Exercise 29.1.12 Derive the PVs of the PO and IO strips based on current-coupon mortgages under constant SMM and zero servicing spread.
Exercise 29.1.11 Derive Eq. (29.11) by using Eqs. (29.2) and (29.4).
Exercise 29.1.10 Verify Eqs. (29.9) and (29.10).
Exercise 29.1.9 Verify that (1) PPi = Bi [SMMi /(1−SMMi ) ] and (2) the actual principal payment Pi +PPi is bi−1(Pi +RBi ×SMMi ) (not bi Pi ).
Exercise 29.1.8 Show that the scheduled monthly mortgage payment at month i is Bi−1(r/m)(1+r/m)n−i+1(1+r/m)n−i+1 −1.
Exercise 29.1.7 Is theSMMassuming 200 PSA twice theSMMassuming 100 PSA?
Exercise 29.1.6 Consider the following PSA numbers:Month 6 12 18 24 30 36 PSA 100 130 154 230 135 125 Compute their equivalent CPRs.
Programming Assignment 29.1.5 Consider an IAS with an amortizing schedule that depends solely on the prevailing k-period spot interest rate. This swap’s cash flow depends on only the prevailing principal amount and the prevailing k-period spot interest rate. Design an efficient algorithm to price
Programming Assignment 29.1.4 Implement the algorithm in Fig. 29.5. The binomial T-bill rate tree and the mortgage rate as a spread over the T-bill rate are parts of the input.
Programming Assignment 29.1.3 Given an n-period binomial short rate tree, design an O(kn2)-time algorithm for generating k-period spot rates on the nodes of the tree. This tree documents the dynamics of the k-period spot rate.
Exercise 29.1.2 Consider two mortgages with identical remaining principals but different mortgage rates. Show that their remaining principal balances after the next monthly payment will be different; in fact, the mortgage with a lower mortgage rate amortizes faster.
Exercise 29.1.1 Derive Eq. (29.4) from Eq. (3.6).
Exercise 28.6.5 Which represents a better deal, refinancing from an 8% loan to a 6% loan or from an 11.5% loan to a 9.5% loan?
Exercise 28.6.4 Consider a mortgagor who refinances every a months with an n-month loan every time. Show that the monthly payment after the ith refinancing is original balance×(1+r )n −(1+r )a(1+r )n −1i r(1+r )n(1+r )n −1, where r is the unchanging monthly mortgage rate.
Exercise 28.6.3 Does it make economic sense to refinance a mortgage if rates have not changed?
Exercise 28.6.2 Given that refinancing involves certain fixed costs, which will tend to prepay faster, mortgage securities backed by 15-year mortgages or 30-year mortgages?
Exercise 28.6.1 There are reasons prepayments arising from lower interest rates increase the return of a pass-through if it was purchased at a discount. What are they?
Exercise 28.5.1 Even without prepayments, the scheduled monthly payment to MBS holders increases slightly over time. Why?
Exercise 28.4.1 Repeat the calculations in Example 28.4.2 under 3% SMM.
Exercise 27.4.12 Assume a flat prevailing spot rate curve and continuous compounding.Prove that for an option-free nonbenchmark bond, any calibrated interest rate tree will compute a spread that equals the yield spread. (The yield spread is the difference between the yields to maturity of benchmark
Exercise 27.4.11 Correct Fig. 27.7(d). Algorithm for computing the OAS of callable bonds: input: P. n, cp[n], r[1..n]. C[0..n], v[1..n], e; P[1..n+1],s; real integer i, j; s=0; // Initial guess. P[1]:= 00; while [P[1] P|> ] { } return s; for (i=1 to n+1) {P[i]=C[n];} // Initialization. for (i=n
Programming Assignment 27.4.10 Implement the algorithm in Fig. 27.8 with the Ridders method.
Programming Assignment 27.4.9 Implement the algorithm in Fig. 27.8 with the secant method.
Programming Assignment 27.4.8 Implement the algorithm in Fig. 27.8 with the differential tree method, which is based on the Newton–Raphson method.
Exercise 27.4.7 Argue that using Monte Carlo simulation to price callable(putable) bonds tends to underestimate (overestimate, respectively) their values.
Exercise 27.4.6 For a putable bond, how does itsOAS behave (1) when the market price decreases, other things being equal, and (2) when the coupon rate decreases, other things being equal and with the market price at par?
Exercise 27.4.5 Explain why the OAS of a callable bond decreases as the interest rate volatility increases, other things being equal.
Exercise 27.4.4 Argue that the OAS does not assume parallel shifts in the term structure.
Exercise 27.4.3 For bonds with embedded options, traditional duration measures such as modified duration lose relevance because of cash flow uncertainties. The duration of a callable bond after the call option is adjusted for is commonly referred to as the option-adjusted duration (OAD) and is
Exercise 27.4.2 Argue that when interest rates rise, the price of a callable bond will not fall as much as the price of its noncallable component.
Exercise 27.4.1 Assume any stochastic discrete-time short rate model. Consider a risky corporate zero that is not currently in default. When the firm defaults, it stays in the default state until the maturity when the investor receives zero dollar. Let pi denote the risk-neutral probability that

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

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