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

Exercise 11.1.7 Why are dividends bad for the bondholders?
Exercise 11.1.6 Suppose that a holding company’s securities consist of 1,000 shares of Microsoft common stock and 55 zero-coupon bonds maturing on the same date as the Microsoft April calls. Figure out the stockholders’ gains (hence the original bondholders’ losses) if the firm issues 5, 10,
Exercise 11.1.5 Repeat the steps leading to Eq. (11.2) except that, this time, the firm issues only five bonds instead of fifteen.
Exercise 11.1.4 If the bondholders can lose money in Eq. (11.2), why do they not demand lower bond prices?
Exercise 11.1.3 Verify the following claims and explain them in simple English:(1) ∂B/∂V > 0, (2) ∂B/∂X> 0, and (3) ∂B/∂τ < 0.
Exercise 11.1.2 Prove Eq. (11.1).
Exercise 11.1.1 Argue that a loan guarantee that makes up any shortfalls in payments to the bondholders is a put with a strike price of B. The tacit assumption here is that the guarantor does not default.
Programming Assignment 10.2.2 Implement the extended binomial tree algorithm for numerical delta and gamma. Compare the results against numerical differentiation and closed-form solutions.
Exercise 10.2.1 Why does the numerical gamma in definition (10.2) not fail for the same reason?
Exercise 10.1.6 Prove that the vega as a function of σ is unimodal for σ >0. A function is unimodal if it is first increasing and then decreasing, thus having a single peak.
Exercise 10.1.5 (1) At what stock price is the theta of a European call smallest?(2) Show that the theta of an American put is always negative.
Exercise 10.1.4 What is the charm, defined as ∂/∂τ , of a European option?
Exercise 10.1.3 Show that at-the-money options have the maximum time value.
Exercise 10.1.2 Prove that ∂ P/∂X= e−rτ N(−x +σ√τ ).
Exercise 10.1.1 Verify Eq. (10.1) and that the delta of a call on a stock paying a continuous dividend yield of q is e−qτ N(x).
Programming Assignment 9.7.3 Carefully implement the diagonal method and benchmark its efficiency against the standard backward-induction algorithm.
Exercise 9.7.2 Verify the validity of Rule 2 under the binomial model.
Exercise 9.7.1 Prove that an American call fetches the same price as an American put after swapping the current stock price with the strike price and the riskless rate with the continuous dividend yield.
Programming Assignment 9.6.9 Implement the binomial tree algorithms for American options on a stock that pays a continuous dividend yield.
Exercise 9.6.8 Give an example whereby the use of risk-neutral probability (9.21)makes early exercise for American calls optimal.
Exercise 9.6.7 (1) Someone argues that we should use [ (ert −d)/(u−d) ] as the risk-neutral probability thus: Because the option value is independent of the stock’s expected return μ−q, it can be replaced with r . Show him the mistakes. (2) Suppose that we are asked to use the original
Exercise 9.6.6 Derive probability (9.21) rigorously by an arbitrage argument.
Exercise 9.6.5 Prove that the put–call parity becomes C = P+ Se−qτ −PV(X)under the continuous-payout model.
Programming Assignment 9.6.4 Implement the ideas described in this subsection.
Programming Assignment 9.6.3 Implement binomial tree algorithms for American options on a stock that pays a known dividend yield.
Exercise 9.6.2 Start with an American call on a stock that pays d dividends. Consider a package of d+1 European calls with the same strike price as the American call such that there is a European call expiring just before each ex-dividend date and a European call expiring at the same date as the
Exercise 9.6.1 Argue that the value of a European option under the case of known dividend yields equals (1−δ)m European option on a non-dividend-paying stock with the strike price (1−δ)−mX.
Programming Assignment 9.5.1 Implement the algorithm in Fig. 9.13 for American puts.
Programming Assignment 9.4.4 Write a program to compute the implied volatility of American options.
Exercise 9.4.3 (Implied Binomial Tree). Suppose that we are given m different European options prices, their identical maturity, their strike prices, their underlying asset’s current price, the underlying asset’s σ, and the riskless rate. (1)What should n be? (2) Assume that the path
Exercise 9.4.2 Solving for the implied volatility of American options as if they were European overestimates the true volatility. Discuss.
Exercise 9.4.1 Calculating the implied volatility from the option price can be facilitated if the option price is a monotonic function of volatility. Show that this is true of the Black–Scholes formula.
Exercise 9.3.8 Here is yet another way to assign u and d:u = eσ√τ/n+(1/n) ln(X/S), d = e−σ√τ/n+(1/n) ln(X/S), q = erτ/n −d u−d.(1) Show that it works. (2)What is special about this choice?
Exercise 9.3.7 Derive Theorem 9.3.4 from Lemma 9.3.3 and Exercise 6.1.6.
Exercise 9.3.6 Prove ∂2P/∂X2 = ∂2C/∂X2 (see Fig. 9.11 for illustration).
Exercise 9.3.5 A binary call pays off $1 if the underlying asset finishes above the strike price and nothing otherwise.3 Show that its price equals e−rτ N(x −σ√τ ).
Exercise 9.3.4 Verify convergence (9.17).
Exercise 9.3.3 Verify the following with the Black–Scholes formula and give heuristicarguments as to why they should hold without invoking the formula.(1) C ≈ S− Xe−rτ if S X. (2) C→S as τ →∞. (3) C→0 as σ →0 if S Xe−rτ . (5) C→S as r →∞.
Exercise 9.3.2 Show that E[ (St − S)/S ]t→μ+ σ2 2, (9.16)where t ≡ τ/n.
Exercise 9.3.1 The price volatility of the binomial model should match that of the actual stock in the limit. As q does not play a direct role in the BOPM, there is more than one way to assign u andd. Suppose we require that q = 0.5 instead of ud = 1.(1) Show that u = expμτn+σ*τn, d =
Programming Assignment 9.2.15 Implement the Monte Carlo method. Observe its convergence rate as the sampling size m increases.
Programming Assignment 9.2.14 Implement the algorithm in Fig. 9.9 and benchmark its speed. Because variables such as b and D can take on extreme values, they should be represented in logarithms to maintain precision.
Exercise 9.2.13 Modify the linear-time algorithm in Fig. 9.9 to price puts.
Programming Assignment 9.2.12 Implement the binomial tree algorithms for calls and puts.
Exercise 9.2.11 Consider a single-period binomial model with two risky assets S1 and S2 and a riskless bond. In the next step, there are only two states for the risky assets, (S1u1, S2u2) and (S1d1, S2d2). Show that this model does not admit a riskneutral probability for certain u1, u2, d1, d2, and
Exercise 9.2.10 Assume the BOPM. (1) Show that a state contingent claim that pays $1 when the stock price reaches Suidn−i and $0 otherwise at time n can be replicated by a portfolio of calls. (2) Argue that continuous trading with bonds and stocks can replicate any state contingent claim.
Exercise 9.2.9 Prove the put–call parity for European options under the BOPM.
Exercise 9.2.8 Inspect Eq. (9.10) under u→d, that is, zero volatility in stock prices.
Exercise 9.2.7 Show that the call’s delta is always nonnegative.
Exercise 9.2.6 Prove that early exercise is not optimal for American calls.
Exercise 9.2.5 The standard arbitrage argument was used in deriving the call value.Use the risk-neutral argument to reach the same value.
Exercise 9.2.4 Suppose that a call costs hS+ B+k for some k = 0 instead of hS+ B. How does one make an arbitrage profit of M dollars?
Exercise 9.2.3 Prove that the call’s expected gross return in a risk-neutral economy is R.
Exercise 9.2.2 Consider two securities,Aand B. In a period, securityA’s price can go from $100 to either (a) $160 or (b) $80, whereas security B’s price can move to$50 in case (a) or $60 in case (b). Price security B when the interest rate per period is 10%.
Exercise 9.2.1 Prove that d < R< u must hold to rule out arbitrage profits.
Exercise 8.4.6 Prove that American options on a non-dividend-paying stock satisfy C − P ≥ S− X. (This and relation (8.3) imply that American options on a nondividend-paying stock satisfy C − S+ X≥ P ≥ C − S+PV(X).)➤ Exercise 8.6.1 Consider the portfolio of puts and put on the
Exercise 8.4.4 Argue that an American put should be exercised when X− S >PV(X).
Exercise 8.4.3 Why is it not optimal to exercise an American put immediately before an ex-dividend date?
Exercise 8.4.2 Prove that if at all times before expiration the PV of the interest from the strike price exceeds the PV of future dividends before the expiration date, the call should not be exercised before expiration.
Exercise 8.4.1 Consider an investor with an American call on a stock currently trading at $45 per share. The option’s expiration date is exactly 2 months away, the strike price is $40, and the continuously compounded rate of interest is 8%.Suppose the stock is deemed overpriced and it pays no
Exercise 8.3.6 Consider a European-style derivative whose payoff is a piecewise linear function passing through the origin. A security with this payoff is called a generalized option. Show that it can be replicated by a portfolio of European calls.
Exercise 8.3.5 A European capped call option is like a European call option except that the payoff is H− X instead of S− X when the terminal stock price S exceeds H. Construct a portfolio of European options with an identical payoff.
Exercise 8.3.4 Prove put–call parity (8.2) for a single dividend of size D∗ at some time t1 before expiration: C = P+ S−PV(X)− D∗d(t1).
Exercise 8.3.3 In a certain world in which a non-dividend-paying stock’s price at any time is known, a European call is worthless if its strike price is higher than the known stock price at expiration. However, Exercise 8.3.2 says that C ≥ S−PV(X), which is positive when S > PV(X). Try to
Exercise 8.3.2 Strengthen Lemma 8.3.1 to C ≥ max(S−PV(X), 0).
Exercise 8.3.1 (1) Suppose that the time to expiration is 4 months, the strike price is $95, the call premium is $6, the put premium is $3, the current stock price is$94, and the continuously compounded annual interest rate is 10%. How to earn a riskless arbitrage profit? (2) An options market
Exercise 8.2.2 Derive a bound similar to that of Lemma 8.2.4 for European puts under negative interest rates. (This case might be relevant when inflation makes the real interest rate negative.)
Exercise 8.2.1 Show that Lemma 8.2.3 can be strengthened for European calls as follows: The difference in the values of two otherwise identical options cannot be greater than the present value of the difference in their strike prices.
Exercise 8.1.2 (Arbitrage Theorem). Consider a world with m states and n securities.Denote the payoff of security j in state i by Di j . Let D be the m×n matrix whose (i, j)th entry is Di j . Formulate necessary conditions for arbitrage freedom.
Exercise 8.1.1 Give an arbitrage argument for d(1) ≥ d(2) ≥ ·· · .
Exercise 7.4.5 Astate contingent claim has a payoff function f such that f (x) = 0 for all x = X and ∞−∞ f (x) dx = 1. Mathematically, f is called the Dirac delta function.Argue that the value of a state contingent claim equals ∂2C/∂X2.
Exercise 7.4.4 Start with $100 and put 100/(1+r ) in the money market earning an annual yield of r . The rest of the money is used to purchase calls. (1) Figure out the payoff of this strategy when the option expires 1 year from now. (2) What is the r that makes the strategy a “90/10” one,
Exercise 7.4.3 Both a protective put on a diversified portfolio and a fire insurance policy provide insurance. What is the essential difference between them?
Exercise 7.4.2 Verify the maximum profit of the cash-secured put.
Exercise 7.4.1 Howwould you characterize buying a call in terms of market outlook and risk posture?
Exercise 6.4.3 (1) Prove that a minimizes the mean-square error E[ (X−a)2 ]when a = E[ X]. (2) Show that the best predictor a of Xk based on X1, X2, . . . , Xk−1 in the mean-square-error sense, that is, with minimum E[ (Xk−a)2 | X1, X2, . . . , Xk−1 ], is the conditional least-squares
Exercise 6.4.2 Verify that the f in Eq. (6.20) is minimized at $X = (1/n)n i=1 xi .
Exercise 6.4.1 Let X1 and X2 be random variables. The random variable Y ≡ (X2 − E[ X2 ])−{α +β(X1 − E[ X1 ]) }
Exercise 6.3.1 Find the estimated regression line for { (1, 1.0), (2, 1.5), (3, 1.7),(4, 2.0) }. Check that the coefficient of determination indeed equals the sample correlation coefficient.
Exercise 6.1.6 Let X be a lognormal random variable such that ln X has mean μand variance σ2. Prove the identity ∞a xf (x) dx = eμ+σ2/2 N(μ−ln aσ+σ).
Exercise 6.1.5 Let Y be lognormally distributed with mean μ and variance σ2.Show that lnY has mean ln[ μ/1+(σ/μ)2 ] and variance ln[ 1+(σ/μ)2 ].
Exercise 6.1.4 Prove that central moments (6.7) are equivalent to
Exercise 6.1.3 Prove that E[ AX] = AE[ X] and that Cov[ AX] = ACov[ X] AT.
Exercise 6.1.2 Prove that if E[ X |Y = y ] = E[ X] for all realizations y, then X and Y are uncorrelated. (Hint: Use the law of iterated conditional expectations.)
Exercise 6.1.1 Prove Eq. (6.3) by using the well-known identity E[i aiXi ] = i ai E[ Xi ].
Exercise 5.8.4 Repeat the above two-bond argument to prove that the claims in Exercise 4.2.8 remain valid under the more general setting here.
Exercise 5.8.3 Empirically, long-term rates change less than short-term ones. To incorporate this fact into duration, we may postulate nonproportional shifts as[ 1+ S(i) ]1+ S(i)= Ki−1[ 1+ S(1) ]1+ S(1)for some K < 1.Show that a t-period zero-coupon bond’s price sensitivity satisfies
Exercise 5.8.2 Verify duration (5.17).
Exercise 5.8.1 Assume continuous compounding. Show that if the yields to maturity of all fixed-rate bonds change by the same amount, then (1) the spot rate curve must be flat and (2) the spot rate curve shift must be parallel. (Hint: The yields of zero-coupon bonds of various maturities change by
Exercise 5.7.5 Show that the market has to expect the interest rate to decline in order for a flat spot rate curve to occur under the liquidity preference theory.
Exercise 5.7.4 The return-to-maturity expectations theory postulates that the maturity strategy earns the same return as the rollover strategy with one-period bonds, i.e., [ 1+ S(n) ]n = E[ { 1+ S(1) } { 1+ S(1, 2) } · · · { 1+ S(n−1, n) } ], n > 1.Show that it is inconsistent with the local
Exercise 5.7.3 (1) Prove that E1{ 1+ S(1) }{ 1+ S(1, 2) } · · · { 1+ S(n−1, n) }= 1[ 1+ S(n) ]n under the local expectations theory. (2) Show that the local expectations theory is inconsistent with the unbiased expectations theory.
Exercise 5.7.2 Show that[ 1+ S(n) ]n = E[ 1+ S(1) ] E[ 1+ S(1, 2) ] · · · E[ 1+ S(n−1, n) ]under the unbiased expectations theory.
Exercise 5.7.1 Prove that an n-period zero-coupon bond sold at time k< n has a holding period return of exactly S(k) if the forward rates are realized.
Exercise 5.6.13 Derive Eq. (5.13).f (i, j ) = [ 1+ j S( j) ][ 1+i S(i) ]−1 −1 j −i . (5.13)
Exercise 5.6.12 (1) Figure out a case in which a change in the spot rate curve leaves all forward rates unaffected. (2) Derive the duration −(∂ P/∂y)/P under the shape change in (1), where y is the short rate S(1).
Exercise 5.6.11 Compute the one-period forward rates from this spot rate curve:S(1) = 2.0%, S(2) = 2.5%, S(3) = 3.0%, S(4) = 3.5%, and S(5) = 4.0%.
Exercise 5.6.10 Derive Eqs. (5.8) and (5.9).S(n) = f (0, 1)+ f (1, 2) + · · · + f (n−1, n)n. (5.8)(i, j ) = j S( j )−i S(i)j −i . (5.9)
Exercise 5.6.9 (1) Prove Eq. (5.6). (2) Define the forward spread for period i , s(i ), as the difference between the instantaneous period-i forward rate f (i −1, i)obtained by riskless bonds and the instantaneous period-i forward rate fc(i −1, i)obtained by corporate bonds. Prove that s(i)
Exercise 5.6.8 Consider the following four zero-coupon bonds:Type Maturity Price Yield Type Maturity Price Yield Treasury 1 year 94 6.28% Treasury 2 year 87 7.09%Corporate 1 year 92 8.51% Corporate 2 year 84 8.91%Compute the probabilities of default and the forward probabilities of default.

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

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