Questions and Answers of Accounting For Financial Instruments

Exercise 14.3.8 Let X(t) be the Ornstein–Uhlenbeck process in Eq. (14.11). Show that the differential for Y(t) ≡ X(t) eκt is dY = σeκt dW. (This implies that Y(t), hence X(t) as well, is
Exercise 14.3.7 Given dY/Y = μdt +σ dW and dX/X= r dt, derive the stochasticdifferential equation for F ≡ X/Y.
Exercise 14.3.6 Verify that U ≡ Y/Z follows dU/U = (a − f −bgρ) dt +bdWy −g dWz, where Y and Z are drawn from Example 14.3.5.
Exercise 14.3.5 Redo Example 14.3.4 except that dY = a dt +b dWy and dZ=f dt +g dWz, where dWy and dWz have correlation ρ.
Exercise 14.3.4 Consider the Ito process U ≡ (Y+ Z)/2, where dY/Y = a dt +bdW and dZ/Z= f dt +g dW. Processes Y and Z share the Wiener process W.Derive the stochastic differential equation for dU/U.
Exercise 14.3.3 (1)What is the stochastic differential equation for the process Wn?(2) Show that t sWn dW = W(t)n+1 −W(s)n+1 n+1− n 2 t sWn−1 dt.(Hint: Use Eqs. (13.15) and (13.16) or apply
Exercise 14.3.2 Let X follow the geometricBrownian motion process dX/X=μdt +σ dW. Show that R≡ ln X+σ2t/2 follows dR= μdt +σ dW.
Exercise 14.3.1 Assume that dX/X= μdt +σ dW. (1) Prove that ln X follows d(ln X) = (μ−σ2/2) dt +σ dW. (2) Derive the probability distribution of ln(X(t)/X(0)).
Exercise 14.2.2 Prove Eq. (14.3) by using Ito’s formula.
Exercise 14.1.5 Prove that stochastic integration reduces to the usual Riemann–Stieltjes form for constant processes.
Exercise 14.1.4 Prove that E[ t 0 WdW] = 0.
Exercise 14.1.3 Verify that W(t)2/2 is not a martingale.
Exercise 14.1.2 Verify that using the following simple stochastic process, Y(s) ≡ W(tk) for s ∈ [ tk−1, tk), k= 1, 2, . . . , n, to approximate W results in t 0 WdW= (W(t)2 +t)/2.
Exercise 14.1.1 Prove Theorem 14.1.1 for simple stochastic processes.
Exercise 13.4.2 Write the Brownian bridge process with B(0) = x and B(T) = y.
Exercise 13.4.1 Prove the following identities: (1) E[ B(t) ] = 0, (2) E[ B(t)2 ] =t −(t2/T), and (3) E[ B(s) B(t) ] = min(s, t)−(st/T).
Exercise 13.3.10 We can prove Eq. (13.17) without using Eq. (13.14). Let fn(X) ≡2 n−1 k=0X(k+1) t 2n− Xkt 2n.(1) Prove that |X( (k+1) t/2n )− X(kt/2n)| has mean 2−n/2√2/π and
Exercise 13.3.9 To see the plausibility of Eq. (13.14), take the expectation of its left-hand side and drop limn→∞ to obtain 2n−1 k=0 EX(k+1) t 2n− Xkt 2n2.Show that the preceding
Exercise 13.3.8 Assume that the stock price follows the geometric Brownian motion process S(t) ≡ eX(t), where { X(t), t ≥ 0 } is a (μ, σ) Brownian motion. (1) Show that the stock price is
Exercise 13.3.7 Let Y(t) ≡ eX(t), where { X(t), t ≥ 0 } is a (μ, σ) Brownian motion.Show that E[Y(t) |Y(s) ] = Y(s) e(t−s)(μ+σ2/2) for s < t.
Exercise 13.3.6 Let dQ represent the probability that the random walk that converges to a (μ, 1) Brownian motion takes the moves X1, X2, . . . . Let dP denote the probability that the symmetric
Exercise 13.3.5 Let { X(t), t ≥ 0 } be a (0, σ) Brownian motion. Prove that the following three processes are martingales: (1) X(t), (2) X(t)2 −σ2t, and(3) exp[ αX(t)−α2σ2t/2 ] for α ∈
Exercise 13.3.4 Let p(x, y; t) denote the transition probability density function of a (μ, σ) Brownian motion starting at x; p(x, y; t) = (1/√2πt σ ) exp[−(y−x −μt)2/(2σ2t) ]. Show that
Exercise 13.3.3 Let { X(t), t ≥ 0 } represent theWiener process. Show that the related process { X(t)− X(0), t ≥ 0 } is a martingale. (X(0) can be a random variable.)
Exercise 13.3.2 Verify that KX(t, s) = σ2 ×min(s, t) if { X(t), t ≥ 0 } is a (μ, σ)Brownian motion.
Exercise 13.3.1 Prove that { (X(t)−μt)/σ, t ≥ 0 } is a Wiener process if { X(t), t ≥ 0 } is a (μ, σ) Brownian motion.
Exercise 13.2.13 Show that for any k> 0 there exists a risk-neutral probability measure π under which the price of any asset C equals its discounted expected future price at time k, that is, C(i) =
Exercise 13.2.12 (1) Prove that [ (1/d)−(1/R) ] [ (ud)/(u−d) ] is the up-move probability for the stock price that makes the relative bond price a martingale under the binomial option pricing
Exercise 13.2.11 Show that identity (13.8) holds under stochastic interest rates.
Exercise 13.2.10 Prove that F = Eπ [ Sn ], where Sn denotes the price of the underlying non-dividend-paying stock at the delivery date of the futures contract, time n.(The futures price is thus an
Exercise 13.2.9 Prove that the discounted stock price S(i)/Ri follows a martingale under the risk-neutral probability; in particular, S(0) = Eπ [ S(i)/Ri ].
Exercise 13.2.8 Assume that one unit of domestic(foreign) currency grows to R(Rf, respectively) units in a period, and u and d are the up and the down moves of the domestic/foreign exchange rate,
Exercise 13.2.7 Verify Eq. (13.5).E C(k)Rk C(i) = C = C Ri , i ≤ k. (13.5)
Exercise 13.2.6 Let { Xn } be a martingale and letCn denote the stake on the nth game. Cn may depend on X1, X2, . . . , Xn−1 and is bounded. C1 is a constant. Interpret Cn(Xn − Xn−1) as the
Exercise 13.2.5 Let { Sn ≡n i=1 Xi , n ≥ 1 } be a random walk, where Xi are independent random variables with E[ Xi ] = 0 and Var[ Xi ] = σ2. Show that { S2 n−nσ2, n ≥ 1 } is a martingale.
Exercise 13.2.4 Consider a martingale { Zn, n ≥ 1 } and let Xi ≡ Zi − Zi−1 with Z0 = 0. Prove Var[ Zn ] =n i=1 Var[ Xi ].
Exercise 13.2.3 Define Zn ≡%n i=1 Xi , n ≥ 1, where X1, X2, . . . , are independent random variables with E[ Xi ] = 1. Prove that { Zn } is a martingale.
Exercise 13.2.2 If the asset return follows a martingale, then the best forecast of tomorrow’s return is today’s return as measured by the minimal mean-square error.Why? (Hint: see Exercise
Exercise 13.2.1 Let { X(t), t ≥ 0 } be a stochastic process with independent increments.Show that { X(t), t ≥ 0 } is a martingale if E[ X(t)− X(s) ] = 0 for any s, t ≥ 0 and Prob[ X(0) = 0 ]
Exercise 13.1.5 Construct two symmetric random walks with correlation ρ.
Exercise 13.1.4 (1) Use Eq. (13.1) to characterize the random walk in Example 13.1.1. (2) Show that the variance of the symmetric random walk’s position after n moves is n.
Exercise 13.1.3 Let X1, X2, . . . , be a sequence of uncorrelated random variables with zero mean and unit variance. Prove that { Xn } is stationary.
Exercise 13.1.2 Let Y1,Y2, . . . , be mutually independent random variables and X0 an arbitrary random variable. Define Xn ≡ X0 +n i=1 Yi for n > 0. Show that{ Xn, n ≥ 0 } is a stochastic
Exercise 13.1.1 Prove that E[ X(t)− X(0) ] = t × E[ X(1)− X(0) ], Var[ X(t) ]−Var[ X(0) ] = t ×{Var[ X(1) ]−Var[ X(0) ] }when { X(t), t ≥ 0 } is a process with stationary independent
Exercise 12.5.3 Derive Eq. (12.17) with a forward exchange rate argument.
Exercise 12.5.2 Redesign the swap with the rates in Example 12.5.1 so that the gains are 1% for A, 0.5% for B, and 0.5% for the bank.
Exercise 12.5.1 Use the numbers in Example 12.5.1 to construct the same effective borrowing rates without the bank as the financial intermediary.
Programming Assignment 12.4.8 Write binomial tree programs to implement the idea of avoiding negative risk-neutral probabilities enunciated above
Programming Assignment 12.4.7 Write binomial tree programs to price futures options and forward options.
Exercise 12.4.6 Start with the standard tree for the underlying non-dividendpaying stock (i.e., a stock price S can move to Su or Sd with (ert −d)/(u−d)as the probability for an up move). (1)
Exercise 12.4.4 (1) Verify that, under the Black–Scholes model, a European futures option is worth the same as the corresponding European option on the cash asset if the options and the futures
Exercise 12.4.3 Prove that Fe−rt − X≤ C − P ≤ F − Xe−rt for American futures options.
Exercise 12.4.2 Prove Theorem 12.4.5 for American forward puts. (Hint: Show that P−(X− F) e−rT > 0 first.)
Exercise 12.4.1 With a conversion, the trader buys a put, sells a call, and buys a futures contract. The put and the call have the same strike price and expiration month. The futures contract has the
Exercise 12.3.8 For a commodity that can be sold short, such as a financial asset, prove that F ≥ S+C −U, where U is the net storage cost for carrying one unit of the commodity to the delivery
Exercise 12.3.7 Prove that F ≤ S+C, where C is the net carrying cost per unit of the commodity to the delivery date.
Exercise 12.3.6 A manufacturer needs to acquire gold in 3 months. The following options are open to her: (1) Buy the gold now or (2) go long one 3-month gold futures contract and take delivery in 3
Exercise 12.3.5 For futures, the cost of carry may be called cash and carry, which is the strategy of buying the cash asset with borrowed funds. (1) Illustrate this point with futures contracts when
Exercise 12.3.4 Do Eqs. (12.9) and (12.10) assume that the stock index involved is not adjusted for cash dividends?
Exercise 12.3.3 Suppose that interest rates are uncertain and that futures prices move in the same direction as interest rates. Argue that futures prices will exceed forward prices. Similarly, argue
Exercise 12.3.2 Complete the proof by considering the f0 < F0 case.
Exercise 12.2.5 All the above cases satisfy the following relation:f = (F − X) e−rτ . (12.7)Prove the preceding identity by an arbitrage argument.
Exercise 12.2.4 (1) Prove that a newly written forward contract is equivalent to a portfolio of one long European call and one short European put on the same underlying asset and expiration date with
Exercise 12.2.3 Show that Eq. (11.6) can be simplified to C = Fe−rτ N(x)− Xe−rτ N(x −σ√τ ), P = Xe−rτ N(−x +σ√τ )− Fe−rτ N(−x)without the explicit appearance of the
Exercise 12.2.2 (1)What does the table in Fig. 12.1 say about the relative interest rates between the United States and Germany? (2) Estimate German interest rates from Fig. 12.1 and Eq. (12.2) if
Exercise 12.2.1 Selling forward DEM10 million as in the text denies the hedger the profits if the German mark appreciates. Consider the “60:40” strategy, whereby only 60% of the German marks are
Programming Assignment 11.7.12 (1) Implement binomial tree algorithms for European and American lookback options. The time complexity should be, at most, cubic. (2) Improve the running time to
Programming Assignment 11.7.11 Implement the algorithm in Fig. 11.14.
Programming Assignment 11.7.10 Implement O(n4)-time binomial tree algorithms for European and American geometric average-rate options.
Exercise 11.7.9 Suppose that there is an algorithm that generates an upper bound on the true option value and an algorithm that generates a lower bound on the true option value. How do we design an
Exercise 11.7.8 Two sources of error were mentioned for the approximation algorithm.Argue that they disappear if all the asset prices on the tree are integers.
Exercise 11.7.7 Derive the quadraticpolynomial y = a +bx +cx2 that passes through three points (x0, y0), (x1, y1), and (x2, y2).
Exercise 11.7.6 Assume that ud = 1. Prove that the difference between the maximum running sum and the minimum running sum at a node with an asset price of S0uidn−i is an increasing function of i
Exercise 11.7.5 Let $r = 0 denote the continuously compounded riskless rate per period. (1) The future value of the average-rate call,
Exercise 11.7.4 Arithmeticaverage-rate options were assumed to be newly issued, and there was no historical average to deal with. Argue that no generality was lost in doing so.
Exercise 11.7.3 Explain why average-rate options are harder to manipulate. (They were first written on stocks traded on Asian exchanges, hence the name “Asian options,” presumably because of
Exercise 11.7.2 Verify that the number of geometricaverages at time n is n(n+1)/2.
Exercise 11.7.1 Achieve “selling at the high and buying at the low” with lookback options.
Programming Assignment 11.6.4 Implement binomial tree algorithms for the four compound options.
Exercise 11.6.3 A chooser option (or an as-you-like-it option) gives its holder the right to buy for X1 at time τ1 either a call or a put with strike price X2 at time τ2.Describe the binomial tree
Exercise 11.6.2 Why is a contingent forex put cheaper than a standard put?
Exercise 11.6.1 Recall our firm XYZ.com in Subsection 11.1.1. It had only two kinds of securities outstanding, shares of its own common stock and bonds. Argue that the stock becomes a compound option
Exercise 11.5.7 (Put–Call Parity). Prove that (1) C = P+ Se−$rτ − Xe−rτ for European options and (2) P ≥ max(Xe−rτ − Se−$rτ , 0) for European puts.
Exercise 11.5.6 Show how bound (11.7) may be approximated by the C in Eq. (11.6).
Exercise 11.5.5 Prove that the European forex call and put are worth the same if S = X and r =$r under the Black–Scholes model.
Exercise 11.5.4 Assume the BOPM. (1) Verify that the risk-neutral probability for forex options is [ e(r−$r )t −d ]/(u−d), where u and d denote the up and the down moves, respectively, of the
Exercise 11.5.3 A participating forward contract pays off at expiration S− X if the exchange rate S exceeds X,α(S− X) if S ≤ X.The purchaser is guaranteed an upper bound on the exchange rate
Exercise 11.5.2 In a conditional forward contract, the premium p is paid at expiration and only if the exchange rate is below a specified level X. The payoff at expiration is thus S− X− p if the
Exercise 11.5.1 A range forward contract has the following payoff at expiration:0 if the exchange rate S lies within [ XL, XH ], S− XH if S > XH, S− XL if S < XL.It guarantees that the effective
Exercise 11.4.1 Verify the following claims: (1) A 1% price change in a lowerpriced stock causes a smaller movement in the price-weighted index than that in a higher-priced stock. (2) A 1% price
Exercise 11.2.6 A rolling call comes with barriers H1 > H2 > · · · > Hn (all below the initial stock price) and strike prices X0 > X1 > · · · > Xn−1. This option starts as a European call
Programming Assignment 11.2.5 (1) Implement binomial tree algorithms for European knock-in and knock-out options with rebates. Pay special attention to convergence (see Fig. 11.5). Here is a solved
Exercise 11.2.4 Areset option is like an ordinary option except that the strike price is set to H when the stock price hits H. Assume that H < X. Create a syntheticreset option with a portfolio of
Exercise 11.2.3 Check that the formulas for the up-and-in and down-and-in options become the Black–Scholes formulas for standard European options when S = H.
Exercise 11.2.2 Does the in–out parity apply to American-style options?
Exercise 11.2.1 (1) Prove that a European call is equivalent to a portfolio of a European down-and-out option and a European down-and-in option with an identical barrier. (2) Prove that a European
Exercise 11.1.11 Complete the proof by showing that it is not optimal to call the CBs when λV > P.
Exercise 11.1.10 (1) Replicate the zero-coupon CB with zero-coupon bonds and European calls on a fraction of the total value of the corporation. (2) Replicate the zero-coupon CB with zero-coupon
Exercise 11.1.9 Replicate the zero-coupon CB with the total value of the corporation and European calls on that value.
Programming Assignment 11.1.8 (1) Write a program to solve Eq. (11.3). (2) Write a binomial tree algorithm to price American warrants.

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