Questions and Answers of Accounting For Financial Instruments

Exercise 18.2.10 Pick β =±1. The success of the scheme now depends solely on the choice of Y. Derive the conditions under which the variance is reduced.
Programming Assignment 18.2.8 Implement the antithetic-variates method for arithmeticaverage-rate calls and puts. Compare it with the Monte Carlo method in Programming Assignment 18.2.3.
Exercise 18.2.7 Justify and extend the procedure in Example 18.2.2.
Programming Assignment 18.2.6 Implement the Brownian bridge approach to generate the sample path of geometricBrownian motion.
Programming Assignment 18.2.5 Simulate dXt = (0.06− Xt ) dt +0.3dWt by using t ≡ 0.01. Explain its dynamics.
Exercise 18.2.4 The Monte Carlo method for Ito processes in Fig. 18.6 may not be the most ideal theoretically. Consider the geometric Brownian motion dX/X=μdt +σ dW. Assume that you have access to
Programming Assignment 18.2.3 Implement the Monte Carlo method for arithmeticaverage-rate calls and puts.
Exercise 18.2.2 Consider the Monte Carlo method that estimates the price of the American call by taking the maximum discounted intrinsic value per simulated path and then averaging them: E[
Exercise 18.2.1 How do we price European barrier options by Monte Carlo simulation?
Programming Assignment 18.1.6 Implement the implicit method for American puts.
Programming Assignment 18.1.5 Implement the explicit method for American puts.
Exercise 18.1.4 Derive the stability conditions for the explicit approach to solve the Black–Scholes differential equation. Assume q = 0 for simplicity.
Exercise 18.1.3 Repeat the steps for American calls.
Exercise 18.1.2 What are the terminal conditions?
Exercise 18.1.1 Sketch the finite-difference version of the Poisson equation in matrix form.
Programming Assignment 17.3.6 Implement the binomial model for the option on the best of two risky assets and cash.
Exercise 17.3.5 (Optimal Hedge Ratio) Derive the optimal number of futures to short in terms of minimum variance to hedge a long stock when the two assets are not perfectly correlated. Assume the
Exercise 17.3.4 (1) Write the combinatorial formula for a European call with terminal payoff max(S1S2 − X, 0). (2) How fast can it be priced?
Exercise 17.3.3 With m assets, how many nodes does the tree have after n periods?
Exercise 17.3.2 It is easy to check that ρ = 2(p1 + p4)−1. Show that this identity holds for any correlated binomial random walk defined by R1(i +1)− R1(i) = μ1 ±σ1 and R2(i +1)− R2(i) =
Programming Assignment 17.2.6 Implement trinomial tree algorithms for barrier options. Add rebates for the knock-out type.
Exercise 17.2.5 Derive combinatorial formulas for European down-and-in, downand-out, up-and-in, and up-and-out options.
Exercise 17.2.4 It was shown in Subsection 10.2.2 that binomial trees can be extended backward in time for two periods to compute delta and gamma. Argue that trinomial trees need to be extended
Programming Assignment 17.2.3 Recall the diagonal method in Section 9.7. Write a program to perform backward induction on the trinomial tree with the diagonal method.
Exercise 17.2.2 The trinomial model no longer supports perfect replication of options with stocks and bonds as in the binomial model. Replicating an option with h shares of stock and $B in bonds
Exercise 17.2.1 Verify the following: (1) ln(S(t)/S) has mean μt, (2) the variance of ln(S(t)/S) converges to σ2t, and (3) S(t)’s mean converges to Sert .
Programming Assignment 17.1.20 Implement a linear-time algorithm for doublebarrier options.
Exercise 17.1.19 Consider a generalized double-barrier option with the two barriers defined by functions f and fh, where f(t) < fh(t) for t ≥ 0. Transform it into a double-barrier option with
Exercise 17.1.18 (1) Formulate the in–out parity for double-barrier options.(2) Replicate the double-barrier option by using knock-in options, knock-out options, and double-barrier options that
Exercise 17.1.17 Prove Eq. (17.9).
Exercise 17.1.16 Apply the reflection principle repetitively to verify that|Ai| =n n+a+b+(i−1) s 2for odd in n+a−b+is 2for even i, |Bi|
Exercise 17.1.15 Prove formula (17.8).
Programming Assignment 17.1.4 Try ≡ τ[ ln(S/H)/( jσ) ]2 instead.
Exercise 17.1.13 In formula (17.6), the barrier H is replaced with the effective barrier ˜H, which is one of the n+1 terminal prices. If the effective barrier is allowed to be one of all possible
Exercise 17.1.12 Explain why Fig. 11.5 shows that the calculated values underestimate the analytical value.
Exercise 17.1.11 How do we efficiently price a portfolio of barrier options with identical underlying assets but different barriers under the binomial model?
Programming Assignment 17.1.10 (1) Implement O(n2)-time algorithms for European lookback options, improving Programming Assignment 11.7.11, part (1).(2) Improve the running time to O(n).
Programming Assignment 17.1.9 Implement O(n3)-time algorithms for European geometric average-rate options with combinatorics, improving Programming Assignment 11.7.6.
Programming Assignment 17.1.8 Design fast algorithms for European barrier options.
Exercise 17.1.7 Derive a pricing formula for the European power option max(S(τ )2 − X, 0).
Exercise 17.1.6 Prove that option value (17.6) converges to value (11.4) with q = 0.
Exercise 17.1.5 Derive a combinatorial pricing formula for the reset call option.
Exercise 17.1.4 Consider the exploding call spread, which has the same payoff as the bull call spread except that it is exercised promptly the moment the stock price touches the trigger price K
Exercise 17.1.3 Use the reflection principle to derive a combinatorial pricing formula for the European lookback call on the minimum.
Exercise 17.1.2 Derive pricing formulas similar to formula (17.6) for the other three barrier options: down-and-out, up-and-in, and up-and-out options.
Exercise 17.1.1 What is the probability that the stock’s maximum price is at least Suk?
Exercise 16.3.8 Explain why shorting a bull call spread can in practice hedge a binary option statically.
Programming Assignment 16.3.7 Implement the delta–gamma–vega hedge for options.
Programming Assignment 16.3.6 Implement the delta–gamma hedge for options.
Exercise 16.3.5 Verify that any delta-neutral gamma-neutral self-financing portfolio is automatically theta-neutral.
Programming Assignment 16.3.4 Implement the delta hedge for options.
Exercise 16.3.3 Abroker claimed the option premium is an arbitrage profit because he could write a call, pocket the premium, then set up a replicating portfolio to hedge the short call.What did he
Exercise 16.3.2 (1) Repeat the calculations in Fig. 16.1 but this time record the weekly tracking errors instead of the cumulative costs. Verify that the following numbers result:Weeks to Net
Exercise 16.3.1 (1) A delta hedge under the BOPM results in perfect replication(see Chap. 9). However, this is impossible in the current context. Why? (2) How should the value of the derivative
Exercise 16.2.3 Redo Example 16.2.4 if the goal is to change the beta to 2.0.
Exercise 16.2.2 Show that if the linear regression of s on f based on the data(S1,F1), (S2,F1), . . . , (St−1,Ft−1)is s = β0 +β1 f , then β1 is an estimator of the hedge ratio in Eq.
Exercise 16.2.1 If the futures price equals the expected future spot price, then hedging may in some sense be considered a free lunch. Give your reasons.
Exercise 15.4.2 A forward-start option is like a standard option except that it becomes effective only at time τ∗ from now and with the strike price set at the stock price then (the option thus
Exercise 15.4.1 Suppose that S1, S2, . . . , Sn pay no dividends and follow dSi /Si =μi dt +σi dWi . Let ρjk denote the correlation between dWj and dWk. Show that∂ f∂t+i(μi −λiσi ) Si∂
Exercise 15.3.18 Suppose that the CB is continuously callable once it becomes callable, meaning that it is callable at any instant after a certain time t∗. Argue that the C(V, t) > P(t) case needs
Exercise 15.3.17 Justify Eq. (15.8).dU U = (rf −qf −ρσsσf) dt +σf dW (15.8)
Exercise 15.3.16 Consider a portfolio consisting of a long call on the foreign asset and X long puts on currency C. The strike prices in U.S. dollars of the call (XA)and put (XC) are such that X=
Exercise 15.3.15 Show that both forex options and foreign domesticoptions are special cases of cross-currency options.
Exercise 15.3.14 Verify that the triangular arbitrage must hold to prevent arbitrage opportunities among three currencies.
Exercise 15.3.13 Suppose that the domesticand the foreign bond prices in their respective currencies with a par value of one and expiring at T also follow geometric Brownian motion processes. Their
Exercise 15.3.12 The dynamics of the foreign asset value in domestic currency, SGf, depends on the correlation between the asset price and the exchange rate (see Example 14.3.5). (1) Why is ρ
Exercise 15.3.11 The formulas of Cf and Pf suggest that a foreign equity option is equivalent to S domestic options on a stock paying a continuous dividend yield of qf and a strike price of
Exercise 15.3.10 Derive Eq. (15.6) from Eq. (15.4).
Exercise 15.3.9 (Put–Call Parity) Prove that V(S2, S1, t)−V(S1, S2, t)+ S2 = S1.
Exercise 15.3.8 Verify variance (15.5) for Margrabe’s formula.
Exercise 15.3.7 (1) Derive Margrabe’s formula from the alternative view that a European exchange option is a put on asset 1 with a strike price equal to the future value of asset 2. (2) Derive the
Exercise 15.3.6 (Euler’s Theorem). Prove that ni=1 xi∂ f (x1, x2, . . . , xn)∂xi= f (x1, x2, . . . , xn)if f (x1, x2, . . . , xn) is homogeneous of degree one in x1, x2, . . . , xn.
Exercise 15.3.5 Consider a call on the minimum of two assets with strike price X.Its terminal value is max(min(S1(T), S2(T))− X, 0). Show that this option can be replicated by a long position in
Exercise 15.3.4 A call on the maximum of two assets pays max(S1(T), S2(T)) at expiration. Replicate it by a position in one of the assets plus an exchange option.
Exercise 15.3.3 (1) Justify Eq. (15.4). (2) Generalize it to n assets.
Exercise 15.3.2 Show that geometricaverage-rate options satisfy
Exercise 15.3.1 Derive the partial differential equation for futures options.
Exercise 15.2.6 Verify that the Black–Scholes differential equation is violated where it is optimal to exercise the American put early; i.e., X− S does not satisfy the equation.
Exercise 15.2.5 Solve the Black–Scholes differential equation for European puts.
Exercise 15.2.4 Explain why the formula e−rτ ∞X(y− X)1σ y√2πτexp−{ ln(y/S)−(r −σ2/2)τ }2 2σ2τdy, is equivalent to the Black–Scholes formula for European calls. (Hint:
Exercise 15.2.3 Outline an argument for the claim that the Black–Scholes differential equation results from the BOPM by taking limits.
Exercise 15.2.2 It seems reasonable to expect the predictability of stock returns, as manifested in the drift of the Ito process, to have an impact on option prices.(One possibility is for the log
Exercise 15.2.1 (1) Verify that the Black–Scholes formula for European calls in Theorem 9.3.4 indeed satisfies Black–Scholes differential equation (15.2). (2) Verify that the value of a forward
Exercise 15.1.1 Verify that the diffusion equation is indeed satisfied by integral(15.1).
Exercise 14.4.13 The continuously compounded rate of return X≡ ln S follows dX= (r −σ2/2) dt +σ dW in a risk-neutral economy. Use this fact to show that u = exp[ (r −σ2/2) t +σ√t ], d =
Exercise 14.4.12 From the above discussions, E[ Xi ]→(r −σ2/2)t and Var[ Xi ]→σ√t .Now Xi+1 = ln(Si+1/Si ), where Si ≡ S0eX1+X2+···+Xi is the stock price at time i . Hence Xi+1 =
Exercise 14.4.11 Verify Eqs. (14.18). (Hint: ex ≈ 1+x +x2/2.)
Exercise 14.4.10 What are the shortcomings of modeling the stock price dynamics by dS = μdt +σ dW with constant μ and σ?
Exercise 14.4.9 Justify using S/(S√t) to estimate volatility.
Exercise 14.4.8 Assume that the volatility σ is stochastic but driven by an independent Wiener process. Suppose that the average variance over the time period[ 0, T ] as defined by 0σ 2 ≡ 1T T 0
Exercise 14.4.7 Show that the simple rate of return [ S(t)/S(0) ]−1 has mean eμt −1 and variance e2μt (eσ2t −1).
Exercise 14.4.6 Suppose the stock price follows the geometric Brownian motion process dS/S = σ dW. Example 14.3.3 says that S(t)/S(0) = eX(t), where X(t) is a(−σ2/2, σ) Brownian motion. In other
Exercise 14.4.5 Prove that E[ S(T) ] = S(0) eμT.
Exercise 14.4.4 (1) Prove that A(t)/L(t) is a convex function of the prevailing interest rate. (2) Then verify A(t) < L(t).
Exercise 14.4.3 Suppose that the current spot rate curve is flat. Under the assumption that only parallel shifts are allowed, what can we say about the parameters μand σ governing the short rate
Exercise 14.4.2 Negative interest rates imply arbitrage profits for riskless bonds.Why?
Exercise 14.4.1 Argue that an investor who has information about the entire future value of the Brownian motion’s driving the stock price will have infinite wealth at any given horizon date. In
Exercise 14.3.11 Show that the transition probability density function p of dX=−(1/2) Xdt +dW satisfies the backward equation∂p∂s=−1 2∂2 p∂x2+ 1 2x∂p∂x.(Hint: X(t) ∼
Exercise 14.3.10 Consider the following processes:dS = μSdt +σ SdW1, dσ = β(σ −σ) dt +γ dW2, where dW1 and dW2 are Wiener processes with correlation ρ. Let H(S, σ, τ) be a function of S,
Exercise 14.3.9 Justify the claim in Eq. (13.13) by showing that Y(t) ≡ e−t W(e2t) is the Ornstein–Uhlenbeck process dY=−Ydt +√2dW. (Hint: Consider Y(t +dt)−Y(t).)

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