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

Programming Assignment 27.3.7 Implement the algorithm in Fig. 27.6.
Exercise 27.3.6 Like warrants, CBs can be converted into newly issued shares; they are in fact equivalent under certain assumptions (see Exercise 11.1.10, part (2)). On a per-share basis, the conversion price plus the final coupon payment acts very much like the strike price. But unlike with
Exercise 27.3.5 Argue that under the binomial CB pricing model of Fig. 27.6, the CB price converges to the stock price as the stock price increases.
Exercise 27.3.4 Prove that, much as with American calls, it never pays to convert the bond when the stock does not pay cash dividends and the interest rate remains constant.
Exercise 27.3.3 Can you find one fault with formula (27.2)?
Exercise 27.3.2 Use an arbitrage argument to show that a CB must trade for at least its conversion value.
Exercise 26.6.1 Show that φ(t) = σ2(1−e−2κt )/(2κ) for the extended Vasicek model.
Exercise 26.5.9 Show that Eqs. (26.22) and (26.25) are equivalent in the limit.
Exercise 26.5.8 Verify Eq. (26.25).t T t+t μ(t, u) du = ln ex +e−x 2 , (26.25)
Exercise 26.5.7 When pricing a derivative on bonds under the HJM model, does the tree have to be built over the life of the longer-term underlying bond or just over the life of the derivative?
Exercise 26.5.6 Verify that the model in Example 26.5.3 converges to the Ho–Lee model as a→0.
Exercise 26.5.5 Derive the Hull–White model’s volatility structure.
Exercise 26.5.4 (1) Price calls on a zero-coupon bond under the Ho–Lee model.(2) Price calls on a zero-coupon bond under the Hull–White model.
Exercise 26.5.3 Consider the forward rate dynamics in Eq. (26.13) and define r (t, τ) ≡ f (t, t +τ ). Verify that dr(r, τ) =∂r (t, τ)∂τ+μ(t, t +τ )dt +σ(t, t +τ )dWt .Note that τ denotes the time to maturity.
Exercise 26.5.2 Prove Eq. (26.16). (Hint: Use Eq. (26.15).)
Exercise 26.5.1 What would μ(t, T) be if σ(t, T) = σe−κ(T−t)?
Programming Assignment 26.4.5 Implement a trinomial tree model for the Black–Karasinski model d ln r = (θ(t)−a ln r ) dt +σ dW.
Programming Assignment 26.4.4 Calibration takes the spot rate curve to reverse engineer the Hull–White model’s parameters. However, just because the curve is matched by no means implies that the true model parameters and the estimated parameters are matched. Call a model stable if the model
Programming Assignment 26.4.3 Implement the algorithm in Fig. 26.7.
Exercise 26.4.2 Verify approximation (26.12).Eπ e−$r (i+1)t $r (i) = r j ≈ e−r jt 1−μi, j (t)2 + σ2(t)3 2 . (26.12)
Exercise 26.4.1 Between a normal and a lognormal model, which overprices outof-the-money calls on bonds and underprices out-of-the-money puts on bonds?
Programming Assignment 26.3.5 Implement a forward-induction algorithm to calibrate the Black–Karasinski model given a constant κ.
Exercise 26.3.4 Show that the variance of r after t is approximately[ r (t) σ(t) ]2t.
Programming Assignment 26.3.3 Calibrate the BDT model with the secant method and evaluate its performance against the differential tree method.
Programming Assignment 26.3.2 Implement the algorithm in Fig. 26.3.
Exercise 26.3.1 Describe the differential tree method with backward induction to calibrate the BDT model.
Exercise 26.2.11 Assess the claim that the problem of negative interest rates can be eliminated by making the short rate volatility time dependent.
Exercise 26.2.10 Prove that−ln1+δt−1 1+δt− 1 2ln δ→σ2t (t)2 if we substitute t/t for t and apply Eq. (26.4). The t, t, and σ above are annualized.(Short rate process (26.6) thus converges to dr = {[ ∂ f (0, t)/∂t ]+σ2t } dt +σ dW.)
Exercise 26.2.9 Prove that−ln1+δn 1+δn+1− 1 2ln δ→σ2(T −t)(t)2 if we substitute t/t for t and (T −t)/t for n in Eq. (26.5) before applying Eq. (26.4). T, t, t, and σ above are annualized. (The forward rate process hence converges to df (t, T) = σ2(T −t) dt +σ dW.)
Exercise 26.2.8 Verify that Var[ ξs ] = σ2.
Exercise 26.2.7 Prove that under a general p, the variance of the one-period return of n-period zero-coupon bonds equals (n−1)2 σ2 and the covariance between the one-period returns of n- and m-period zero-coupon bonds equals (n−1)(m−1) σ2.
Programming Assignment 26.2.6 Write a linear-time program to calibrate the original Ho–Lee model. The inputs are t, the current market discount factors, and the short rate volatility σ, all annualized.
Exercise 26.2.5 Consider a portfolio of one zero-coupon bond with maturity T1 and β zero-coupon bonds with maturity T2. Find the β that makes the portfolio instantaneously riskless under the Ho–Lee model.
Exercise 26.2.4 Prove that Eqs. (26.3) and (26.3) become Pd(t +1, t +n) = P(t, t +n)P(t, t +1)exp[ v2 + · · · + vn ]p+(1− p)×exp[ v2 + · · · + vn ], Pu(t +1, t +n) = P(t, t +n)P(t, t +1)1 p+(1− p)×exp[ v2 + · · · + vn ]for general risk-neutral probability p.
Exercise 26.2.3 Show that a rate rise followed by a rate decline produces the same term structure as that of a rate decline followed by a rate rise.
Exercise 26.2.2 Show that vi = [ (i −1) κi −(i −2) κi−1) ]/p(1− p) .
Exercise 26.2.1 Verify Eq. (26.1).Pu(t +1, t +n) = Pd(t +1, t +n) e−(v2+···+vn) (26.1)
Exercise 26.1.1 Is the equilibrium or no-arbitrage model more appropriate in deciding which government bonds are overpriced?
Exercise 25.5.2 Repeat the preceding argument for European puts on coupon bonds and show that the payoff equalsn i=1 ci ×max(Xi − P(r (T), T, ti ), 0).
Exercise 25.5.1 Suppose that the spot rate curve r (r,a, b, t, T) ≡ r +a(T −t) +b(T −t)2 is implied by a three-factor model.Which of the factors, r ,a, andb, affects slope, curvature, and parallel moves, respectively?
Exercise 25.4.1 Two methods were mentioned for calibrating the Black–Scholes option pricing model: historical volatility and implied volatility. Which corresponds to the time-series approach and which to the cross-sectional approach?
Exercise 25.3.1 To construct a combining binomial tree for the CKLS model, what function of r should be modeled?
Exercise 25.2.14 (1) Show that the binomial tree for the untransformed CIR model does not combine. (2)Showthat the binomial tree for the geometric Brownian motion dr = rμdt +rσ dW does combine even though its volatility is not a constant.
Exercise 25.2.13 Verify that the transformation x(y, t) ≡ yσ(z, t)−1 dz turns the process dy = α(y, t) dt +σ(y, t)dW into one for x whose diffusion term is one.
Exercise 25.2.12 (1) The binomial short rate tree as described requires (n2)memory space. How do we perform backward induction on the tree with only O(n)space? (2) Describe a scheme that needs only O(n2) space for the bond price tree.
Exercise 25.2.11 Suppose we want to calculate the price of some interest-ratesensitive security by using the binomial tree for the CIR or the Vasicek model.Assume further that we opt for the Monte Carlo method with antitheticvariates .One difficulty with the standard paradigm covered in Subsection
Programming Assignment 25.2.10 Write a program to implement the binomial short rate tree and the bond price tree for the CIR model. Compare the results against Eq. (25.3).
Exercise 25.2.9 Show that p(r ) = (1/2)+(1/2)m(x(r ))√t .
Exercise 25.2.8 Derive E[ r (k+1)−r (k) ] and Var[ r (k+1)−r (k) ].
Programming Assignment 25.2.7 Implement the implicit method in Exercise 24.4.3 for zero-coupon bonds under the CIR model.
Exercise 25.2.6 Consider a yield curve option with payoff max(0, r (T, T1)−r (T, T2)) at expiration T, where T < T1 and T < T2. The security is based on the yield spread of two different maturities, T1 −T and T2 −T. Assume either the Vasicek or the CIR model. Show that this option is
Exercise 25.2.5 (1) Write the bond price formula in terms of φ1 ≡ γ , φ2 ≡ (β +γ )/2, and φ3 ≡ 2βμ/σ2. (2) How do we estimate σ, given estimates for φ1, φ2, andφ3?
Exercise 25.2.4 (Affine Models) For any short rate model dr = μ(r, t) dt +σ(r, t)dW that produces zero-coupon bond prices of the form P(t, T) = A(t, T) e−B(t,T) r (t), show that the spot rate volatility structure is the curve σ(r, t) B(t, T)/(T −t).
Exercise 25.2.3 Verify that dP/P = r dt − B(t, T) σ√r dW is the bond price process for the CIR model.
Exercise 25.2.2 Show that Eq. (25.3) satisfies the term structure equation.
Exercise 25.2.1 Show that the long rate is 2βμ/(β +γ ), independent of the short rate.
Programming Assignment 25.1.8 Write a program to implement the binomial tree for the Vasicek model. Price zero-coupon bonds and compare the results against Eq. (25.1).
Exercise 25.1.8 Show that discretizing the Vasicek model directly by (14.7) does not result in a combining binomial tree.
Exercise 25.1.7 Prove that E[ r (k+1)−r (k) ]t=β[μ−r (k) ], if 0 ≤ p(r (k)) ≤ 1σ/√t, if p(r (k)) < 0−σ/√t, if 1 < p(r (k))and Var[ r (k+1)−r (k) ]→σ2t.
Exercise 25.1.6 Verify that the variance of ln P(t, T) is σ2
Exercise 25.1.5 Show that the Ito process for the instantaneous forward rate f (t, T) under the Vasicek model with β = 0 is df = σ2βe−β(T−t)1−e−β(T−t) dt +σe−β(T−t) dW.
Exercise 25.1.4 Verify that dP/P = r dt − B(t, T) σ dW is the bond price process for the Vasicek model.
Exercise 25.1.3 Show that Eq. (25.1) satisfies the term structure equation.
Exercise 25.1.2 (1) Show that the long rate is μ−σ2/(2β2), independent of the current short rate. (2) Derive the liquidity premium for the β = 0 case.
Exercise 25.1.1 Connect the Vasicek model with the AR(1) process.
Exercise 24.7.2 Consider a call on a zero-coupon bond with an expiration date that coincides with the bond’s maturity. Does the call premium depend on the interest rate movements between now and the expiration date?
Exercise 24.7.1 Assume that interest rates cannot be negative. (1) Why should a call on a zero-coupon bond with a strike price of $102 be worth zero given a par value of $100? (2) The Black–Scholes formula gives a positive call value. Why?
Exercise 24.6.10 Verify that the value of a European put, like that of the IO, declines with increasing p.
Exercise 24.6.9 Explain why all the securities covered up to now have the same 1-year return of 4% in a risk-neutral economy.
Exercise 24.6.8 (Dynamic Immunization) Explain why the replication idea solves the problem of arbitrage opportunities in immunization against parallel shifts raised in Subsection 5.8.2.
Exercise 24.6.7 Verify the risk-neutral probability p = 0.319 with Eq. (24.18)instead.
Exercise 24.6.6 We could not have obtained the unique risk-neutral probability had we not imposed a prevailing term structure that must be matched. Explain.
Exercise 24.6.5 To use the objective probability q in pricing, we should discount by the risk-adjusted discount factor, 1+r +λ$σ = 1+$μ. Prove this claim.
Exercise 24.6.4 Consider the symmetricrandom walk for modeling the short rate, r (t +1) = α +ρr (t)±σ. Let V denote the current value of an interest rate derivative, Vu its value at the next period if rates rise, and Vd its value at the next period if rates fall. Define u ≡ eα+ρr (t)+σ
Exercise 24.6.3 Assume in a period that the bond price can go from $1 to Pu or Pd and that the value of a derivative can go from $1 to Vu or Vd. (1) Show that a portfolio of $1 worth of bonds and (Pd − Pu)/(Vu −Vd) units of the derivative is riskless. (2) Prove that these many derivatives are
Exercise 24.6.2 Prove that the market price of risk is independent of bond maturity.(Hint: Assemble two bonds in such a way that the portfolio is instantaneously riskless.)
Exercise 24.6.1 Verify Eq. (24.18).
Exercise 24.5.2 What should σp(t, T, P) be like for df (t, T)’s diffusion term to have the functional form ψ(t) f (t, T)? The dependence of σp on P(t, T) is made explicit here.
Exercise 24.5.1 Justify Eq. (24.15) directly.df (t, T) = σ(t, T) T t σ(t, s) ds dt −σ(t, T)dW, (24.15)
Exercise 24.4.4 Consider a futures contract on a zero-coupon bond with maturity date T1. The futures contract expires at time T. Let F(r, t) denote the futures price that follows dF/F = μf dt +σf dW. Prove that F satisfies−∂F∂T+[μ(r, t)−λ(r, t) σ(r, t) ] ∂F∂r+ 1 2σ(r, t)2
Exercise 24.4.3 Describe an implicit method for term structure equation (24.14).You may simplify the short rate process to dr = μ(r ) dt +σ(r )dW. Assume thatμ(r ) ≥ 0 and σ(0) = 0 to avoid negative short rates.
Exercise 24.4.2 Argue that European options on zero-coupon bonds satisfy the term structure equation subject to appropriate boundary conditions.
Exercise 24.4.1 Suppose a liability has been duration matched by a portfolio.What can we say about the relations among their respective time values and convexities?
Exercise 24.3.8 Use the risk-neutral methodology to price interest rate caps, caplets, floors, and floorlets as fixed-income options.
Exercise 24.3.7 Argue that a forward interest rate swap is equivalent to a portfolio of one long payer swaption and one short receiver swaption. (The situation is similar to Exercise 12.2.4, which said that a forward contract is equivalent to a portfolio of European options.)
Exercise 24.3.6 Consider an amortizing swap in which the notional principal decreases by 1/n dollar at each of the n reset points. The initial principal is $1. Write a formula for the swap rate.
Exercise 24.3.5 The price of a consol that pays dividends continuously at the rate of $1 per annum satisfies the following expected discounted-value formula:P(t) = Eπt ∞t e− T t r (s) ds dT.Compare this equation with Eq. (24.9) and explain the difference.
Exercise 24.3.4 Show that, under the unbiased expectations theory, P(t, T) = 1[ 1+r (t) ]{ 1+ Et [ r (t +1) ] } · · · { 1+ Et [ r (T −1) ] }in discrete time and P(t, T) = e− T t Et [ r (s) ] ds in continuous time. (The preceding equation differs from Eq. (24.8), which holds under the local
Exercise 24.3.3 Show that the calibrated binomial interest rate tree generated by the ideas enumerated in Subsection 23.2.2 (hence the slightly more general tree of Exercise 23.2.3 as well) satisfies the local expectations theory. How about the uncalibrated tree in Fig. 23.6?
Exercise 24.3.2 Under the local expectations theory, prove that the forward rate f (t, T) is less than the expected spot rate Et [ r (T) ] provided that interest rates tend to move together in that Et [ r (T) | T t r (s) ds ] is increasing in T t r (s) ds. (The unbiased expectations theory is
Exercise 24.3.1 Assume that the local expectations theory holds. Prove that the(T −t)-time spot rate at time t is less than or equal to Et [ T t r (s) ds ]/(T −t), the expected average interest rate between t and T, with equality only if there is no uncertainty about r (s).
Exercise 24.2.10 Suppose that the effective annual interest rate follows dre re= μ(t) dt +σ(t)dW.Prove that drc(t)1−e−rc(t)=μ(t)− 1 21−e−rc(t) σ(t)2dt +σ(t)dW.(The continuously compounded rate is approximately lognormally distributed when rc(t) = o(dt) as 1−e−rc(t) ≈
Exercise 24.2.9 Show that P(t, T)M(t)= 1 M(T)in a certain economy, where M(t) ≡ e t 0 r (s) ds is the money market account. (Hint:Exercise 5.6.6.)
Exercise 24.2.8 Derive the liquidity premium and the forward rate for the Merton model. Verify that the forward rate goes to minus infinity as the maturity goes to infinity.
Exercise 24.2.7 Prove the following continuous-time analog to Eq. (5.9):f (t, T,M−t) = (M−t) r (t,M)−(T −t) r (t, T)M−T.(Hint: Eq. (24.3).)
Exercise 24.2.6 Verify that f (t, T, L) = 1 LP(t, T)P(t, T + L)−1is the analog to Eq. (24.2) under simple compounding.
Exercise 24.2.5 Show that the τ -period spot rate equals (1/τ )τ−1 i=0 f (t, t +i)(average of forward rates) if all the rates are continuously compounded.
Exercise 24.2.4 Prove Eq. (24.6) from Eq. (24.2).
Exercise 24.2.3 Prove Eq. (24.4) from Eq. (5.11).
Exercise 24.2.2 Suppose we sell one T-time zero-coupon bond and buy P(t, T)/P(t,M) units of M-time zero-coupon bonds at time t. Proceed from here to justify Eq. (24.1).

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

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