Math 623 (IOE 623), Fall 2008: Final examName:Student ID:This is a closed book exam. You may bring up to ten one sided A4 pages of notes to the exam. You may alsouse a calculator but not its memory function.Please write all your solutions in this exam booklet (front and back of the page if necessary). Keep yourexplanations concise (as time is limited) but clear. State explicitly any additional assumptions you make.Time is counted in years, prices in USD, and all interest rates are continuously compounded unless otherwisestated.You are obliged to comply with the Honor Code of the College of Engineering. After you have completed theexamination, please sign the Honor Pledge below. A test where the signed honor pledge does not appear may notbe graded.I have neither given nor received aid, nor have I used unauthorized resources, on this examination.Signed:1(1) Suppose I wish to compute the value of a European Straddle option on a stock. The option pays Φ(S) ifexercised when the stock price is S, whereΦ(S) =(52 − S if S ≤ 52S − 52 if S ≥ 52.The current value of the stock is 53 and it expires 9 months from today. The value of the option isto be obtained by numerically solving a terminal-boundary value problem for a PDE. The PDE is theBlack-Scholes PDE transformed by the change of variable S = ex, where S is the stock price.(a) The value of the option as a function of the variables x and t with the units of t in years, is u(x, t)where u(x, t) satisfies the PDE,∂u∂t+ λ∂u∂x+ .0392∂2u∂x2− .035u = 0.Find the value of the coefficient λ in the PDE.(b) The PDE is to be numerically solved in some region a < x < b, 0 < t < T , using the explicit Eulermethod. Setting u(n∆x, m∆t) = umnthen umnsatisfies the recurrence equationum−1n= Aumn+ Bumn−1+ Cumn+1, m ≤ M.Taking ∆t = 0.006, ∆x = 0.02 find numerical values for M and uM200.(c) Calculate the numerical value of the coefficient A in the scheme given in (b).(d) Using the value of A you computed in (c), determine whether the scheme in (b) is stable. Explainyour answer.2(2) We wish to find the value of an American put option which expires 8 months from today. The put optionis on a stock which has current price 31 and volatility 0.34. The strike price for the option is 28 andthe risk free rate of interest is 4.2%. The value of the option is to be obtained by numerically solvinga terminal-boundary value problem for a PDE. The PDE is the Black-Scholes PDE transformed by thechange of variable S = ex, where S is the stock price. Let u(x, t) be the value of the option correspondingto the stock price x = log(S) and time t < T , where T is the expiration date of the option.(a) You have some numerical code which solves the PDE in u(x, t) for x in the finite interval 1 < x < 8and t < T . Write down the boundary conditions for u(x, t) when x = 1 and x = 8.(b) A colleague suggests to you that the numerical code can be made more efficient without loss ofaccuracy by reducing the interval 1 < x < 8 to a smaller interval 2.5 < x < 4. Would you agree ordisagree? Explain why.(c) Suppose the numerical code stores the values of u(0.01n, 0.004m) as umn. Write an algorithm todetermine and plot the location of the no-exercise boundary for the American option.3(3) I wish to use the Monte-Carlo method to find the value of an option on a stock Stwhich evolves accordingto geometric Brownian motion with volatility σ = 0.27. The risk free rate of interest is r = 0.041 andthe expiration date of the option is 6 months from today. The payoff Φ(S) of the option is given by theformula,Φ(S) =10 if S ≤ 2535 − S if 25 ≤ S ≤ 350 if S ≥ 35.(a) Suppose today’s stock price is 29 and ξ = −0.1814 is a random value of the standard normal variable.Find the value of the option in the Monte Carlo method which corresponds to this random value ofξ.(b) Explain why it is appropriate to use the variance reduction technique of antithetic variables to valuethe option.(c) Find the value of the option corresponding to the random value of ξ in (a) when one uses the variancereduction method of (b).4(4) The current value of a stock is 24. The Heston stochastic volatility model for the evolution of the priceStof the stock is used to estimate the value of a stock option with expiration 9 months from today. TheHeston model is governed by the system of equations:(dYt= [0.31 − 2.8 Yt]dt + β√YtdWtdStSt= 0.032 dt +√Yt(ρ dWt+p1 − ρ2dZt) .Here Wtand Ztare independent (uncorrelated) Brownian motions under the risk neutral measure Q.(a) In order to price the option the system of equations is to be solved by the explicit Euler methodapplied to stochastic differential equations. Setting Sm= Sm∆t, Ym= Ym∆t, m = 0, 1, 2..., writedown appropriate values for S0, Y0. Explain your answer.(b) Suppose ρ = −0.3, β = 0.5, ∆t = 0.01, and you have computed Sm, Ymfor m = 0, 1, ...20, withS20= 18, Y20= 0.19. You are given the two values ξ = 0.2316, η = 0.3517 of independentlysampled standard normal variables. Find the corresponding values of S21, Y21.(c) For what range of parameter values β is this Heston model appropriate to use to value stock options?Explain your answer.5(5) The Hull-White model with σ = 0.0087, a = 0.14, ∆t = 1/16, and J the largest integer smaller thanp3/(2a∆t), is to be used to find the value of some interest rate derivatives. The lattice sites for themodel are (m, j), m = 0, 1, 2.., |j| ≤ min{m, J}. A lattice site (m, j) corresponds to time t = m∆t andinterest rate rmj.(a) Suppose we know that r245= 0.023. Find the value of r242.(b) Write down formulas for the transition probabilities pu(j), ps(j), pd(j) when j = 13. Give anexplanation of why the formulas have the given values.(c) The HW model is to be used to find the value of a swaption. The swaption consists of a prepaymentoption after 6 years on a 15 year loan with a notional principle of 100, where the rate is 4.5% payableannually. Let V (m, j) denote the value of the swaption corresponding to the lattice site (m, j), sotoday’s value of the prepayment option is V (0, 0). Taking r245to have the value in part (a), findV (24, 5) given that V (25, 5) = 1.5473, V (25, 6) = 1.5362, V (25, 4) =