Math 623 (IOE 623), Fall 2007: 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 put option on a stock. The option has strike price23 and it expires 4 months from today. The current value of the stock is 24. Its volatility (in unitsof years−1/2is 0.29. Assume that the continuous rate of interest over the lifetime of the option is 4.34percent. The value of the option is to be obtained by numerically solving a terminal-boundary valueproblem for a PDE. The PDE is the Black-Scholes PDE transformed by the change of variable S = ex,where S is the stock price.(a) Write down the PDE for the value of the option as a function of the variables x and t, where theunits of t are in years.(b) The PDE is to be numerically solved in the region a < x < b, 0 < t < T . What are suitable valuesfor a, b, T ? Explain your answer.(c) What should the terminal and boundary conditions for the PDE be? Explain your answer.(d) Assume you decide to numerically solve the terminal-boundary value problem for the PDE by usingthe explicit Euler method. For the x increment you take ∆x = 1/100. Find an appropriate value forthe t increment ∆t. How accurate would you expect your computed value for the option to be i.e.how small the error from the exact Black-Scholes price? Explain your answers.Note: In your answers to the above you should use actual numerical values, not letters denotingparameters (like T, σ etc).2(2) We have written some computer code to find the value of a European put option with strike price 37,and wish to alter the code to find the value of the corresponding American option. If u(x, t) denotes thevalue of the European option at time t (in years) when the stock price is S = ex, then the numericallycomputed value un,mapproximates the value of u(0.02n, 0.004m). The explicit Euler method yields thealgorithm:un,m−1= 0.37482 un,m+ 0.313875 un+1,m+ 0.311125 un−1,m.(a) Find the values of the continuous interest rate and the stock volatility (with the time unit beingyears) which have been used to price the option. Explain your answer.(b) Show how to alter the algorithm above to price the American option. Explain your answer.(c) Suppose we have stored the values of un,m. Write an algorithm to determine the location of theno-exercise boundary for the American option. Explain your answer.3(3) We have been given some data from a million i.e. 106simulations of the Monte-Carlo method to computethe value of an option. The sum of the million values we obtained is 1340389, and the sum of the squaresof the million values we obtained is 7660517.(a) Find the estimated value of the option from the million simulations.(b) Estimate the percentage error for the option value given in (a).(c) Does it seem reasonable that the data given above came from a million Monte Carlo simulations?Explain your answer.4(4) We wish to estimate the value of a basket option on three stocks. The payoff on the option depends onlyon the arithmetic average of the three stocks at the expiration of the option, which is 6 months fromtoday. The risk neutral evolution of the three stocks is modelled as follows:dS1,tS1,t= 0.047 dt + 0.29 dW1,t,dS2,tS2,t= 0.047 dt + 0.34 dW2,t,dS3,tS3,t= 0.047 dt + 0.28 dW3,t,where the W1,t, W2,t, W3,t, t > 0, are correlated Brownian motions with correlation matrix ρ = [ρi,j],and ρi,jdt = E[dWi,tdWj,t]. The units of time t are in years. The values of the three stocks today areS1,0= 24, S2,0= 29, S3,0= 21.(a) The matrix ρ is given by the formula:ρ =∗ −0.54 ∗∗ ∗ 0.280.37 ∗ ∗Find the numerical values of the entries in the matrix which are denoted by ‘∗’. Explain your answer.(b) We use a control variate for a variance reduction method for the Monte-Carlo simulation to estimatethe price of the basket option. The control variate is the geometric average of the three stocks atthe expiration of the option. Show that the control variate is the exponential of a Gaussian variableand compute the mean and standard deviation of this Gaussian variable.(c) Suppose that for the standard Monte-Carlo simulation the value of the arithmetic option is 1.3718,the value of the geometric option is 1.1197, and β = 1.0984. Find the improved value of the arithmeticoption if the exact value of the geometric option is 1.1232.(d) If in addition you are given that the variance of the arithmetic payoff is 5.7139 and the varianceof the geometric payoff is 4.5155, find the coefficient of correlation for the two random variablescorresponding to the arithmetic payoff and the geometric payoff.5(5) Today’s bond prices with face value 100 and maturities from 1 up to 10 years is given by the followingtable:Maturity 1 2 3 4 5 6 7 8 9 10Price 95.24 91.31 87.31 82.92 78.89 75.35 71.83 68.27 64.71 61.51The values of interest rate derivatives are to be computed using the Hull-White tree with σ = 0.014,a = 0.11 and the bond data above. We use linear interpolation of the bond prices above to estimate thevalues of bonds with maturity which is not an integer number of years.(a) Find the value of ∆r corresponding to ∆t = 1/8.(b) Find all the possible values for the interest rate r on the Hull-White tree corresponding to the timest = 0 and t = 1/8.(c) Suppose that we wish to find the value of an interest rate cap of 7 percent on a 10 year loan withprincipal 100. Let αm= 0.049 correspond to time t = m∆t where ∆t = 1/8 and m = 16. Find allnodes (m, j), |j| ≤ m, m = 16, for which the caplet corresponding to interest over the time intervalfrom 2 years to 2 and 1/8 years is