Math 623 (IOE 623), Winter 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 Butterfly Spread option on a stock. The option paysΦ(S) if exercised when the stock price is S, whereΦ(S) =0 if S ≤ 40S − 40 if 40 ≤ S ≤ 5060 − S if 50 ≤ S ≤ 600 if S ≥ 60.The current value of the stock is 48 and it expires 6 months from today. Its volatility (in units ofyears−1/2is 0.34. Assume that the continuous rate of interest over the lifetime of the option is 3.75percent. 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 . Find suitable numericalvalues for a, b, T and explain your reasoning for this choice..(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. How large can you take ∆t/(∆x)2and the numerical scheme remainstable? Justify your answer.2(2) We wish to find the value of an Asian option on a zero dividend stock which expires 9 months from today.The payoff on the option is the excess of the stock price at expiration over the continuous average ofthe stock price during the 9 month life span of the option. The current price of the stock is 35 and itsvolatility is 0.28 per annum. The risk free interest rate is 0.038 per annum.(a) Write down the two variable partial differential equation for a function w of (t, ξ) on intervals0 < t < 3/4, 0 < ξ < ξmax, together with boundary and terminal conditions, which one needsto solve to compute the value of the Asian option. Find a suitable numerical value for ξmax.(b) The solution w(t, ξ) of the equation can be written as an expectation valuew(t, ξ) = E[Φ(ξ(T )) | ξ(t) = ξ], where ξ(t) is the solution of a stochastic differential equation. Writedown a formula for the function Φ(ξ) and also the stochastic equation which ξ(t) must satisfy.(c) Suppose I wanted to find the the value of the option using the Monte-Carlo method. Explain carefullyhow I could do this, either using the representation in (b) or an alternative method.3(3) I wish to find the value of an option on a stock Stwhich evolves according to geometric Brownian motionwith volatility σ = 0.26. The risk free rate of interest is r = 0.038 and the expiration date of the optionis 6 months from today. The payoff Φ(S) of the option is given by the formula,Φ(S) =15 if S ≤ 3045 − S if 30 ≤ S ≤ 450 if S ≥ 45.(a) Suppose today’s stock price is 40. We wish to find the value of the option by implementing theMonte Carlo method. Thus we write the value V of the option as V = E[Ψ(ξ)] where ξ is a standardnormal variable. Find a formula for the function Ψ(ξ).(b) Suppose that today’s stock price is 70. Why does it make sense to look for a variance reductionmethod to find the value of the option?(c) If we use importance sampling as a variance reduction technique then V = E[Ψ1(ξ)] for some suitablefunction Ψ1different from Ψ. Obtain a formula for Ψ1.4(4) We wish to use the Hull-White model to find the value of an interest rate cap on a 15 year loan with anotional principle of 100, where the cap is 5.5%. The parameters in the model are σ = 0.015, a = 0.2.The model has been calibrated to today’s yield curve by taking ∆t = 1/32 and using interpolation. Theresulting α values corresponding to the time m∆t are denoted αm. The value of the cap is given byV (0, 0), where V (m, j) satisfies the recurrenceV (m, j) = exp[−r(m, j)]{Cap(m, j)++ pu(j)V (m + 1, j + 1) + ps(j)V (m + 1, j) + pd(j)V (m + 1, j − 1)},if |j| ≤ min(m, J − 1), and V (M, j) = 0, |j| ≤ min(M, J).(a) Find the value of M and the sum pu(j) + ps(j) + pd(j).(b) Obtain a formula for r(m, j) which can be explicitly computed once we know m, j, αm.(c) Obtain a formula for Cap(m, j) which can be explicitly computed once we know m, j, αm.(d) Suppose that today’s yield curve varies with a minimum of 2.5% and a maximum of 6% over the 15year period. Estimate how likely it is that the spot rate in the Hull-White model ever exceeds 10%.5(5) We wish to find the value of a bond call option using the BDT interest rate model. The expiration dateof the option is 6 years from today. The payoff is the excess over 55 of the value then of the bond withface value 100 and maturity 15 years from today. We take ∆t = 1/16 in the model and assume the modelhas been calibrated to today’s yield curve and volatilities, giving parameter values rm0, βm, m = 0, 1, ..corresponding to the time m∆t.(a) Write down an algorithm which we can use to compute the value of the bond call option.(b) Suppose the parameter values corresponding to m = 80 are rm0= 0.0025 βm= 0.1605. Find fromthis the mean of the random variable r(t) for some suitable t.(c) With the same data as in part (b) find the variance of log