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W8finalsolns - Math 623(IOE 623 Winter 2008 Final exam Name...

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Math 623 (IOE 623), Winter 2008: Final exam Name: 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 also use 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 your explanations 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 otherwise stated. You are obliged to comply with the Honor Code of the College of Engineering. After you have completed the examination, please sign the Honor Pledge below. A test where the signed honor pledge does not appear may not be graded. I have neither given nor received aid, nor have I used unauthorized resources, on this examination . Signed: 1
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(1) Suppose I wish to compute the value of a European Butterfly Spread option on a stock. The option pays Φ( S ) if exercised when the stock price is S , where Φ( S ) = 0 if S 40 S - 40 if 40 S 50 60 - S if 50 S 60 0 if S 60 . The current value of the stock is 48 and it expires 6 months from today. Its volatility (in units of years - 1 / 2 is 0 . 34. Assume that the continuous rate of interest over the lifetime of the option is 3 . 75 percent. The value of the option is to be obtained by numerically solving a terminal-boundary value problem for a PDE. The PDE is the Black-Scholes PDE transformed by the change of variable S = e x , 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 the units of t are in years. (b) The PDE is to be numerically solved in the region a < x < b, 0 < t < T . Find suitable numerical values 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 using the explicit Euler method. How large can you take Δ t/ x ) 2 and the numerical scheme remain stable? Justify your answer. Solution (a): ∂u ∂t + 1 2 σ 2 2 u ∂x 2 + ( r - 1 2 σ 2 ) ∂u ∂x - ru = 0 . Solution (b): Evidently T = 0 . 5 and σ = 0 . 34. Writing X ( t ) = log S ( t ), then X ( T ) is a Gaussian variable with mean X (0) + ( r - 1 2 σ 2 ) T and variance σ 2 T . The option is in the money at the expiration date T if log 40 < X ( T ) < log 60. Hence if X (0) < log 40 - 3 σ T or X (0) > log 60 + 3 σ T , then for the option to be in the money at time T requires X ( T ) to be 3 standard deviations from its mean.
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