STAT0018: Stochastic Methods in Finance 2
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STAT0018: Stochastic Methods in Finance 2
Practice ICA
February 2023
Instructions
● Answer ALL questions.
● You have 60 minutes to complete this paper.
● After the 60 minutes have elapsed, you have 20 minutes to upload your solutions.
● You may submit only one answer to each part-question.
● The numbers in square brackets indicate the relative weights attached to each part question.
● Marks are awarded not only for the final result but also for the clarity of your
answer.
Administrative details
● This is an open-book exam. You may use your course materials to answer questions.
● You may not contact the course lecturer with any questions, even if you want to clarify something or report an error on the paper. If you have any doubts about a question, make a note in your answer explaining the assumptions that you are making in answering it. You should also fill out the exam paper query form online.
● Some part-questions may require text-based answers; many of these will indicate the maximum number of sentences you may write. You must adhere to this or you will lose marks.
● Some questions may ask you to approach a problem in a particular way; please take note of this. Failure to do so may result in marks being deducted.
Formatting your solutions for submission
● You can hand-write or type your solutions.
● Your solutions should be presented in the same order as the (part-) questions.
● You should submit ONE pdf document that contains your solutions for all part- questions. Please follow UCL’s guidance on combining text and photographed/ scanned work should you need to do so.
● Make sure that your handwritten solutions are clear and are readable in the document you submit.
Plagiarism and collusion
● You must work alone. In particular, any discussion of the paper with anyone else is not acceptable. You are encouraged to read the Department of Statistical Science’s advice on collusion and plagiarism.
● Parts of your submission will be screened to check for plagiarism and collusion.
● If there is any doubt as to whether the solutions you submit are entirely your own work you may be required to participate in an investigatory viva to establish authorship.
Question [0 marks]
Suppose that you are deciding whether to build a factory that would produce widgets.
The factory produces one widget per year forever and this one unit can be sold at the
price Pt in year t. The output price, Pt , follows a geometric Brownian motion (GBM), i.e. dP = αPdt + σPdz , where α is the annual drift parameter, σ > 0 is the annualised variance parameter, and dz is the increment of a Wiener process. The investment cost for this factory is I > 0 and the variable cost of production is C > 0 per widget. The risk-free rate is r, with r > α > 0. Assume that the option to invest into the factory is perpetual and that an active factory can be permanently abandoned (zero salvage value) at any time, i.e. the project cannot be resumed once abandoned.
(a) What is the expected value of the widget factory, G(P), at price, P > 0? [0]
(b) Using contingent claims and considering an optimal threshold price level, C* , below which to abandon the factory, find the value of the project, V (P).
(c) Using value-matching (VM) and smooth-pasting (SP) conditions at the optimal threshold price level, find B2 and C* . Subsequently, using dynamic programming parameters, i.e. δ 三 ρ _ α and r 三 ρ, verify that B2 > 0 and C* > 0. Compare these two constants with those obtained for a project that has infinitely many embedded operational options and comment on the meaning of your findings in no more than two sentences. Recall that for a project with temporary suspension options, we know that B2(so) = 
8 
_ 
、 and the suspension threshold is C . [0]
(d) Applying contingent claims analysis, show that the value of the option to invest
is F (P) = A1 Pβ 1 , where β1 > 1 is the positive root of the characteristic quadratic
o (.) (without determining the parameters). Determine VM and SP conditions
between F (P) and the appropriate branch of V (P). Finally, using VM and SP conditions, derive the nonlinear equation for the optimal investment threshold P* , i.e. (β1 _ β2 )B2 (P* )β2 + (β1 _ 1) 
_ β1 ( 
+ I) = 0, and compare P* (analytically or graphically, but not numerically) with the thresholds for a project with suspension options (P*,so ) and no flexibility (P*,nf ). [0]
2023-02-22