UNIVERSITY COLLEGE LONDON EXAMINATION FOR INTERNAL STUDENTS MODULE CODE : ASSESSMENT : PATTERN MODULE NAME : LEVEL: : DATE : TIME : STAT0013 STAT0013A6UA, STAT0013A6UB, STAT0013A6UC STAT0013A7UC STAT0013A7PC STAT0013 - Stochastic Methods in Finance Undergraduate Undergraduate (Masters Level) Postgraduate 12/05/2021 10:00 

 This paper is suitable for candidates who attended classes for this module in the following academic year(s): 2018/19, 2019/20, 2020/21 TURN OVER STAT0013 Examination Paper 2020/2021 Page 1 STAT0013 – Stochastic Methods in Finance (2021) • Answer ALL questions. • You have three hours to complete this paper. • After the three hours has elapsed, you have one additional hour to upload your solutions. • You may submit only one answer to each question. • Section A carries 40% of the total marks and Section B carries 60%. The relative weights attached to each question are as follows: A1 (9), A2 (9), A3 (10), A4 (6), A5 (6), B1 (5), B2 (13), B3 (9), B4 (12), B5 (14), B6 (7). • 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. Formatting your solutions for submission • Some part-questions require you to type your answers instead of handwriting them. These questions state [Type] at the start of the part-question. You must follow this instruction. Failure to do so may result in marks being deducted. For questions without the [Type] instruction, you may choose to type or hand-write your answer. • You should submit ONE pdf document that contains your solutions for all questions/part- questions. Please follow UCL’s guidance on combining text and photographed/ scanned work. • 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 via Turnitin 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. TURN OVER STAT0013 Examination Paper 2020/2021 Page 2 Particular Instructions for STAT0013 • Marks will not only be given for the final (numerical) answer but also for the accuracy and clarity of the answer. So make sure to write down workings, e.g. formulas, calculations, reasoning. • Show your full working for all questions. Do not write formulas alone without any comment about what you are calculating. • Except where otherwise stated, interest is compounded continuously and there are no transac- tion charges or buy-sell spreads. Assume that a positive risk-free interest rate always exists, and is the same for all maturities and is constant over time unless otherwise stated. All risk-free rates are expressed on an annualised basis. • All data in this exam are fictional. CONTINUED STAT0013 Examination Paper 2020/2021 Page 3 A1 Section A (a) Give the mathematical expression of the gamma and vega of a portfolio of derivatives. Explain briefly the meaning of these two quantities. [3] (b) A portfolio consists of stocks in the Asian markets. It is known that the portfolio has a negative delta. Suppose there is a small increase in the value of the stocks. Explain briefly the effect of this increase in the total value of the portfolio. [2] (c) Find the value of delta of a 9-month European call option on a stock with a strike price equal to the current stock price (t = 0). The interest rate is 8%. The volatility is