UNIVERSITY COLLEGE LONDON EXAMINATION FOR INTERNAL STUDENTS MODULE CODE : STAT0013 ASSESSMENT : STAT0013A6UA, STAT0013A6UB, STAT0013A6UC PATTERN STAT0013A7UC STAT0013A7PC MODULE NAME : STAT0013 - Stochastic Methods in Finance LEVEL: : Undergraduate Undergraduate (Masters Level) Postgraduate DATE : 12/05/2021 TIME : 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 Section A A1 (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 σ = 0.10. [4] A2 A 9-months forward contract on a non-dividend paying stock is entered into when the stock price is £40 and the risk-free interest rate is 5%. (a) What are the forward price and the value of the forward contract at the initial time? [4] (b) Suppose that at the initial time the forward contract is being bought and sold in the market for £5, instead of being traded at the fair price. Outline a simple trading strategy to take advantage of the arbitrage opportunity, stating how much risk-free profit will be made per contract. [5] A3 (a) Write down the Black-Scholes-Merton partial differential equation. State the boundary condition for the case of a derivative with payoff S2T + 2ST + 3. [2] (b) Consider a European put option and a European call option on the same underlying asset, with the same strike price K = $5, and same expiry date T = 1. The put is traded at $3, the call at $10, and the underlying asset at $6. The interest rate is r = 0.05. i. Verify that put-call parity does not hold in this case. [3] ii. Describe a portfolio that exploits the arbitrage opportunity. State the risk-free profit that the portfolio will provide. [5] A4 Consider the Geometric Brownian Motion (GBM) process dSt = µStdt+ σStdBt, S0 = 1. A stock price follows the above GBM, so that for the first two years, µ = 4 and σ = 2, and for the next two years, µ = 0 and σ = 2. Express the probability P [S4 < s ], for any s > 0, as a function of the cumulative distribution function, N(·), of the standard normal distribution. Hint: You can make use of the equivalent expression St = S0 exp((µ− σ22 )t+ σBt). [6] TURN OVER STAT0013 Examination Paper 2020/2021 Page 4 A5 Consider a European call option on a stock. The call option will expire in 6 months. The current stock price is $30, and the strike price of the call option is $20. At the expiration date, the stock price can either be $35 or it can be $25. The risk free interest rate is 4%. What is the value of the European call option today? [6] CONTINUED STAT0013 Examination Paper 2020/2021 Page 5 Section B B1 Consider a given portfolio with delta equal to 2, 000 and vega equal to 60, 000. We plan to create a new portfolio that is both delta and vega neutral by adding to the given portfolio: i) units of the underlying stock, and ii) units of a traded option with delta equal to 0.5 and vega equal to 10. How many units of the underlying stock and the traded option will we need? [5] B2 A stock price St follows the usual model dSt = µStdt+ σStdBt with expected return µ = 0.16 and volatility σ = 0.35. The current price is £38. (a) What is the probability that a European call option on the stock with an exercise price of £40 and a maturity date in 6 months will be exercised? [5] (b) Using the Black- Scholes formula, find the price of the call option if the risk-free interest rate is µ. Hint: You might have already calculated quantity d2 (required for finding the price of the call) in your answer in part (a). This can save you a lot of calculations. [4] (c) What is the probability that a European put option on the stock with the same exercise price and maturity date will be exercised? Find the price of the put option, using put-call parity. [4] B3 Consider a binomial model with T = 2 periods, S = 100, u = 1.6 and d = 0.6. The interest rate is r = 0.1. What is the price at time t = 0 for an American put option that has exercise price K = 97? Hint: To avoid too many calculations, you are given that the risk-neutral probability of one-step up movement is pˆ = 0.505, and that Su = 160, Sd = 60, Su2 = 256, Sd2 = 36, Sud = 96. [9] B4 (a) Assume in this question that all the derivatives have the same maturity date and the same underlying asset. Write explicitly the payoff function for a portfolio consisting of a short position in two European put options with exercise price £20 and a long position in three European put options with exercise price £25. [4] (b) Consider the following payoff function at maturity T and strike price K: Payoff = min(ST −K, 4). i. Draw the payoff diagram, that is the plot of payoff at date of maturity versus price of underlying asset. [4] ii. By using 1 unit of forward contract (in a long or short position) and 1 unit of call option (long or short position), create a portfolio with the payoff function given above. The strike prices of these two derivatives need not be equal to K. [4] TURN OVER STAT0013 Examination Paper 2020/2021 Page 6 B5 (a) Assume that Bt corresponds to a standard Brownian motion. i. Use Itoˆ’s formula to find the SDE satisfied by the process Yt such that Yt = sin(Bt) t 2 + 5. Indicate clearly the Itoˆ’s correction term, the drift term and the variance term in the SDE you have obtained. [4] ii. We know that Xt satisfies the SDE: dXt = dt+ 2 √ XtdBt, X0 = 0. Solve this SDE to obtain an analytic expression for Xt. Hint: The solution is of the form Xt = Bpt , for an appropriate positive integer p ≥ 2. [4] (b) Assume that (Bt, t ≥ 0) and (Wt, t ≥ 0) are independent standard Brownian motions. i. Let Xt =Wt +Bt. Compute the covariance of Xt and Xs, for s < t. [3] ii. Show that the process Yt such that Yt = B2t − t is a fair game process. [3] B6 Consider a portfolio that consists of: a long position on 8 shares of the underlying asset; a short position on 5 units of risk-less bond; a long position on 6 units of European puts on the above underlying. The risk free interest rate is r = 5%. The volatility is σ = 0.1. The current value of a share of the underlying is $10. The strike price for the call is K = $12 and the expiry date is T = 1 year. (a) Find the delta of this portfolio. [4] (b) Find the rho of a portfolio which is as the one described above, with the difference that it contains 0 units of European puts instead of 6 units of European puts. [3] END OF PAPER
欢迎咨询51作业君
