MA323 Computational Methods in Financial Mathematics Assessed Coursework (2023) 02/03/2023 1 Guidelines 1.1 Submission Your coursework must be submitted by Monday 17th April 2023, 4pm (UK time). • All consequences regarding late submission can be found on the School’s website https://info.lse.ac.uk/Staff/Divisions/Academic-Registrars-Division/ Teaching-Quality-Assurance-and-Review-Office/Assets/Documents/Calendar/ GeneralAcademicRegulations.pdf Note in particular: Five marks (out of 100) will be deducted for coursework submitted late within 24 hours of the deadline and a further five marks will be deducted for each subsequent 24-hour period until the coursework is submitted. After five working days, coursework will only be accepted with the permission of the Chair of the Sub-Board of Examiners. • Please submit your complete report together with all files to
[email protected]. Include “MA323 submission” in the subject line. • For the submission please only submit one zip-file that contains all your files except the Plagiarism Statement, which you should submit in the same email. Your zip- file should have the following name: If your candidate number is 123456, then name your zip-file MA323 CN123456.zip. Your zip-file must contain your Jupyter Notebook (please provide an ipynb file, which the examiners can run, and a pdf version of it.) You should name these files according to your course code and candidate number, e.g., MA323 CN123456report.ipynb. • The content of your work must remain anonymous, so do not write your name on anything ex- cept the Plagiarism Statement. Instead, you must identify your work with your Examination Candidate Number. You can check your candidate number on ‘LSE for You’. 1.2 Academic integrity • When you submit the coursework you must submit a completed and signed copy of the Plagiarism Statement (available on Moodle). You are required to read the information on plagiarism on the following website: https://info.lse.ac.uk/Staff/Divisions/Academic-Registrars-Division/ Teaching-Quality-Assurance-and-Review-Office/Assets/Documents/Calendar/ RegulationsAssessmentOffences-Plagiarism.pdf 1 Note in particular the first paragraph on this website: “All work for classes and seminars (which could include, for example, written assignments, group work, presentations, and any other work, including computer programs) must be the student’s own work. Direct quotations from other work must be placed properly within quotation marks or indented and must be cited fully. All paraphrased material must be clearly acknowledged. Infringing this requirement, whether deliberately or not, or passing off the work of others as the student’s own work, whether deliberately or not, is plagiarism.” • You must take this assessment completely alone and not show or discuss it with anyone else. • This is an open book assessment and you are allowed to use any material made available on Moodle for MA323 and any academic literature as long as you cite the material that you use fully. • You are not permitted to: consult any other person or AI tool about the content of the assessment; allow any other person or AI tool to edit or proofread your work; submit any ideas or phrasing that are not your own (without appropriate citation). • You should not include any writing of your own that has been submitted for a different summative assessment. • The examiners may conduct vivas to check that you were the author of your submitted assessment. 1.3 Specific guidelines on report • The coursework consists of four problems. Your answers to all four problems will count towards the final mark. • Write a report and not just a question-answer style exercise set solution to answer the ques- tions. Your report should contain all results and their derivation, interpretation and discus- sion. Use complete sentences throughout. Give detailed arguments to explain your ideas and carefully justify your answers. • The only acceptable programming languages are Python 3.7, 3.8, 3.9, 3.10, or 3.11. • Please provide only one notebook. Separate answers to the different questions clearly in this notebook. • Your submitted Jupyter notebook should run completely without any error messages. • In particular, note that your Jupyter notebook should NOT ask the user to enter variables needed for the computation. Choose reasonable default parameters yourself and make clear in your instructions what the meaning and the names of the variables are such that the examiners can test several examples. • Add appropriate comments to your code to explain what your code is doing. • Your figures should be well formatted, with good axis labelling and appropriate titles. 1.4 Assessment The coursework will be marked in line with the departmental assessment criteria which are available on pages 19/20 of your student handbook: https://www.lse.ac.uk/Mathematics/assets/documents/Handbooks/2022-23/ 22-0201-UG-Mathematics-Handbook-Final.pdf © LSE 2023 2 2 Coursework Description If you use a random number generator for any of the problems below, seed the generator so that the results are reproducible. Problem 1. Consider the two integrals I1 = ∫ 2 0 cos(x)exdx; I2 = ∫ ∞ 0 cos(x5)e−x/2dx and the sum S = ∞∑ k=0 ( e−1 cos(k) k! ) . (a) Describe a numerical method to approximate the value of I1 such that the approximation error is guaranteed to be bounded by 1/100. Implement this method in Python and provide the value of the approximation. (b) Explain how one can approximate the integral I2 using a Monte-Carlo estimator. Implement the Monte-Carlo estimator in Python and provide a figure that plots a Monte-Carlo estimate against the number of samples (as we have done in the lectures and programming sessions). Describe a variance-reduction technique that can be applied here and discuss how well this technique works in this example. (c) Develop and implement a Monte-Carlo estimator in Python to approximate the sum S. Problem 2. Consider the function f : R→ R given by f(x) = { 3 2(x− 1)2, if x ∈ (0, 2), 0, if x /∈ (0, 2). (a) Suppose you would like to generate a sample from f using von Neumann’s acceptance-rejection algorithm. Specify a probability density function g ̸= f that can be used for this purpose and describe in detail how you can obtain a sample from f by sampling from g using von Neumann’s acceptance-rejection algorithm. For your choice of g what is the best possible proportion of numbers that your algorithm accepts? Implement von Neumann’s acceptance- rejection algorithm in Python to obtain 10000 samples from f and plot a histogram of the samples. (b) An alternative to von Neumann’s acceptance-rejection algorithm from part (a) for sampling from f would be to use the inverse transform method. Implement it in Python and draw again a histogram of the samples. Which of these two methods do you think is more suitable for generating a sample from f and why? Hint: If you get a Python runtime warning, the following discussion might help: https://stackoverflow.com/questions/45384602/ numpy-runtimewarning-invalid-value-encountered-in-power © LSE 2023 3 Problem 3. We consider a financial market consisting of three assets (one riskless asset and two stocks). The riskless asset has time-t price Bt = e rt, where r ≥ 0 is the constant interest rate, and the two stocks have time-t prices SAt = S A 0 exp (( r − σ 2 2 ) t+ σWAt ) ; SBt = S B 0 exp (( r − η 2 2 ) t+ ηWBt ) . Here SA0 > 0 and S B 0 > 0 are the initial stock prices, σ > 0 and η > 0 are the volatilities, and (WAt )t≥0 and (WBt )t≥0 are independent Brownian motions under the risk-neutral measure. (In Latex, η is written as \eta.) Consider a European option with payoff H at the maturity date T > 0 given by H = max{SAT , SBT }. (a) Write down a Monte-Carlo estimator for the time-0 price of this option. Justify your answer. (b) Explain in detail how one can generate the random variables that are used in your Monte- Carlo estimator in part (a). (c) Write Python code that computes the approximation of the time-0 price of the option using a Monte-Carlo estimator. (d) Use your Python code to compute the time-0 price of the option for the model parameters SA0 = 4, S B 0 = 3, r = 0.01, σ = 0.2, η = 0.4, T = 2 using the Monte-Carlo estimator. © LSE 2023 4 Problem 4. We now consider a financial market consisting of two assets (one riskless asset and one stock). We still assume that the time-t price of the riskless asset is given by Bt = e rt with r ≥ 0. We now assume that the dynamics under the risk-neutral measure of the risky asset are given by dSt = rStdt+ σ(St − 1)dWt, where (Wt)t≥0 is again a Brownian motion under the risk-neutral measure and σ > 0. We assume S0 > 1. (a) Explain how you can generate a sample path of S = (St)t≥0 on the discrete time grid 0 < h < 2h < . . . < nh for h > 0, n ∈ N. Write Python code that provides samples of ST for T > 0. Create a plot with ten sample paths of (St) for S0 = 4, r = 0.01, σ = 0.2, and T = 1 with h = 1/250. (b) Consider a European call option with strike K = 4erT , i.e, an option whose payoff H at the maturity date T > 0 is given by H = (ST − 4erT )+. Write down a Monte-Carlo estimator for the price of this option. Justify your answer. (c) Write Python code that computes the approximation of the time-0 price of the option using a Monte-Carlo estimator together with an asymptotic confidence interval. Use your Python code to compute the time-0 price of the option for the model parameters S0 = 4, r = 0.01, σ = 0.2, T = 1 using the Monte-Carlo estimator. Provide a 95%-asymptotic confidence interval for the time-0 price of the option. Discuss your results. (d) Specify a variance-reduction technique for approximating the time-0 price of the option in part (b) and implement it in Python. Discuss your findings. © LSE 2023 5
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