MATH40082 (Computational Finance)
Assignment No. 1: Monte Carlo Methods
计算金融代写 1 Background 1.1 Stock Options The trader has calibrated a specialised risk neutral process for some underlying stock price.
1.1 Stock Options
The trader has calibrated a specialised risk neutral process for some underlying stock price. Given the current stock is S0, market prices indicate the risk-neutral distribution of the stock price at time t is given by:
St ∼ N(f(S0, t), v2(S0, t)) (1)
for some calibrated functions f and v2.
If (1) describes the risk neutral distribution, then the formula to value a call option V with payoff
at time t = T is given by
V (S0, 0) = e−rT EQ[g(ST )]. (2)
Then to carry out a Monte Carlo valuation of an option, we may use samples from a standard random normal distribution
φ ∼ N(0, 1) (3)
to write the equation
ST = f(S0, T) + v(S0, T)φ. (4)
Equation (5) then generates a single random path, from which we can value a payoff
V i = e−rT g( f(S0, T) + v(S0, T)φi) . (5)
1.2 Path Dependent Options
Now assume that the stochastic process follow a standard Geometric Brownian motion governed by
dS = (µ − D0)Sdt + σSdW.
Then the following options will depend on S(tk) which are the share prices at K + 1 equally spaced sampling times t0, t1, …, tK with t0 = 0 and tK = T (unlike part (a), the computation cannot proceed from t = 0 to t = T in one step). Full details are given the the lecture notes – but the important point to note is that
where f is the payoff function depending the type of option. 计算金融代写
There are different classes of Asian option, resulting in different payoff conditions. In this coursework we look at simple European style call or put options. A fixed strike call option will have the payoff
f(S, A) = max(A − X, 0)
where X is the strike price and a floating strike call option would be
f(S, A) = max(S − A, 0).
where A is sometimes calles the average strike price.
A fixed strike put option will have the payoff
f(S, A) = max(X − A, 0)
where X is the strike price and a floating strike put option would be
f(S, A) = max(A − S, 0).
where A is the strike price.
where f is the payoff function depending the type of option.
There are different classes of Lookback option, resulting in different payoff conditions. In this coursework we look at simple European style call or put options. We can either have a floating strike S or a fixed strike X. For example a floating strike Lookback call option would give
f(S, A) = max(S − A, 0)
where A must be the minimum, and a floating strike Lookback put option would be
f(S, A) = max(A − S, 0).
where A must be the maximum.
A fixed strike call option will have the payoff
f(S, A) = max(A − X, 0)
where X is the strike price and A must be the maximum. and a fixed strike put option will have the payoff
f(S, A) = max(X − A, 0)
where X is the strike price and A must be the minimum.
The discretely sampled knock-out barrier option will be knocked out (and return a value of zero) if the a barrier asset price B is crossed before the maturity date.
The option will be an “up” option if the knock out condition is on S > B, or a “down” option if the condition is on S < B. So for example a up-and-out knockout barrier call option has the conditions
2 Tasks 计算金融代写
2.1 Stock Options
• Write a program to calculate the value of the put option V using the parameters given at t = 0 using Monte Carlo simulation. You need only include the code in the appendix of your report. (Coding 3 marks)
• Plot a single figure showing the value of the option with t = 0 at S0 with increasing N (N = 1000, 2000, . . ., 50000, or more!) alongside the exact value from the analytical formula. Comment on the appearance of your result. (Understanding 4 marks)
• Use antithetic variables, moment matching or the Halton sequence to extend the Monte-Carlo method and analyse its benefit when valuing this option. Produce at most 2 plots or tables of your results, and comment on the accuracy, speed and efficiency of the chosen method(s). (Understanding 4 marks, Originality/Initiative 3 marks)
2.2 Path Dependent Options
For this task you are required to value a discrete maximum fixed-strike lookback call option with the following parameters. The option matures at T = 1, the interest rate is r = 0.02, the dividend rate D0 = 0.02 and volatility is σ = 0.33. The stock price, currently at S0 = 6.1, will be observed on K = 25 plus one equally spaced dates throughout the lifetime of the option, where t0 = 0 and tK = T. The fixed strike price is X = 6.1. You should use Monte-Carlo simulation to value this option.
• Code up the path dependent option. (Coding 2 marks)
• Using the given parameters at t = 0, produce at most 2 plots or tables to investigate the value of the path dependent option with different values of N. Comment on your results, what value are they converging towards? (Understanding 6 marks)
• Extend your method using antithetic or moment matching, and produce at most 2 plots or tables to investigate the speed, accuracy and efficiency of the method. Comment on your results. (Understanding 2 marks, Originality/Initiative 5 marks)
3 Instructions 计算金融代写
This assignment will account for 40% of your final mark in this module. The total number of marks in this assessment are 40, and they will be awarded as follows:
5 for working codes;
|0-50%||Little or no attempt, codes not working|
|50%-70%||One or two bugs in the code are affecting the results|
|70%-100%||Results in 2.1 and 2.2 from the codes appear correct|
5 for the presentation of your written report;
|0-50%||Poorly presented work. Significant amount of text unreferenced. Graphs and tables poorly labelled making it difficult to interpret them.|
|50%-70%||Good presentation. Text is readable. Graphs are ok, maybe missing labels and not always referenced correctly. Report is overly long and unnecessarily repeats the same (or similar) results.|
|70%-100%||Excellent presentation, well written and well referenced. Graphs are clear, tables used when appropriate. Report keeps within the page limit.|
20 for the understanding of the problems involved;
|0-50%||Results are poorly presented or they are without supporting text. The methods are described but are not shown to be implemented through results. The student is unable to demonstrate they can correctly interpret results.|
|50%-70%||Demonstrates a good understanding of the standard methods. Is able to generate standard results and discuss them. Results are well presented.|
|70%-100%||Student is able to correctly interpret standard results and evaluate the efficiency of the standard methods.|
10 for originality/initiative.
|0-50%||Little or no attempt at implementing any of the new methods. Those that have been implemented have poorly presented results or the student is unable to demonstrate they can correctly interpret results.|
|50%-70%||Demonstrates a good understanding of new or alternative methods. Is able to implement new methods, present results and discuss them. Results are well presented.|
|70%-100%||Has implemented difficult algorithms not detailed in the course. Presentation of the results is excellent. Student is able to correctly interpret results and compare methods in a coherent way.|
Please see individual bullet points in the task section for a break down of the marks. 计算金融代写
Reports should be prepared electronically using either MS Word, LaTeX, or similar, and must be submitted without your name, but with your university ID number online through the TurnItIn system. Please include the program files used to generate results for the report in an appendix as plain text. Your report should be written in continuous prose in the form of a technical report and should be approximately 8 – 10 pages long (excluding appendices). Any programming language may be used. The deadline for this assignment is 11am on Monday 28th March.
THIS DEADLINE MUST BE STRICTLY ADHERED TO – Reports handed in AFTER 11am Monday 28th March will be docked 4 marks plus an additional 4 marks each day thereafter until a mark of zero is reached. Reports handed in after 5pm Friday 8th April will be awarded a mark of zero and will not be marked.
In order that your report conforms to the standards for a technical report, you should use the following structure: 计算金融代写
• give a brief introduction stating the problem you are solving and the parameters you are using (from the model or method),
• present your results in the form of figures and tables, using the order of items in the bullet points as a guide as to the order of your document
• absolutely NO screenshots of running code need to be included,
• do not include overly long tables – a table should never cross over a page,
• present the results for any methods you have implemented, there is no credit for a discussion of a method that has not been shown to be implemented by you (through results) for your problem
• refer to and discuss each of your results in the text, part of the marks available in each bullet point are for interpreting the results
• try to keep to the page limit, removing any unnecessary results from the main text
• number and caption your figures and tables and refer to them by their number (not their position in the text),
• number any equations to which you refer,
• use consistent internal (and external) referencing.