SDE Solution Form Analysis MTHM006 Assignment

MTHM006 John Thuburn

2024/25 Summative Assignment 2

Full marks achieved in this assignment will contribute 10% of the final module mark. Please submit your solutions via the module ELE page by the deadline 12:00 noon on Thursday 20 March, 2025. Usual penalties apply for late submission unless mitigation is approved.

Please submit

• a pdf file of your answers to the theoretical questions, including the theoretical and dis- cussion parts of Q5, e.g., as a scan of handwritten answers or captured from a tablet;

• the computer codes that you write for Q5;

• the plots that you produce for Q5;

• your declaration on the use of GenAI (see below);

all zipped into a single file. Please write down your answers clearly, quoting numerical values to 2 decimal places, unless otherwise stated. Plots should be clearly and legibly labelled, and computer codes should be thoroughly commented to demonstrate your understanding.

You are expected to work independently and you are reminded that plagiarism is a disciplinary offence.

Marking criteria and notes on the use of GenAI are given below the questions.

Questions

1. Determine whether there are any functions g(t) and real constants ? and ? for which the stochastic differential equation

dX_t = ?(t + ?W_t)² dt + g(t) ? W^? dW_t

has a solution of the form X_t = f (t, W_t). Find the solutions when they exist. [10]

2. Consider the stochastic process X_t = e^8t sin(aW_t), with a ? R. For which values of a, if any, is X_t a martingale process? Justify your In particular, you should check that

E[|X_t|] < ? where necessary. [10]

3. Suppose that (W_t)_t?0 is a standard Brownian Motion under the market probability P with natural filtration (F_t)_t?0. Let (?_t)_t?0 be a stochastic process adapted to (F_t)_t?0, and define

L_t = exp

t 1 t

?_s dW_s ?

0 0

?² ds .

Find the stochastic differential equation satisfied by L_t.

You are given that the probability density function p(x, t) for some stochastic process X_t under the market probability P is related to the probability density function p_?(x, t) for the same stochastic process under an alternative probability measure P_? by

p_?(x, t) = L_t p(x, t).

For the case ?_t ? 1, show that the stochastic process X_t = W_t + t is normally distributed under P_?, and determine its mean and variance under P_?. [20]

4. Let (W_t)_t?0 be a standard Brownian Compute E[e^?Wt ] in two ways:
   
   
   • by integrating over the normal distribution of W_t;
   
   
   • by writing the stochastic differential equation satisfied by e^?Wt , and hence the differ- ential equation satisfied by its expectation.
   
   
   

Hence evaluate

"? _t

se?Ws/2 dWs

2#

You may quote the Ito isometry formula, if needed. [10]

5. In this question you will investigate the use of the binomial algorithm for pricing different options. You may freely use and adapt the provided Matlab function m or the Python function BinomialCall.py, which use the binomial algorithm to compute the fair price of a European call option. When you are asked to discuss your results you should refer to appropriate mathematical and financial ideas.

For this question use the following parameter values:

Browse the code for the function BinomialCall.m or BinomialCall.py and check that you understand how it works. Call the function with initial asset price S₀ = 100, N = 100 steps, and other parameters as above, and check that you get the answer 6.9702 for the (time zero) option price

a) Modify the function so that it computes the price of a European put option and use the same parameters as above to compute that Verify that your results satisfy Put-Call Parity. (5)

b) Write a script that will call your European put pricing function for a range of initial asset prices S₀ ? [0, 200]. Plot the (time zero) option price versus the initial asset price. Discuss whether the shape of the curve is what you expect? (5)

c) Write a function that takes input parameters r, mu, sigma, S0, E, and T, and returns the exact value of the time zero price for the European put option as given by the Black-Scholes equation and discussed in lectures. You might find it helpful to express the cumulative normal function in terms of Matlabs erf function. For the same values of S₀ as in part (b), plot the Black-Scholes option prices on the same graph as the binomial algorithm option prices; choose your line styles / colours / symbols carefully so that both sets of results are clearly visible.

On a second plot, plot the differences between the two sets of results. Comment on the accuracy of the binomial algorithm.

Let the number of steps N take the values N = 2^q, q = 1, 2, . . . , 10. For each value of N , compute ?, the maximum absolute value of the error, i.e., the difference between the binomial algorithm price and the Black-Scholes price over the range S₀ ? [0, 200].

Plot ? versus N on log-log axes. What can you say about how quickly the binomial algorithm converges to the exact answer as we expend more computational effort? (15)

d) Create a modified version of your European put pricing function that computes the price of an American put option. For the same values of S₀ as in part (b), plot the (time zero) option price for the American put option versus the initial assetm Compare the time zero price of the American put option with the time zero price of the European put option. Discuss whether the differences are what you expect. (10)

e) Add some code to your American put pricing function that will determine the exercise boundary S_B(t), that is, the value of S(t) above or below which it is preferable to exercise immediately rather than continue to hold the For N = 500, plot the curve S_B(t) and discuss its features. Trying other values of N might help you to

understand the behaviour. (15)

[50]

[Total marks 100]

Assessment Criteria

20 marks out of 100 are allocated for demonstrating understanding by clearly communicating the methods used and, in Q5, discussing the results obtained. It is important that you give a few words of explanation in your mathematical calculations.

To achieve a pass mark, you need to demonstrate basic understanding of the subject, together with a basic ability in the analytical and numerical techniques taught in the module. There may be many mistakes, but there will be clear evidence of the relevant knowledge.

To achieve a first-class mark, you will have demonstrated a very sound understanding of the mathematical techniques and relevant concepts in the subject through correct calculations and concise but clear explanations. There may be a few minor errors but there should not be any major gaps or errors in knowledge or understanding.

Use of Generative AI

This assessment is AI-supported. This is because some uses of GenAI tools may help you to complete the assessment without compromising your ability to demonstrate that you have achieved the intended learning outcomes. See below for guidance about appropriate uses of GenAI tools.

You should note that GenAI tools are currently rather weak at distinguishing fact from fiction, understanding mathematical correctness, and writing bug-free code. Note, also, that markers will be looking for evidence of human understanding, not just correct calculations, so you should include some words of explanation within your calculations and sufficient comments within your code.

When submitting your assessment, you must include the declaration below, listing all the ways in which you have used GenAI tools for this assessment. You must also reference the use of GenAI outputs within your assessment, in line with the Universitys referencing guidelines. You should also keep a record of which GenAI tools you use, including the prompts and outputs, in case you are asked to present this at a viva.

Submitting your work without an accompanying declaration, or one with no uses listed, will be considered a declaration that you have not used GenAI tools in preparing your work. If a declaration cannot be uploaded then by submitting your assessment you are confirming that you have followed the instructions for the assessment and the guidelines about using GenAI tools.

Student declaration

This assessment is AI-supported. I acknowledge the following uses of GenAI tools in this assess- ment.

Replace the following examples with your own list.

• I have used GenAI tools to help me to correct my computer

• I have used GenAI tools to suggest section headings for my

• I have used GenAI tools to help me to correct my grammar or

• I have used GenAI tools to suggest topics to discuss in my literature

I declare that I have referenced the use of GenAI outputs within my assessment, in line with the Universitys referencing guidelines.