Please make sure you carefully read and understand the question or task. If you have

unanswered questions, please post these on the course Moodle Discussion Forum, and we’ll respond.

1. Question

This assessment is group-based in order to test the implementation of techniques in  computational finance, such as simulation of asset prices, pricing options using stochastic models, Monte Carlo methods as applied to complex derivatives using modelling software  packages and programming languages. Question Paper at the end of this document.

2. Further Information

a.    Not more than 3 students per group are allowed.

b.    Students not able to form a group to be allocated randomly into groups of 2 or 3 each.

c.    The students in a group are supposed to work together to create a coherent piece of code which addresses the question.

d.   This is a substantive piece of work which involves reading an academic paper and textbooks. The

group is supposed to write a code in MATLAB and summarise their findings in a separate short report.

3. Assessment Rubric/Criteria

If you make use of AI at any point in your research or writing process, no matter at what

stage, you must acknowledge the use of that source/platform as you would any other piece of evidence/material in your submission.

Turnitin

Your coursework will be processed through Turnitin for similarity checking.  You can submit a draft of your coursework to Turnitin before submitting your final copy.  You will find information about using Turnitin in the Student Information Point Moodle [USIP/PSIP]

6. Generative AI

Generative AI offers many new opportunities for learning and the development of academic   skill although, like any technology, it must be used judiciously.  Students should consider the data protection and privacy issues that can be caused by using AI.  Consider how your personal information will be used before signing up to AI tools and ensure you read any data protection policies before interacting with AI.  You should not feel pressured into using AI tools if you are uncomfortable with the data protection or privacy issues.  Bear in mind that responses to AI queries can be biased due to the inherent biases present in their training data.  This can lead to unfair and discriminatory responses.

Copying (including paraphrasing) AI responses to queries would be considered as plagiarism, as it would for copying the response from any internet search.

Further information can be foundhere.

7. Non-submission with good cause

Good cause for non-submission and late submission

We understand that during your studies, events that you cannot control (e.g., death of a  family member, personal circumstances, physical and mental ill health, etc.) may impact your ability to perform well in or complete assessments.

If you are experiencing such circumstances, you can submit a good cause claim in MyCampus.

You have five working days from the assessment deadline date to submit your good cause  claim. If you are prevented from submitting your claim within five days for good reason, you must detail this in your claim. You will receive an acknowledgement on MyCampus when

you submit. After you have submitted your claim, you have five working days to retract it.

If you have any questions, please contact your subject team:

business-accounting-finance@glasgow.ac.uk

business-economics@glasgow.ac.uk

[email protected]

The problems involve the development of functional Matlab code.  All the problems bear equal weight. Please explain carefully the technical challenges faced while programming and comment on the final results obtained in a short report.  The short report and the final MATLAB code should be submitted as a single ZIP folder on Moodle. The template to summarise the results is provided separately.

Topic :  Pricing Asian Options under Heston’s Stochastic Volatility Model

We consider the price of an asset S t  whose dynamics under the risk-neutral measure is described by the following system of stochastic differential equations:

dS(t) = S(t) (rdt + 4ν(t)dW(t)) ,      S(0) = S0 ,

dν(t) = κ(θ – ν(t))dt + σ4ν(t)dZ(t),      ν(0) = ν0 .

Here W and Z are correlated Brownian motions, that is,

dW(t)dZ(t) = ρdt ,

r is the interest rate, κ, θ and σ are positive constants satisfying 2κθ ≥ σ 2 .

Problem 1: Use the formula derived in Theorem 4.1 of the article by Kim and Wee [2] to compute the

prices of geometric fixed-strike Asian call options. The payoff function of the option is given as max(G[0,T] – K)+ ,    G[0,T] = exp 0T lnS(u )du) .

Use the following model parameters:  S0  =  100, ν0  = 0.09, t  =  0, r  =  0.05, θ  = 0.348, σ  =  0.39, κ = 1.15, ρ = –0.64. In the analytical formula, use n = 10, 20, 30 terms in the infinite series expansion and use 105  as the upper bound in the infinite integral.  Illustrate the results as in Table 1 of the article by Kim and Wee [2] for T = 0.5, T = 1.0, T = 2.0 with K = 90, 100, 110 used for each T value.

Problem  2:   Using  the  parameter  values  as in  Problem  1,  use  appropriate  discretisation  schemes  - Euler and Milstein - to estimate the prices of arithmetic fixed-strike Asian call options via Monte Carlo simulation. The payoff function of the option is given as

max(A[0,T] – K)+ ,    A[0,T] = 0T S(u )du.

Use different levels of discretisation step ∆t = 10–3 , 10–4 , 10–5  and illustrate the results in a table for T = 0.5, T = 1.0, T = 2.0 with K = 90, 100, 110 used for each T value.  The results must be produced for number of sample paths 50, 000 and 100, 000.

References

[1]  Heston, S. (1993) A closed-form solution for options with stochastic volatility with applications to bonds and currency options. Review of Financial Studies. 6(2), 327-343.

[2]  Kim, B. and Wee, I.S. (2014) Pricing of geometric Asian options under Heston’s stochastic volatility model. Quantitative Finance. 14:10, 1795-1809.