Topic outline

  • Announcements

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    • Student Forum
    • Announcements Forum
    • Student forum

      This forum is available for everyone to post messages to. If you have any questions about the module, please ask them here.

    • Tutorial group allocation File

      Please note that:

      This group allocation is for tutorial.


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    • Response for module evaluation File

      Thanks very much for all the nice words. They are very important to me and I am very touched. Even the feedback on the areas for improvement was politely and gently raised. It's always great to receive constructive feedback that not only acknowledges my strengths but also offers gentle guidance on areas to improve.

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  • Syllabus

    1. Stochastic models for security prices
    Discuss the continuous time log-normal model of security prices and the empirical evidence for or against the model.
    Explain the definition and basic properties of standard Brownian motion or Wiener process.
    Demonstrate a basic understanding of stochastic differential equations, the Itô integral, diffusion and mean-reverting processes.
    State Itô’s Lemma and be able to apply it to simple problems.
    Write down the stochastic differential equation for geometric Brownian motion and show how to find its solution
    Write down the stochastic differential equation for the Ornstein-Uhlenbeck process and show how to find its solution
    2. Models for the term structure of interest rates
    Explain the principal concepts and terms underlying the theory of a term structure of interest rates.
    Describe the desirable characteristics of models for the term structure of interest rates.
    Apply the term structure of interest rates to modelling various cashflows, including calculating the sensitivity of the value to changes in the term structure.
    Describe, as a computational tool, the risk-neutral approach to the pricing of zero-coupon bonds and interest-rate derivatives for a general one-factor diffusion model for the risk-free rate of interest.
    Describe, as a computational tool, the approach using state-price deflators to the pricing of zero-coupon bonds and interest-rate derivatives for a general one-factor diffusion model for the risk-free rate of interest.
    Demonstrate an awareness of the Vasicek, Cox-Ingersoll-Ross and Hull-White models for the term structure of interest rates.
    Discuss the limitations of these one-factor models and show an awareness of how these issues can be addressed.
    3. Simple models for credit risk
    Define the terms ‘credit event’ and ‘recovery rate’.
    Describe the different approaches to modelling credit risk: structural models, reduced form models, intensity-based models.
    Demonstrate a knowledge and understanding of the Merton model.
    Demonstrate a knowledge and understanding of a two-state model for credit ratings with a constant transition intensity.
    Describe how the two-state model can be generalised to the Jarrow-Lando-Turnbull model for credit ratings.
    Describe how the two-state model can be generalised to incorporate a stochastic transition intensity.
    4. Option pricing and valuations
    Demonstrate an understanding of the Black–Scholes derivative-pricing model:
    Show how to use the Black–Scholes model in valuing options and solve simple examples.
    Discuss the validity of the assumptions underlying the Black–Scholes model.
    Describe and apply in simple models, including the binomial model and the Black–Scholes model, the approach to pricing using deflators and demonstrate its equivalence to the risk-neutral pricing approach.


  • Week 1

    In Week 1, we will discuss Wiener Process, Brownian Motion, and Geometric Brownian Motion.

    We will have a lecture on Tuesday (14:00-15:00) at B.R.:3.01, and a tutorial (15:00 - 16:00) at Maths: MB-204. On Thursday we will have a lecture (13:00 - 15:00) at Engineering: 216. If you can not attend in person, please watch the recording.

  • Week 2

  • Module Description

    • This module covers advanced techniques in financial mathematics for actuaries, building on the foundational material in Financial Mathematics 1.
      We revisit the discrete-time binomial model, introducing some more formal concepts such as conditional expectations that allow us to express our earlier results in a more elegant form.

      Then we look at continuous time models such as Black-Scholes model. The Black-Scholes equation is a partial differential equation, widely used by market practitioners, that models the price evolution of an option. Its solution, roughly speaking, provides the option price that eliminates risk when trading the option in a financial market without transactional costs.
      Using stochastic calculus techniques, we will derive the Black-Scholes equation and then solve it explicitly for the prices of European call and put options.

      We also consider some more advanced applications, such as models for stock prices involving jumps and stochastic volatility, as well as interest rate models and credit risk models.

  • Where to get help

    • There will undoubtedly be times during the term when you get stuck with revising the material or solving exercises. This is normal and an important part of the learning process. Nevertheless, if you feel overwhelmed or simply need some pointers to help you understand some of the concepts, there are various ways to reach out.

      Who to contact for what:

      • Student forum: it is encouraged that you discuss the material and exercises with your fellow students. You can post any kinds of questions in the student forum. The module organiser will also monitor any posts and answer queries.
      • Module organiser: is always willing to help either during the office hour or by email. Please contact us if you need any kind of more personal support/advice or have questions that you don't want to post in the forum. Your queries might be answered through the forum if they are of interest to all students in the class.
      • For problems beyond that: QMUL  counselling service   https://www.qmul.ac.uk/welfare/

  • Assessment information

    • The module will be assessed by:

      • Final exam in the May/June examination period: 70% weighting. All materials and courseworks discussed in the module is examinable unless otherwise stated. Please note that the exam will be an on-campus three-hour exam. You will be allowed a non-programmable calculator and 3 sheets of handwritten A4 notes  (= 6 sides) to bring to the exam. More details will follow.
      • Assessed coursework 1 to be completed in Week 7 : 15% weighting. You will complete an analysis with the software Excel.
      • Assessed coursework 2 to be completed in Week 12 : 15% weighting. You will complete an analysis with the software Excel.

      If you fail the module based on these assessments, you will have the opportunity to obtain a pass mark by resitting the final exam during the Late Summer Examination Period (typically August).

      You will find written feedback on the Assessed Coursework on QMPlus before the final exam. Please feel free to join the office hours or contact the module organisers if you want to discuss the result.

      Past exam papers can be found below.

  • Assessed Coursework 1

    Available: 11:00 am, 28 February (Wednesday) 2024

    Deadline: 12:00 noon, 6 March (Wednesday) 2024

  • Assessed Coursework 2

    Available: 11:00 am, 3 April (Wednesday) 2024

    Deadline: 12:00 noon, 10 April (Wednesday) 2024


  • Tips on preparing the 3-page notes for the semi-open-book format exams

    You are allowed 3 pages (double sided) A4 notes during the final exam. Below are some suggestions on how to prepare the notes:

    - Making the summary is a useful revision task in itself so do this yourself rather than using a friend's, and write it out rather than just printing existing material.
    - You don't need to condense the whole module onto 3 pages. Pick out the things you find hard to remember.
    - One approach is to attempt a past paper without any notes and see which places you felt a reminder would have been useful.
    - More guidance on how to prepare the 3-page notes will be / have already been covered during the lectures. Please refer to the QReview recordings.

  • General course materials

    • Add information here.

  • Past exam papers

  • Q-Review

  • Online Reading List

  • Week 3 & 4

    In Week 3 and 4, we try to answer: Why does the GBM describe the behaviour of the prices? We further discuss Black-Scholes model, and Greeks.

  • Week 5

    Dividends

  • Week 6 (& Assessed coursework 1)

    • Call-put parity
    • Indices and the diversification of risk
    • Volatility
    • Stochastic Calculus

    Please see Assesses coursework 1 under the tab "Assessment".

  • Week 7 - Reading Week, no lectures or tutorial

  • Week 8

    Stochastic process, Ito's Lemma, Ito's formula.

  • Week 9

    Interest rate term structure

  • Week 10

    Credit risk models

    • The Merton model
    • Two-state intensity-based model
    • THe Jarrow-Lando-Turnbull model

    Please see Assesses coursework 2 under the tab "Assessment".

  • Week 11 (& Assessed coursework 2)

    Advanced probability theory, conditional expectation, Martingales

  • Week 12 - Revision and industrial speakers from Aon