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  • Quantitative Models for Financial Risk
    This Course Guide has been taken from the most recent presentation of the course. It would be useful for reference purposes but please note that there may be updates for the following presentation.

    MATH S390
    Quantitative Models for Financial Risk

    1. Introduction

    Welcome to MATH S390 Quantitative Models for Financial Risk. We hope that you will find your study of this course enjoyable and rewarding.

    This Course Guide contains information about the content and components of MATH S390 Quantitative Models for Financial Risk that you should be aware of before starting to study the course. Like most Open University courses, MATH S390 combines the study of written materials with other activities.

    2. What this course is about

    This course will help you to develop professional skills in using quantitative models, as well as their applications for analysis of financial risk and for critically evaluating the statistical and physical properties of Brownian motion in pricing stock options.

    MATH S390 starts with the basic principles and assumptions for options -- the concept of no arbitrage -- and teaches you the valuation of European and American options through the binomial model and the Black-Scholes option pricing model. After completing the course, you will be able to apply the stochastic and continuous-time models to compute the pricing of financial options and other derivative securities for the assessment of risk.

    2.1 Prerequisites

    You are advised to have studied a basic statistics course such as MATH S280 and a mathematics course such as MATH S221, or to have achieved an equivalent level of mathematical maturity before studying this course.

    2.2 Course aims

    MATH S390 Quantitative Models for Financial Risk aims to:

    • provide you with a basic understanding of various option pricing formulae, hedging techniques, bond models and interest rates;

    • introduce different types of financial risks and the common quantitative methods used to set up financial derivative models, and to analyse and interpret various problems and risks arising in financial engineering;

    • introduce basic statistical and mathematical theory, in particular the stochastic and continuous-time differentiation models required for computing the pricing of financial options and other derivative securities for the assessment of risk;

    • enhance your ability to elaborate on the assumptions for developing options models, and to equip you with the ideas of forwards and options and the concept of non-arbitrage in order to consider the pricing of financial derivatives;

    • equip you to use the binomial model and the Black-Scholes option pricing model to interpret the valuation of European and American options;

    • develop your professional skills in stochastic calculus and its applications for risk analysis and finance;

    • teach you the properties of Brownian motion and how to apply them to evaluate the price of stock options problems; and

    • develop quantitative financial risk models through the multi-variable calculus and differential equations.

    2.3 Course learning outcomes

    Upon completing MATH S390, you should be able to:

    • Define and explain basic terminology in the financial risk and quantitative process.

    • Determine the option value involving one or more sets of cash flows at specified times, and apply the binomial tree model to price options.

    • Apply the stochastic calculus for modelling Brownian motion, and use Ito’s Lemma to derive the Stochastic Differential Equations.

    • Apply the non-arbitrage principle and risk neutral (martingale) pricing.

    • Derive a partial differential equation for the Black-Scholes option pricing model, solve the Black-Scholes differential equation, and interpret the computational results.

    • Construct a hedge for a variety of risk positions, derive and apply the Delta Hedging Equation, and develop numerical methods for pricing financial products and hedging strategies for a portfolio of options. 

    • Perform calculations of yields, continuous compounding, and par, forward and zero yield curves.

    2.4 Course organization

    Units Weeks Assessment
    1 Introduction to financial risk and quantitative process 2  
    2 Tree models for stocks and options 3 Assignment 1
    3 Mathematical methods for the Black-Scholes model 2  
    4 Risk models in hedging 2 Assignment 2
    5 Quantitative methods for bond models and interest rate options 3  
    6 Financial risk models in practice 3 Assignment 3
    Revision 1  
    3. Components of the course

    A typical fortnight’s work will consist mainly of reading a course unit and working through the exercises that it contains.

    Each unit has continuous assessment associated with it. There are three assignments, of which the best two will count towards your final score. The course ends with a three-hour examination.

    3.1 Course units

    There are six course units. Each unit is divided into four to seven sections. A typical section might be studied in a single session (in an evening, for example). Each unit begins with an introductory section that asks you to recall what you learned in a prerequisite course or from previous units as you begin to study the current one, and also gives advice on which sections may be most or least time-consuming. Towards the end of the unit, you will find a short Summary section that reminds you of the operations you should be able to do as a result of having studied the unit. This will help ensure that you have mastered the contents of one unit before moving on to the next.

    Examples and self-tests

    The course units contain various types of question for you to work through as part of your study.

    First, there are examples, which show you how to carry out some technique or method. The solution to an example is given in the text immediately below it; your task is to read and to follow the workings of the problem in order to learn how to apply that technique yourself.

    Next, self-tests are designed to give you practice in achieving what the preceding text has taught you. You should attempt to solve these by yourself, consulting the solutions (which are given at the back of the unit) only in order to check that your own answer is correct. If you are stuck, look at the solution as a last resort, but look at just enough of it to see how to proceed, before returning to complete the rest of the solution of the problem for yourself. It cannot be emphasized too strongly that doing self-tests in this way is an essential part of studying mathematics; nobody learns much mathematics just by reading texts.

    You should try each example within a section as you come to it. At the end of most sections you will find additional self-tests that provide extra practice if you need it; these self-tests may be a little more demanding than the majority of the exercises within the sections. You may regard these self-test exercises as optional, but it is recommended that you do as many of them as you can find the time for.

    3.2 MATH S390 Handbook

    This is designed as a work of reference, and provides a convenient source of basic definitions and formulas for use throughout the course. In addition, you will be given a handbook in the examination room.

    The Handbook has two main components: a collection of useful formulas and definitions, many of which you will have come across already in prerequisite courses, and summaries of the main concepts, definitions and techniques in each of the course units. Two pages at the back of the Handbook summarize particular formulas from the course which need to be called upon regularly, for rapid reference.

    It is a good idea to start using the Handbook right from the beginning of the course so that you become familiar with its contents.

    3.3 Stop press notices

    The stop presses act as a course newsletter containing useful and often essential information such as errata and details of tutorial arrangements. It is important that you read each stop press as soon as you receive it. These notices are also posted in the Online Learning Environment (OLE).

    3.4 The Online Learning Environment (OLE)

    The Online Learning Environment User Guide (http://ole.ouhk.edu.hk/help.html) explains to you the hardware and software requirements for you to access the course electronically. It also helps you to use the components in the OLE. Through the OLE, you can get more information on the course and communicate with other students and tutors of the course.

    3.5 Academic Timetable

    This gives the starting date for each unit, the dates when assignments are due and weekends when tutorials are scheduled.

    3.6 Tutorials

    Dates for tutorials are given in the Presentation Schedule. Other details, such as tutorial venues and exact timing, will be given to you through emails and the OLE. Attend tutorials to meet your tutor and the other students on the course. Be active in sharing your views in tutorials.

    3.7 Assessment

    The course has both a continuous assessment and examination assessment component.

    The distribution of marks on these assessments towards the overall course score is set out in the following table.

    Assessment type Weighting of the course score
    Tutor-marked assignments 30%
    Final Examination 70%
    Total 100%

    You will be awarded the full five credits for MATH S390 if you can get at least 40 marks on both the OCAS and the examination. Read the Student Handbook for information on the awarding of course results.


    There are three assignments for the course, of which the best two results will count towards your final score. Since these assignments are important for you to secure the concepts you have learned in the study units, you will be required to submit all three of the assignments. Upon receiving assignments from the students, tutors will be required to mark the assignments and return them to students with their comments and feedback.

    The assignments will require you to:

    • perform the derivation of a continuous model;
    • apply a theorem and solve the given model;
    • analyse case studies in order to value a option; and
    • complete a computer project using Excel.

    The marks for the best two assignments will be distributed as follows:

      Coverage Weighting of the assignments
    Assignment 1 This assignment covers Units 1-2. There will be 2-3 problem-solving exercises. 15%
    (best 2 of 3 assignments)
    Assignment 2 This assignment covers Unit 3. There will be 2-3 problem-solving exercises.
    Assignment 3 This assignment covers Units 4-5. There will be 2-3 problem-solving exercises.


    The three-hour final examination for MATH S390 will be ‘closed book’, with the exception of the course Handbook. The examination is worth 70% of the total marks for the course. The exam paper will be divided into two parts:

    • Part I will contain some short questions that assess your general knowledge of the course material from all units.
    • Part II will comprise more challenging long questions based on a problem-solving approach. The questions will assess your skills in quantitative modeling; in applying certain models to measure the financial risk problems; and in concluding results for recommendation.

    Questions in assignments and in the examination carry both accuracy marks and method marks. You should therefore, as a general practice, show all your work to solve each problem.

    We expect you to leave numbers like π andas they are, but you should simplify expressions such as sin (π/2). If you need to use decimal fractions at any time, two decimal places will normally be sufficient.

    3.8 Calculators and mathematics software

    You can use a calculator, mathematics software such as Scientific Notebook, or spreadsheet software such as Microsoft Excel to evaluate mathematical functions such as exponential, logarithmic, trigonometric (and their inverses) and hyperbolic (and their inverses) functions when you study the course.

    Calculators are allowed in the examination. The University has a List of Approved Models of Calculators so that students realize what types of calculators are allowed in the examination. This List is updated according to the types of calculators approved by the Hong Kong Examinations and Assessment Authority. You will receive the List from Registry before the examination.

    You are not allowed to use a non-approved calculator or a calculator without the ‘HKEA/HKEAA Approved’ label in the examination. For your early information, a copy of the approved calculator list is attached at the back of this guide.

    4. Developer profiles

    Professor James Caldwell obtained his BSc(Hons) and MSc Degrees from the Queen’s University of Belfast in 1964 and 1966, respectively. In 1974 he obtained his PhD in Applied Mathematics from Teesside University. He took up various teaching posts in the UK before moving to Australia as Head of Mathematics at the University of Southern Queensland. He then returned to teach at Sunderland University and worked in lecturing and research posts at a number of UK universities. In 1990 he joined the City University of Hong Kong as Adjunct Professor in the Department of Mathematics.

    Professor Caldwell was awarded his first higher doctorate (DSc) from Queen’s University of Belfast in 1985, and his second DSc from Teesside University in 2007 in recognition of his scholarly work in Mathematical Modelling.

    In parallel with his academic career, Professor Caldwell worked for a number of large organizations, including as Head of Modelling for Unilever Research UK. Through his research work, he has published hundreds of articles and conference papers, and more than a dozen textbooks and theses. As a result of scientific publications, he has had extensive experience in editorial work involving Mathematics. Furthermore, he has had extensive experience in course development work in Mathematics at a number of universities.

    Dr Douglas Kei-shing Ng received his BSc (1st class Hons) and MPhil degrees in Applied Mathematics from the City University of Hong Kong in 2001 and 2003, respectively. Between 2003 and 2006, he was a teacher in a secondary school. He joined the Hong Kong Polytechnic University in 2006 to undertake his PhD research on medical informatics. He has been a part-time tutor of Mathematics at the Open University of Hong Kong since 2006 and is now a full-time Lecturer for the OUHK. His research interests include Nonlinear Partial Differential Equations, Mathematical Modelling, Computer Aided Detection Methods for Cerebrovascular Diseases, and Meshless Computational Methods.

    Dr Chi-wang CHAN obtained his BSc (1st class Hons) and MPhil degrees in Physics from the Chinese University of Hong Kong in 2000 and 2003, respectively. From 2002 to 2003, he was a Physics and Mathematics subject teacher in a secondary school. From 2003 to 2009, he went to New York University for his PhD studies in Statistical Physics, with specialization in Transports and Brownian motion in Confined Systems. He has been a part-time tutor of Mathematics at the Open University of Hong Kong since 2010. His academic interests are in the fields of Stochastic Process, Quantitative Finance, Astrophysics and Science education.

    5. List of approved calculators for examinations

    An updated list will be sent to you before the examination.

    In addition to the following models, calculators bearing the 'HKEA/HKEAA Approved' labels are also allowed.


    SC-801       SC-802       SC-809       SC-813


    AC‑688       AC‑689       AC‑690       AC‑692
    AC‑693       AC‑694       AT‑1       AT‑105
    AT‑106 A       AT‑108 A       AT‑168       AT‑208 N/B
    AT‑231 A/B/C/D       AT‑232 /S       AT‑233       AT‑241 T
    AT‑244 H       AT‑256 H       AT‑268       AT‑281 /S
    AT‑282       AT‑283       AT‑368       AT‑508
    AT‑510       AT‑512       AT‑518       AT‑520
    AT‑522       AT‑601 A       AT‑620 A       AT‑630
    AT‑687       AT‑2129 A/B       AT‑6120       AT‑6320
    AT‑9300       BD‑1       BD‑2       D‑8 /N
    D‑10 /N       D‑12 N       SC-170       SC-180
    SC-200       SC-500                


    B300       B500       B600       B700


    BT-206       BT-2016-12       BT-2018-12       DC-308-8S/12
    DC-318-8S/12       DC-338-8S/12       DC-408       DC-508


    BS‑100       BS-102       BS‑120       BS 122
    BS-123       BS‑200       BS‑300       BS-1200TS
    CB II BK/G       CB III       F‑45       F‑65
    F‑73 /P       F‑402       F-500       F-502
    F-600       F‑602       F-604       F-612
    F-700       F‑800 P       F‑802 P       FC-4 S
    FC-42 S       FINANCIAL/II       FS-400       FS-600
    HS-20H       HS-100       HS-102H       HS-120L
    HS-1200RS/T/TV/TS       KC-20       KS-10       KS‑20
    KS‑30       KS‑80       KS-100       KS-102
    KS‑120       KS-122       KS-123       L‑20 II W AD
    L‑30 II W AD       L-813 II       L‑1011       L‑1214II/AD
    L‑1218       LC-22       LC‑23       LC‑34 /T
    LC-44       LC‑63       LC‑64 T       LC-101
    LC-500H       LC‑1016       LC-1222       LC-1620H
    LS‑8       LS‑21       LS-25H II       LS‑31 II
    LS-32       LS-39H       LS‑41 II       LS‑42
    LS‑43 B/S       LS-51       LS‑52 BK/W       LS‑54 W
    LS‑61       LS-62 BK/W       LS‑80/H       LS-81 Z
    LS‑82 H/Z       LS-88Hi/V       LS‑100 II/H/TS       LS-102 Z
    LS-120H/L/RS/V       LS-151       LS‑500       LS‑510
    LS‑550 G/B1       LS‑552       LS‑553       LS‑560
    LS-562       LS‑563       LS-566H       LS-716H
    LS-1000H       LS-1200H       M‑10       M‑20
    M-30       OS‑1200       PS‑8 BK/W       PS‑10BK/W
    SK-100H       T-14BK/G/W       T‑19       TR-10H
    TR-1200H       TS‑81/H       TS‑83       TS-85H
    TS-101       TS‑103       TS-105H       TS-120TL
    TX-1210Hi       WS‑100       WS‑120       WS-121H
    WS-200H       WS-220H       WS-1200H       WS-1210Hi
    WS-2222       WS-2224       WS-2226        


    AZ-45F       BF‑80       BF‑100       CV‑700
    D-20A       D-20D/M       D-40D       D‑100 W/L/LA
    D-120 L/W/T/LA/TE       DF-10L       DF-20L       DF-120TE
    DJ-120       DN‑10       DN‑20       DN‑40
    DS‑1 B/L       DS‑2 B/L       DS‑3/L       DS‑8 E
    DS‑10E/L/G       DS‑20 E/L/G       DS‑120       FC‑100
    FN‑10       FN‑20       FX‑8       FX‑10 F
    FX‑39       FX‑50 F       FX-55       FX‑61 F
    FX‑68 /B       FX‑78       FX‑82/B/C/D/L/LB/SUPER/SX/W       FX‑85 /M/N/V
    FX‑100/A/B/C/V/D       FX‑115 /M/N/V/D       FX‑120       FX‑135
    FX‑140       FX‑210       FX‑350/A/C/D/H/HA/W       FX‑451 M
    FX-500 /A       FX‑550 /S       FX‑570 A-/C‑/V/D/S       FX-911S/SA
    FX‑991/M/N/V/D/H/S       FX‑992 V/VB/S       FX‑3400 P       FX‑3600 P/V/A/PV
    FX-3650P       FX‑3800 P       FX-3900PV       FX-3950P
    HL‑100 L       HL‑122/L       HL‑812 /E/L       HL-820 A/LU/D
    HS-4A       HS‑8 G/L/LU/D       HS‑9       HS‑88
    HS‑90       J-10 A/D       J‑20       J‑30 C
    J‑100W/L/LA       J‑120 L/W/T       JE‑2       JE‑3
    JF-100/TE       JF-120TE       JL‑210       JN‑10
    JN‑20       JN‑40       JS‑8 C       JS‑10 /C/M/L/LA
    JS‑20/C/M/L/LA       JS‑25       JS-40 L/LA       JS‑110
    JS‑120       JS‑140       LC-401A       LC‑403 C/E/L/LU/LB
    LC-700       LC-710       LC‑787 G/GU       LC‑797 G/GU
    LC‑798 G       LC‑1000 /L       LC‑1210       MC‑40 S
    MC‑801 S       MJ‑20       MJ-120       MS-5A
    MS-6       MS‑7/LA       MS‑8 W/A       MS‑9
    MS-10 W/L       MS-20W/TE       MS‑70 L       MS‑100 A/TE/V
    MS‑120 A/TE/V       MS‑140 A       MS‑170 L/LA       MS180
    MS‑270 L/LA       MS-470 L       NS‑3       NS-10L
    NS-20L       RC‑770       S‑1       S‑2
    S‑20 L       SJ‑20       SL‑80 E       SL‑100 A/B
    SL‑110 A/B       SL‑120 A/B       SL-200       SL‑210
    SL‑220       SL‑240/L       SL‑300H/J/L/LH/LU/LB       SL‑310 M
    SL‑330       SL‑350       SL‑450       SL‑510 /A
    SL‑704       SL‑720 /L       SL‑760 A/C/LU/LB       SL-787
    SL-790L       SL-797       SL-805A       SL‑807 /A/L/LU
    SL-817 L       SL-850       SL-910L       SL‑1000 M
    SL-1200L       SL‑1510       SL-1530T       SL‑2000 M
    US‑20       US‑100       WD-100L       WD-120L
    WJ‑10       WJ‑20       WJ-100L       WJ120L


    CT-500       CT‑600       ELS-301       ELS-302
    ELS-501       F‑908 /N       F‑920       F-940 N
    F‑950       FT‑200       LC‑505       LC‑508 N
    LC‑510 N       LC‑516 N       LC‑531       LC-5001
    LH-700       LH-830       SB‑741 P       SDC‑810
    SDC‑814       SDC‑826       SDC‑830       SDC‑833
    SDC‑834       SDC-836       SDC‑839       SDC-848
    SDC‑850       SDC‑865       SDC-868       SDC‑875
    SDC-878       SDC‑880       SDC-888       SDC-8001
    SDC-8150       SDC-8360       SDC-8401       SDC-8460
    SDC-8480       SDC-8780/L       SDC-8890       SLD‑702
    SLD‑705 B       SLD‑707       SLD-708       SLD‑711 /N
    SLD‑712 /N       SLD‑720       SLD‑722       SLD‑723
    SLD‑725       SLD‑732       SLD‑735       SLD-737
    SLD‑740       SLD-742       SLD‑750       SLD-760
    SLD-767       SLD‑781       SLD-7001       SLD-7401
    SR‑30       SR‑35       SR‑70       SR-260
    SRP‑40       SRP‑45       SRP‑60       SRP‑65
    SRP‑75       SRP-80       SRP-285II        


    HP-6S       HP-6S Solar       HP-9S       HP‑10 B/BII
    HP‑11 C       HP‑12 C       HP‑15 C       HP‑16 C
    HP‑20 S       HP‑21 S       HP-30S        


    KC-107       KC-117       KC-119       KC121
    KC127       KC-153       KC159        


    EL-231C/L       EL-233G       EL-240C       EL-310A
    EL-326L/S       EL-330A       EL331A       EL-334H/A
    EL-337M       EL338A       EL-344G       EL-354L
    EL-373       EL376G       EL386L       EL387L
    EL-480G       EL-501V       EL-506A/G/R/V       EL‑509G/D/S/L/R/V
    EL‑520 D/G/L/R/V       EL‑530 A       EL-531 GH/H/P/LH/RH/VH       EL‑546D/G/L
    EL-556G/L       EL‑731       EL-733A       EL-771C
    EL-782C       EL-792C       EL-879L       EL-2125
    EL-2128H       EL-2135       EL‑5020        


    BA-III       BA-SOLAR       BA-II/PLUS       BA-35
    TI-25X SOLAR       TI-30 /Xa/Xa Solar/XIIB       TI-31       TI-32
    TI-34 /II       TI-35 /X       TI‑36 /X Solar       TI‑60


    101 /A       102       103       105
    106       107       P-127       SC-106A
    SC-107B/C/F       SC-108       SC-109 /X       SC-110 /X
    SC-111 /X       SC-118 /A/B       SC-128        

    [End of calculator list]

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