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 BlackScholes option pricing model. After completing the course, you will be able to apply the stochastic and continuoustime 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 continuoustime 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 nonarbitrage in order to consider the pricing of financial derivatives;

equip you to use the binomial model and the BlackScholes 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 multivariable 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 nonarbitrage principle and risk neutral (martingale) pricing.

Derive a partial differential equation for the BlackScholes option pricing model, solve the BlackScholes 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 
TMA 1 
3 
Mathematical methods for the BlackScholes model 
2 

4 
Risk models in hedging 
2 
TMA 2 
5 
Quantitative methods for bond models and interest rate options 
3 

6 
Financial risk models in practice 
3 
TMA 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 tutormarked assignments (TMAs), of which the best two will count towards your final score. The course ends with a threehour 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 timeconsuming. 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 selftests
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, selftests 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 selftests 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 selftests that provide extra practice if you need it; these selftests may be a little more demanding than the majority of the exercises within the sections. You may regard these selftest 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 
Tutormarked 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.
Tutormarked assignments (TMAS)
There are three tutormarked assignments (TMAs) 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 TMAs. Upon receiving TMAs from the students, tutors will be required to mark the assignments and return them to students with their comments and feedback.
The TMAs 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 tutormarked assignments will be distributed as follows:

Coverage 
Weighting of the TMAs 
TMA 1 
This TMA covers Units 12. There will be 23 problemsolving exercises. 
15%
+
15%
(best 2 of 3 TMAs) 
TMA 2 
This TMA covers Unit 3. There will be 23 problemsolving exercises. 
TMA 3 
This TMA covers Units 4 5. There will be 23 problemsolving exercises. 
Examination
The threehour 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 problemsolving 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 p and Ö2 as they are, but you should simplify expressions such as sin (p/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 nonapproved 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 Keishing 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 parttime tutor of Mathematics at the Open University of Hong Kong since 2006 and is now a fulltime 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 Chiwang 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 parttime 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.
A.MAX
SC801 



SC802 



SC809 



SC813 
ATABA/AURORA
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 



SC170 



SC180 
SC200 



SC500 








BISTEC
BLT
BT206 



BT201612 



BT201812 



DC3088S/12 
DC3188S/12 



DC3388S/12 



DC408 



DC508 
SC183 












CANON
BS‑100 



BS102 



BS‑120 



BS 122 
BS123 



BS‑200 



BS‑300 



BS1200TS 
CB II BK/G 



CB III 



F‑45 



F‑65 
F‑73 /P 



F‑402 



F500 



F502 
F600 



F‑602 



F604 



F612 
F700 



F‑800 P 



F‑802 P 



FC4 S 
FC42 S 



FINANCIAL/II 



FS400 



FS600 
HS20H 



HS100 



HS102H 



HS120L 
HS1200RS/T/TV/TS 



KC20 



KS10 



KS‑20 
KS‑30 



KS‑80 



KS100 



KS102 
KS‑120 



KS122 



KS123 



L‑20 II W AD 
L‑30 II W AD 



L813 II 



L‑1011 



L‑1214II/AD 
L‑1218 



LC22 



LC‑23 



LC‑34 /T 
LC44 



LC‑63 



LC‑64 T 



LC101 
LC500H 



LC‑1016 



LC1222 



LC1620H 
LS‑8 



LS‑21 



LS25H II 



LS‑31 II 
LS32 



LS39H 



LS‑41 II 



LS‑42 
LS‑43 B/S 



LS51 



LS‑52 BK/W 



LS‑54 W 
LS‑61 



LS62 BK/W 



LS‑80/H 



LS81 Z 
LS‑82 H/Z 



LS88Hi/V 



LS‑100 II/H/TS 



LS102 Z 
LS120H/L/RS/V 



LS151 



LS‑500 



LS‑510 
LS‑550 G/B1 



LS‑552 



LS‑553 



LS‑560 
LS562 



LS‑563 



LS566H 



LS716H 
LS1000H 



LS1200H 



M‑10 



M‑20 
M30 



OS‑1200 



PS‑8 BK/W 



PS‑10BK/W 
SK100H 



T14BK/G/W 



T‑19 



TR10H 
TR1200H 



TS‑81/H 



TS‑83 



TS85H 
TS101 



TS‑103 



TS105H 



TS120TL 
TX1210Hi 



WS‑100 



WS‑120 



WS121H 
WS200H 



WS220H 



WS1200H 



WS1210Hi 
WS2222 



WS2224 



WS2226 




CASIO
AZ45F 



BF‑80 



BF‑100 



CV‑700 
D20A 



D20D/M 



D40D 



D‑100 W/L/LA 
D120 L/W/T/LA/TE 



DF10L 



DF20L 



DF120TE 
DJ120 



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 



FX55 



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 
FX500 /A 



FX‑550 /S 



FX‑570 A/C‑/V/D/S 



FX911S/SA 
FX‑991/M/N/V/D/H/S 



FX‑992 V/VB/S 



FX‑3400 P 



FX‑3600 P/V/A/PV 
FX3650P 



FX‑3800 P 



FX3900PV 



FX3950P 
HL‑100 L 



HL‑122/L 



HL‑812 /E/L 



HL820 A/LU/D 
HS4A 



HS‑8 G/L/LU/D 



HS‑9 



HS‑88 
HS‑90 



J10 A/D 



J‑20 



J‑30 C 
J‑100W/L/LA 



J‑120 L/W/T 



JE‑2 



JE‑3 
JF100/TE 



JF120TE 



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 



JS40 L/LA 



JS‑110 
JS‑120 



JS‑140 



LC401A 



LC‑403 C/E/L/LU/LB 
LC700 



LC710 



LC‑787 G/GU 



LC‑797 G/GU 
LC‑798 G 



LC‑1000 /L 



LC‑1210 



MC‑40 S 
MC‑801 S 



MJ‑20 



MJ120 



MS5A 
MS6 



MS‑7/LA 



MS‑8 W/A 



MS‑9 
MS10 W/L 



MS20W/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 



MS470 L 



NS‑3 



NS10L 
NS20L 



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 



SL200 



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 



SL787 
SL790L 



SL797 



SL805A 



SL‑807 /A/L/LU 
SL817 L 



SL850 



SL910L 



SL‑1000 M 
SL1200L 



SL‑1510 



SL1530T 



SL‑2000 M 
US‑20 



US‑100 



WD100L 



WD120L 
WJ‑10 



WJ‑20 



WJ100L 



WJ120L 
CITIZEN
CT500 



CT‑600 



ELS301 



ELS302 
ELS501 



F‑908 /N 



F‑920 



F940 N 
F‑950 



FT‑200 



LC‑505 



LC‑508 N 
LC‑510 N 



LC‑516 N 



LC‑531 



LC5001 
LH700 



LH830 



SB‑741 P 



SDC‑810 
SDC‑814 



SDC‑826 



SDC‑830 



SDC‑833 
SDC‑834 



SDC836 



SDC‑839 



SDC848 
SDC‑850 



SDC‑865 



SDC868 



SDC‑875 
SDC878 



SDC‑880 



SDC888 



SDC8001 
SDC8150 



SDC8360 



SDC8401 



SDC8460 
SDC8480 



SDC8780/L 



SDC8890 



SLD‑702 
SLD‑705 B 



SLD‑707 



SLD708 



SLD‑711 /N 
SLD‑712 /N 



SLD‑720 



SLD‑722 



SLD‑723 
SLD‑725 



SLD‑732 



SLD‑735 



SLD737 
SLD‑740 



SLD742 



SLD‑750 



SLD760 
SLD767 



SLD‑781 



SLD7001 



SLD7401 
SR‑30 



SR‑35 



SR‑70 



SR260 
SRP‑40 



SRP‑45 



SRP‑60 



SRP‑65 
SRP‑75 



SRP80 



SRP285II 




HEWLETT‑PACKARD
HP6S 



HP6S Solar 



HP9S 



HP‑10 B/BII 
HP‑11 C 



HP‑12 C 



HP‑15 C 



HP‑16 C 
HP‑20 S 



HP‑21 S 



HP30S 




KARCE
KC107 



KC117 



KC119 



KC121 
KC127 



KC153 



KC159 




SHARP
EL231C/L 



EL233G 



EL240C 



EL310A 
EL326L/S 



EL330A 



EL331A 



EL334H/A 
EL337M 



EL338A 



EL344G 



EL354L 
EL373 



EL376G 



EL386L 



EL387L 
EL480G 



EL501V 



EL506A/G/R/V 



EL‑509G/D/S/L/R/V 
EL‑520 D/G/L/R/V 



EL‑530 A 



EL531 GH/H/P/LH/RH/VH 



EL‑546D/G/L 
EL556G/L 



EL‑731 



EL733A 



EL771C 
EL782C 



EL792C 



EL879L 



EL2125 
EL2128H 



EL2135 



EL‑5020 




TEXAS INSTRUMENTS
BAIII 



BASOLAR 



BAII/PLUS 



BA35 
BUSINESSEDGE 



FINANCIALINVESTMENTANALYST 



MATH EXPL0RER 



TICOLLEGIATE 
TI25X SOLAR 



TI30 /Xa/Xa Solar/XIIB 



TI31 



TI32 
TI34 /II 



TI35 /X 



TI‑36 /X Solar 



TI‑60 
TI‑65 












TRULY
101 /A 



102 



103 



105 
106 



107 



P127 



SC106A 
SC107B/C/F 



SC108 



SC109 /X 



SC110 /X 
SC111 /X 



SC118 /A/B 



SC128 




[End of calculator list]