Introductory description
This module is available for students on a course where it is a listed option (subject to restrictions*) and as an Unusual Option to students who have completed the prerequisite modules.
Prerequisites:
ST401 Stochastic Methods in Finance or ST403 Brownian Motion or ST908 Probability and Stochastic Processes (non Statistics students)
*Students who are not enrolled on the MSc in Mathematical Finance may take at most two of;
ST909 Application of Stochastic Calculus in Finance,
ST958 Advanced Trading Strategies,
ST420 Statistical Learning and Big Data.
Module web page
Module aims
To give a thorough understanding of how stochastic calculus is used in continuous time finance.
To develop an indepth understanding of models used for various asset classes.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
Option Pricing and Hedging in Continuous Time
 Pricing Europeans via equivalent martingale measures, numeraire, fundamental valuation formula, arbitrage and admissible strategies
 Pricing Europeans via PDEs (brief review)
 Completeness for the Black Scholes economy
 Implied volatility, market implied distributions, Dupire
 Stochastic volatility and incomplete markets
 Pricing a vanilla swaption, Black's formula for a PVBPdigital swaption
 Multicurrency Economy
 BlackScholes economy with dividends
 Economy with possibility of default CVA, DVA of a vanilla swap
Applications across Asset classes
Interest Rates: Term Structure Models  Short rate models. Introduction to main examples, implementation of HullWhite
 Market Models (Brace, Gaterek and Musiela approach), specification in terminal and spot measure
 Pricing callable interest rate derivatives with market models, drift approximation and
separability, implementation via LongstaffSchartz  Greeks via Monte Carlo for market models, pathwise method, likelihood ratio method.
 Markovfunctional models
 Practical issues in choice of model for various exotics, Bermudan swaptions
 Calibration: global versus local
 Stochastic volatility models, SABR
Credit  Description of main credit derivative products: CDS, Firsttodefault swaps, CDOs
 Extension of integration by parts, Ito's formula, Doleans exponential to cover jumps
 Martingale characterization of single jump processes, Girsanov's Theorem
 State variable, default and enlarged filtrations
 Filtration switching formula
 Intensitycorrelation versus defaultevents correlation
 Conditional Jump Diffusion approach to modelling of default correlation
FX  Stochastic local volatility models, calibration,
 Gyongy's Theorem
 Barrier options
Time permitting
Equity  Dividends
 Volatility as an asset class, variance swaps, volatility derivatives
 Heston model
Learning outcomes
By the end of the module, students should be able to:
 Demonstrate an advanced theoretical knowledge of the main models currently used across asset classes in the market, an appreciation of calibration and implementation issues concerning these models and a sufficient grounding in the tools of stochastic calculus to be able to keep abreast of new advances.
 Appreciate the practical issues in the implementation of models in the commercial setting and sufficient familiarity with the main models to enable implementation to be carried out.
 Critically assess the suitability of a particular model for a given product.
 Research new advances in modelling which is an important skill in the fast changing market setting.
 Carry out relevant calculations using knowledge of stochastic calculus when faced with implementing an unfamiliar model.
Indicative reading list
 Bergomi L (2016) Stochastic volatility modelling, Chapman and Hall
 Buehler H (2009) Volatility Markets: Consistent Modeling, Hedging and Practical Implementation of Variance Swap Market Models VDM Verlag Dr. Muller
 Elouerkhaoui, Y (2017), Credit Correlation: Theory and Practice, Macmillan.
 Hunt PJ and Kennedy JE, (2004), Financial Derivatives in Theory and Practice, second edition, Wiley.
 Homescu, C, Local Stochastic Volatility Models: Calibration and Pricing (2014)
 Available at SSRN: https:fissrn.com/abstract=2448098 or
htto://dx.doi.org/10.2139/ssrn.2448098  Pelsser A, (2000), Efficient Methods for Valuing Interest Rate Derivatives, Springer.
 Glasserman P, (2004), Monte Carlo Methods in Financial Engineering, Springer.
 Gatheral J, (2006) The Volatility Surface: A Practitioners Guide, Wiley
Subject specific skills
Demonstrate an advanced theoretical knowledge of the main models currently used across asset classes in the market, an appreciation of calibration and implementation issues concerning these models and a sufficient grounding in the tools of stochastic calculus to be able to keep abreast of new advances.
Appreciate the practical issues in the implementation of models in the commercial setting and sufficient familiarity with the main models to enable implementation to be carried out.
Critically assess the suitability of a particular model for a given product.
Research new advances in modelling which is an important skill in the fast changing market setting.
Carry out relevant calculations using knowledge of stochastic calculus when faced with implementing an unfamiliar model.
Transferable skills
TBC
Study time
Type 
Required 
Lectures 
30 sessions of 1 hour (20%)

Tutorials 
10 sessions of 1 hour (7%)

Private study 
110 hours (73%)

Total 
150 hours 
Private study description
Weekly revision of lecture notes and materials, wider reading, practice exercises and preparing for examination.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Students can register for this module without taking any assessment.
Assessment group D3

Weighting 
Study time 
Class Test 1

10%


This class test will take place during a lecture in week 8 of term 2.

Class Test 2

10%


This class test will take place during a lecture in week 10 of term 2.

Locally Timetabled Examination

80%


The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade.

Assessment group R1

Weighting 
Study time 
Locally Timetabled Examination  Resit

100%


Feedback on assessment
Feedback on class tests will be returned after 4 weeks, following each test.
Solutions and cohort level feedback will be returned for the examinations.
Examination scripts are retained for the external examiners and will not be returned to you.
Past exam papers for ST909
Courses
This module is Core for:

Year 1 of
TIBSN3G1 Postgraduate Taught Financial Mathematics
This module is Optional for:

USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

Year 3 of
G300 Mathematics, Operational Research, Statistics and Economics

Year 4 of
G300 Mathematics, Operational Research, Statistics and Economics
This module is Option list A for:

Year 4 of
USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

Year 5 of
USTAG301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated

Year 4 of
USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)

Year 5 of
USTAG1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)
This module is Option list B for:

Year 4 of
USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

Year 5 of
USTAG301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated
This module is Option list D for:

USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

Year 4 of
G30C Master of Maths, Op.Res, Stats & Economics (Operational Research and Statistics Stream)

Year 4 of
G30C Master of Maths, Op.Res, Stats & Economics (Operational Research and Statistics Stream)

Year 5 of
USTAG301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated
This module is Option list E for:

Year 4 of
USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

Year 5 of
USTAG301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated