ST90815 Stochastic Calculus for Finance
Introductory description
This module runs in Term 1 and is core for students on the MSc in Mathematical Finance.
PhD students interested in taking the module should consult the module leader.
This module is not available to undergraduate students.
Module aims
This module provides a thorough introduction into discretetime martingale theory, Brownian motion, and stochastic calculus, illustrated by examples from Mathematical Finance.
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.
1 Conditional expectations
(a) Elementary conditional expectations
(b) Measuretheoretic conditional expectations
(c) Properties of conditional expectations
2 Martingale Theory
(a) Stochastic processes and filtrations
(b) Martingales, submartingales, and supermartingales
(c) Discrete stochastic integral
(d) Stopping times and stopping theorem
(e) Martingale convergence theorems
(1) Applications to Finance (option pricing in complete markets)
3 Markov Processes
(a) Markov processes and Markov property
(b) Strong Markov property
4) Brownian motion and continuous local martingales
(a) Definition and fundamental properties of Brownian
(b) Quadratic variation
(c) Continuous local martingales and semimartingales
5) Stochastic calculus
(a) Integration with respect to local martingales
(b) Finite variation processes and LebesgueStieljes integration
(c) Integration with respect to semimartingales
(d) Ito's formula
(e) Levy's characterisation of Brownian motion
(f) Stochastic exponentials and Novikov's condition
(g) Girsanov's theorem
(h) Ito representation theorem
(i) FeynmanKac formula
(j) Applications to Finance (Black Scholes model)
6) Stochastic differential equations
(a) Strong solutions and Lipschitztheory
(b) Examples (0Uprocesses, CIR processes, etc.)
Learning outcomes
By the end of the module, students should be able to:
 Explain and apply the concept of measuretheoretic conditional expectations
 Demonstrate an understanding of discrete time martingale theory and apply the theory to option pricing
 Understand the basic properties of Brownian motions
 Explain the main steps in the construction of the stochastic integral
 Be proficient in applying Ito's formula and Girsanov's theorem in problems arising in Mathematical Finance
 Solve standard SDEs appearing in Mathematical Finance
Indicative reading list
View reading list on Talis Aspire
Subject specific skills
Explain and apply the concept of measure theoretic conditional expectations
Show an understanding of martingales and the connection with gains from trade
Understand the Markov property and the strong Markov property and apply it to examples
Demonstrate the ability to perform calculations involving martingales and stochastic integrals
Be proficient in applying Ito's formula and Girsanov's theorem to problems in Mathematical finance
Demonstrate the ability to translate problems from mathematics to finance and viceversa
Transferable skills
Demonstrate problem solving skills involving concepts from the module
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 revising of lecture notes and materials, solving of problem sheets, and preparing for class tests and the final exam.
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 D1
Weighting  Study time  

Class test 1 (20 minute synchronous online assessment)  10%  
This class test takes place in the middle of the term during a lecture. 

Class test 2 (20 minute synchronous online assessment)  10%  
This class test takes place at the end of the term during a lecture. 

Online Examination  80%  
The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade. ~Platforms  Moodle

Feedback on assessment
Solutions and written cohort level feedback will be provided for the final exam. Oral cohort level feedback will be provided for the class tests.
Scripts are retained for external examiners and will not be returned to you.
Postrequisite modules
If you pass this module, you can take:
 ST95815 Advanced Trading Strategies
Courses
This module is Core for:
 Year 1 of TIBSN3G1 Postgraduate Taught Financial Mathematics