Course: B8.3 Mathematical Models of Financial Derivatives (2025-26)

Lecturer: Profile: Sam Howison

Course information

General prerequisites:

B8.1 Probability, Measure and Martingales (previously named Martingales Through Measure Theory) would be good background. Part A Probability is a prerequisite. Part A Integration is also good background, though not a prerequisite.

Course term: Hilary

Course lecture information: 16 lectures

Course weight: 1

Course level: H

Assessment type: Written Examination

Course overview:

The course aims to introduce students to derivative security valuation in financial markets. At the end of the course the student should be able to formulate a model for an asset price and then determine the prices of a range of derivatives based on the underlying asset using arbitrage free pricing ideas.

Learning outcomes:

Students will have a familiarity with the mathematics behind the models and analytical tools used in Mathematical Finance. This includes being able to formulate a model for an asset price and then determining the prices of a range of derivatives based on the underlying asset using arbitrage free pricing ideas.

Course synopsis:

Overview of asset types, financial markets (including Limit Order Books) and derivative contracts
Discounting, arbitrage, and other basic financial arguments
Discrete-time models (binomial trees)
Introduction to Brownian motion and stochastic calculus; Ito's Lemma; conditional expectations and the Feynman-Kac formula
Hedging in continuous times and the Black-Scholes model
European-style option valuation: the Black-Scholes formulae as discounted expectations and as solutions of the Black-Scholes PDE. More general payoffs
American-style options
Mistage, barrier and other exotic options
Hedging and the Greeks: implied volatility. Extensions to the Black-Scholes model

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Section outline

Select section General

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General

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- Select activity Announcements
  
  Announcements Forum
- Select activity Reading List
  
  Reading List URL
- Select activity Discussion Forum
  
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Course Materials
- Select activity B8.3 Problem Sheet 1
  
  B8.3 Problem Sheet 1 Assignment
- Select activity B8.3 Problem Sheet 2
  
  B8.3 Problem Sheet 2 Assignment
- Select activity B8.3 Problem Sheet 3
  
  B8.3 Problem Sheet 3 Assignment
- Select activity B8.3 Problem Sheet 4
  
  B8.3 Problem Sheet 4 Assignment
- Select activity B8.3 Slides from lectures
  
  B8.3 Slides from lectures Assignment
  
  The dark-mode slides as used in lectures. NOT SUITABLE FOR PRINTING. And the same in normal light-mode. Take your pick.
- Select activity Printable version of the slides
  
  Printable version of the slides Assignment
  
  Printable version of slides, three landscpe slides per A4 portrait side. Some images may have dark backgrounds. (The slides used in lectures will have dark backgrounds.)
- Select activity MMSC special topic suggestions
  
  MMSC special topic suggestions File
- Select activity Slides from 23/24
  
  Slides from 23/24 File
  
  Slides from 23/24 (as the course was given by Alvaro Cartea, they take a slightly different approach in places; hence only for background/alternative perspectives).
- Select activity Notes
  
  Notes File
  
  Lecture notes from 2023/24 by Alvaro Cartea. They are close to the course but may take a different approach in places.
- Select activity Barchart.com S&P 500 options
  
  Barchart.com S&P 500 options URL
  
  The website shows a lot of option prices: have a look around to familiarise yourself with them.
- Select activity Binance live BTC LOB
  
  Binance live BTC LOB URL
  
  Live evolution of the Bitcoin price as seen in Lecture 1. Note the order book and trade list feeds.
- Select activity Hyperliquid blockckain-based exchange
  
  Hyperliquid blockckain-based exchange URL
Select section Intercollegiate Classes

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Intercollegiate Classes
- Select activity Class Signup
  
  Class Signup Class registration
  
  Registration start: Friday, 16 January 2026, 12:00 PM
  
  Registration end: Friday, 13 February 2026, 12:00 PM
- Select activity Class Tutor's Comments
  
  Class Tutor's Comments Assignment
  
  Class tutors will use this activity to provide overall feedback to students at the end of the course.
  
  Not available unless: You have signed up to a class

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Intercollegiate Classes