Mathematical Theory of Option Pricing - 2024 entry

On this module, you will be expected to study stochastic models of finance, including the Black Scholes option pricing model. You will have the opportunity to study numerical methods in order to solve partial differential equations. The module applies the mathematical and computational material from complementary modules to a central problem in finance - that of option pricing.

Pre-requisite modules: MTH3024 Stochastic Processes, or MTHM002 Methods for Stochastics and Finance

By taking this module, you will gain an understanding of the theoretical assumptions on which the mathematical models underlying option pricing depend, and of the methods used to obtain analytic or numerical solutions to a variety of option pricing problems.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 Comprehend the mathematical theories needed to set up the Black-Scholes model;

2 Understand the role of Ito's calculus in the Black-Scholes PDE;

3 Transform the Black-Scholes PDE to the heat diffusion equation;

4 Analyse and derive the solution of the Black-Scholes PDE for the standard European put/call options;

Discipline Specific Skills and Knowledge:

5 Derive rigorously a quantitative model from a set of basic assumptions;

6 Use the solution to the mathematical model to predict the behaviour of the option price;

7 Relate and transform one PDE to a simpler type;

Personal and Key Transferable/ Employment Skills and Knowledge:

8 Demonstrate enhanced problem solving skills and the ability to apply numerical methods coded in MATLAB or Python.

- Financial concepts and assumptions: risk free and risky asset;

- Options;

- No arbitrage principle;

- Put-call parity;

- Discrete-time asset price model: option pricing by binomial method;

- Interpretation in terms of risk-neutral valuation;

- Continuous time stochastic processes: Brownian motion, stochastic calculus;

- Ito’s lemma and construction of the Ito integral;

- Black-Scholes theory: geometric Brownian motion;

- Derivation of the Black-Scholes PDE; transformation to the heat equation;

- Explicit formulae for vanilla European call and put options;

- Extensions, e.g. to dividend-paying assets and American options;

- Numerical methods: finite difference schemes for the Black-Scholes PDE, including comparison of stability and accuracy;

- Overview of risk-neutral valuation approach in continuous time.

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Reading list for this module: