Contents

1. Introduction 2

1.1. Reading List 2

2. The Geometric Brownian Motion Universe 4

2.1. Analytic Values of European Puts and Calls 7

2.2. The Black–Scholes Equation in log-space 9

2.3. Asian Options 10

2.4. Multivariate Geometric Brownian Motion 11

2.5. Gaussian Integrals on a wedge 13

3. The Binomial Model Universe 14

4. The Partial Differential Equation Approach 20

4.1. The Diffusion Equation 20

4.2. Finite Difference Methods for the Diffusion Equation 22

4.3. The Fourier Transform and the von Neumann Stability Test 29

4.4. Stability and the Fourier Transform 31

4.5. Option Pricing via the Fourier transform 33

4.6. Fourier Transform Conventions 35

5. Mathematical Background Material 37

5.1. Probability Theory 37

5.2. Differential Equations 39

5.3. Recurrence Relations 40

5.4. Mortgages – a once exotic instrument 43

5.5. Pricing Mortgages via lack of arbitrage 44

5.6. Exponential Growth and Population Rhetoric 45

6. Numerical Linear Algebra 46