Financial Mathematics Lecture Notes
COURSE NOTES Financial Mathematics MTH3251 Modelling in Finance and Insurance ETC 3510.
Lecturer: Prof. Fima Klebaner School of Mathematical Sciences Monash University Semester 1, 2012
Contents
1 Lecture 1. Introduction. 1.1 Prices of stocks as functions of time . . . . . . . . . . . . . . . . . . 1.2 Simulated functions of time that look like stock prices . . . . . . . . 1.3 Example of modelling using Random Variables . . . . . . . . . . . . 2 Lecture 2. Random variables. Revision. 2.1 Random Variables and Their Distributions. General. . . . 2.2 Expected value= mean, Variance=( Standard Deviation)2 . 2.3 Common models. Specific Probability distributions. . . . . 2.4 Realization of Random Variables. Using Excel . . . . . . . 2.5 Expectation of a function of a random variable . . . . . . . 3 Lecture 3. Multivariate distributions. 3.1 Independence . . . . . . . . . . . . . . 3.2 Covariance and Correlation . . . . . . 3.3 Summary of Expectation, Variance and 3.4 Multivariate Normal distribution . . . . . . . . . . . . . . . . . Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 2 4 4 4 5 8 8 10 10 11 12 12 15 15 16 17 17 18 18 19 20 20 20 22 23 23 23 24 25 26
4 Lecture 4. Tools for distributions. Transforms 4.1 Moment generating functions . . . . . . . . . . 4.2 Characteristic functions . . . . . . . . . . . . . 4.3 Properties of transforms. . . . . . . . . . . . . . 4.4 Moments and moment generating functions. . .
5 Lecture 5. Conditional Expectation 5.1 Conditional Expectation and Conditional Distribution . . . . . . . . 5.2 Properties of Conditional Expectation . . . . . . . . . . . . . . . . 6 Lecture 6. Conditional Expectation as the best predictor 6.1 Expectation as best predictor . . . . . . . . . . . . . . . . . 6.2 Conditional Expectation, Best Possible
Lecturer: Prof. Fima Klebaner School of Mathematical Sciences Monash University Semester 1, 2012
Contents
1 Lecture 1. Introduction. 1.1 Prices of stocks as functions of time . . . . . . . . . . . . . . . . . . 1.2 Simulated functions of time that look like stock prices . . . . . . . . 1.3 Example of modelling using Random Variables . . . . . . . . . . . . 2 Lecture 2. Random variables. Revision. 2.1 Random Variables and Their Distributions. General. . . . 2.2 Expected value= mean, Variance=( Standard Deviation)2 . 2.3 Common models. Specific Probability distributions. . . . . 2.4 Realization of Random Variables. Using Excel . . . . . . . 2.5 Expectation of a function of a random variable . . . . . . . 3 Lecture 3. Multivariate distributions. 3.1 Independence . . . . . . . . . . . . . . 3.2 Covariance and Correlation . . . . . . 3.3 Summary of Expectation, Variance and 3.4 Multivariate Normal distribution . . . . . . . . . . . . . . . . . Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 2 4 4 4 5 8 8 10 10 11 12 12 15 15 16 17 17 18 18 19 20 20 20 22 23 23 23 24 25 26
4 Lecture 4. Tools for distributions. Transforms 4.1 Moment generating functions . . . . . . . . . . 4.2 Characteristic functions . . . . . . . . . . . . . 4.3 Properties of transforms. . . . . . . . . . . . . . 4.4 Moments and moment generating functions. . .
5 Lecture 5. Conditional Expectation 5.1 Conditional Expectation and Conditional Distribution . . . . . . . . 5.2 Properties of Conditional Expectation . . . . . . . . . . . . . . . . 6 Lecture 6. Conditional Expectation as the best predictor 6.1 Expectation as best predictor . . . . . . . . . . . . . . . . . 6.2 Conditional Expectation, Best Possible