Intermediate Mathematics: Probability And Stochastic Processes
Course Outline: The use of Probability theory in financial modelling can be traced back to the work on Bachelier at the beginning of last century with advanced probabilistic methods being introduced for the first time by Black, Scholes and Merton in the seventies. The modern financial quantitative analysts make use of sophisticated mathematical concepts, such as martingales and stochastic integration, in order to describe the behaviour of the markets or to derive computing methods.
Who The Course is For:
Quantitative analysts, financial engineers, researchers, risk managers, structurers, market analysts and product controllers. Past participants have included: Chief investment officers, Asset Managers, Strategists, Private Banks, Relationship Managers
Prior Knowledge:
Delegates should have a good understanding of Elementary Probability Theory, Calculus and Linear Algebra (covered in Maths Refresher).
Course Programme*
Day One
Probability Theory
Random variables, independence and conditional independence. Discrete random variables: mass density, expectation and moments calculation
Conditional discrete distributions, sums of discrete random variables
Continuous random variables; Probability density function, cumulative probability density function; Expectation and moments calculation; Conditional distributions and conditional expectation; Functions of random variables
Examples: Normal distribution, gamma distribution, exponential distribution, Poisson distribution
Exercise: Properties of the gamma distribution and the log-normal distribution
Workshop: Multivariate normal distributions. Linear transformations. Counter-example
Generating functions. Moment generating functions. Characteristic functions
Convergence theorems: the strong law of large numbers, the central limit theorem
Examples: Characteristic functions of Bernoulli, binomial, exponential distributions
Exercise: Moment generating functions and characteristic functions of Poisson, normal and multivariate normal distributions
Markov Chains
Discrete time Markov chains, the Chapman-Kolmogorov equation
Recurrence and transience. Invariance
Discrete martingales. Martingale representation theorem. Convergence theorems
Examples: Random walks: simple, reflected, absorbed
Workshop: Pricing European options within the Cox-Ross-Rubinstein model
Continuous time Markov chains. Generators
Forward/backward equations. Generating functions
Examples: The Poisson process
Exercise: Superposition of Poisson Processes. Thinning
Day Two
Stochastic Calculus
The Wiener process. Path properties. Monte Carlo simulation
Gaussian processes. Diffusion processes
Examples: The Wiener process with drift. The Brownian Bridge
Exercise: The Geometric Brownian Motion. Properties of its distribution (moments)
Semi-martingales. Stochastic integration
Ito's formula. Integration by parts formula
Workshop: The Ornstein-Uhlenbeck process. Properties of its distribution (mean variance, covariance). Monte Carlo simulation
Stochastic Differential Equations
Stochastic differential equations. Existence and uniqueness of solutions. Equations with explicit solutions
The Markov property. Girsanov's theorem
Exercise: The Vasicek model. Connection with the O-U process. Mean. Variance. Covariance. Pricing zero-coupon bonds
Workshop: The Cox Ingersoll Ross Model. Connection with the O-U process. Properties of its distribution (mean variance, covariance). Pricing zero-coupon bonds
* Subject to change. Please consult provider for more information.
This program is eligible for 16 Continuing Education credit hours from the CFA Institute. If you are a CFA Institute member, CE credit for your participation in this program will be automatically recorded in your CE Diary.
