INTERMEDIATE MATHEMATICS:
Understanding Stochastic Calculus
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
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Prior Knowledge:
Delegates should have a good understanding of Elementary Probability Theory, Calculus and Linear Algebra (covered in Maths Refresher).
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.
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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
