Stochastic Calculus

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.

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.

This course bridges the gap between mathematical theory and financial practice by providing a hands-on approach to probability theory, Markov chains and stochastic calculus. Participants will practice all relevant concepts through a batch of Excel based exercises and workshops.

• 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

Delegates should have a good understanding of Elementary Probability Theory, Calculus and Linear Algebra (covered in Maths Refresher).

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

Example: The Poisson process

Exercise: Superposition of Poisson Processes. Thinning

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: Multivariate normal distributions. Linear transformations. Counter-example

Examples: Characteristic functions of Bernoulli, binomial, exponential distributions

Exercises: Moment generating functions and characteristic functions of Poisson, normal and multivariate normal distributions

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