Stochastic processes play a key role in analytical finance and insurance, and in financial engineering. This course presents the basic models of stochastic processes such as  Markov chains, Poisson processes and Brownian motion. It provides an application of stochastic processes in finance and insurance. These topics are oriented towards applications of stochastic models in real-life situations.

Prerequisite Modules: MA2500 Foundations of Probability and Statistics

• Explain different models stochastic processes (random walk, Markov chains with discrete and continuous time, Brownian motion and Poisson process) and appreciate and use modern methods of stochastic processes for finance and insurance.
• Use the Cox-Ross-Rubinstein and Black-Scholes option pricing formulae in finance.
• Apply formulas for evaluation of various distributional parameters, prices for financial contracts, and other characteristics connected to stochastic processes.
• Appreciate and apply Markov chains to No Claim Discounting in Motor Insurance.
• Use the fundamental theorem in risk theory and the Cramer-Lundberg approximation.

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

Skills:

Mathematical formulation and modelling.

Option pricing using binomial and Black-Scholes models.

Solution of ruin problem and problems of the risk theory.

Transferable Skills:

Mathematical modelling of various types of problem in Finance.

An appreciation of the use of binomial model and Black-Scholes for option pricing.

Mathematical modelling of various types of problem in Insurance.

An appreciation of the use of compound Poisson process in risk business of insurance company.

• Stochastic processes.
  • Discrete and continuous time Markov models.
  • Martingales.  
  • The binomial no-arbitrage pricing model. 
  • The arbitrage-free price and hedging strategy. 
  • The risk-neutral probabilities. 
  • The Cox-Ross-Rubinstein binomial option pricing formula for puts and calls. 
  • Drift and volatility. 
  • Implementation of binomial trees. 
  • Continuous-time finance. 
  • The Brownian motion and geometric Brownian motion. 
  • The Black-Scholes formula. 
  • Volatility and stochastic volatility. 
  • Put-call parity. 
  • Exotic options. 
  • Hedging strategy.

• Markov chains and transition probabilities
  • The Chapman-Kolmogorov equations.
  • Transition matrix.
  • Transient behaviour.
  • Long-run behaviour.
  • Simple random walks.
  • Gambler`s ruin.
  • Applications to No Claim Discounting in motor insurance.

• Introduction to jump processes and applications.
  • Poisson process and compound Poisson process.
  • Risk theory. 
  • Risk processes.
  • Ruin problem.
  • Risk business of insurance company.
  • Cramer-Lundberg approximation.