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.

54 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 100 hours private study including preparation of solutions for selected exercises.

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.

Formative assessment is carried out by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during examples lectures.  

The in-course element of summative assessment consists of assessed exercises (some to be attempted at home, some in examination conditions) similar in form to the tutorial exercises. This allows students to demonstrate a level of knowledge and skills appropriate to that stage in the module.

The major component of summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of four from five equally weighted questions.

• 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.

Quantitative Modelling of Derivative Securities, Avellaneda, M., & Laurence, P., Chapman and Hall, London, 2000

Elementary Stochastic Calculus with Finance in View, Mikosch, T., World Scientific, 1998

Futures, Options and Swaps, Kolb, R. W., Blackwell Publishing, 2003

Models and Analysis of Stochastic Systems, Kalkukarni, V. G., Chapman and Hall, 1995

Insurance Risk and Ruin, D. C. M. Dickson, Cambridge University Press, 2005

Ruin Probabilities, Asmussen, S., & Albrecher, H., World Scientific, 2010

Modern Actuarial Risk Theory, Kaas, R., Goovaerts, M., Dhaene, J., & Denuit, M., Springer, 2008

Not applicable.