Handbook

You will use the concept of random variables and their uses as  models of uncertain future  payoffs. An important concept for analysis is the conditional expectation. Special attention is given to normal distribution and multivariate normal distribution, in which explicit calculations are possible. Systems evolving in time encorporating uncertainty are modelled as stochastic (random) processes. Examples of such in discrete time are Random Walk and Martingales. As an application we look at the Risk model in insurance and obtain the bound on Ruin probability. Models in continuous time are based on Brownian motion. Stochastic analysis uses the novel concepts of  Ito integral and Ito's formula. Applications in finance include the Black-Scholes model and the Ornstein-Uhlenbeck process. Simple stochastic differential equations are introduced. Another application to interest rates is Vasicek's stochastic differential equation. A new mathematical technique Change of probability measure is introduced. Girsanov theorem gives the change of measure for Brownian motion and related processes. To manage financial risks the  Fundamental theorems of Asset pricing are stated and applied to various models, such as the . Binomial and Black-Scholes models. Important concepts of arbitrage, replicating portfolios are used for  pricing and hedging options and other financial contracts.