Applied Probabilistic Calculus for Financial Engineering

2Probabilistic Calculus for Modeling Financial Engineering

2.1 Introduction to Financial Engineering

To establish useful and realistic mathematical models for financial analysis, with the objectives of assets allocation and the concomitant gains, or losses, it is useful to consider both discrete models and continuous models.

2.1.1 Some Classical Financial Data

Consider the following two typical sets of financial data:

1. Dow Jones – 100 year historical chart (Figure 2.1)

   Interactive chart of the Dow Jones Industrial Average (DJIA) stock market index for the last 100 years.

   This historical data is inflation-adjusted using the headline CPI and each data point represents the month-end closing value. The current month is updated on an hourly basis with today's latest value. The current value of the DJIA as of 08:48 p.m. EDT on June 3, 2016 is $17,838.56.

2. Apple, Inc., AAPL – 5-year historical prices (Figure 2.2)

Figure 2.1 Dow Jones Industrial Average (DJIA) –100 year historical chart.

Figure 2.2 Apple, Inc., AAPL (a 5-year historical prices).

2.2 Mathematical Modeling in Financial Engineering

2.2.1 A Discrete Model versus a Continuous Model

In general, a discrete model calls for analysis and solutions using finite difference techniques, whereas a continuous model, using partial ...