Mathematical Modeling in Finance with Stochastic Processes

3 First Step Analysis for Stochastic Processes

2. We callruinthe state of going broke before reaching a fortune goal.

Mathematical Ideas

Reasons for Modeling with a Coin Flipping Game

We need a better understanding of the paths that risky securities take. We

shall make and investigate a greatly simplified model. For our model, we

assume:

1. Time is discrete, occurring att 0 = 0,t 1 ,t 2 ,....


2. There are no risk-free investments available, (i.e. no bonds on the mar-
   ket).


3. There are no options, and no financial derivatives.


4. The only investments are risk-only, that is, our fortune at any time is
   a random variable:
   Tn+1=Tn+Yn+1
   whereT 0 is our given initial fortune, and for simplicity,

Yn=

{

+1 probability 1/ 2

−1 probability 1/ 2

Our model is commonly called “gambling” and we will investigate the prob-

abilities of making a fortune by gambling.

Some Humor

An Experiment

1. Each person should have a chart for recording the outcomes of each
   game (see below) and a sheet of graph paper.


2. Each person should use a fair coin to flip, say a penny.


3. Each “gambler” flips the coin, and records a +1 (gains $1) if the coin
   comes up “Heads” and records −1 (loses $1) if the coin comes up
   “Tails”. On the chart, the player records the outcome of each flip