Mathematical Modeling in Finance with Stochastic Processes

100 CHAPTER 3. FIRST STEP ANALYSIS FOR STOCHASTIC PROCESSES

Corollary 3.The probability of ultimate ruin of a gambler with initial capital
T 0 playing against an infinitely rich adversary is

qT 0 = 1, p≤q

and
qT 0 = (q/p)T^0 , p > q.

Proof.Leta→∞in the formulas. (Check it out!)

Remark.This corollary says that the probability of “breaking the bank at
Monte Carlo” as in the movies is zero, at least for the simple games we are
considering.

Some Calculations for Illustration

p q T 0 a Prob of Ruin Prob of Success Exp Gain Duration

0.5 0.5 9 10 0.1000 0.9000 0 9

0.5 0.5 90 100 0.1000 0.9000 0 900

0.5 0.5 900 1,000 0.1000 0.9000 0 90,000

0.5 0.5 950 1,000 0.0500 0.9500 0 47,500

0.5 0.5 8,000 10,000 0.2000 0.8000 0 16,000,000

0.45 0.55 9 10 0.2101 0.7899 -1 11

0.45 0.55 90 100 0.8656 0.1344 -77 766

0.45 0.55 99 100 0.1818 0.8182 -17 172

0.4 0.6 90 100 0.9827 0.0173 -88 441

0.4 0.6 99 100 0.3333 0.6667 -32 162

Why do we hear about people who actually win?

We often hear from people who consistently make their “goal”, or at least
win at the casino. How can this be in the face of the theorems above?
A simple illustration makes clear how this is possible. Assume for conve-
nience a gambler who repeatedly visits the casino, each time with a certain
amount of capital. His goal is to win 1/9 of his capital. That is, in units of
his initial capitalT 0 = 9, anda= 10. Assume too that the casino is fair so
thatp= 1/2 =q, then the probability of ruin in any one year is:

qT 0 = 1− 9 /10 = 1/ 10.