Mathematical Modeling in Finance with Stochastic Processes

4.4. THE ABSOLUTE EXCESS OF HEADS OVER TAILS 147

1. What is the probability of an excess of a fixed number of heads over
   tails or tails over heads at some fixed time in the coin-flipping game?


2. What is the probability that the number of heads exceeds the number
   of tails or the number of tails exceeds the number of heads by some
   fixed fraction of the number of tosses?

Using the Central Limit Theorem, we will be able to provide precise answers
to each question and then to apply the ideas to interesting questions in
gambling and finance.

The Half-Integer Correction to the Central Limit Theorem

Often when using the Central Limit Theorem to approximate a discrete dis-
tribution, especially the binomial distribution, we adopt thehalf-integer
correction, also called thecontinuity correction. The correction arises
because the binomial distribution has a discrete distribution while the stan-
dard normal distribution has a continuous distribution. For any integers
and real valueh with 0 ≤ h < 1 the binomial random variable Sn has
P[|Sn|≤s] =P[|Sn|≤s+h], yet the corresponding Central Limit Theorem
approximation with the standard normal cumulative distribution function,
P[|Z|≤(s+h)/

√

n] increases ashincreases from 0 to 1. It is customary to
takeh= 1/2 to interpolate the difference. This choice is also justified by
looking at the approximation of the binomial with the normal.
Symbolically, the half-integer correction to the Central Limit Theorem is

P[a≤Sn≤b]≈

∫((b+1/2)−np)/√npq

((a− 1 /2)−np)/√npq

1

√

2 π

exp(−u^2 /2)du

=P[((a− 1 /2)−np)/

√

npq≤Z≤((b+ 1/2)−np)/

√

npq]

for integersaandb.

The absolute excess of heads over tails

Consider the sequence of independent random variablesYiwhich take values
1 with probability 1/2 and−1 with probability 1/2. This is a mathematical
model of a fair coin flip game where a 1 results from “heads” on theith
coin toss and a−1 results from “tails”. LetHnandLnbe the number of
heads and tails respectively innflips. ThenTn=

∑n

i=1Yi=Hn−Lncounts