Frequently Asked Questions In Quantitative Finance

Chapter 3: The Most Popular Probability Distributions 235

Variance

a.

Chi square Bounded below and unbounded above. It has

two parametersa≥0, the location;ν, an integer, the

degrees of freedom. Its probability density function is

given by

e−(x+a)/^2

2 ν/^2

∑∞

i= 0

xi−^1 +ν/^2 ai

22 ij!(i+ν/2)

x≥0,

where(·) is the Gamma function. The chi-square distri-

bution comes from adding up the squares ofνnormally

distributed random variables. The chi-square distribu-

tion with one degree of freedom is the distribution of

the hedging error from an option that is hedged only

discretely. It is therefore a very important distribution

in option practice, if not option theory.

Chi Square

0

0.05

0.1

0.15

0.2

0.25

0.3

00.511.522.533.5 4

a = 0

b = 3