Springer Finance

3.4 Quadratic Variation 101

n- 1

'E if'<t;)i(tj+l-tj),

i=O

which is a Riemann sum for the integral of the function (^1) /'(t)j. Therefore,
n-^1 T

FVT(/) = lim L if'(tj)i(ti+l-tj) =

r

if'(t)i dt,

IIIIII-+O i=O lo

and we have rederived (3.4.2).

3.4.2 Quadratic Variation

Definition 3.4.1. Let f(t) be a function defined forO :S t :S T. The quadratic

variation off up to .time T is

n- 1

f, f = lim �)f(tH1)-f(ti)]^2 ,

IIIIII-+Oj=O

(3.4.5)

where II= {to,tt, ... ,tn} and 0 =to< t 1 < ·· · < tn = T.

Remark :1 .4 .. 2. Suppose the function f has a continuous derivative. Then

n- 1 n- 1 n- 1

'Elf(tj+l)-f(tiW = L if'(tj)i^2 (tj+1-tj)^2 :s IIIIII· L if'(tj)i^2 (tj+1 -tj),

i=O

and thus

i=O i=O

f,f :S lim [IIIIII· I: l/'(tj)l^2 (ti+1-ti)l

IIIIII-+^0 i=O

n- 1

= lim IIIIII· lim L if'(tj)i^2 (ti+l-ti)

IIIIII-+^0 IIIIII-+^0 i=O

= lim IIIIII·

rT

l!'(t)i^2 dt = o.

IIIIII-+^0 Jo

In the last step of this argument, we use the fact that f'(t) is continuous to

ensure that f 0 T lf'(t)j^2 dt is finite. If J: lf'(t)j^2 dt is infinite, then

leads to a 0 · oo situation, which can be anything between 0 and oo. 0