Mathematical Modeling in Finance with Stochastic Processes

5.1 Intuitive Introduction to Diffusions

random walk on the axis, we then need to know the size ofδTn. Now

E[δ·Tn] = (p−q)·δ·n

and
Var [δ·Tn] = 4·p·q·δ^2 ·n.

We wantnto be large (about 1 million) and to see the walk on the screen
we want the expected end place to be comparable to the screen size, say 30
cm. That is,
E[δ·Tn] = (p−q)·δ·n < δ·n≈30cm

soδmust be 3· 10 −^5 cm = 0.0003mm to get the end point on the screen. But
then the movement of the walk measured by the standard deviation

√

Var [δ·Tn]≤δ·

√

n= 3× 10 −^2 cm

will be so small as to be indistinguishable. We will not see any random
variations!

Trying to use the variance to derive the limit

Let us turn the question around: We want to see the variations in many-step
random walks, so the standard deviations must be a reasonable fractionD
of the screen size

√

Var [δ·Tn]≤δ·

√

n≈D·30 cm.

This is possible ifδ=D· 3 × 10 −^2 cm.We still want to be able to see the
ending expected position which will be

E[δ·Tn] = (p−q)·δ·n= (p−q)·D· 3 × 104 cm.

To be consistent with the variance requirement this will only be possible
if (p−q) ≈ 10 −^2. That is, p−q must be comparable in magnitude to
δ= 3× 10 −^2.

The limiting process

Now generalize these results to visualize longer and longer walks in a fixed
amount or time. Sinceδ→0 asn→ ∞, then likewise (p−q)→0, while