mfdc2d

7/27/2019 mfdc2d

http://slidepdf.com/reader/full/mfdc2d 1/5

2. Preliminaries for Differentiation in Stochas-

tic Environments

In ordinary calculus

df = f xdx,

where f x is the derivative of f (x) w.r.t x. Or-dinary differential equations (ODE) are based

on the rules of the usual differentiation rules.

1

The question is: How differentiation can be

extended to stochastic analysis?

Although, as we will see, derivative does not

(usually) exist in stochastic analysis, we get

a reasonable definition for differential equa-

tion for stochastic variables. These are calledstochastic differential equations (SDE)

2

7/27/2019 mfdc2d


With SDEs we can analyze the dynamics of

a stochastic process, S t, such that

dS t = a(S t, t) dt + b(S t, t) dW t,(1)

where dW t is an unpredictable innovation term

during the infinitesimal interval dt.

The terms a(S t, t) and b(S t, t) represent drift

and diffusion (amount of randomness), re-

spectively.

Thus the change has the representation

change = mean change + unpredictable innovation.

3

Let us next consider a discrete approximation

of (1) in time interval [0, T ] with t0 = 0 <

t1 < · · · < tn = T .

Assume further that ∆t = tk − tk−1 = T /n,

and for simplicity that a(S t, t) ≡ a and b(S t, t) =

b are constants.

Then the changes of (1) can be approxi-

mated as

∆S k = a∆t + b∆W k,(2)

where ∆S k = S tk − S tk−1

and ∆W k = W tk −

W tk−1.

Given S 0, we get a ”solution” for (2) as

S t = S 0 + at + b

nk=1

∆W k.(3)

4

7/27/2019 mfdc2d


In order to model S t as a stochastic process,

we must impose some restrictions to the ran-

dom innovations ∆W k.

First, technically it is reasonable to assume

that E[∆W k] = 0 for all k and for all n. De-

note the variances as

V k = E[∆W k]2,

and

V =n

k=1

V k.

To obtain a reasonable model for stock price

dynamics, we introduce the following assump-

tions

5

Assumption 1: Positive lower bound for volatil-

ity

V > A1 > 0,

where A1 is independent of n.

Assumption 2: Finite upper bound

V < A2 < ∞,

where A2 is independent of n.

Assumption 3: Uncertainty is not concen-

trated

V k

V max > A3, 0 < A3 < 1,

with A3 independent of n, and

V max = maxk

V k, k = 1, . . . , n .

This assumption guarantees that whenever

markets are open there is at least some volatil-ity.

6

7/27/2019 mfdc2d


Proposition 2.1: (Merton 1990) Under assumptions

1, 2, and 3, the variance of ∆W k is

E [∆W k]2 = σ2k h,

where h = ∆t = T /n, and σk is a finite constant that

may depend on information at time tk−1.

7

Although the differential ∆W k is reasonable,

there is no such concept like derivative of W t!

Another implication of the model is that the

sample path of W t becomes increasingly ir-

regular. That is, although the sample path

is continuous, it is nowhere differentiable!

The first claim is easily seen, because if there

existed a stochastic process At such that

W t+h − W t

h

q .m .→ At

then

limh→0

E

W t+h − W t

h − At

2

= 0.

8

7/27/2019 mfdc2d


But then (assume h > 0, and that

E[∆W t+h]2 = h, i.e., σt = 1)

E

∆W t+h

h −A(t)

2

= E

∆W t+h

h

2

− 2E (∆W t+hAt) + E[At]2.

Now

E

∆W t+hAt

= E

AtEt(∆W t+h)

= 0,

because

Et[∆W t+h] = E[W t+h] = 0

(unpredictable).

However

E∆W t+h

h

2

= h

h2

= 1

h

→ ∞ as h → 0.

thus no limit exist, and no derivative can be

defined!

9

Nevertheless, as the differential concept is

reasonable and useful, the concept must be

defined via (stochastic) integral, which we’ll

deal with later.

10

mfdc2d

Documents

Transcript of mfdc2d