derivation of Black-Scholes formula in martingale form

derivation of Black-Scholes formula in martingale form

Contents:

0.1

Assumptions

0.1.1

Asset price

0.1.2

Money-market account

0.1.3

Portfolio process

0.2

Derivation

0.2.1

Change of probability measure

0.2.2

Discounted portfolio process is a martingale

0.2.3

Portfolio process as a conditional expectation

0.3

Existence of solutions

0.3.1

Proposed construction

0.3.2

Verification

This entry derives the

Black-Scholes formula

in

martingale

form.

The portfolio process

V

t

representing a stock option
will be shown to

satisfy

:

V

t

=

e

-

r

⁢

(

T

-

t

)

𝔼

ℚ

[

V

T

∣

ℱ

t

]

.

(1)

(The quantities appearing here are defined precisely,
in the

section

on “

Assumptions

” below.)

Equation (

1

)
can be used in practice to calculate

V

t

for all times

t

,
because from the

specification

of a financial contract,
the value of the portfolio at time

T

,
or in other words, its pay-off at time

T

,
will be a known

function

.
Mathematically speaking,

V

T

gives the

terminal

condition

for the solution of a

stochastic differential equation

.

0.1

Assumptions

0.1.1

Asset price

The asset or stock price

X

t

is to be modelled by the stochastic
differential equation:

d

⁢

X

t

=

μ

⁢

X

t

⁢

d

⁢

t

+

σ

⁢

X

t

⁢

d

⁢

W

t

,

(2)

where

μ

and

σ

>

0

are

constants

.

The

stochastic process

W

t

is a standard

Brownian motion

adapted

to the

filtration

{

ℱ

t

}

.

See the main article
on the Black-Scholes formula (

http://planetmath.org/BlackScholesFormula

)
for an explanation and justification of this modelling assumption.

0.1.2

Money-market account

The money-market account accumulates

interest

compounded continuously

at a rate of

r

.
It satisfies the stochastic differential equation:

d

⁢

M

t

=

r

⁢

M

t

⁢

d

⁢

t

.

(3)

This happens to take the same form as an

ordinary differential equation

,
for the process

M

t

has no randomness in it at all,
under the assumption of a fixed

interest rate

r

.

The solution is to equation (

3

) with

initial condition

M

0

is

M

t

=

M

0

⁢

e

r

⁢

t

.

0.1.3

Portfolio process

The price of the option is derived by following a

replicating portfolio

consisting of

Δ

t

units of the stock

X

t

and

Θ

t

units of the money-market
account. If

V

t

denotes the value of this portfolio at time

t

,
then

V

t

=

Δ

t

⁢

X

t

+

Θ

t

⁢

M

t

.

(4)

A certain “self-financing condition” on the portfolio requires that

V

t

also satisfy the stochastic differential equation:

d

⁢

V

t

=

Δ

t

⁢

d

⁢

X

t

+

Θ

t

⁢

d

⁢

M

t

.

(5)

This condition essentially says that we cannot input extra amounts
of money out of thin air into our portfolio; we must start with what we have.

Equation (

5

) is not a mathematically proven statement,
but another modelling assumption, justified by an analogous equation
governing trading in discretized time periods.

0.2

Derivation

We first manipulate the
stochastic differential equation (

4

)
for the portfolio process

V

t

,
to express it in terms of the Brownian motion

W

t

.

d

⁢

V

t

=

Δ

t

⁢

d

⁢

X

t

+

Θ

t

⁢

r

⁢

M

t

⁢

d

⁢

t

from eq. (

5

) and (

3

)

=

Δ

t

⁢

d

⁢

X

t

+

r

⁢

(

V

t

-

Δ

t

⁢

X

t

)

⁢

d

⁢

t

from eq. (

4

)

=

Δ

t

⁢

(

μ

⁢

X

t

⁢

d

⁢

t

+

σ

⁢

X

t

⁢

d

⁢

W

t

)

+

r

⁢

(

V

t

-

Δ

t

⁢

X

t

)

⁢

d

⁢

t

from eq. (

2

)

=

r

⁢

V

t

⁢

d

⁢

t

+

Δ

t

⁢

X

t

⁢

(

(

μ

-

r

)

⁢

d

⁢

t

+

σ

⁢

d

⁢

W

t

)

rearrangement

0.2.1

Change of probability measure

Define the Brownian motion with drift

λ

:

W

~

t

=

λ

⁢

t

+

W

t

,

λ

=

μ

-

r

σ

;

(6)

so that

d

⁢

W

~

t

=

λ

⁢

d

⁢

t

+

d

⁢

W

t

, and

d

⁢

V

t

=

r

⁢

V

t

⁢

d

⁢

t

+

σ

⁢

Δ

t

⁢

X

t

⁢

d

⁢

W

~

t

.

(7)

The

introduction

of the process

W

~

t

is not merely for notational convenience but is mathematically
meaningful. If the

probability space

we are working in
is

(

Ω

,

ℱ

T

,

ℙ

)

,
and

W

t

, for

0

≤

t

≤

T

, is a
standard

Wiener process

on

(

Ω

,

ℱ

T

,

ℙ

)

,
then

W

~

t

will not be a standard Wiener process
on

(

Ω

,

ℱ

T

,

ℙ

)

, but it

will

be a standard Wiener process
under

(

Ω

,

ℱ

T

,

ℚ

)

with a different

probability measure

ℚ

.

The probability measure

ℚ

is obtained by Girsanov’s

theorem

.
The exact form for

ℚ

can be calculated,
but it will not be needed in this derivation.

In finance,

ℚ

is known as the

risk-neutral measure

,
and the quantity

λ

is the

market price of risk

.

0.2.2

Discounted portfolio process is a martingale

From equation (

7

),
we see that the value of the portfolio grows at the risk-free
interest rate of

r

,
apart from the randomness associated due to the stochastic
differential

d

⁢

W

~

t

.

It is thus reasonable to expect that,
if we normalize the portfolio value amount
by the amount that cash grows due to accumulation of risk-free
interest,
the resulting process,

V

t

/

M

t

, should have a zero growth rate.
That this is indeed the case can be verified
by a computation with Itô’s

formula

— more specifically, the
for Itô

integrals

:

d

⁢

(

V

t

M

t

)

=

d

⁢

(

V

t

⋅

1

M

t

)

=

(

d

⁢

V

t

)

⁢

1

M

t

+

V

t

⁢

d

⁢

(

1

M

t

)

=

r

⁢

V

t

M

t

⁢

d

⁢

t

+

σ

⁢

Δ

t

⁢

X

t

M

t

⁢

d

⁢

W

~

t

+

V

t

⁢

d

⁢

(

1

M

t

)

from eq. (

7

)

=

r

⁢

V

t

M

t

⁢

d

⁢

t

+

σ

⁢

Δ

t

⁢

X

t

M

t

⁢

d

⁢

W

~

t

+

V

t

M

t

⁢

d

⁢

t

from

1

M

t

=

e

-

r

⁢

t

M

0

.

Thus,

d

⁢

(

V

t

M

t

)

=

σ

⁢

Δ

t

⁢

X

t

M

t

⁢

d

⁢

W

~

t

.

Or, in integral form:

V

t

1

M

t

1

=

V

t

0

M

t

0

+

∫

t

0

t

1

σ

⁢

Δ

t

⁢

X

t

M

t

⁢

𝑑

W

~

t

,

0

≤

t

0

≤

t

1

≤

T

.

(8)

Assuming

Δ

t

is a

ℱ

t

-adapted process
— where

{

ℱ

t

}

is the filtration

generated by

the Brownian
motion

W

t

(or equivalently

W

~

t

) —
the Itô integral in equation (

8

)
is a martingle under the probability space

(

Ω

,

ℱ

T

,

ℚ

)

.

0.2.3

Portfolio process as a conditional expectation

Then by the

definition

of a martingale,
we have

V

t

0

M

t

0

=

𝔼

ℚ

[

V

t

1

M

t

1

∣

ℱ

t

0

]

,

0

≤

t

0

≤

t

1

≤

T

,

where

𝔼

ℚ

[

⋅

∣

ℱ

t

]

denotes the

conditional
expectation

, of a

random variable

on the

measurable space

(

Ω

,

ℱ

T

)

,
under
the probability measure

ℚ

.

In particular, setting

t

0

=

t

≤

T

and

t

1

=

T

,
and rearranging the factors of

M

t

=

e

r

⁢

t

,
we obtain the desired result, equation (

1

).

0.3

Existence of solutions

So far,
we have derived the form of the solution for the portfolio value process

V

t

,

assuming that it exists

.
Actually, if we were to take only
equations (

4

) and (

5

)
as the problem to solve mathematically,
without any reference to the financial motivations,
it is possible to work backwards and deduce the

existence

of the solution.

0.3.1

Proposed construction

Let

ℚ

be the risk-neutral probability measure,
and let

U

be any given

𝐋

1

⁢

(

Ω

,

ℱ

T

,

ℚ

)

random variable,
representing the

terminal condition

.
Define the family of random variables dependent on time,

V

t

=

e

-

r

⁢

(

T

-

t

)

𝔼

ℚ

[

U

∣

ℱ

t

]

,

0

≤

t

≤

T

.

(9)

It is easy to verify that, for any

U

,
the process

V

t

⁢

e

-

r

⁢

t

is a martingale
with respect to

ℱ

t

, the filtration generated
by the Wiener process

W

~

t

under the probability
measure

ℚ

.

0.3.2

Verification

We now invoke the martingale representation theorem for Itô processes:
for any martingale

Z

t

, with respect to

ℱ

t

under
the probability measure

ℚ

,
there exists a

ℱ

t

-adapted process

G

t

such that

Z

t

has the representation:

Z

t

1

-

Z

t

0

=

∫

t

0

t

1

G

t

⁢

𝑑

W

~

t

.

Letting

Z

t

=

V

t

⁢

e

-

r

⁢

t

and
comparing with equations (

8

)
and (

4

),
we are motivated to

define

the

ℱ

t

-adapted processes:

Δ

t

=

G

t

⁢

e

r

⁢

t

σ

⁢

X

t

,

Θ

t

=

V

t

-

Δ

t

⁢

X

t

M

t

=

Z

t

-

G

t

/

σ

M

0

.

Then the process

V

t

constructed
by equation (

9

)
trivially satisfies equation (

4

).
And it is a simple matter to check that
equation (

5

) holds as well:

d

⁢

V

t

=

d

⁢

(

Z

t

⁢

e

r

⁢

t

)

=

e

r

⁢

t

⁢

d

⁢

Z

t

+

r

⁢

e

r

⁢

t

⁢

Z

t

⁢

d

⁢

t

Itô’s product rule

=

e

r

⁢

t

⁢

G

t

⁢

d

⁢

W

~

t

+

r

⁢

V

t

⁢

d

⁢

t

=

σ

⁢

Δ

t

⁢

X

t

⁢

d

⁢

W

~

t

+

r

⁢

V

t

⁢

d

⁢

t

=

Δ

t

⁢

X

t

⁢

(

r

⁢

d

⁢

t

+

σ

⁢

d

⁢

W

~

t

)

add and subtract

+

r

⁢

(

V

t

-

Δ

t

⁢

X

t

)

the

d

⁢

t

term

=

Δ

t

⁢

d

⁢

X

t

+

r

⁢

Θ

t

⁢

M

t

⁢

d

⁢

t

,

where in the last equality we have used the SDE for

X

t

in terms of

d

⁢

W

~

t

in place of

d

⁢

W

t

:

d

⁢

X

t

=

r

⁢

X

t

⁢

d

⁢

t

+

σ

⁢

X

t

⁢

d

⁢

W

~

t

,

obtained by substituting
in equation (

2

),
the differential of equation (

6

).

References

1

Bernt Øksendal.

Stochastic Differential Equations,
An Introduction with Applications

, 5th edition. Springer, 1998.

2

Steven E. Shreve.

Stochastic Calculus for Finance II:
Continuous-Time Models

. Springer, 2004.

Title

derivation of Black-Scholes formula in martingale form

Canonical

name

DerivationOfBlackScholesFormulaInMartingaleForm

Date of creation

2013-03-22 17:20:37

Last modified on

2013-03-22 17:20:37

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

5

Author

stevecheng (10074)

Entry type

Derivation

Classification

msc 60H10

Classification

msc 91B28

Related topic

BlackScholesFormula

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