Black-Scholes PDE

Black-Scholes PDE

Contents:

0.1

Derivation from martingale form

0.1.1

Using the Feynman-Kac formula

0.1.2

Proof of Markov property

0.1.3

Using Itō’s formula

0.2

Comparison of the martingale and PDE forms

0.3

Analytic solution

0.4

Quantity of stock needed to replicate option

0.5

Generalizations

The

Black-Scholes partial differential equation

is the partial differentiation equation:

r

⁢

f

=

∂

⁡

f

∂

⁡

t

+

r

⁢

x

⁢

∂

⁡

f

∂

⁡

x

+

1

2

⁢

σ

2

⁢

x

2

⁢

∂

2

⁡

f

∂

⁡

x

2

,

f

=

f

⁢

(

t

,

x

)

,

(1)

on the domain

0

≤

x

<

∞

,

0

≤

t

≤

T

.
Its solution gives the price

function

of a stock option (or any other contingent claim on a tradable asset)
under the

assumptions

of the Black-Scholes model for prices.

The

parameters

r

,

σ

>

0

and

T

>

0

are, respectively, the prevailing

continuously-compounded

risk-free
rate of

interest

, the volatility of the stock,
and the time to maturity of the stock option.

More generally, the term

Black-Scholes partial differential equation

can refer to other

partial differential equations

similar

in form to equation (

1

), derived
in similar ways under more
general modelling assumptions than the most basic Black-Scholes model.

0.1

Derivation from martingale form

The

Black-Scholes formula

, in its

martingale

form,
gives the theoretical value of the stock option

V

(

t

)

=

e

-

r

⁢

(

T

-

t

)

𝔼

ℚ

[

V

(

T

)

∣

ℱ

t

]

,

0

≤

t

≤

T

,

(2)

where

V

⁢

(

T

)

is a given pay-off function
or

terminal

condition, assumed to be a

𝐋

1

⁢

(

ℚ

)

random variable

;

ℱ

t

is the

filtration

generated by

the the standard

Wiener
process

W

~

⁢

(

t

)

under the

probability measure

ℚ

(the “risk-neutral measure”).

Equation (

2

)
gives the solution to the

stochastic differential equations

:

V

⁢

(

t

)

=

Δ

⁢

(

t

)

⁢

X

⁢

(

t

)

+

Θ

⁢

(

t

)

⁢

M

⁢

(

t

)

,

M

⁢

(

t

)

=

e

r

⁢

t

,

(3)

d

⁢

V

⁢

(

t

)

=

Δ

⁢

(

t

)

⁢

d

⁢

X

⁢

(

t

)

+

Θ

⁢

(

t

)

⁢

d

⁢

M

⁢

(

t

)

,

d

⁢

M

⁢

(

t

)

=

r

⁢

M

⁢

(

t

)

⁢

d

⁢

t

,

(4)

d

⁢

X

⁢

(

t

)

=

r

⁢

X

⁢

(

t

)

⁢

d

⁢

t

+

σ

⁢

X

⁢

(

t

)

⁢

d

⁢

W

~

⁢

(

t

)

.

(5)

From the martingale form of the result,
there are two ways to derive the PDE form (

1

).

0.1.1

Using the Feynman-Kac formula

One way is to appeal to the

Feynman-Kac formula

,
which states in this case, that
under certain

regularity

conditions

1

1

The standard assumption is that

ψ

is twice continuously-differentiable
and

𝔼

ℚ

⁢

[

|

ψ

⁢

(

X

⁢

(

T

)

)

|

]

<

∞

.
However, for the particular PDE we consider,
these assumptions can be relaxed.

on the
function

ψ

:

(

0

,

∞

)

→

ℝ

, the
function

f

:

[

0

,

T

]

×

(

0

,

∞

)

→

ℝ

defined by

f

(

t

,

x

)

=

e

-

r

⁢

(

T

-

t

)

𝔼

ℚ

[

ψ

(

X

(

T

)

)

∣

X

(

t

)

=

x

]

,

(6)

satisfies

the partial differential equation (

1

)
with the terminal condition

f

⁢

(

T

,

x

)

=

ψ

⁢

(

x

)

.

(The

stochastic process

X

⁢

(

t

)

is to satisfy equation (

5

).)

Now, equations (

6

) and (

2

)
look quite similar,
but with an important

difference

:
the

conditional expectation

on equation (

6

)
is with respect to the event

{

X

⁢

(

t

)

=

x

}

,
while equation (

2

)
is conditioning on the filtration

ℱ

t

.
We claim that these are actually the same thing —
that is, for fixed

t

, the random variable

𝔼

ℚ

[

V

(

T

)

∣

ℱ

t

]

can be written as a function of

X

⁢

(

t

)

—

provided that the terminal condition is itself a function of

X

⁢

(

T

)

:

V

⁢

(

T

)

=

ψ

⁢

(

X

⁢

(

T

)

)

.

We will demonstrate this claim later. Assuming that the claim holds,
we may thus set

V

⁢

(

t

)

=

f

⁢

(

t

,

X

⁢

(

t

)

)

,
and the Black-Scholes partial differential equation follows from
the Feynman-Kac formula as explained earlier.

0.1.2

Proof of Markov property

Formally, any

ℱ

t

-

measurable

stochastic process

X

⁢

(

t

)

that
ensures

𝔼

[

ψ

(

X

(

T

)

)

∣

ℱ

t

]

is always a

measurable function

of

X

⁢

(

t

)

,
for any Borel-measurable

ψ

,
is called a

Markov process

,
or is said to have the

Markov property

.
Thus we want to show that the stock process

X

⁢

(

t

)

has
the Markov property.
For this,
we will need the following explicit
solution for it at time

T

in terms of an

initial condition

at time

t

:

X

⁢

(

T

)

=

X

⁢

(

t

)

⁢

exp

⁡

(

(

r

-

1

2

⁢

σ

2

)

⁢

(

T

-

t

)

+

σ

⁢

(

W

~

⁢

(

T

)

-

W

~

⁢

(

t

)

)

)

:=

X

⁢

(

t

)

⁢

g

⁢

(

W

~

⁢

(

T

)

-

W

~

⁢

(

t

)

)

.

Then

𝔼

ℚ

[

V

(

T

)

∣

ℱ

t

]

=

𝔼

ℚ

[

ψ

(

X

(

T

)

)

∣

ℱ

t

]

=

𝔼

ℚ

[

ψ

(

X

(

t

)

g

(

W

~

(

T

)

-

W

~

(

t

)

)

)

∣

ℱ

t

]

.

Since

X

⁢

(

t

)

is

ℱ

t

-measurable,
it may be treated as a

constant

x

while taking conditional
expectations with respect to

ℱ

t

.
And

W

~

⁢

(

T

)

-

W

~

⁢

(

t

)

is a random variable

independent

of

ℱ

t

(and hence of

X

⁢

(

t

)

).
Therefore the

expression

that appears on the right-hand side
above is a function of

x

=

X

⁢

(

t

)

,
and would be unchanged if the conditioning
is changed from

ℱ

t

to

X

⁢

(

t

)

.

0.1.3

Using Itō’s formula

There is also a more direct method
of deriving equation (

1

)
using Itō’s

formula

for Itō processes.

Set

V

⁢

(

t

)

=

f

⁢

(

t

,

X

⁢

(

t

)

)

as before.
Make the assumption that

f

is twice

continuously differentiable

,
to be checked later; then we can apply Itō’s formula to expand

d

⁢

V

⁢

(

t

)

:
(

partial derivatives

evaluated at

x

=

X

⁢

(

t

)

)

d

⁢

V

⁢

(

t

)

=

∂

⁡

f

∂

⁡

t

⁢

d

⁢

t

+

∂

⁡

f

∂

⁡

x

⁢

d

⁢

X

⁢

(

t

)

+

1

2

⁢

∂

2

⁡

f

∂

⁡

x

2

⁢

d

⁢

X

⁢

(

t

)

⁢

d

⁢

X

⁢

(

t

)

=

(

∂

⁡

f

∂

⁡

t

+

1

2

⁢

σ

2

⁢

X

⁢

(

t

)

2

⁢

∂

2

⁡

f

∂

⁡

x

2

)

⁢

d

⁢

t

+

∂

⁡

f

∂

⁡

x

⁢

d

⁢

X

⁢

(

t

)

.

(7)

By

theorems

on the uniqueness of solutions to stochastic differential
equations, the

coefficients

of the
correponding

d

⁢

t

and

d

⁢

X

⁢

(

t

)

terms in equations (

4

) and (

7

)
must be equal almost surely.
Equating the coefficients, we find:

Δ

⁢

(

t

)

=

∂

⁡

f

∂

⁡

x

,

(8)

r

⁢

(

V

⁢

(

t

)

-

Δ

⁢

(

t

)

⁢

X

⁢

(

t

)

)

=

Θ

⁢

(

t

)

⁢

r

⁢

M

⁢

(

t

)

=

∂

⁡

f

∂

⁡

t

+

1

2

⁢

σ

2

⁢

X

⁢

(

t

)

2

⁢

∂

2

⁡

f

∂

⁡

x

2

.

(9)

Equation (

9

)
is essentially equation (

1

),
except that

x

has been replaced by

X

⁢

(

t

)

in some places.
However,
because

X

⁢

(

t

)

is a random variable that takes on all values on

(

0

,

∞

)

,
and

f

and its

derivatives

are assumed to be

continuous

,
equation (

9

) must hold for arbitrary values

x

substituted for

X

⁢

(

t

)

. Hence we obtain equation (

1

).

To verify
the initial assumption that

f

is twice continuously differentiable,
we write formula (

2

)
in a more explicit form (using the Markov property for

X

⁢

(

t

)

as before):

f

⁢

(

t

,

x

)

=

e

-

r

⁢

(

T

-

t

)

𝔼

ℚ

[

ψ

(

X

(

T

)

)

∣

X

t

=

x

]

=

∫

-

∞

∞

e

-

r

⁢

(

T

-

t

)

⁢

ψ

⁢

(

y

)

⁢

p

⁢

(

x

,

y

,

T

-

t

)

⁢

𝑑

y

,

where

p

⁢

(

x

,

y

,

τ

)

is the transition

density

of the
stock process from

x

to

y

over a time interval of length

τ

.
In fact, from the solution for

X

⁢

(

t

)

,
the density

p

⁢

(

x

,

⋅

,

τ

)

is the density for the log-normal random variable
whose

logarithm

has mean

log

⁡

x

+

(

r

-

1

2

⁢

σ

2

)

⁢

τ

and

variance

σ

2

⁢

τ

.
By

differentiation

under the

integral sign

,
we see that

f

⁢

(

t

,

x

)

must be twice continously

differentiable

for

0

≤

t

<

T

and

0

<

x

<

∞

.

0.2

Comparison of the martingale and PDE forms

The martingale formulation of the result,
in equation (

2

)
is more general than the PDE formulation (

1

):
the latter imposes extra regularity conditions,
and more importantly,
it can only describe prices

V

⁢

(

t

)

=

f

⁢

(

t

,

X

⁢

(

t

)

)

that are

functions only of the time

t

and the current stock price

X

⁢

(

t

)

.
In the case of the call option (

ψ

⁢

(

x

)

=

x

-

K

for some

K

>

0

),
this assumption is true;
but there are other sorts of contingent claims
whose pay-off depend on the history of the stock process

X

⁢

(

t

)

—
for example, a common kind of contingent claim has a final
pay-off that depends on an

average

of the stock price

X

⁢

(

t

)

over time

t

∈

[

0

,

T

]

.

However, the PDE form of the solution is often more
amenable to a numerical solution.
While the

expectation

in equation (

2

)
can be approximated numerically
with Monte-Carlo

simulation

,
for low-dimensional problems,
solutions based on finite-difference approximations
of PDEs are often quicker to compute than those based on Monte-Carlo.

0.3

Analytic solution

Though the solution for the Black-Scholes PDE
is already known using the martingale representation (

2

),
we can also solve the PDE directly using classical analytical methods (

http://planetmath.org/AnalyticSolutionOfBlackScholesPDE

).

Actually, the classical solution can be quite instructive
in that it shows more or less the physical meaning
behind the PDE, and also the behavior of solutions
as the terminal condition is varied. (For example,
the PDE is a

transformation

of a diffusion equation,
and consequently its solutions are always infinitely smooth,
as long as the terminal condition satisfies
some local-integrability

properties

.)

0.4

Quantity of stock needed to replicate option

Another important result that
must not be left unmentioned is
equation (

8

), derived during
the course of showing the Black-Scholes PDE.

In the article on the Black-Scholes pricing formula,
a solution for

V

⁢

(

t

)

was produced
that satisfied (

4

),
but the

ℱ

t

-

adapted process

δ

⁢

(

t

)

that appears in that stochastic differential equation was
obtained by appealing to an

existence theorem

.
However, under the assumption that

V

⁢

(

t

)

=

f

⁢

(

t

,

X

⁢

(

t

)

)

,
equation (

8

) gives an actual

computable

formula:

Δ

⁢

(

t

)

=

∂

⁡

f

⁢

(

t

,

x

)

∂

⁡

x

|

x

=

X

⁢

(

t

)

.

which is crucial in practice,
for the theoretical “price”
of a stock option
would be useless if one cannot produce in reality that option
for the stated price.

0.5

Generalizations

(To be written.)

Title

Black-Scholes PDE

Canonical

name

BlackScholesPDE

Date of creation

2013-03-22 16:31:08

Last modified on

2013-03-22 16:31:08

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

6

Author

stevecheng (10074)

Entry type

Result

Classification

msc 91B28

Classification

msc 60H10

Synonym

Black-Scholes partial differential equation

Synonym

Black-Scholes equation

Related topic

AnalyticSolutionOfBlackScholesPDE

Related topic

ExampleOfSolvingTheHeatEquation

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