Black-Scholes formula

Black-Scholes formula

Contents:

0.1

Black-Scholes pricing formula

0.2

General representation of option price

0.3

The risk-neutral measure

0.4

Model assumptions

0.4.1

Stock price

0.4.2

Interest on cash

The

Black-Scholes formula

gives the theoretical “arbitrage-free” price of a stock option
at any time

t

before the maturity time

T

.

In financial parlance, an “arbitrage” is a riskless
profit. So the prices being “arbitrage-free”
means that it is impossible to construct
a

strategy

1

1

Strictly

speaking, the strategies described here must be construed as the

limiting case

, in

continuous

time, of no-arbitrage trading
strategies in discrete time, as the time period between trades become
small. Otherwise, it is not even

obvious

what trading
in “continuous time” means, and how to actually execute such trades.

of trading the stock option and the underlying stock,
at the prices given,
that starts with no money at time

0

,
ends with a balance at time

T

that
is non-negative almost surely,
and is

positive

with some positive

probability

.

A stock

call option

is a contract, in which the vendor gives
the buyer a right to purchase some shares of stock in the future,
at a fixed

strike price

agreed upon the writing of the contract.
However, the right need not be exercised by the buyer.
In fact it would be disadvantageous to do so if the stock price
evolves below the strike price since the writing of the contract;
if the buyer still wants the stock, it would be cheaper to just
buy it from the market.

0.1

Black-Scholes pricing formula

Let

•

K

be the

strike price

of the call option

•

T

be the

maturity date

of the call option

•

r

be the prevailing

interest rate

, assumed to be constant

•

σ

>

0

be the volatility of the stock price (per unit square-root time)

•

X

⁢

(

t

)

be the

stochastic process

representing the price of the stock itself at time

t

Then the fair price

V

⁢

(

t

)

, at time

t

≤

T

, of the
call option with the above contract

parameters

,
under the Black-Scholes modelling assumptions,
is given by:

V

⁢

(

t

)

=

x

⁢

Φ

⁢

(

d

+

⁢

(

τ

,

x

)

)

-

K

⁢

e

-

r

⁢

τ

⁢

Φ

⁢

(

d

-

⁢

(

τ

,

x

)

)

,

d

±

⁢

(

τ

,

x

)

=

log

⁡

(

x

/

K

)

+

(

r

±

1

2

⁢

σ

2

)

⁢

τ

σ

⁢

τ

,

τ

=

T

-

t

,

x

=

X

⁢

(

t

)

,

(1)

where

Φ

is the

cumulative distribution function

of a standard normal random variable.

The result (

1

)
is commonly referred to as the

Black-Scholes pricing formula

or just as the

Black-Scholes formula

,
after Fischer Black and Myron Scholes, who derived it in the 1970s.

F. Black and M. Scholes first arrived at the solution for

V

⁢

(

t

)

not with our probabilistic calculations,
but by describing

V

⁢

(

t

)

in terms of a

partial differential equation

, and solving the PDE analytically.
That PDE, now called, appropriately,
the

Black-Scholes partial differential equation

,
gives an important alternative formulation of the results here.

0.2

General representation of option price

V

⁢

(

t

)

in equation (

1

) is a stochastic process itself,
but it happens to depend on

X

⁢

(

t

)

only
for its random part.

More generally, for any financial contract, not necessarily
a call option, that pays an amount

V

⁢

(

T

)

,
the Black-Scholes model shows that its fair price

V

⁢

(

t

)

at time

t

must be given by:

V

(

t

)

=

e

-

r

⁢

(

T

-

t

)

𝔼

ℚ

[

V

(

T

)

∣

ℱ

t

]

.

(2)

for some

probability measure

ℚ

,
and

{

ℱ

t

}

is the filtration
describing the information available about the stock price up to time

t

.

If we set

V

⁢

(

T

)

=

max

⁡

(

X

⁢

(

T

)

-

K

,

0

)

, then
equation (

1

)
follows from a straightforward
calculation starting with equation (

2

).

Intuitively, (

2

) states that
the option price at the

current

time is the

expected value

of the promised pay-off amount at the future time,
discounted back to the current time at the rate of

interest

in force.

0.3

The risk-neutral measure

The probability measure

ℚ

appearing in (

2

),
is called the

risk-neutral measure

.
It depends on the underlying risk factors that drive the contract pay-off,
but it does not depend on the contract itself.
This probability measure is distinct from the real-world or historical
probability measure.
It does not actually describe “probabilities” in the intuitive sense
of the word, but is a mathematical tool to express the solutions
to the underlying

stochastic differential equations

.

Equation (

2

) is equivalent to
saying that

V

⁢

(

t

)

⁢

e

-

r

⁢

t

is a

martingale

under the probability
measure

ℚ

.

0.4

Model assumptions

The Black-Scholes formula is derived under the following
model assumptions.

In the real world, of course, all of these simplifying assumptions
are wrong in one way or another; however,
the Black-Scholes formula is often used
to give ballpark figures when quoting option prices.

0.4.1

Stock price

The stock
(or any other underlying tradable asset)
has a price that follows a

geometric Brownian motion
process

described by the stochastic differential equation:

d

⁢

X

⁢

(

t

)

=

μ

⁢

X

⁢

(

t

)

⁢

d

⁢

t

+

σ

⁢

X

⁢

(

t

)

⁢

d

⁢

W

⁢

(

t

)

,

(3)

where

•

X

⁢

(

t

)

is the price of the stock at time

t

≥

0

; the quantity

X

⁢

(

t

)

is
a

random variable

.

(That is, for each

t

,

X

⁢

(

t

)

is a function of

ω

,
for

ω

ranging over the underlying

sample space

; but the implicit
notation is more convenient and widely used.)

•

μ

>

0

is the growth rate of the stock. The quantity

μ

has units
of

reciprocal

time.

μ

is assumed to be a constant (

independent

of time and the state
of the system).

•

σ

>

0

is the volatility of the stock.
It can be thought as the

standard deviation

normalized by time;

σ

has units of the reciprocal square root of time.

σ

is assumed to be constant.

•

W

⁢

(

t

)

is a standard

Brownian motion

, also called a

Wiener process

.

The assumption of

geometric

Brownian motion means
that the

relative

price movements

X

⁢

(

t

+

Δ

⁢

t

)

/

X

⁢

(

t

)

after time

t

are independent of the stock price history before time

t

.
In general,
relative changes in price
are more financially significant and useful to model
(e.g. “Google stock has gone up 40% this year”)
than absolute changes in price.

We also assume that the stock does not pay

dividends

,
that there are no on the trading of any asset,
or borrowing or lending of any amount of cash at the risk-free interest rate,
and no transaction costs.
Trading is also assumed to be conducted continuously
in perfectly divisible amounts,
at a single price at each point in time.

0.4.2

Interest on cash

Moreover, cash in the model economy is assumed to grow
with time at a

risk-free

continuously-compounded

interest rate

of

r

,
and

r

is constant. Thus the amount

M

⁢

(

t

)

in the “money-market account”
or “bank account”
at time

t

,
with an initial infusion of amount

M

⁢

(

0

)

at time

0

,
is given by

M

⁢

(

t

)

=

M

⁢

(

0

)

⁢

e

r

⁢

t

.

Title

Black-Scholes formula

Canonical

name

BlackScholesFormula

Date of creation

2013-03-22 16:29:59

Last modified on

2013-03-22 16:29:59

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

15

Author

stevecheng (10074)

Entry type

Topic

Classification

msc 60H10

Classification

msc 91B28

Synonym

Black-Scholes pricing formula

Related topic

AnalyticSolutionOfBlackScholesPDE

Related topic

DerivationOfBlackScholesFormulaInMartingaleForm

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