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The evolvement of r and S over timeƒ 226

In the Black-Scholes option pricing mo

del (1973), there are two securities,

a

money market account which offers a

constant risk-free interest rate

and a stock

(just like in the binomial asset pricing model).

ƒ

The

money market account follows a deterministic process

such as:

where

r

is the riskless interest rate,

dt

is a small time step, and

dB

is called the t

increment of

B

over the time interval

[t,t+dt]

.

ƒ

The

stock follows a geometric Brownian motion (GBM)

such as:

where

μ

is the constant mean of

S,

dt

is a small time step,

σ

is the constant

standard deviation of

S,

dS

(dWt

)t

is called the increment of

S (W)

over the time

interval

[t,t+dt]

, and

W

is a Wiener process

. For any fixed time interval

[t,t+dt]

the increment

dS

(dWt

)t

is a stochastic variable!

t

t t

dW

dt

dS S

σ

μ

+

=

rdt

dB B

t t

=

Derivative securities: Options - Black-Scholes model