108156.pdf

70 Mathematics for Finance

5.W(t) has a normal distribution with mean 0 and variancet,thatis,with

density√ 21 πte−x

2

2 t. This is related to the distribution ofwN(t). The latter is

not normal, but approaches the normal distribution in the limit according

to the Central Limit Theorem.

An important difference betweenW(t)andwN(t)isthatW(t) is defined for
allt≥0, whereas the time inwN(t) is discrete,t=n/Nforn=0, 1 , 2 ,....
The price process obtained in the limit fromSN(t)asN →∞will be
denoted byS(t). WhileSN(t) satisfies the approximate equation (3.8) with the
appropriate substitutions, namely

SN(t+

1

N

)−SN(t)≈

(

m+

1

2

σ^2

)

SN(t)

1

N

+σSN(t)(wN(t+

1

N

)−wN(t)),

the continuous-time stock pricesS(t) satisfy an equation of the form

dS(t)=

(

m+^1

2

σ^2

)

S(t)dt+σS(t)dW(t). (3.9)

Here dS(t)=S(t+dt)−S(t)anddW(t)=W(t+dt)−W(t) are the increments
ofS(t)andW(t) over an infinitesimal time interval dt. The explicit formulae
for the solutions are also similar,

SN(t)=SN(0) exp(mt+σwN(t))

in the discrete case, whereas

S(t)=S(0) exp(mt+σW(t))

in the continuous case.

Figure 3.11 Density of the distribution ofS(10)

SinceW(t) has a normal distribution with mean 0 and variancet, it follows
that lnS(t) has a normal distribution with mean lnS(0) +mtand varianceσ^2 t.
Because of this it is said that the continuous-time price processS(t) has thelog