A Brief Introduction to Continuous Time Stochastic Processes
Consider the following stochastic differential equation:
dx = a dt + σ dz .
(1)
Here a and σ are parameters and dz denotes a normally-distributed random variable with
a mean of 0 and a variance of dt. The value of dz for any infinitesimal time period dt is
independent of is value in any other infinitesimal time period. This kind of random process is
referred to as a Weiner process or as Brownian motion. Three important properties of a Weiner
process are:
1. dz dt = dt dz = 0.
2. (dz)2 = dt .
3.
∫
t
t +h
(This is weird.)
dz = ( zt+h - zt ),
and ( zt+h - zt ) is normally-distributed with mean 0
and variance h .
What Equation (1) says is that the value of variable x changes in (infinitesimal) amounts
over very small increments of time as a result of two factors. The first is a deterministic drift
factor, a dt , which causes x to rise ( a > 0) or fall ( a < 0) by a constant amount per unit of time.
The second factor is the stochastic term, σ dz, which imparts randomness to movements in x.
It is obvious that E[dx] = a dt and Var[dx] = σ 2 dt . It is also obvious that
∫
T
0
dx = xT - x0 = aT + σ ( zT - z0 ) .
(2)
Here ( zT - z0 ) has mean 0 and variance T ; thus, xT is normally-distributed with a mean
of (x0 + aT ) and a variance of σ2 T .
Now let S denote the price of some asset which pays no dividends. The usual continuous
time representation is
dS
S = µ dt + σ dz .
(3)
Equation (3) says that the proportionate rate of change in the value of this asset (which is
its instantaneous rate of return) follows a Weiner process with an expected value of µ dt and an
T dS
instantaneous variance σ dt . One might expect that
would be equal to [ln ST - ln S0 ],
0
S
∫
1
dS
is a function of a random variable S and the ‘usual’ rules of calculus
S
do not apply in this case. The rule that does apply is something called Ito’s Lemma, which is
stated as follows:
but that is not the case.
(Ito’s Lemma)
Consider a stochastic process of the very general form dx = g(x,t) dt + k(x,t) dz ,
and consider a variable y that is a function of x and time : y = F(x,t) . Then y
satisfies the following stochastic differential equation:
dy = ( MF/Mx) dx + ( MF/Mt) dt + ½ (M2F/ Mx2) (dx)2.
(4)
This looks pretty frightening but it is really quite easy to apply. Let’s apply it to
determine the stochastic behaviour of the logarithm of security price S described in Equation (3).
First, to verify that dS satisfies the general form required (for dx) in Ito’ Lemma, re-write
Equation (3) as
dS = µS dt + σS dz .
(5)
The function under consideration here is F(S) = ln S , so the three partial differentials on
the RHS of Equation (4) are readily seen to be
( MF/MS) = 1/S
( MF/Mt) = 0
(M2F/ Mx2) = - 1/S2 .
Assemble these according to Equation (4) to get
d ln S = (1/S) dS - ½ (1/S2) (dS)2 .
(6)
dS is given by the RHS of Equation (5). To find (dS)2 we have to square the RHS. This
yields (dS)2 = (µS )2(dt)2 + 2 µ σS2 dt dz + σ2 S2 (dz)2 . But from Properties 1. and 2. for Weiner
processes, dt dz = 0 , (dz)2 = dt , and (dt)2 always equals 0 in calculus. Therefore (dS)2 = σ2 S2 dt.
Now put all of the pieces together
d ln S = (µ - ½ σ 2 ) dt + σ dz .
(7)
Now we integrate Equation (7) over the finite interval [0,T] to find the value of ln ST .
ln ST = ln S0 + (µ - ½ σ 2 )T + σ ( zT - z0 ).
(8)
ln ST is normally distributed with mean S0 + ( µ - ½ σ 2 )T and variance σ 2T . ST has a
lognormal distribution with an expected value of S0 e µT .
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This stuff is really quite simple. Now let’s set up the Black-Scholes option pricing
formula for the price of a call option. This is easy to do in a risk-neutral world in which all
assets earn the instantaneous risk-free rate of return, rf , and the price of any asset is simply the
expected value of its payoff, discounted back to the present at the risk-free rate.
Let C denote the price at date 0 of a European call on a security with current price S0.
The call has an exercise price of X and an exercise date of T. C is simply the discounted value of
the expected payoff at date T :
C = exp (- rf T )
∫
+∞
X
(S - X ) f (S ) dS ,
(9)
where f (S ) is the probability density function for the share price. We must remember that this is
a risk-neutral world, so ln ST = ln S0 + (rf - ½ σ 2 )T + σ ( zT - z0 ); that is, the expected
(instantaneous) rate of return on the share is the risk-free rate rf .
The solution to Equation (9) is the Black-Scholes formula. To see this, consult the
Supplement pertaining to lognormal probabilities .
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