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4 Stochastic Processes
4.2 Wiener Process
From the discussion in the previous section it should be obvious that the price of shares
S and options have an average drift upwards rate ρ and a superimposed randomly
fluctuating part, characterized by the rms spread or volatility σ. This directly implies
to write the temporal evolution of S as a stochastic process and express it as
d S = ρ Sdt + σ SdW (t)
or
ds = ρsdt + σ sdW (t) ,
(4.7)
where we introduce s = S/S 0 and use the initial share value S 0 to normalize. dW
is a white-noise (Wiener) process. An equation of this type, in physics often an
equation of motion subject to additional random forces, is called a Langevin equation.
Remarkably, one of the first examples appears in Bachelier’s thesis [2] on finance,
which precedes Einstein’s famous analysis of Brownian motion [3] by a few years.
Note that in the absence of noise (σ = 0) we recover a purely exponential growth
with rate ρ. Equation 4.7 is an example of a stochastic differential equation. Since
working with them is not everyday fare, we will discuss their ingredients and the
rules of manipulating them in some detail. Let us start with the source of randomness
in (4.7), the Wiener process dW itself.
In order to discuss the Wiener process dW let us consider the simplified Langevin
equation dx = σ dW , which basically means that the particle with coordinate x
receives a sequence of random kicks σ ξ, where ξ has rms amplitude unity. The
values of ξ are sampled from a Gaussian distribution
φ(ξ ) =
1
√
2π
e
−ξ
2 /2
.
(4.8)
In order to calculate the expected value g(ξ ) of a function g(ξ ) that depends on the
random variable ξ , we need to average over many realizations of random numbers,
denoted by g(ξ ) =
g(ξ )φ(ξ )dξ . By direct integration of g(ξ ) = ξ and g(ξ ) = ξ
2
we find
ξ = 0
and
ξ
2
= 1 ,
(4.9)
which shows us that the mean is zero and the rms amplitude is unity. If we introduce
a discretized form of the equation of motion, we obtain
x(t n ) = x(t n−1 ) + σ ξ(t n )
or
x(t n ) = σ
n
i=1
ξ(t i ) ,
(4.10)
where we assume that time is discretized in small steps t = t n − t n−1 . The second
equation tells us that the value of x at time t n is just the sum of n random numbers
with the properties specified above. It is easy to see that for average position x(t n )
and second moment averaged over many realizations of the random numbers, are
given by
4 Stochastic Processes
4.2 Wiener Process
From the discussion in the previous section it should be obvious that the price of shares
S and options have an average drift upwards rate ρ and a superimposed randomly
fluctuating part, characterized by the rms spread or volatility σ. This directly implies
to write the temporal evolution of S as a stochastic process and express it as
d S = ρ Sdt + σ SdW (t)
or
ds = ρsdt + σ sdW (t) ,
(4.7)
where we introduce s = S/S 0 and use the initial share value S 0 to normalize. dW
is a white-noise (Wiener) process. An equation of this type, in physics often an
equation of motion subject to additional random forces, is called a Langevin equation.
Remarkably, one of the first examples appears in Bachelier’s thesis [2] on finance,
which precedes Einstein’s famous analysis of Brownian motion [3] by a few years.
Note that in the absence of noise (σ = 0) we recover a purely exponential growth
with rate ρ. Equation 4.7 is an example of a stochastic differential equation. Since
working with them is not everyday fare, we will discuss their ingredients and the
rules of manipulating them in some detail. Let us start with the source of randomness
in (4.7), the Wiener process dW itself.
In order to discuss the Wiener process dW let us consider the simplified Langevin
equation dx = σ dW , which basically means that the particle with coordinate x
receives a sequence of random kicks σ ξ, where ξ has rms amplitude unity. The
values of ξ are sampled from a Gaussian distribution
φ(ξ ) =
1
√
2π
e
−ξ
2 /2
.
(4.8)
In order to calculate the expected value g(ξ ) of a function g(ξ ) that depends on the
random variable ξ , we need to average over many realizations of random numbers,
denoted by g(ξ ) =
g(ξ )φ(ξ )dξ . By direct integration of g(ξ ) = ξ and g(ξ ) = ξ
2
we find
ξ = 0
and
ξ
2
= 1 ,
(4.9)
which shows us that the mean is zero and the rms amplitude is unity. If we introduce
a discretized form of the equation of motion, we obtain
x(t n ) = x(t n−1 ) + σ ξ(t n )
or
x(t n ) = σ
n
i=1
ξ(t i ) ,
(4.10)
where we assume that time is discretized in small steps t = t n − t n−1 . The second
equation tells us that the value of x at time t n is just the sum of n random numbers
with the properties specified above. It is easy to see that for average position x(t n )
and second moment averaged over many realizations of the random numbers, are
given by