The Wiener process, also known as Brownian motion, is the basic process of continuous-time financial modeling.
To visualize a Wiener process, consider a sequence of up and down moves of the price process, St. The up or down moves are determined at times tk tk–1 + t,k 0, ,n. At each tk, the up or down amount is
Y = E[X Ft] = Et[ ]X
X = Et[ ]X
E E[ [X H] G] = E[X G]
t = t s ,
Et[ ]Ss = t
= = …
E[E [ ]X t£s
E[ ]X ]
algebras are elements of a filtration F. In this case, since F ⊂F, t s, we£
Assume that S is a stochastic process adapted to F, 0£t T£ . The process S
S, 0 t s T
£
£ £
determined by sampling from a normal distribution with mean 0 and vari- ance t:
(2.13) Here, Z is a standard normal distribution and is a sample point in the sample space . The sample point represents a sequence of up and down moves along the trajectory of the Wiener process. We would get a standard Wiener process by letting t 0. The properties of the Wiener process are the following:
■ For each sample ,W(t,) is a continuous function of t.
■ The initial condition of a Wiener process is W(t 0,) 0 a.s.
Almost surely (a.s.) means that the probability of W(0,) 0 is 1.
■ The increments of the Wiener process are normal and independent.
This means
(2.14)
For simplicity, we omit reference to the sample point , and use sub- scripts for the time dependence.
A Wiener process is adapted to a -algebra Ft. The filtration can be the one generated by the Wiener process itself, or it can be one generated by the Wiener process as well as other processes, as long as the other processes don’t reveal information about future movements of the Wiener process.
The following are additional properties of the Wiener process:
■ The Wiener process is Markovian. This means that for 0 t s, con- ditional on Ft, everything random about Ws(such as the mean, vari- ance, and so forth) depends only on Wt.
■ The statement above implies that the Wiener process is a martingale:
Et[Ws] Wt, t s.
■ The Wiener process is an infinitely “wiggly” function that does not have a defined slope or tangent. The mathematical concept that cap- tures the infinite wiggliness of a process is called second variation or quadratic variation. Differentiable functions have zero second varia- tion. The Wiener process has finite second variation.
To motivate the notion of second variation (SV) as a characterization for the Wiener process, we will first discuss the first variation (FV) and the second variation of a differentiable function.
W t( k+1, ) = W t( k, )+ tZ
→
∈
= =
=
E0[Ws–Wt] = 0, t£s, var[Ws–Wt] = s–t, t£s, cov[(Ws–Wt),(Wv–Wu)] = 0, t£ £ £s u v
Ft { }t 0
=
‡
£ £
£
First Variation of a Differentiable Function
The first variation of a differentiable function is finite, whereas the first variation of the Wiener process is infinite.
Define points in time
(2.15) and define
(2.16) The first variation, also called simply the variation of a function f(t), is defined as
(2.17)
If the function f(t) is differentiable, the mean value theorem and a little algebra give
(2.18) This means that, in general, the first variation of a differentiable func- tion will be different from zero.
First Variation of the Wiener Process
The first variation of the Wiener process is infinite. The reason for this will become clear after we discuss the second variation of the Wiener process.
Second Variation of a Differentiable Function The second, or quadratic, variation is defined as
(2.19)
Applying the mean value theorem and some algebra, we find
(2.20) 0 = t0£ £t1 ... £tn = T
= max(ti+1–ti), 0£i<n
FV f( ) f t( i+1)–f t( )i i=0
i=n lim0
=
FV f( ) df t( ) t ---d dt t=0
= t=T
SV f( ) f t( i+1)–f t( )i 2 i=0
i=n lim0
=
SV f( ) df t( ) t ---d 2dt t=0
t=T lim0
=
Since in a differentiable function the integrand is bounded, the second variation of a differentiable function is zero:
SV(f) 0 (2.21)
Second Variation of the Wiener Process
Some algebra shows that the second variation of the Wiener process is equal to its variance:
SV(W(t T)) T (2.22)
This also says that the first variation of the Wiener process is infinite:
(2.23)
Since the second variation is finite and the Wiener process is continu- ous, this shows that the first variation is infinite.
Products of Infinitesimal Increments of Wiener Processes
In using stochastic calculus as a practical tool, the product of two infinites- imal increments of Wiener processes is not a stochastic quantity.
First Practical Result: dWdW=dt
From the derivation of the second variation of the Wiener process we found that (2.24) The quantity (W(t+t) W(t))2 t is a stochastic process whose variance vanishes like t2 as t 0. We also know that E[(W(t+t) W(t))2] t, or equivalently, E[(W(t+t) W(t))2 t] 0. This means that (W(t+t) W(t))2 t tends to zero as t 0. We can write,
(2.25)
=
= =
FV W T( ( )) W t( i+1)–W t( )i i=0
i=n
∑
lim→0
= 1 max(W)
--- W t( i+1)–W t( )i 2
i=0 i=n
∑
lim→0
‡
SV W T( ( )) max(W) ---
‡
var[(W t( +t)–W t( ))2–t] = 2t2
– –
→ –
= – – =
– – →
W t( +t)–W t( )
( )2→t as t→0
Using differential notation, this means
dWdW dt (2.26)
This result is striking in that, to lowest order, the product of these two random quantities is a deterministic quantity.
Second Practical Result: dW1111dW2222= 0= 0= 0= 0 If If If If W1111 and and and and W2222 Are Independent Are Independent Are Independent Are Independent
Corollary: dW1111dW2222=dtIfIfIfIfW1111andandandandW2222AreAreAreAre CorrelatedCorrelatedCorrelatedCorrelated
This is a straightforward consequence of the last result. Assume that Z1 and Z2 are independent Wiener processes. We can construct correlated Wiener processes W1 and W2 as follows:
(2.27) (2.28) It is straightforward to verify that the correlation between dW1 and dW2 have variance dt and correlation coefficient :
(2.29) (2.30)
(2.31) Here we made use of dZ1dZ2 0. The correlation coefficient between dW1 and dW2 is
(2.32)
The product of the increments of two correlated Wiener processes is (2.33)
=
dW1( )t = dZ1( )t
dW2( )t = dZ1( )t + 1–2dZ2( )t
var[dW1( )t ] = var[dZ1( )t ] = dt var[dW2( )t ] = var[dZ1( )t ]+var[dZ2( )t ]
2dt+(1–2)dt
= dt
=
cov[dW1( )dWt 2( )t ] = dZ1dZ2+ 1–2dZ1dZ2 dt
=
=
cov[dZ1, dZ2] var[dZ1] var[dZ2] ---
= dt dt dt ---
=
=
dW1( )dWt 2( )t dZ12
( )t + 1–2dZ2( )dZt 1( )t
= dt
=