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Next: Itô's formula Up: Modelling: Continuous state space Previous: Modelling: Continuous state space Contents

Brownian motion in continuous time

The main component of the asset price in this context is a Brownian motion in continuous time. This can be intuitively thought of as the limit of the random walks

$\displaystyle \Delta _{\delta }B\left( t\right) =B\left( t+\delta \right) -B\le... ...ght) =\sqrt{\delta }\varepsilon \left( t+\delta \right) +o\left( \delta \right)$ ,

where the innovations $\varepsilon \left( t\right)$ satisfy the usual conditions, and the limit is taken as $\delta \rightarrow 0$ . For example one can visualize how the convergence is achieved when the number of steps go from

in figure 7.1. The mean is taken to be equal to one, and the volatility equal to 2.

**Figure 7.1:** Brownian and Geometric Brownian Motion paths. One can also observe the expected value and the $\pm 40 \%$ intervals. Samples of 2000 and 40 points are generated to show the convergence from discrete to continuous time.
$\begin{figure}\centering %% \epsfig{file=c:/miktex/mymiktex/derivatives/v2003/NE... ...=c:/miktex/mymiktex/derivatives/v2003/NET/Brown04.eps,height=5cm}%% \end{figure}$

Now the process $B\left( t\right)$ , thought of as a function of time has two very important properties:

$B\left( t\right)$ is almost surely continuous for all , and
$B\left( t\right)$ is almost surely nowhere differentiable for all .

The Brownian motion $B\left( t\right)$ is generating a filtration $%% \left\{ \mathscr{F}\left( t\right) \right\} _{t\geq 0}$ in a similar way that a discrete time random walk generated the filtration $\left\{ \mathscr{F}_{t}\right\} _{t=0}^{n}$ . It includes the information that one obtains by observing the Brownian motion up to time . It is easy to show that $B\left( t\right)$ forms a martingale, $\mathbf{E}\left[ B\left( t+s\right) \vert\mathscr{F}\left( t\right) \right] =B\left( t\right)$ . In addition,

$\displaystyle B\left( t\right) -B\left( s\right) \sim \mathcal{N}\left( 0,\left\vert t-s\right\vert \right)$ .

Using the Brownian motion one can define other simple processes, called stochastic differential equations [SDEs], of the form

$\displaystyle dX\left( t\right) =\alpha dt+\beta dB\left( t\right)$ .

The parameter $\alpha$ is the drift while the parameter $\beta$ is the volatility of the process.

If one wants to solve the above process assuming $X\left( 0\right) =0$ , and to find the value of $X\left( t\right)$ , elementary calculus would suggest a solution of the form

$\displaystyle X\left( t\right) =\int_{0}^{t}\alpha ds+\int_{0}^{t}\beta dB\left( s\right)$ .

At this point the problems start to begin: since the function $B\left( t\right)$ is discontinuous everywhere, one cannot approach the second integral of the above expression in the Riemann sense

^7.1. Such integrals called Itô integrals; their general form is

$\displaystyle I\left( t\right) =\int_{0}^{t}\eta \left( s\right) dB\left( s\right)$ ,

where $\eta \left( t\right)$ is $\mathscr{F}\left( t\right)$ -measurable [or adapted] and square integrable $\int_{0}^{t}\eta ^{2}\left( s\right) ds<\infty$ [just a technical condition]. Itô integrals have the following properties:

Adapted-ness: $I\left( t\right)$ is $\mathscr{F}\left( t\right)$ adapted;
Linearity: For every $\alpha _{1},\alpha _{2}\in \mathbbm{R}$ , and $%% \eta _{1}\left( t\right) ,\eta _{2}\left( t\right)$ $\mathscr{F}\left( t\right)$ -measurable functions $\int_{0}^{t}\left[ \alpha _{1}\eta _{1}\left( s\right) +a_{2}\eta _{2}\left( s... ...t( s\right) +\alpha _{2}\int_{0}^{t}\eta _{2}\left( s\right) dB\left( s\right)$ ;
Martingale: $I\left( t\right)$ is a martingale [w.r.t. its filtration $\mathscr{F}\left( t\right)$ ]
Continuity: $I\left( t\right)$ is a continuous function of the upper limit of the integration, .
Itô isometry: $\mathbf{E}I^{2}\left( t\right) =\mathbf{E}%% \int_{0}^{t}\eta ^{2}\left( s\right) ds$ [hence the technical condition].

Example 41 The process $X\left( t\right)$ above will have

$\displaystyle \mathbf{E}X\left( t\right)$	$\displaystyle =$	$\displaystyle \mathbf{E}\left[ \int_{0}^{t}\alpha ds+\int_{0}^{t}\beta dB\left( s\right) \right]$
	$\displaystyle =$	$\displaystyle \alpha t$ , and

$\displaystyle \mathbf{E}\left[ X\left( t\right) -\alpha t\right] ^{2}$	$\displaystyle =$	$\displaystyle \mathbf{E}%% \int_{0}^{t}\beta ^{2}ds$
	$\displaystyle =$	$\displaystyle \beta ^{2}t$ .

We have already said that if $t\geq s$ we have $\mathbf{E}\left[ B\left( t\right) -B\left( s\right) \right] ^{2}=t-s$ [which is just the variance]. We can also show that $\mathbf{Var}\left[ B\left( t\right) -B\left( s\right) \right] ^{2}=2\left[ t-s\right] ^{2}$ . This can be rewritten as

$\displaystyle \mathbf{E}\left[ \Delta _{\delta }B\left( t\right) \right] ^{2}$	$\displaystyle =$	$\displaystyle \delta$ , while
$\displaystyle \mathbf{Var}\left[ \Delta _{\delta }B\left( t\right) \right] ^{2}$	$\displaystyle =$	$\displaystyle o\left( \delta \right)$ .

This can also be expressed as $\left[ \Delta _{\delta }B\left( t\right) %% \right] ^{2}=\delta +o\left( \delta \right)$ . Now if we take the limit $%% s\rightarrow t$ [or $\delta \rightarrow 0$ ] we will have

$\displaystyle \mathbf{E}\left[ dB\left( t\right) \right] ^{2}$	$\displaystyle =$	$\displaystyle dt$ , while
$\displaystyle \mathbf{Var}\left[ dB\left( t\right) \right] ^{2}$	$\displaystyle =$	0.

Therefore although $dB\left( t\right)$ represents an instantaneous random movement, $\left[ dB\left( t\right) \right] ^{2}$ turns out to be deterministic, and in fact equal to

. We write this relationship informally as

$\displaystyle dB\left( t\right) \times dB\left( t\right) =dt$ .

Kyriakos 2003-03-17