In a previous article on the site we have introduced stochastic calculus in the context of its role in quantitative finance. The Markov and Martingale properties have also been defined in order to prepare us for the necessary mathematical tools used to model asset price paths.
In both of these articles it was stated that Brownian motion would provide a model for path of an asset price over time. In this article Brownian motion will be formally defined and its mathematical analogue, the Wiener process, will be explained. It will be shown that a standard Brownian motion is insufficient for modelling asset price movements and that a geometric Brownian motion is more appropriate.
In the previous discussion on the Markov and Martingale properties a discrete coin toss experiment was carried out with an arbitrary number of time steps. The current goal is to work towards a continuous-time random walk, which will provide a more sophisticated model for the time-varying price of assets. In order to achieve this, the number of time steps will need to be increased. However, the manner in which they are increased must occur in a specific fashion, so as to avoid a nonsensical (infinite) result.
Consider a continuous real-valued time interval $[0,T]$, with $T > 0$. In this interval $N$ coin tosses will be carried out, which each take a time $T/N$. Hence the coin tosses will be spaced equally in time. Concurrently the payoff returned from each coin toss will be modified. The sequence of discrete random variables representing the coin toss is $Z_i \in \{-1,1\}$. A further sequence of discrete random variables, $\tilde{Z}_i \in \{\sqrt{T/N},-\sqrt{T/N}\}$, can also be defined. This definition of such a sequence of discrete random variables is used to provide a very specific quadratic variation of the coin toss.
The quadratic variation of a sequence of DRVs is defined as the sum of the squared differences of the current and previous terms:
\begin{eqnarray*} \sum^i_{k=1}\left(S_k-S_{k-1}\right)^2 \end{eqnarray*}For $Z_i$, the first coin toss random variable sequence, the quadratic variation is given by:
\begin{eqnarray*} \sum^i_{k=1}\left(S_k-S_{k-1}\right)^2 = i \end{eqnarray*}For $\tilde{Z}_i$ the quadratic variation of the partial sums $\tilde{S}_i$ is:
\begin{eqnarray*} \sum^N_{k=1}\left(\tilde{S}_k-\tilde{S}_{k-1}\right)^2 = N \times \left(\sqrt{\frac{T}{N}}\right)^2 = T \end{eqnarray*}Thus, by construction, the quadratic variation of the amended coin toss $\tilde{Z}_i$ is simply the total duration of all tosses, $T$.
Importantly, note that both the Markov and Martingale properties are retained by $\tilde{Z}_i$. As $N\rightarrow \infty$ the random walk coin toss does not diverge. If the value of the asset at time $t$, with $t\in[0,T]$, is given by $S(t)$, then its conditional expectation at the end of the interval, given that $S(0) = 0$, is $\mathbb{E}(S(T))=0$ with a variance of $\mathbb{E}(S(T)^2) = T$.
Although the technical details will not be discussed, as the number of steps $N$ becomes infinite, the Wiener process is obtained, more commonly called a standard Brownian motion, which will be denoted by $B(t)$. Formally, the definition is given by:
Definition: Wiener Process/Standard Brownian Motion
A sequence of random variables $B(t)$ is a Brownian motion if $B(0)=0$, and for all $t,s$ such that $s < t$, $B(t)-B(s)$ is normally distributed with variance $t-s$ and the distribution of $B(t)-B(s)$ is independent of $B(r)$ for $r \leq s$.
Properties of Brownian Motion
Standard Brownian motion has some interesting properties. In particular:
- Brownian motions are finite. The construction of $\tilde{Z}_i$ was chosen carefully in order that in the limit of large $N$, $B$ was both finite and non-zero.
- Brownian motions have unbounded variation. This means that if the sign of all negative gradients were switched to positive, then $B$ would hit infinity in an arbitrarily short time period.
- Brownian motions are continuous. Although Brownian motions are continuous everywhere they are differentiable nowhere. Essentially this means that a Brownian motion has fractal geometry. This has important implications regarding the choice of calculus methods used when Brownian motions are to be manipulated.
- Brownian motions satisfy both the Markov and Martingale properties. The conditional distribution of $B(t)$ given information until $s < t$ is dependent only on $B(s)$ and, given information until $s < t$, the conditional expectation of $B(t)$ is $B(s)$.
- Brownian motions are strongly normally distributed. This means that, for $s < t$, $s,t\in[0,T]$, that $B(t)-B(s)$ is normally distributed with mean zero and variance $t-s$.
Note that the last property is NOT the same as having B(t) normally distributed with mean zero and variance $t$ - this is a weaker property.
Brownian motions are a fundamental component in the construction of stochastic differential equations, which will eventually allow derivation of the famous Black-Scholes equation for contingent claims pricing.
