Financial Engineering

3. Brownian Motion

3.1 Introduction

We work on a filtered probability space $(\Omega, \sigmaF, \probP),$ where $\Omega \subset \R$ is the sample space, $\sigmaF$ a $\sigma$-algebra consisting of events and $\probP$ a probability measure.

Definition 3.1 (Filtration): Let $I$ be an index set with a total order. The non-decreasing family of $\sigma$-algebras $(\sigmaF_i)_{i\in I}$ lying in $\sigmaF,$ i.e. $\forall i \in I : \sigmaF_i \subset \sigmaF,$ is called a filtration.

For us, $I$ will denote time and therefore be a subset of $\R_{+}.$ Thus, we endow the probability space with a filtration $(\sigmaF_i)_{i\in I}$ with $I \subset \R_{+}.$

Definition 3.2 (Stochastic process): A real-valued stochastic process is a collection of $\R$-valued random variables usually written as $\qty{X_i \mid i \in I}.$
Definition 3.3 (Sample path): A sample path or trajectory describes a single outcome of a stochastic process, i.e. it is a map $X_{(\cdot)}(\omega) : I \to \R$ for some fixed $\omega.$
Definition 3.4 (Increment): An increment is defined as the difference between two random variables of a certain stochastic process, i.e. $X_{t_i} - X_{t_j}.$

3.1.1 Special Stochastic Processes

Definition 3.5 (Stationary stochastic process): A stochastic process is stationary if all its random variables are identically distributed, i.e. for all $n \in \N$ the random variables $X_{t_1}, \ldots, X_{t_n}$ have the same probability distribution.
Definition 3.6 (Lévy process): A Lévy process is a stochastic process with stationary and independent increments. So for every $n \in \N$ and $t_1 < t_2 < \ldots < t_n,$ the $n-1$ increments $X_{t_2} - X_{t_1}, \ldots, X_{t_n} - X_{t_{n - 1}}$ are independent of each other and the distribution of an increment is already determined by its time difference $\Delta t.$
Definition 3.7 (Markov property): A stochastic process satisfies the Markov property if its value in the future depends on the current value but is conditionally independent of its previous behaviour, i.e. for every $n$ and $t_1 < t_2 < \ldots < t_n,$ it holds $$ \probP[X_{t_n} \leq x_n \mid X_{t_1} = x_1, \ldots, X_{t_{n-1}} = x_{n-1}] = \probP[X_{t_n} \leq x_n \mid X_{t_{n-1}} = x_{n-1}] $$
Definition 3.8 (Martingale): A martingale is a stochastic process where the conditional expectation of a future value $X_t$ given the current values and previous values $\sigmaF_s$ equal the current value, i.e. $\E_{\probP}[X_t \mid \sigmaF_s] = X_s$ for all $s < t.$
Note: In addition, all random variables of the martingale must be integrable, i.e. $\E_{\probP}[\abs{X_t}] < + \infty$ for all $t \in I.$

3.1.2 Convergence of Random Variables

We recall the convergence of random variables.

Definition 3.9 (Convergence of random variables): Let $X_1, X_2, \ldots$ be a sequence of real-valued random variables and $X$ be a random variable as well. Then the sequence converges
  1. almost surely if $\probP[\lim_{n\to \infty} X_n = X] = 1$
  2. in the $L^r$-norm if $\E[\abs{X_n}^r]$ and $\E[\abs{X}^r]$ exist and $\lim_{n \to \infty} \E[\abs{X_n - X}^r] = 0$
  3. in probability if for all $\epsilon > 0$ $\lim_{n \to \infty} \probP[\abs{X_n - X} > \epsilon] = 0$
  4. weakly/in distribution if $\lim_{n \to \infty} F_n(x) = F(x)$ for all $x \in \R$
Note: 1. and 2. imply 3., which in turn implies 4. Moreover, $L^p$ convergence implies $L^q$ convergence for $p > q \geq 1.$

3.1.3 Quadratic variation and covariation

Let $P$ be a partition of $[0,T]$ and its norm given by its mesh size.

Definition 3.10 (Quadratic variation): Let $X_t$ be a process. The quadratic variation is $$ [X_t]_T = \lim_{\norm{P} \to 0} \sum_{k=1}^n (X_{t_k} - X_{t_{k-1}})^2 $$
Definition 3.11 (covariation): Let $X_t$ and $Y_t$ be processes. The covariation is $$ [X_t,Y_t]_T = \lim_{\norm{P} \to 0} \sum_{k=1}^n (X_{t_k} - X_{t_{k-1}})(Y_{t_k} - Y_{t_{k-1}}) $$
Note: These limits are defined as convergence in probability.

3.1.4 Mathematical Formulation of Brownian Motion

$W_t$ is the position at time $t$ in one dimension of a physical particle. The collection of these random variables indexed by the continuous-time parameter $t$ is a stochastic process with the properties mentioned below.

Theorem 3.12 (Brownian motion): A Brownian motion $(W_t)_{t \geq 0}$ with respect to the probability measure $\probP$ and to a filtration $(\sigmaF_t)_{t\geq0}$ is a real-valued stochastic process such that $W_t$ is $\sigmaF_t$-adapted at every time $t$ and satisfies the following conditions:
  • $W_0 = 0$
  • The increments $W_t - W_s$ for all $t$ and $s$ such that $s\leq t$ are independent of $\sigmaF_s$ under $\probP$ and follow the normal distribution $W_t - W_s \sim \lawN(0, t-s)$
  • $W$ has continuous trajectories, i.e. the function $t \to W_t(\omega)$ is continuous $\probP$-almost surely
Note: $\probP$-almost surely means for $\omega \in A$ s.t. $A \in \sigmaF_{\infty}$ and $P(A) = 1.$
Proposition 3.13 (Probability density of Brownian motion): The probability density of the position of a Brownian motion at the end of time period $[0, t]$ is $$ p(x) = \frac{1}{\sqrt{t} \sqrt{2\pi}} \exp(-\frac{x^2}{2t}) $$
Proposition 3.14 (Probability distribution of Brownian motion): The probability distribution of the increment $W_{t+u} - W_t$ is $$ \probP[W_{t+u} - W_t \leq a] = \int_{-\infty}^a \frac{1}{\sqrt{u} \sqrt{2\pi}} \exp(-\frac{x^2}{2u}) \dd{x} $$
Note: As the probability distribution of $W_t$ is normal with standard deviation $\sqrt{t},$ it is the same as $\sqrt{t} Z,$ $Z \sim \lawN(0,1).$ When calculating expectations of Brownian motions, it is convenient to use $\sqrt{t}Z.$

3.2 Stock Price Dynamics

Our goal is to formulate a mathematical object $S$ that can describe the movement of stock prices through time. We define $S: [0,T] \times \Omega \to \R$ on a filtered probability space $(\Omega, \qty{\sigmaF_t}_{t\in[0,T]}, \probP).$ If we had three wishes for free, we would like $S_t$ to satisfy

  1. The trajectories should be almost surely “erratic”, i.e. they should not be differentiable - and hence predictable - with respect to time.
  2. If we know the present, then the future is independent of the past, i.e. $S$ should be a Markov process.
  3. The trajectories should be continuous.

With this wishlist we can define the concept of diffusion processes as follows.

Definition 3.15 (Diffusion process): A one-dimensional Markov process $S : [0,T] \times \Omega \to \R$ on $(\Omega, \sigmaF, \probP)$ is called diffusion process if:
  • $\probP[\abs{S_{t+h} - S_t} \leq \epsilon \mid S_t] = 1 - o(h)$
  • $\E_{\probP}[S_{t+h} - S_t \mid S_t] = \mu(S_t, t) h + o(h),$ i.e. the local expectation of the changes are approximately proportional to the length $h$ of the time interval $[t, t+h]$
  • $\E_{\probP}[(S_{t+h} - S_t)^2 \mid S_t] = \sigma^2 (S_t, t) h + o(h),$ i.e the loval variance is also approximately proportional to the length $h.$

Choosing Brownian motions as a diffusion process for stock prices seems like a good fit. To get a first intuition, we look at a discrete time model, the Random Walk, which will also show us why the variance of Brownian motions is proportional to time.

3.3 Random Walk

For the Random Walk we assume that the time $t$ is discrete, i.e. $t \in {t_0, t_1, \ldots, t_n}$ with layer spacing $\Delta t = t_{k+1} - t_k = \frac{T}{n}.$

Definition 3.16 (Binomial Process): A binomial process $\qty{W_k}_{k=0}^n$ on $[0,T]$ is defined as
  • $W_0 = 0$
  • up and down moves have size $\sqrt{\Delta t}$ with probability $p = \frac{1}{2}$

Hence, if $\qty{X_1, X_2, \ldots}$ is a sequence of independent binomial random variables taking values $\pm \sqrt{\Delta t}$ with equal probability, then the position of the particle at time time $T$ is $W_n = \sum_{i=1}^n X_i.$

Note: We call the process $W_n$ a Random Walk.
Proposition 3.17 (Properties of the Random Walk): The Random Walk $W_n$ has the following properties:
  1. $\E[W_n] = 0$
  2. $\Var[W_n] = T$
  3. $W_n$ is a Markov process
  4. $W_n$ can reach $n+1$ values
  5. $\lim_{n\to \infty} W_n \stackrel{\text{CLT}}{\sim} \lawN(0,T)$

We will show 5. of Proposition 3.17. For that, recall the definition of the moment generating function.

Definition 3.18 (Moment Generating Function): The moment generating function or MGF of a random variable $X$ is defined as $$ m_X(\theta) = \E[\exp(\theta X)] $$ for all values $\theta$ for which the expectation exists.
Proof: The probability distribution of $W_n$ is determined uniquely by its MGF, i.e. $$ m_{W_n}(\theta) = \E[\exp(\theta W_n)] = \prod_{k=1}^n \E[\exp(\theta X_k)] = \E[\exp(\theta X)]^n $$ Since $\E[\exp(\theta X)] = \frac{1}{2} \exp(\theta \sqrt{\Delta t}) + \frac{1}{2} \exp(- \theta \sqrt{\Delta t}) \approx 1 + \frac{1}{2} \theta^2 \Delta t$ for small $\Delta t$ using power series expansion, we have $$ m_{W_n}(\theta) \approx (1 + \frac{1}{2} \theta^2 \Delta t)^n $$ We write the MGF in log form $$ \ln m_{W_n}(\theta) \approx n \ln(1 + \frac{1}{2} \theta^2 \Delta t) \approx n \frac{1}{2} \theta^2 \Delta t = \frac{1}{2} \theta^2 T $$ where we have used $\ln(1+y) \approx y$ for small $y.$ Hence $m_{W_n}(\theta) \xrightarrow{n \to \infty} \exp(\frac{1}{2} \theta^2 T)$ which is the MGF of the normal distribution $\lawN(0,T).$ Thus in the continuous-time limit of the discrete-time framework, the probability density of the terminal position $W_n$ is the same as that of a Brownian motion that has run an amount of time $T.$

Finally, we prove that 2. of Proposition 3.17, i.e. $\Var[W_n] = T,$ arises only if we choose $\pm \sqrt{\Delta t}$ as the size of the Random Walk increment.

Proof: Let $y$ be the increment over time step $\Delta t$ such that $X_k = \pm y.$ Then $$ \Var(W_n) = n \Var(X_k) = ny^2 = \frac{T}{\Delta t} y^2 = T \frac{y^2}{\Delta t} $$ In order to ensure that $\Var(W_n)$ stays finite as $\Delta t \to 0$ and $y \to 0$ we choose $y^2 = c \Delta T$ where $c > 0$ is a constant. Then $\Var(W_n) = Tc$ and since time units are arbitrary there is no sense in choosing $c$ differently from $1.$ Hence, $y = \pm\sqrt{\Delta t}.$

This legitimizes our choice of $\pm \sqrt{\Delta t}$ as the increment.

3.4 Three Key Properties

3.4.1 Non-Differentiability

For the Random Walk, the slope of the path is $\frac{\abs{W_{k+1}-W_k}}{\Delta t}=\frac{1}{\sqrt{\Delta t}}$ which becomes infinite as $\Delta t \to 0.$ Recall the definition of differentiability.

Definition 3.19 (Differentiable function): A continuous function $f$ is differentiable at $x$ if below and above limits exist and they coincide, i.e. $$ \lim_{h \uparrow 0}\frac{f(x+h)-f(x)}{h} = \lim_{h \downarrow 0} \frac{f(x+h)-f(x)}{h} $$

We prove that Brownian motion paths are nowhere differentiable.

Proof: For the Brownian motion path, we consider the rate of change over $[t,t + \Delta t],$ i.e. $X_{\Delta t} = \frac{W_{t + \Delta t} - W_t}{\Delta t} \sim \lawN\qty(0, \frac{1}{\Delta t}).$ Hence, we can write $X_{\Delta t} \sim \frac{Z}{\sqrt{\Delta t}}$ with $Z$ standard normal. Take any interval $A = (a, b],$ then $\probP[X_{\Delta t} \in A] = \probP[a \sqrt{\Delta t} < Z \leq b \sqrt{\Delta t}].$ As $\Delta t \to 0$ we get $\probP[X_{\Delta t} \in A] = 0.$ As $A$ can be chosen arbitrarily, we conclude that the rate of change is not finite. And since $t$ is arbitrary, we also conclude that the Brownian motion path is nowhere differentiable.

3.4.2 Continuity

We prove that Brownian motion paths are continuous.

Proof: We calculate \begin{align*} \probP[\abs{W_{t + \Delta t} - W_t} < \epsilon] &= \frac{1}{\sqrt{2\pi \Delta t}} \int_{-\epsilon}^{\epsilon} \exp(- \frac{x^2}{2 \Delta t}) \dd{x} \\ &= \frac{1}{\sqrt{2\pi}} \int_{-\frac{\epsilon}{\sqrt{\Delta t}}}^{\frac{\epsilon}{\sqrt{\Delta t}}} \exp(- \frac{u^2}{2}) \dd{u} \end{align*} For $\Delta t \to 0$ we have $$ \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \exp(- \frac{u^2}{2}) \dd{u} = 1 $$ Hence, the Brownian motion path is continuous.

3.4.3 Variability

Proposition 3.20 (Quadratic variation of Brownian motion): The quadratic variation of a Brownian motion $W_t$ after time $T$ is $[W_t]_T = T.$

We prove this property by showing that $[W_t]_T$ converges in $L^2$-norm to $T,$ which implies convergence in probability by Definition 3.9.

Proof: We consider $[0,T]$ with partition $\Delta = \frac{T}{n}$ and $t_k = k \Delta t.$ Then \begin{align*} & \E\qty[\qty(\sum_{k=0}^{n-1} (W_{t_{k+1}} - W_{t_k})^2 - T)^2] \\ =& \E\qty[\qty(\sum_{k=0}^{n-1} ((W_{t_{k+1}} - W_{t_k})^2 - \Delta t))^2] \end{align*} Expanding the square results in full square terms $\qty((W_{t_{k+1}} - W_{t_k})^2 - \Delta t)^2$ and cross-terms \begin{align*} &\qty((W_{t_{k+1}} - W_{t_k})^2 - \Delta t)\qty((W_{t_{j+1}} - W_{t_j})^2 - \Delta t) = \Delta t^2 \qty(Z_k^2 - 1)\qty(Z_m^2 - 1) \end{align*} where $Z_k, Z_m$ are iid $\lawN(0,1).$ Taking the expectation of the cross-terms results in $$ \Delta t^2 \qty(\E[Z_k^2] - 1) \qty(\E[Z_m^2] - 1) = 0 $$ as $Z_k^2, Z_m^2$ are iid $\lawChi_1$ and have mean $\E[Z_k^2] = \E[Z_m^2] = 1.$ Hence \begin{align*} & \E\qty[\qty(\sum_{k=0}^{n-1} (W_{t_{k+1}} - W_{t_k})^2 - T)^2] \\ =& \E\qty[\sum_{k=0}^{n-1} \qty((W_{t_{k+1}} - W_{t_k})^2 - \Delta t)^2] \\ =& \sum_{k=0}^{n-1} \E\qty[\qty((W_{t_{k+1}} - W_{t_k})^2 - \Delta t)^2] \end{align*} Expanding the square, results in \begin{align*} & \E\qty[\qty(W_{t_{k+1}}-W_{t_k})^4]-2 \E\qty[\qty(W_{t_{k+1}}-W_{t_k})^2] \Delta t+\Delta t^2 \\ = & \E\qty[\qty(\sqrt{\Delta t} Z)^4]-2 \E\qty[\qty(\sqrt{\Delta t} Z)^2] \Delta t+\Delta t^2 \\ = & \Delta t^2(3-2+1) \\ = & 2\frac{T^2}{n^2} \end{align*} which tends to $0$ as $n \to \infty.$ Hence $[W_t]_T = T$ in $L^2$-norm.
Proposition 3.21 (First variation of Brownian motion): A Brownian motion has unbounded first variation, i.e. given a partition $P$ of $[0,T]$ the limit $\lim_{\norm{P} \to 0} \sum_{k=1}^n \abs{X_{t_k} - X_{t_{k-1}}}$ is unbounded.
Note: The first variation is also called total variation. In general, any continuous process with non-zero quadratic variation has unbounded first variation.

3.4.4 Limit of Random Walk

We have already demonstrated that if the mesh size tends to zero, the Random Walk converges to the Brownian motion. That is why Brownian motions are the most widely used processes for modelling randomness in finance.