Financial Engineering

3. Brownian Motion

3.1 Introduction

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

Definition 3.1 (Filtration): Let T\set{T} be an index set with a total order. The non-decreasing family of σ\sigma-algebras (Ft)t∈T(\sigmaF_t)_{t\in \set{T}} lying in F\sigmaF, i.e. ∀t∈T:Ft⊂F\forall t \in \set{T} : \sigmaF_t \subset \sigmaF, is called a filtration.

For us, T\set{T} will denote time and therefore be a subset of R+\R_{+}.

Definition 3.2 (Stochastic process): A real-valued stochastic process is a collection of R\R-valued random variables usually written as {Xt:Ω→R∣t∈T}\cb{X_t : \Omega \to \R \mid t \in \set{T}}.
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(⋅)(ω):T→RX_{(\cdot)}(\omega) : \set{T} \to \R for some fixed ω∈Ω\omega \in \Omega.
Definition 3.4 (Increment): An increment is defined as the difference between two random variables of a certain stochastic process, i.e. Xti−XtjX_{t_i} - X_{t_j} where ti,tj∈Tt_i, t_j \in \set{T}.

3.1.1 Special Stochastic Processes

Definition 3.5 (Markov process): 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 nn and t1<t2<…<tnt_1 < t_2 < \ldots < t_n, it holds P(Xtn≤xn∣Xt1=x1,…,Xtn−1=xn−1)=P(Xtn≤xn∣Xtn−1=xn−1) \P(X_{t_n} \leq x_n \mid X_{t_1} = x_1, \ldots, X_{t_{n-1}} = x_{n-1}) = \P(X_{t_n} \leq x_n \mid X_{t_{n-1}} = x_{n-1})
Note: We can write the Markov property for continuous-time stochastic processes as P(Xt+s∈A∣Ft)=P(Xt+s∈A∣Xt)\P(X_{t+s} \in A \mid \sigmaF_t) = \P(X_{t+s} \in A \mid X_t) for all s>0s > 0.
Example (Markov process): We will later see that a simple Random Walk is a Markov process.
Definition 3.6 (Martingale): A martingale is a stochastic process where the conditional expectation of a future value XtX_t given the current value XsX_{s} and previous values Fs\sigmaF_s equals the current value, i.e. EP[Xt∣Fs]=Xs\E_{\P}[X_t \mid \sigmaF_s] = X_s for all s<ts < t.

In other words, our expected gain in the process is zero at all times. We can also view this definition as a mathematical formalization of a game of chance being fair.

Note: For the martingale definition to be valid, the first moment of the random variables of the martingale must exist, i.e. EP[∣Xt∣]<∞\E_{\P}[\abs{X_t}] < \infty for all t∈Tt \in \set{T}.

A process can be a martingale without being a Markov process and vice versa. This is because a Markov process's definition is about the conditional distribution of future values depending only on the current value, while a martingale's definition is about the conditional expectation of future values being equal to the current value.

Definition 3.7 (Stationary stochastic process): A stochastic process is stationary if all its random variables are identically distributed, i.e. for all n∈Nn \in \N the random variables Xt1,…,XtnX_{t_1}, \ldots, X_{t_n} have the same probability distribution.
Example (Stationary stochastic process): The process {Xt}\cb{X_t} with Xt∼N⁡(0,σ2)X_t \sim \lawN(0, \sigma^2) for all t∈Tt \in \set{T} is stationary.
Definition 3.8 (Lévy process): A Lévy process is a stochastic process with stationary and independent increments, i.e. for every n∈Nn \in \N and t1<t2<…<tnt_1 < t_2 < \ldots < t_n, the n−1n-1 increments Xt2−Xt1,…,Xtn−Xtn−1X_{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 fully determined by its time difference Δt\Delta t.
Note:
  • All Lévy processes are Markov processes.
  • A Lévy process is a martingale if and only if its mean is zero, i.e. E[X0]=0\E[X_0] = 0.
Example (Lévy process): We will later see that Brownian motion is a Lévy process.

3.1.2 Convergence of Random Variables

We recall the convergence of random variables.

Definition 3.9 (Convergence of random variables): Let X1,X2,…X_1, X_2, \ldots be a sequence of real-valued random variables and XX be a random variable as well. Then the sequence converges
  1. almost surely if P[lim⁡n→∞Xn=X]=1\P[\lim_{n\to \infty} X_n = X] = 1
  2. in the LrL^r-norm if E[∣Xn∣r]\E[\abs{X_n}^r] and E[∣X∣r]\E[\abs{X}^r] exist and lim⁡n→∞E[∣Xn−X∣r]=0\lim_{n \to \infty} \E[\abs{X_n - X}^r] = 0
  3. in probability if for all ε>0\epsilon > 0 lim⁡n→∞P[∣Xn−X∣>ε]=0\lim_{n \to \infty} \P[\abs{X_n - X} > \epsilon] = 0
  4. weakly/in distribution if lim⁡n→∞Fn(x)=F(x)\lim_{n \to \infty} F_n(x) = F(x) for all x∈Rx \in \R
Note: 1. and 2. imply 3., which in turn implies 4. Moreover, LpL^p convergence implies LqL^q convergence for p>q≥1p > q \geq 1.

3.1.3 Quadratic variation and covariation

Let PP be a partition of [0,T][0,T] and its norm ∥P∥\norm{P} given by its mesh size max⁡1≤k≤n∣tk−tk−1∣\max_{1 \leq k \leq n} \abs{t_k - t_{k-1}}.

Definition 3.10 (Quadratic variation): Let XtX_t be a process. The quadratic variation is [Xt]T=Plim⁡∥P∥→0∑k=1n(Xtk−Xtk−1)2 [X_t]_T \peq \lim_{\norm{P} \to 0} \sum_{k=1}^n (X_{t_k} - X_{t_{k-1}})^2
Note: The quadratic variation vanishes for differentiable functions. For stochastic processes this is not necessarily the case as we will see later.
Definition 3.11 (Covariation): Let XtX_t and YtY_t be processes. The covariation is [Xt,Yt]T=Plim⁡∥P∥→0∑k=1n(Xtk−Xtk−1)(Ytk−Ytk−1) [X_t,Y_t]_T \peq \lim_{\norm{P} \to 0} \sum_{k=1}^n (X_{t_k} - X_{t_{k-1}})(Y_{t_k} - Y_{t_{k-1}})
Note: Both the quadratic variation and the covariation are defined as convergence in probability.

3.1.4 Mathematical Formulation of Brownian Motion

Let WtW_t denote the position at time t∈Tt \in \set{T} in one dimension of a physical particle. The collection of these random variables indexed by the continuous-time parameter tt is a stochastic process with the properties mentioned below.

Theorem 3.12 (Brownian motion): The Brownian motion {Wt}\cb{W_t} is the real-valued stochastic process that satisfies the following:
  • starting point: W0=0W_0 = 0
  • Lévy process: non-overlapping increments Wt−WsW_t - W_s for all t,s∈Tt,s \in \set{T} such that s≤ts\leq t are independent and follow the normal distribution Wt−Ws∼N⁡(0,t−s)W_t - W_s \sim \lawN(0, t-s)
  • continuous paths: WW has continuous paths, i.e. t→Wt(ω)t \to W_t(\omega) are continuous almost surely

From the definition of Brownian motion, we derive the probability density and probability distribution.

Proposition 3.13 (Probability density of Brownian motion): The probability density of the position xx of a Brownian motion at the end of the time period [0,t][0, t] is p(x)=12πtexp⁡(−x22t) p(x) = \frac{1}{\sqrt{2\pi t}} \exp{-\frac{x^2}{2t}}
Proposition 3.14 (Probability distribution of Brownian motion): The probability distribution of the increment Wt+u−WtW_{t+u} - W_t is P[Wt+u−Wt≤a]=∫−∞a12πuexp⁡(−x22u) dx \P[W_{t+u} - W_t \leq a] = \int_{-\infty}^a \frac{1}{\sqrt{2\pi u}} \exp{-\frac{x^2}{2u}} \dd{x}
Note: As the probability distribution of WtW_t is normal with standard deviation t\sqrt{t}, it is the same as tZ\sqrt{t} Z, Z∼N⁡(0,1)Z \sim \lawN(0,1). When calculating expectations of Brownian motions, it is convenient to use tZ\sqrt{t}Z.

3.2 Stock Price Dynamics

Our goal is to formulate a mathematical object SS that can describe the movement of stock prices through time. We define {St}\cb{S_t} as a real-valued continuous-time process on a filtered probability space (Ω,{Ft}t∈[0,T],P)(\Omega, \cb{\sigmaF_t}_{t\in[0,T]}, \P). We would like it to satisfy the following:

  1. If we know the present, then the future is independent of the past, i.e. {St}\cb{S_t} should be a Markov process.
  2. The paths should be continuous.
  3. The paths should be almost surely “erratic”, i.e. they should not be differentiable, and hence predictable, with respect to time.

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

Definition 3.15 (Diffusion process): A diffusion process {St}\cb{S_t} is a real-valued stochastic process that satisfies the following:
  • Markov process: P(St+h∈A∣Ft)=P(St+h∈A∣St)\P(S_{t+h} \in A \mid \sigmaF_{t}) = \P(S_{t+h} \in A \mid S_{t})
  • continuous paths: P[∣St+h−St∣≤ε∣St]=1−o(h)\P[\abs{S_{t+h} - S_t} \leq \epsilon \mid S_t] = 1 - o(h)
  • local expectation: EP[St+h−St∣St]=μh+o(h)\E_{\P}[S_{t+h} - S_t \mid S_t] = \mu h + o(h), i.e. the local expectation of the changes are approximately proportional to the length hh
  • local variance: EP[(St+h−St)2∣St]=σ2h+o(h)\E_{\P}[(S_{t+h} - S_t)^2 \mid S_t] = \sigma^2 h + o(h), i.e the local variance of the changes is approximately proportional to the length hh

The return of a stock is the percentile difference between the prices at consecutive times, i.e. St+Δt−StSt\frac{S_{t+\Delta t} - S_t}{S_t}. Written infinitesimally, this is  dStSt=μ dt+σ dWt \frac{\dd S_t}{S_t} = \mu \dd t + \sigma \dd W_t where we model the returns with a percentage drift μ\mu and percentage volatility σ\sigma. We chose the Brownian motion for the erratic volatility term to fulfill the requirements imposed on {St}\cb{S_t}. The next chapter will cover how to solve this stochastic differential equation.

3.3 Random Walk

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. We assume that the time tt is discrete, i.e. T={t0,t1,…,tn}\set{T} = \cb{t_0, t_1, \ldots, t_n} with layer spacing Δt=tk+1−tk=Tn\Delta t = t_{k+1} - t_k = \frac{T}{n}.

Definition 3.16 (Random Walk): The Random Walk {Wk}\cb{W_k} is the discrete-time real-valued process that satisfies the following:
  • starting point: W0=0W_0 = 0
  • evolution: up and down moves have size Δt\sqrt{\Delta t} with probability p=12p = \frac{1}{2}

Hence, if {X1,X2,…}\cb{X_1, X_2, \ldots} is a sequence of independent binomial random variables taking values ±Δt\pm \sqrt{\Delta t} with equal probability, then the position at time TT is Wn=∑i=1nXiW_n = \sum_{i=1}^n X_i.

Proposition 3.17 (Properties of the Random Walk): The Random Walk WnW_n has the following properties:
  1. E[Wn]=0\E[W_n] = 0
  2. Var⁡[Wn]=T\Var[W_n] = T
  3. {Wk}\cb{W_k} is a Markov process
  4. WnW_n can reach n+1n+1 values
  5. lim⁡n→∞Wn∼CLTN⁡(0,T)\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.

Recap (Moment Generating Function): The moment generating function or MGF of a random variable XX is defined as mX(θ)=E[exp⁡(θX)] m_X(\theta) = \E[\exp{\theta X}] for all values θ\theta for which the expectation exists.
Proof: The probability distribution of WnW_n is determined uniquely by its MGF, i.e. mWn(θ)=E[exp⁡(θWn)]=∏k=1nE[exp⁡(θXk)]=E[exp⁡(θX)]n 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⁡(θX)]=12exp⁡(θΔt)+12exp⁡(−θΔt)≈1+12θ2Δt\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 Δt\Delta t using Taylor series expansion, we have mWn(θ)≈(1+12θ2Δt)n m_{W_n}(\theta) \approx \pa{1 + \frac{1}{2} \theta^2 \Delta t}^n We write the MGF in logarithmic form ln⁡(mWn(θ))≈nln⁡(1+12θ2Δt)≈n12θ2Δt=12θ2T \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)≈y\ln{1+y} \approx y for small yy. Hence mWn(θ)→n→∞exp⁡(12θ2T)m_{W_n}(\theta) \xrightarrow{n \to \infty} \exp{\frac{1}{2} \theta^2 T} which is the MGF of the normal distribution N⁡(0,T)\lawN(0,T).

Thus in the continuous-time limit of the discrete-time framework, the probability density of the terminal position of the Random Walk WnW_n is the same as that of a Brownian motion that has run an amount of time TT. In other words, if the mesh size tends to zero, the Random Walk converges to the Brownian motion.

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

Proof: Let yy be the increment over time step Δt\Delta t such that Xk=±yX_k = \pm y. Then Var⁡(Wn)=nVar⁡(Xk)=ny2=TΔty2=Ty2Δt \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⁡(Wn)\Var(W_n) stays finite as Δt→0\Delta t \to 0 and y→0y \to 0 we choose y2=cΔTy^2 = c \Delta T where c>0c > 0 is a constant. Then Var⁡(Wn)=Tc\Var(W_n) = Tc and since time units are arbitrary there is no sense in choosing cc differently from 11. Hence, y=±Δty = \pm\sqrt{\Delta t}.

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

3.4 Three Key Properties

We will prove three key properties of the Brownian motion.

3.4.1 Non-Differentiability

Recall the definition of differentiability.

Recap (Differentiable function): A continuous function ff is differentiable at xx if the below and above limits exist and they coincide, i.e. lim⁡h↑0f(x+h)−f(x)h=lim⁡h↓0f(x+h)−f(x)h \lim_{h \uparrow 0}\frac{f(x+h)-f(x)}{h} = \lim_{h \downarrow 0} \frac{f(x+h)-f(x)}{h}

For the Random Walk, the slope of the path is ∣Wk+1−Wk∣Δt=1Δt\frac{\abs{W_{k+1}-W_k}}{\Delta t}=\frac{1}{\sqrt{\Delta t}} which becomes infinite as Δt→0\Delta t \to 0. This gives us a hint that Brownian motion paths are nowhere differentiable. We prove this in the following.

Proof: For the Brownian motion path, we consider the rate of change over [t,t+Δt][t,t + \Delta t], i.e. XΔt=Wt+Δt−WtΔt∼N⁡(0,1Δt)X_{\Delta t} = \frac{W_{t + \Delta t} - W_t}{\Delta t} \sim \lawN\pa{0, \frac{1}{\Delta t}}. Hence, we can write XΔt∼ZΔtX_{\Delta t} \sim \frac{Z}{\sqrt{\Delta t}} with ZZ standard normal. Take any interval A=(a,b]A = (a, b], then P[XΔt∈A]=P[aΔt<Z≤bΔt]\P[X_{\Delta t} \in A] = \P[a \sqrt{\Delta t} < Z \leq b \sqrt{\Delta t}]. As Δt→0\Delta t \to 0 we get P[XΔt∈A]=0\P[X_{\Delta t} \in A] = 0. As AA can be chosen arbitrarily, we conclude that the rate of change is not finite. And since tt 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 P[∣Wt+Δt−Wt∣<ε]=12πΔt∫−εεexp⁡(−x22Δt) dx=12π∫−εΔtεΔtexp⁡(−u22) du\begin{align*} \P[\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 Δt→0\Delta t \to 0 we have 12π∫−∞∞exp⁡(−u22) du=1 \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \exp{- \frac{u^2}{2}} \dd{u} = 1 Hence, the Brownian motion path is continuous in probability.

3.4.3 Variability

Proposition 3.18 (Quadratic variation of Brownian motion): The quadratic variation of a Brownian motion WtW_t after time TT is [Wt]T=PT[W_t]_T \peq T.

We prove this property by showing that [Wt]T[W_t]_T converges in L2L^2-norm to TT, which implies convergence in probability by Definition 3.9.

Proof: We consider [0,T][0,T] with partition Δ=Tn\Delta = \frac{T}{n} and tk=kΔtt_k = k \Delta t. Then E[(∑k=0n−1(Wtk+1−Wtk)2−T)2]=E[(∑k=0n−1((Wtk+1−Wtk)2−Δt))2]\begin{align*} & \E\bk{\pa{\sum_{k=0}^{n-1} (W_{t_{k+1}} - W_{t_k})^2 - T}^2}\\ ={}& \E\bk{\pa{\sum_{k=0}^{n-1} \pa{(W_{t_{k+1}} - W_{t_k})^2 - \Delta t}}^2} \end{align*} Expanding the square results in full square terms ((Wtk+1−Wtk)2−Δt)2\pa{(W_{t_{k+1}} - W_{t_k})^2 - \Delta t}^2 and cross-terms ((Wtk+1−Wtk)2−Δt)((Wtj+1−Wtj)2−Δt)=Δt2(Zk2−1)(Zm2−1) \pa{(W_{t_{k+1}} - W_{t_k})^2 - \Delta t}\pa{(W_{t_{j+1}} - W_{t_j})^2 - \Delta t} = \Delta t^2 \pa{Z_k^2 - 1}\pa{Z_m^2 - 1} where Zk,ZmZ_k, Z_m are iid N⁡(0,1)\lawN(0,1). Taking the expectation of the cross-terms results in Δt2(E[Zk2]−1)(E[Zm2]−1)=0 \Delta t^2 \pa{\E[Z_k^2] - 1} \pa{\E[Z_m^2] - 1} = 0 as Zk2,Zm2Z_k^2, Z_m^2 are iid χ12\lawChi_1 and have mean E[Zk2]=E[Zm2]=1\E[Z_k^2] = \E[Z_m^2] = 1. Hence E[(∑k=0n−1(Wtk+1−Wtk)2−T)2]=E[∑k=0n−1((Wtk+1−Wtk)2−Δt)2]=∑k=0n−1E[((Wtk+1−Wtk)2−Δt)2]\begin{align*} & \E\bk{\pa{\sum_{k=0}^{n-1} (W_{t_{k+1}} - W_{t_k})^2 - T}^2} \\ ={}& \E\bk{\sum_{k=0}^{n-1} \pa{(W_{t_{k+1}} - W_{t_k})^2 - \Delta t}^2} \\ ={}& \sum_{k=0}^{n-1} \E\bk{\pa{(W_{t_{k+1}} - W_{t_k})^2 - \Delta t}^2} \end{align*} Expanding the square, results in E[(Wtk+1−Wtk)4]−2E[(Wtk+1−Wtk)2]Δt+Δt2=E[(ΔtZ)4]−2E[(ΔtZ)2]Δt+Δt2=Δt2(3−2+1)=2T2n2\begin{align*} & \E\bk{\pa{W_{t_{k+1}}-W_{t_k}}^4}-2 \E\bk{\pa{W_{t_{k+1}}-W_{t_k}}^2} \Delta t+\Delta t^2 \\ ={}& \E\bk{\pa{\sqrt{\Delta t} Z}^4}-2 \E\bk{\pa{\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 00 as n→∞n \to \infty. Hence [Wt]T=L2T[W_t]_T \lneq{2} T and thus [Wt]T=PT[W_t]_T \peq T.
Proposition 3.19 (First variation of Brownian motion): A Brownian motion has unbounded first variation, i.e. given a partition PP of [0,T][0,T] the limit lim⁡∥P∥→0∑k=1n∣Xtk−Xtk−1∣\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.