The Discrete-Time Random Walk model

The random walk model is a basic model that describes stock prices as a sum of multiple random IID variables, where IID stands for “Independent and identically distributed”.

Independent means the variables do not depend on each other. For instance, if each variable represents the result of the toss of a fair coin, the probability of getting a head or tail for the current toss does not depend on the previous toss.

Identically distributed means the variables follow the same probability distribution (e.g. they all follow a normal distribution).

In the sections below, we’ll first consider a random walk model that consists of multiple standardized random IID variables, before generalizing the model to non-standardized IID variables.

Using Standardized Random IID variables

A standardized random IID variable is a random variable that has the following properties:

Mean = 0
Variance = 1
Correlation = 0

In other words, 

E[z_t]
= Mean
= 0

E[z_t²] 
= E[z_t²] − (E[z_t])²  (here, we are subtracting 0 since E[z_t] = 0)
= Variance
= 1

E[z_tz_t’]
= E[z_t].E[z_t’] (cos z_t and z_t’ are independent)
= 0

Important Values If zt ~ iid(0, 1) E[zt] = 0 E[zt2] = 1 E[ztzt’] = 0

Now, consider a random walk model that consists of a sum of these random IID variables. That is,

S_T = z₁ + z₂ + z₃ + …. + z_T

E[S_T]
= E[z₁] + E[z₂] + …. E[z_T] (because expectation is a linear operator)
= 0

Var[S_T]
= E[S_T²] − E[S_T]²
= E[S_T²] − 0²
= E[(z₁ + z₂ + …. z_T)²]
= (sum of E[z_t²] for t from 0 to T) + 2*(sum of E[z_tz_t’] for t < t’) 
= (sum of 1 from 0 to T) + 0
= T ———————- (1)

Solution of Model If ST = z1 + z2 + z3 + …. + zT    where z ~ iid(0,1) E[ST] = 0 Var[ST] = T

Generalized Random Walk Model

Let’s generalize the model by introducing two constant parameters so that the mean and variance is no longer 0 and 1

Let r_t = σ*z_t + μ, where σ measures risk and μ measures return.

E[r_t] 
= E[σ*z_t + μ]
= E[σ*z_t] + E[μ]
= σ*E[z_t] + E[μ]
= μ

Variance
= E[(r_t − μ)²]
= E[(σ*z_t + μ − μ)²]
= E[(σ*z_t)²]
= σ²*E[z_t²]
= σ²

Variance
= E[r_t²] − (E[r_t])²
= E[r_t²] − μ²

Therefore, E[r_t²] = μ² + σ²

Covariance
= E[(r_t − μ)(r_t’ − μ)], where t not equals t’
= E[σ*z_t * σ*z_t’]
= σ² * E[z_tz_t’]
= 0

Covariance
= E[(r_t − μ)(r_t’ − μ)]
= E[r_tr_t’] − μ*E[r_t] − μ*E[r_t’] + μ²
= E[r_tr_t’] − μ²
= 0

Therefore, E[r_tr_t’] = μ²

Important Values If rt = σ*zt + μ     where z ~ iid(0,1) E[rt] = μ E[rt2] = μ2 + σ2 E[rtrt’] = μ2

Let’s consider a generalized random walk model where X_T = r₁ + r₂ + …. r_T

Mean
= E[X_T]
= E[r₁] + E[r₂] + … + E[r_T]
= T*μ

Variance
= E[X_T²] − E[X_T]²
= E[(r₁ + r₂ + …. r_T)²] − T²μ²
= (sum of E[r_t²] for t from 0 to T) + 2*(sum of E[r_tr_t’] for t < t’) − T²μ²
= T*(σ² + μ²) + T(T−1)μ² − T²μ²  (because there are ^TC₂ E[r_tr_t’] terms)
= T*σ²

Or

Variance
= E[(X_T − T*μ)²]
= E[(r₁ − μ + r₂ − μ + … + r_T − μ)²]
= E[(σ*z₁ + σ*z₂ + … + σ*z_T)²]
= σ²*E[(z₁ + z₂ + …. + z_T)²]    (refer to (1) above)
= T*σ²

Solution of Model If XT = r1 + r2 + …. rT Mean = T*μ Variance = T*σ2