Stochastic Processes

What:

• A stochastic process can be thought of as a single random variable that evolves dynamically with time.
• Can be discrete/continuous time: drawn from either a discrete or continuous probability distribution.
• Only discrete value and discrete time observations for actual asset prices.

Markov Process

• The probability distribution of the price at and particular point in the future is independent of the path followed by previous values.
• Consitent with weak form efficiency.

Wiener Process

• Type of Markov stochastic process with \(\mu=0\) and \(\sigma^2=1\)

A random variable follows a Wiener process if: - During a short time interval \(\Delta t\)

\[\Delta Z(t) = Z(t) - Z(t-\Delta t) = \varepsilon \sqrt{\Delta t}\] \(\varepsilon \sim N(0,1)\)

• The values of \(\Delta Z(t)\) for any two short time intervals \(\delta t\) are independent.
• As \(\Delta t \to 0\), \(\Longrightarrow dZ = \varepsilon \sqrt{dt}\)

Moments of the Wiener Process

The first two moments of \(\Delta Z(t)\)

\[E\left(\Delta Z(t)\right) = E\left(\varepsilon \sqrt{\Delta t}\right) = \sqrt{\Delta t} E\left(\varepsilon\right)= 0\]
\[Var\left(\Delta Z(t)\right) = Var\left(\varepsilon \sqrt{\Delta t}\right) = \Delta t Var\left(\varepsilon\right) = \Delta t\]
\[Sd\left(\Delta Z(t)\right) = Sd\left(\varepsilon \sqrt{\Delta t} \right) = \sqrt{\Delta t} Sd\left(\varepsilon\right) = \sqrt{\Delta t}\] \

Wiener Process Over Long Horizon

Evolution of \(Z(t)\) over a longer time period \(T\) - Divide \(T\) into \(N\) small intervals of length \(\Delta t\). - \(N=\frac{T}{\Delta t}\)
\[Z(T)-Z(0)=\sum^N_{i=1}\varepsilon_i \sqrt{\Delta t}\] where \[\varepsilon \sim {\scriptstyle i.i.d.} N(0,1)\]

• The resulting moments are:

\[E\left[Z(T) - Z(0)\right] = \sum_{i=1}^N \sqrt{\Delta t} E[\varepsilon_i] = 0\]

\[Var\left(Z(T) - Z(0)\right) = \sum_{i=1}^N \Delta t Var(\varepsilon_i) = T\]
\[Sd\left(Z(T) - Z(0)\right) = \sum_{i=1}^N \sqrt{\Delta t} Sd(\varepsilon_i) = \sqrt{T}\] \

Generalized Wiener Process

• Generalized Wiener process for \(x\) in terms of \(dz\) \[dx=adt+bdz\] Where: \(a\) is the drift rate and \(b^2\) is the variance.
• Without the \(bdz\) term, \(dx=adt\), which implies \(\frac{dx}{dt}=a\)
• integrating w.r.t time: \[ x=x_0+at\]
• Over period of time length \(T\), \(x\) changes by an amount \(aT\).
• \(bdz\) term adds variability or noise to the path of \(x\).

• In discrete time, for \(\Delta t\), \[\Delta x = a\Delta t+b\varepsilon \sqrt{\Delta t}\] So moments of \(x\) are: \[\mu=a \Delta t\] \[\sigma=b \sqrt{\Delta t}\]
  \[\sigma^2=b^2 \Delta t\]

Ito’s Process

• A generalization of a generalized Wiener process:

\[dX = a(X,t) dt + b(X,t) dZ\] - Drift rate and variance rate parameters are now functions of the value of the underlying variable x at time \(t\) - In discrete time \(t\rightarrow t+\Delta t\), \(X\rightarrow X+\Delta X\) \[\Delta X = a(X,t) \Delta t + b(X,t) \varepsilon \sqrt{\Delta t}\]

Application to asset prices:

• Ito process is used as model for asset prices: \[dS = \mu S dt + \sigma S dZ\] \[\mu S=a(S, t)\]
• Now drift rate increases proportionaly with asset price and is independent of time. \[\sigma^2 S^2=b^2(S, t)\]
• Volatility increases proportionally woth asset price and is independent of time.
• The magnitude of the price changes, i.e. the drift, changes with the level of the price.
• mirrors behaviour of variance, which also changes with the level of asset prices.
• Performs better a as a model for asset prices than generalized Wiener process \(dx=adt+bdz\).

Geometric Brownian Motion

\[\frac{dS}{S} = \mu dt + \sigma dZ\]

• \(\frac{dS}{S}\) is the return.
• \(\mu\) and \(\sigma\) are the expected return and volatility.

Discrete time \[ \frac{\Delta S}{S} = \mu \Delta t + \sigma \varepsilon \sqrt{\Delta t}\] \[{\frac{\Delta S}{S} \sim N(\mu \Delta t, \sigma \sqrt{\Delta t})}\]

Ito’s Lemma

If \(X\) follows the Ito process: \[dX = a(X,t) dt + b(X,t) dZ\]

According to Ito’s Lemma, for some function \(G(X,t)\)

\[ dG = \left(\frac{\partial G}{\partial X} a(X,t) + \frac{\partial G}{\partial t} + \frac{1}{2} \frac{\partial^2 G}{\partial X^2} b(X,t)^2 \right) dt + \frac{\partial G}{\partial X} b(X,t) dZ\]

so \(G(X, t)\) has drift and variance rates \[ \frac{\partial G}{\partial X} a(X,t) + \frac{\partial G}{\partial t} + \frac{\partial^2 G}{\partial X^2} b(X,t)\] \[\left(\frac{\partial G}{\partial X}\right)^2 b(X,t)^2 \]

So, for stock price model

\[ dS = \mu S dt + \sigma S dZ \] - According to Ito’s Lemma, \(G(S, t)\) follows \[dG = \left(\frac{\partial G}{\partial S} \mu S + \frac{\partial G}{\partial t} + \frac{1}{2} \frac{\partial^2 G}{\partial S^2} \sigma^2 S^2 \right) dt + \frac{\partial G}{\partial S} \sigma SdZ.\]

Lognormal Distribution of Prices

• Given returns are normal, \(\frac{dS}{S}\)
• Using Ito’s Lemma, \(G(S,t) = \ln(S)\) \[\frac{\partial G}{\partial S} = \frac{1}{S}\] \[\frac{\partial^2 G}{\partial S^2} = -\frac{1}{S}\] \[\frac{\partial G}{\partial t} = 0\] \[\Rightarrow dG = \left(\mu - \frac{\sigma^2}{2} \right) dt + \sigma dZ\] \(\ln(S)\) follows a generalized Wiener process with drift rate \(\mu - \frac{\sigma^2}{2}\) and variance rate \(\sigma^2\)

The discrete analog to the process for \(\Delta \ln(S)\) is

\[\Delta \ln(S) = \left(\mu - \frac{\sigma^2}{2} \right) \Delta t + \sigma \varepsilon \sqrt{\Delta t}\]

For \(\Delta t = T, \Delta \ln(S) =\ln(S_T) - \ln(S_0)\) \[\ln(S_T) - \ln(S_0) \sim N\left(\left(\mu -\frac{\sigma^2}{2}\right)T, \sigma \sqrt{T}\right)\] \[\Rightarrow \ln(S_T) \sim N\left(\ln(S_0) + \left(\mu -\frac{\sigma^2}{2}\right)T, \sigma \sqrt{T}\right)\] - This means that prices are lognormally distributed.

Lognormal Distribution

Suppose \(S_T \sim LN\left(\ln(S_0) + \left(\mu -\frac{\sigma^2}{2}\right)T, \sigma \sqrt{T}\right)\)

• By defintion \(\ln(S_T) \sim N\left(\ln(S_0) + \left(\mu - \frac{\sigma^2}{2}\right)T, \sigma \sqrt{T}\right)\)

• So the mean and variance of \(S_T\) \[E[S_T] = S_0 e^{\mu T}\] \[Var(S_T) = S_0^2 e^{2\mu T} \left(e^{\sigma^2 T}-1\right)\]