Modeling Equity Prices in Discrete Time

Stochastic Foundations for Finance

Lorenzo Naranjo

Fall 2024

Basic Time-Series Analysis

White Noise

  • The white noise process is defined as \varepsilon_{t} \sim \text{i.i.d.} \,\, \mathcal{N}(0, \sigma_{\varepsilon}^{2}).
  • We then have that \begin{aligned} \operatorname{E}(\varepsilon_{t}) & = \operatorname{E}_{t-1}(\varepsilon_{t}) = 0, \\ \operatorname{E}(\varepsilon_{t} \varepsilon_{t-j}) & = \operatorname{Cov}(\varepsilon_{t}, \varepsilon_{t-j}) = 0, \\ \operatorname{Var}(\varepsilon_{t}) & = \operatorname{Var}_{t-1}(\varepsilon_{t}) = \sigma_{\varepsilon}^{2}. \end{aligned}
  • Thus, for white noise there is no predictability or serial correlation, and the conditional variance is constant, i.e., the process is conditionally homoskedastic.

Conditional Expectations

  • Consider the following MA(1) process x_{t} = \varepsilon_{t} + \phi \varepsilon_{t-1}, \tag{1} where \{\varepsilon_{t}\} is a white noise process.
  • We denote by \operatorname{E}_{t}(\cdot) = \operatorname{E}(\cdot \mid \text{all information at } t).
  • Thus, \begin{aligned} \operatorname{E}_{t-2}(x_{t}) & = \operatorname{E}_{t-2}(\varepsilon_{t}) + \phi \operatorname{E}_{t-2}(\varepsilon_{t-1}) = 0, \\ \operatorname{E}_{t-1}(x_{t}) & = \operatorname{E}_{t-1}(\varepsilon_{t}) + \phi \operatorname{E}_{t-1}(\varepsilon_{t-1}) = \phi \varepsilon_{t-1}, \\ \operatorname{E}_{t}(x_{t}) & = \operatorname{E}_{t}(\varepsilon_{t}) + \phi \operatorname{E}_{t}(\varepsilon_{t-1}) = \varepsilon_{t} + \phi \varepsilon_{t-1}. \end{aligned}
  • We define \operatorname{E}(x_{t}) = \operatorname{E}(x_{t} \mid \text{no information about other variables}).

The figure plots the evolution of x_t = ε_t + φε_{t-1} where \varepsilon_{t} is white noise with unit variance and φ = 0.7.

Autocovariance and Autocorrelation

  • The autocovariance function of x_{t} is defined as \gamma_{n} = \operatorname{Cov}(x_{t}, x_{t-n}) \quad \text{for $n \geq 0$}.
    • Note that \gamma_{0} = \operatorname{Var}(x_{t}).
  • Thus, the autocorrelation functions is defined as \rho_{n} = \frac{\gamma_{n}}{\gamma_{0}}.
    • We always have \rho_{0} = 1.
    • For the white noise process we also have that \rho_{n} = 0 for n \geq 1.

Autocorrelation Function of the MA(1) Process

  • For x_{t} defined by (1) we have that \gamma_{n} = \begin{cases} \sigma_{\varepsilon}^{2} (1 + \phi^{2}) & \text{if $n = 0$}, \\ \phi \sigma_{\varepsilon}^{2} & \text{if $n = 1$}, \\ 0 & \text{if $n \geq 2$}. \end{cases}
  • The autocorrelation function of the MA(1) process is \rho_{n} = \begin{cases} 1 & \text{if $n = 0$}, \\ \dfrac{\phi}{1 + \phi^{2}} & \text{if $n = 1$}, \\ 0 & \text{if $n \geq 2$}. \end{cases}

The figure plots the evolution of x_{t} = \varepsilon_{t} + \phi \varepsilon_{t-1} where \varepsilon_{t} is white noise with unit variance and \phi = 0.7. Note that in theory \rho_{1} = 0.470.

Modeling Stock Prices

Percentage Changes of the Stock Price

  • We start by defining the percentage price change of the stock price over one period as \frac{\Delta S_{t}}{S_{t}} = \frac{S_{t+1} - S_{t}}{S_{t}} = \frac{S_{t+1}}{S_{t}} - 1.
  • We note that \ln(x) \approx x - 1, so that \Delta \ln(S_{t}) = \ln(S_{t+1}) - \ln(S_{t}) = \ln\left(\frac{S_{t+1}}{S_{t}}\right) \approx \frac{S_{t+1}}{S_{t}} - 1 = \frac{\Delta S_{t}}{S_{t}}.
    • Log changes are good approximations for percentage changes.

Modelling Log-Stock Prices

  • Assume that \Delta \ln(S_{t}) = \ln(S_{t+1}) - \ln(S_{t}) = \left(\mu - \frac{1}{2} \sigma^{2}\right) + \sigma \varepsilon_{t+1}, where \varepsilon_{t} is a white noise process with unit variance.
  • We’ll see in a moment that the term \frac{1}{2} \sigma^{2} is just a convenience to make sure that the expected stock price grows at a rate of \mu per unit of time.

Solving for the Stock Price

  • Solving forward, \begin{aligned} \ln(S_{t+1}) & = \ln(S_{t}) + \left(\mu - \frac{1}{2} \sigma^{2}\right) + \sigma \varepsilon_{t+1}, \\ \ln(S_{t+2}) & = \ln(S_{t+1}) + \left(\mu - \frac{1}{2} \sigma^{2}\right) + \sigma \varepsilon_{t+2} \\ & = \ln(S_{t}) + 2 \left(\mu - \frac{1}{2} \sigma^{2}\right) + \sigma(\varepsilon_{t+1} + \varepsilon_{t+2}), \\ & \hspace{0.55em} \vdots \\ \ln(S_{t+n}) & = \ln(S_{t}) + n \left(\mu - \frac{1}{2} \sigma^{2}\right) + \sigma(\varepsilon_{t+1} + \varepsilon_{t+2} + \ldots + \varepsilon_{t+n}). \end{aligned}
  • Note that \operatorname{Var}(\varepsilon_{t+1} + \varepsilon_{t+2} + \ldots + \varepsilon_{t+n}) = n.

Log-Stock Price Moments

  • We can compute the conditional expectation and variance of s_{t+n} at time t, \begin{aligned} \operatorname{E}_{t}(\ln(S_{t+n})) & = \ln(S_{t}) + n \left(\mu - \frac{1}{2} \sigma^{2}\right), \\ \operatorname{Var}_{t}(\ln(S_{t+n})) & = n \sigma^{2}. \end{aligned}
  • Thus, we can conclude that \ln(S_{t+n}) \sim \mathcal{N}_{t}(\ln(S_{t}) + n \left(\mu - \frac{1}{2} \sigma^{2}\right), n \sigma^{2}).
  • This is very useful since we can now compute things like \operatorname{E}_{t}(S_{t+n}), or \operatorname{P}(S_{t+n} \leq K) = \operatorname{P}(\ln(S_{t+n}) \leq \ln(K)).
  • Indeed, \operatorname{E}_{t}(S_{t+n}) = \operatorname{E}_{t}(e^{\ln(S_{t+n})}) = e^{\operatorname{E}_{t}(\ln(S_{t+n})) + \frac{1}{2} \operatorname{Var}_{t}(\ln(S_{t+n}))} = S_{t} e^{n \mu}.

Example 1 (Computing Stock Price Probabilities) Time is measured in years. Suppose that the price S_{t} of a stock is such that log-changes are i.i.d. normally distributed such that \operatorname{E}_{t} \Delta \ln(S_{t}) = 20\% and \sigma_{t} \Delta \ln(S_{t}) = 30\% per year. If the stock price today is $100, what is the probability that the stock price is greater than $110 in 5 years from now? \begin{aligned} \operatorname{P}(S_{5} \leq 110) & = \operatorname{P}(\ln(S_{5}) \leq \ln(110)) \\ & = \operatorname{\Phi}\left(\frac{\ln(110) - (\ln(100) + 0.2 \times 5)}{0.3 \sqrt{5}} \right) \\ & = 8.87 \%. \end{aligned} Thus, \operatorname{P}(S_{5} > 110) = 1 = 0.0887 =91.13 \%.

Thinking About Time

  • In the previous example time is measured in years, and we computed the probability of the price being greater than $110 in 5 years from now.
  • What if we want to know the probability of the price being greater of $110 in 4.5 years from now?
  • From now on we measure time in years, and all parameters are expressed per year.
  • We still assume that time is discrete but now each time increment is equal to \Delta t.
    • If we think that the stock price changes twice per year, \Delta t = 0.5.
  • Thus, we define \Delta \ln(S_{t}) = \ln(S_{t + \Delta t}) - \ln(S_{t}).

Generalizing The Model

  • We can write our model of stock prices as \Delta \ln(S_{t}) = \left(\mu - \frac{1}{2} \sigma^{2} \right) \Delta t + \sigma \sqrt{\Delta t} \varepsilon_{t + \Delta t}, \tag{2} where \varepsilon_{t} is a white noise process such that \varepsilon_{t} \sim \mathcal{N}(0, 1).
    • We can do this standardization because the normal distribution is divisible.
    • If we divide one year in four or just one piece does not affect our probabilistic view of the stock price after a year.

The Noise Term

  • Let’s denote the noise term in (2) as \Delta z_{t} = z_{t + \Delta t} - z_{t} = \sqrt{\Delta t} \varepsilon_{t + \Delta t}.
    • We’ll see later that as \Delta t \rightarrow \infty the term z_{t} defines Brownian motion or Wiener process.
  • If we split the time interval [t, T] in n parts so that \Delta t = \frac{T - t}{n}, we have that conditional on time t information, \begin{aligned} z_{T} - z_{t} & = \Delta z_{t} + \Delta z_{t +\Delta t} + \ldots + \Delta z_{t + (n - 1) \Delta t} \\ & = \sqrt{\Delta t} (\varepsilon_{t + \Delta t} + \varepsilon_{t + 2 \Delta t} + \ldots + \varepsilon_{t + n \Delta t}). \end{aligned} is normally distributed with mean zero and variance T - t.

The figure plots simulated paths of z_{t} = \sqrt{\Delta t} \sum_{i=1}^{5000} \varepsilon_{i} where \{\varepsilon_{i}\} is a white noise process with unit variance and \Delta t = 10 / 5000 = 0.002 years.

The Stock Price Distribution

  • We can write (2) as \begin{aligned} \ln(S_{T}) - \ln(S_{t}) & = \Delta \ln(S_{t}) + \Delta \ln(S_{t +\Delta t}) + \ldots + \Delta \ln(S_{t + (n - 1) \Delta t}) \\ & = \left(\mu - \frac{1}{2} \sigma^{2} \right) (T - t) + \sigma \sqrt{\Delta t} (\varepsilon_{t + \Delta t} + \varepsilon_{t + 2 \Delta t} + \ldots + \varepsilon_{t + n \Delta t}). \end{aligned}
  • Thus, conditional on time t information, \ln(S_{T}) is normally distributed with mean and variance given by \begin{aligned} \operatorname{E}_{t}(\ln(S_{T})) & = \ln(S_{t}) + \left(\mu - \frac{1}{2} \sigma^{2} \right) (T - t), \\ \operatorname{Var}_{t}(\ln(S_{T})) & = \sigma^{2} (T - t). \end{aligned}
    • The stock price follows a lognormal distribution.

Example 2 (Calculating a Confidence Interval on the Stock Price) Consider a stock whose price follows a lognormal distribution. The expected return of the stock price returns is 12% per year and the volatility is 25% per year. The current stock price is $25. If T = 0.5, we have that: \begin{aligned} \operatorname{E}(\ln(S_{T})) & = \ln(25) + \left(0.12 - 0.5(0.25)^{2}\right)(0.5) = 3.2633, \\ \operatorname{SD}(\ln(S_{T})) & = 0.25 \sqrt{0.5} = 0.1768. \end{aligned} Using the 95% confidence interval for \ln(S_{T}) we can compute [e^{3.2633 - 1.96(0.1768)}, e^{3.2633 + 1.96(0.1768)}] = [18.48, 36.96]. Therefore, there is a 95% probability that the stock price in 6 months will lie between $18.48 and $36.96.

Calculating the Moments of the Stock Price

  • Some algebra reveals that the expectation and standard deviation of S_{T} is \begin{aligned} \operatorname{E}(S_{T}) & = S_{0} e^{\mu T}, \\ \operatorname{SD}(S_{T}) & = \operatorname{E}(S_{T}) \sqrt{e^{\sigma^{2} T} - 1}. \end{aligned}

Example 3 Consider a stock whose price follows a lognormal distribution. The expected return of the stock price returns is 12% per year and the volatility is 25% per year. If the current stock price is $25, the expected price and standard deviation 6 months from now are: \begin{align*} \operatorname{E}(S_{T}) & = 25 e^{0.12 (0.5)} = \$26.55 \\ \operatorname{SD}(S_{T}) & = 26.55 \sqrt{e^{0.25^{2} (0.5)} - 1} = \$4.73 \end{align*}

The figure plots simulated paths for the stock price S_{t} = S_{0} \exp\left(\left(\mu - \frac{1}{2} \sigma^{2} \right) (T - t) + \sigma \sqrt{\Delta t} \sum_{i=1}^{5000} \varepsilon_{i}\right) where S_{0} = 100, \mu = 0.20, \sigma = 0.20, \{\varepsilon_{i}\} is a white noise process with unit variance and \Delta t = 10 / 5000 = 0.002 years. The dashed line denotes \operatorname{E}\left( S_{t} \right) = S_{0} e^{\mu t}.

It’s Irrelevant How You Split the Time Interval

  • In our stock price model we have that \operatorname{E}_{t}(S_{T}) = S_{t} e^{\mu (T - t)}.
    • It does not matter what \Delta t you choose, you always get the same forecast.
  • Another way to see this is to think about how to estimate \mu from observed data: \hat{\mu} = \frac{1}{T - t} (\ln(S_{T}) - \ln(S_{t})).
    • It does not matter if you sample the data more frequently.
    • The only thing that matters is for how long you have been collecting data.