Modeling Stock Prices in Continuous Time

Stochastic Foundations for Finance

Lorenzo Naranjo

Fall 2024

Brownian Motion

Stochastic Process

  • A stochastic process describes the evolution of a random variable over time.
  • In finance we use stochastic processes to model the evolution of stock prices, interest rates, volatility, foreign exchange rates, commodity prices, etc.
  • We distinguish between:
    • Discrete-time processes: The values of the process \left\{ S_{n} \right\} are allowed to change only at discrete time intervals, i.e. n \in \{ 0, 1, 2, \ldots, N \} or n \in \mathbb{N}.
    • Continuous-time processes: The stochastic process \left\{ S_{t} \right\} is defined for all t \in [0, T].

Brownian Motion

  • A very useful process can be defined as follows: z_{t + \Delta t} = z_{t} + \sqrt{\Delta t} e_{t + \Delta t} where z_{0} = 0 and \left\{ e_{t} \right\} are i.i.d. such that e_{t} \sim N(0, 1).
  • Note that here time increases each step by \Delta t.
  • Letting \Delta t \rightarrow 0, the resulting process \left\{ z_{t} \right\} for t \in [0, T] is called a Brownian motion or Wiener process.

Properties of Brownian Motion

  • The sample paths of a Brownian motion are continuous.
  • For s < t, the increment z_{t} - z_{s} \sim N(0, t - s), i.e. is normally distributed with mean 0 and variance t - s.
  • Increments are independent of each other.
  • In particular, note that z_{t} \sim N(0, t) for 0 < t \leq T.

Brownian Motion Simulations

The figure plots simulated paths of z_{n \Delta t} = \sqrt{\Delta t} \sum_{i=1}^{n} \varepsilon_{i} where n = 1, 2, \ldots, 5000, \{\varepsilon_{i}\} is a white noise process with unit variance and \Delta t = 10 / 5000 = 0.002 years.

Geometric Brownian Motion

  • Now we turn our attention to modeling stock prices \left\{ S_{t} \right\}.
    • We need to be careful, though, as stock prices cannot be negative.
    • We also would like to allow the model to display a certain drift \mu and volatility \sigma.
  • To achieve this, we model the percentage change of a stock price between t and t + \Delta t as: \frac{\Delta S_{t}}{S_{t}} = \mu \Delta t + \sigma \Delta z_{t}
  • Note that the percentage change in price over an interval \Delta t is normally distributed with mean \mu \Delta t and variance \sigma^{2} \Delta t.
  • This process is called a geometric Brownian motion (GBM).

Geometric Brownian Motion Simulations

The figure plots simulated paths for the stock price S_{n \Delta t} = S_{0} \exp\left(\left(\mu - \frac{1}{2} \sigma^{2} \right) (T - t) + \sigma \sqrt{\Delta t} \sum_{i=1}^{n} \varepsilon_{i}\right) where n = 1, 2, \ldots, 5000, 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}.

Preliminary Results on Brownian Motion

  • The Brownian motion increment can be approximated as: \Delta z_{t} = z_{t + \Delta t} - z_{t} = \sqrt{\Delta t} e_{t + \Delta t}
  • If we define \xi = (\Delta z_{t})^{2}, we have that: \begin{aligned} \operatorname{E}(\xi) & = \Delta t \\ \operatorname{Var}(\xi) & = \operatorname{E}(\xi^{2}) - \left(\operatorname{E}(\xi)\right)^{2} = 3 (\Delta t)^{2} - (\Delta t)^{2} = 2 (\Delta t)^{2} \approx 0 \end{aligned}
  • Similarly, if we define \zeta = (\Delta t)(\Delta z_{t}), we have that: \begin{aligned} \operatorname{E}(\zeta) & = 0 \\ \operatorname{Var}(\zeta) & = \operatorname{E}(\zeta^{2}) - \left(\operatorname{E}(\zeta)\right)^{2} = (\Delta t)^{2} \operatorname{E}(\xi) = (\Delta t)^{3} \approx 0 \end{aligned}
  • Hence, (\Delta z_{t})^{2} \approx \Delta t and (\Delta t)(\Delta z_t) \approx 0 for small \Delta t.

The Square Change of the Stock Price

  • Of the most surprising results of stochastic calculus is that the squared changes matter.
  • Using the results derived before: \begin{aligned} (\Delta S_{t})^{2} & = (\mu S_{t} \Delta t + \sigma S_{t} \Delta z_{t})^{2} \\ & = (\mu S_{t})^{2} \underbrace{(\Delta t)^{2}}_{\approx 0} + 2 \mu \sigma (S_{t})^{2} \underbrace{(\Delta t)(\Delta z_{t})}_{\approx 0} + (\sigma S_{t})^{2} \underbrace{(\Delta z_{t})^{2}}_{\approx \Delta t} \\ & \approx \sigma^{2} S_{t}^{2} \Delta t \end{aligned}
  • This says that the square change in the stock price is almost deterministic.

Ito’s Lemma

Intuitive Ito’s Lemma

  • Consider a GBM process \left\{S_{t}\right\} and a smooth function f(\cdot).
  • A second order Taylor approximation around S_{t} implies: f(S_{t} + \Delta S_{t}) \approx f(S_{t}) + f'(S_{t}) (\Delta S_{t}) + \frac{1}{2} f''(S_{t}) (\Delta S_{t})^{2}
  • We can finally conclude that: \Delta f(S_{t}) \approx \left( \mu S_{t} f'(S_{t}) + \frac{1}{2} \sigma^{2} S_{t}^{2} f''(S_{t}) \right) \Delta t + \sigma S_{t} f'(S_{t}) \Delta z_{t}

Ito’s Lemma

  • The continuous-time analog of the previous analysis is as follows.
  • As before, we consider a GBM process \left\{S_{t}\right\} given by: dS = \mu S dt + \sigma S dz and a smooth function F(\cdot).
  • Define a new process \left\{X_{t}\right\} as X_{t} = F(S_{t}) for all t \in [0, T].
  • Ito’s lemma states that: dF = \left( \mu S F'(S) + \frac{1}{2} \sigma^{2} S^{2} F''(S) \right) dt + \sigma S F'(S) dz

Ito Calculus Rules

  • It is usually more convenient to use the following results when working with stochastic processes defined through Brownian motions: \begin{align*} (dt)^{2} & = 0 \\ (dt)(dz) & = (dz)(dt) = 0 \\ (dz)^{2} & = dt \end{align*}
  • Ito’s Lemma can then be restated as: dF = F'(S) dS + \frac{1}{2} F''(S) (dS)^{2} where (dS)^{2} = (\mu S dt + \sigma S dz)^{2} = \sigma^{2} S^{2} dt

Solving for GBM

  • Define X = \ln(S), which implies S = e^{X}.
  • We have that F'(S) = 1 / S and F''(S) = -1 / S^{2}, which implies that: dX = \left( \mu - \frac{1}{2} \sigma^{2} \right) dt + \sigma dz
  • We can then solve for X_{T}: X_{T} - X_{0} = \int_{0}^{T} dX = \int_{0}^{T} \left( \mu - \frac{1}{2} \sigma^{2} \right) dt + \int_{0}^{T} \sigma dz = \left( \mu - \frac{1}{2} \sigma^{2} \right) T + \sigma z_{T}.
  • Thus, S_{T} = S_{0} e^{\left( \mu - \frac{1}{2} \sigma^{2} \right) T + \sigma z_{T}}.

Properties of Stock Prices Following a GBM

  • The previous result can be rewritten as: \ln(S_{T}) = \ln(S_{0}) + \left( \mu - \frac{1}{2} \sigma^{2} \right) T + \sigma z_{t}
  • We can conclude that \ln(S_{T}) \sim N(m, s^{2}), where: \begin{align*} m & = \ln(S_{0}) + \left( \mu - \frac{1}{2} \sigma^{2} \right) T \\ s & = \sigma \sqrt{T} \end{align*}
  • In other words, S_{T} is lognormally distributed with mean m and variance s^{2}.

A Generalized Form of Ito’s Lemma

  • Most derivatives not only depend on the underlying asset but also depend on time since they have fixed expiration dates.
  • The analysis we did before for Ito’s Lemma generalizes easily to handle this case.
  • Consider a non-dividend paying stock that follows a GBM: dS = \mu S dt + \sigma S dz and a smooth function F(S, t).
  • Ito’s Lemma in this case applies in the following form: dF = \frac{\partial F}{\partial S} dS + \frac{1}{2} \frac{\partial^{2} F}{\partial S^{2}} (dS)^{2} + \frac{\partial F}{\partial t} dt where (dS)^{2} = \sigma^{2} S^{2} dt.

Martingales

Discrete-Time Martingales

  • A discrete-time martingale \left\{Z_{n}\right\}_{n \geq 0} is a stochastic process such that: \operatorname{E}\left(Z_{n+1} \;\middle|\; Z_{1}, Z_{2}, \ldots, Z_{n}\right) = Z_{n}
  • Note that a martingale need not be a random walk.
    • For example, consider the process \left\{ Z_{n} \right\}: Z_{n+1} = Z_{n} \varepsilon_{n+1} where \left\{ \varepsilon_{n} \right\} is an i.i.d. sequence such that \operatorname{E}\left(\varepsilon_{n}\right) = 1 for all n \geq 1.
    • It is a martingale since: \operatorname{E}\left(Z_{n+1} \;\middle|\; Z_{1}, Z_{2}, \ldots, Z_{n}\right) = \operatorname{E}\left(Z_{n} \varepsilon_{n+1} \;\middle|\; Z_{n}\right) = Z_{n} \operatorname{E}\left(\varepsilon_{n+1} \;\middle|\; Z_{n}\right) = Z_{n}.
  • Random walks are martingales, though.

Continuous-Time Martingales

  • A continuous-time process M_{t} is a martingale if \operatorname{E}(M_{T} \mid \text{all information at $t$}) = M_{t}.
  • If M(T) = M(t) + \int_{t}^{T} \theta(s) dz(s), then dM = \theta dz.
  • The process M(t) is martingale for t \in [0, T] as long its expected quadratic variation over [0, T] is finite, i.e. \operatorname{E}\left(\int_{0}^{T} \theta(s)^{2} ds\right) < \infty. \tag{1}
  • Thus, a process M(t) with zero drift is a martingale if equation (1) holds, otherwise the process is a local martingale.