Modeling Stock Prices in Continuous-Time

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2025

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].

Random Walk

• A random walk \left\{ X_{n} \right\} is a stochastic process defined as: \begin{aligned} X_{0} & = x_{0} \\ X_{n + 1} & = X_{n} + e_{n + 1} \end{aligned} where \left\{ e_{n} \right\} are independent and identically distributed (i.i.d.) random variables such that \operatorname{E}(e_{n}) = 0 for all n \geq 1.
• Note that e_{n} need not be normally distributed.
• For example, e_{n} could be such: \operatorname{P}(e_{n} = 1) = \operatorname{P}(e_{n} = -1) = 0.5

Random Walk Simulation

The figure plots simulated paths for the random walk defined as X_{0}= 0, X_{n + 1} = X_{n} + e_{n + 1}, where \{e_{n}\} is an i.i.d sequence taking the values 1 and -1 with equal probability, and n \leq 5000.

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.

Brownian Motion

• A very useful random walk can be defined as follows: W_{t + \Delta t} = W_{t} + \sqrt{\Delta t} e_{t + \Delta t} where W_{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\{ W_{t} \right\} for t \in [0, T] is called a Wiener process or Brownian motion.

Properties of Brownian Motion

• The sample paths of a Brownian motion are continuous.
• For s < t, the increment W_{t} - W_{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 W_{t} \sim N(0, t) for 0 < t \leq T.

Brownian Motion Simulation

The figure plots simulated paths for \left\{W_{t}\right\} where t \in [0, 10].

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 W_{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 Simulation

The figure plots simulated paths for a geometric Brownian motion \left\{S_{t}\right\} where t \in [0, 10], S_{0} = 100, \mu = 0.20, and \sigma = 0.20. The dashed line denotes \operatorname{E}\left( S_{t} \right) = S_{0} e^{\mu t}.

Preliminary Results on Wiener Processes

• The Wiener process increment can be approximated as: \Delta W_{t} = W_{t + \Delta t} - W_{t} = \sqrt{\Delta t} e_{t + \Delta t}
• If we define \xi = (\Delta W_{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 W_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 W_{t})^{2} \approx \Delta t and (\Delta t)(\Delta W_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 square changes matter.
• Using the results derived before: \begin{aligned} (\Delta S_{t})^{2} & = (\mu S_{t} \Delta t + \sigma S_{t} \Delta W_{t})^{2} \\ & = (\mu S_{t})^{2} \underbrace{(\Delta t)^{2}}_{\approx 0} + 2 \mu \sigma (S_{t})^{2} \underbrace{(\Delta t)(\Delta W_{t})}_{\approx 0} + (\sigma S_{t})^{2} \underbrace{(\Delta W_{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.

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 W_{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 dW 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) dW

Ito Calculus Rules

• It is usually more convenient to use the following results when working with stochastic processes defined through Brownian motions: \begin{aligned} (dt)^{2} & = 0 \\ (dt)(dW) & = (dW)(dt) = 0 \\ (dW)^{2} & = dt \end{aligned}
• 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 dW)^{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 dW
• We can then solve for X_{T}: \begin{aligned} X_{T} - X_{0} & = \int_{0}^{T} dX = \int_{0}^{T} \left( \mu - \frac{1}{2} \sigma^{2} \right) dt + \int_{0}^{T} \sigma dW \\ & = \left( \mu - \frac{1}{2} \sigma^{2} \right) T + \sigma W_{T} \\ S_{T} & = S_{0} \exp\left( \left( \mu - \frac{1}{2} \sigma^{2} \right) T + \sigma W_{T} \right) \end{aligned}

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 W_{T}
• We can conclude that \ln(S_{T}) \sim N(m, s^{2}), where: \begin{aligned} m & = \ln(S_{0}) + \left( \mu - \frac{1}{2} \sigma^{2} \right) T \\ s & = \sigma \sqrt{T} \end{aligned}
• In other words, S_{T} is lognormally distributed with mean m and variance s^{2}.

Calculating a Confidence Interval on the Stock Price

Example 1 Consider a stock whose price at time t is given by S_{t} and that follows a GBM. The expected return is 12% per year and the volatility is 25% per year. The current spot price is $25. If we denote X_{T} = \ln(S_{T}) and take T = 0.5, we have that: \begin{aligned} \operatorname{E}(X_{T}) & = \ln(25) + \left(0.12 - 0.5(0.25)^{2}\right)(0.5) = 3.2633 \\ \operatorname{SD}(X_{T}) & = 0.25 \sqrt{0.5} = 0.1768 \end{aligned} Hence, the 95% confidence interval for S_{T} is given by: [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 the expectation and standard deviation of S_{T}: \begin{aligned} \operatorname{E}(S_{T}) & = S_{0} e^{\mu T} \\ \operatorname{SD}(S_{T}) & = E(S_{T}) \sqrt{e^{\sigma^{2} T} - 1} \end{aligned}

Example 2 Consider a stock whose price at time t is given by S_{t} and that follows a GBM. The expected return is 12% per year and the volatility is 25% per year. The current spot price is $25. The expected price and standard deviation 6 months from now are: \begin{aligned} \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{aligned}

Computing Partial Expectations

• Since \ln(S_{T}) \sim \mathcal{N}(m, s^{2}). Then we have that: \begin{aligned} \operatorname{E}\left(S_{T} \large\mathbb{1}_{\{S_{T} > K\}}\right) & = e^{m + \frac{1}{2}s^{2}} \mathop{\Phi}\left(\frac{m + s^{2} - \ln(K)}{s}\right) \\ & = S_{0} e^{\mu T} \mathop{\Phi}\left( \frac{\ln(S_{0}/K) +(\mu + \frac{1}{2} \sigma^{2})T}{\sigma \sqrt{T}} \right) \\ \operatorname{E}\left(K \large\mathbb{1}_{\{S_{T} > K\}}\right) & = K \mathop{\Phi}\left(\frac{m - \ln(K)}{s}\right) \\ & = K \mathop{\Phi}\left( \frac{\ln(S_{0}/K) +(\mu - \frac{1}{2} \sigma^{2})T}{\sigma \sqrt{T}} \right) \end{aligned}
• It turns out that these results are everything we need in order to derive the Black-Scholes pricing formulas!

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 dW 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.