Lesson 43

Stochastic Processes and Option Pricing

Big question

How can continuous-time uncertainty be simulated and translated into a no-arbitrage option value?

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Big question
Concept
Activity
Quiz

Learning objectives

  • Simulate Brownian motion and geometric Brownian motion.
  • Explain Ito’s lemma intuitively.
  • Calculate European option payoffs and Black-Scholes prices.
  • Use Monte Carlo and sensitivity analysis.
  • Prerequisites: Probability, logarithms, derivatives, and Chapter 22.
  • Key terms: Brownian motion, diffusion, Ito’s lemma, geometric Brownian motion, Black-Scholes, Greek.

Simple explanation

A standard Brownian motion begins at zero, has independent normally distributed increments, and variance that grows with elapsed time. Its paths are continuous but nowhere classically differentiable. A generalized diffusion adds drift and a volatility scale. Geometric Brownian motion multiplies both by the current price, keeping simulated prices positive under the exact lognormal solution.

Key terms

Simulate Brownian motion and geometric Brownian motion
A core idea in Chapter 43 that students apply carefully in economic analysis.
Ito’s lemma intuitively
A core idea in Chapter 43 that students apply carefully in economic analysis.
Calculate European option payoffs and Black-Scholes prices
A core idea in Chapter 43 that students apply carefully in economic analysis.
Monte Carlo and sensitivity analysis
A core idea in Chapter 43 that students apply carefully in economic analysis.
Prerequisites: Probability, logarithms, derivatives, and Chapter 22
A core idea in Chapter 43 that students apply carefully in economic analysis.
Key terms: Brownian motion, diffusion, Ito’s lemma, geometric Brownian motion, Black-Scholes, Greek
A core idea in Chapter 43 that students apply carefully in economic analysis.

Chapter 43 model

Theexactlog−priceincrementisnormalwithdrift(mu−sigma2/2)dtandvariancesigma2dt.The exact log-price increment is normal with drift (mu-sigma^2/2)dt and variance sigma^2 dt.

Interpret the expression in words and units before using it in a claim.

Example

Interpretation. One path is a possible model realization, not a forecast. Many paths are needed for a distribution or Monte Carlo estimate.

Prerequisites

  • Complete the preceding course chapters or review their summaries as needed.

Full theory and examples

43.2

Brownian motion accumulates random increments

A standard Brownian motion begins at zero, has independent normally distributed increments, and variance that grows with elapsed time. Its paths are continuous but nowhere classically differentiable. A generalized diffusion adds drift and a volatility scale. Geometric Brownian motion multiplies both by the current price, keeping simulated prices positive under the exact lognormal solution.

43.3

Ito’s lemma adds a variance correction

Ordinary calculus ignores squared infinitesimal changes, but a Brownian increment has magnitude proportional to the square root of time, so its square contributes at order dt. Ito’s lemma therefore adds a second-derivative term. Applied to log price under geometric Brownian motion, it produces drift mu minus one-half sigma squared. The result explains why expected arithmetic return and expected log growth differ.

43.4

Black-Scholes is a model, not a market law

The Black-Scholes-Merton framework derives a European option price by constructing a locally hedged portfolio under assumptions including continuous trading, frictionless markets, constant volatility and rate, and a diffusion price process (Black and Scholes 1973; Merton 1973). Real markets violate these assumptions. The formula remains a useful benchmark, and sensitivity measures such as delta and vega reveal how the benchmark changes when inputs move.

43.5

Core equations

Geometric Brownian motion

The exact log-price increment is normal with drift (mu-sigma^2/2)dt and variance sigma^2 dt.

European call

The Black-Scholes call price under the standard no-dividend assumptions.

43.6

Python demonstrations

43.6.1

Demonstration 24.1: Simulate geometric Brownian motion

Verified output

Interpretation. One path is a possible model realization, not a forecast. Many paths are needed for a distribution or Monte Carlo estimate.

43.6.2

Demonstration 24.2: Black-Scholes call price

Verified output

Interpretation. The value is conditional on the model inputs and assumptions. It is not a trading recommendation or a guarantee of market price.

43.7

Visual evidence

43.8

Reference table

Why This Matters

Stochastic-process models connect return dynamics, derivative valuation, simulation, and risk management.

Common Mistake

Using an annual volatility with time measured in days, or a daily volatility with time measured in years.

Ceteris LAB Tip

Write all time and rate units beside the formula before calculating an option value.

R-to-Python / Source Bridge

The source option-pricing lecture develops Brownian motion, Ito’s lemma, geometric Brownian motion, hedging, and Black-Scholes. Every R simulation is replaced by reproducible NumPy code and original sensitivity graphics (Tsay 2013).

Table 24. Chapter reference.
InputEffect on call priceCaution
spot Susually positivemoneyness matters
strike Kusually negativecontract definition
time Toften positivetheta can vary
rate rusually positive for callsyield conventions
volatility sigmapositive vegaimplied volatility varies by strike and maturity

Visual evidence

Figure 35. Six simulated standard Brownian-motion paths.
Figure 35. Six simulated standard Brownian-motion paths.
Figure 36. Forty one-year geometric Brownian-motion price paths.
Figure 36. Forty one-year geometric Brownian-motion price paths.
Figure 37. European call and put payoffs at a strike of 100.
Figure 37. European call and put payoffs at a strike of 100.
Figure 38. Black-Scholes European call prices across stock prices and volatilities.
Figure 38. Black-Scholes European call prices across stock prices and volatilities.

Additional Python demonstrations

Live Python

Source demonstration 2

Source demonstration 2

Stdout

Run Python to see results here.

Status / stderr

Ready to run Python in your browser.

Line-by-line guide

  1. Line 1Load a Python library needed for data work or regression.
  2. Line 3Run this Python instruction as part of the lesson workflow.
  3. Line 4Run this Python instruction as part of the lesson workflow.
  4. Line 6Run this Python instruction as part of the lesson workflow.
  5. Line 7Create or update a Python object used in the analysis.
  6. Line 8Create or update a Python object used in the analysis.
  7. Line 9Run this Python instruction as part of the lesson workflow.
  8. Line 11Create or update a Python object used in the analysis.
  9. Line 12Display a result so students can inspect the output.

Verified source output

59.88
9.4134

Interpretation. One path is a possible model realization, not a forecast. Many paths are needed for a distribution or Monte Carlo estimate.

Interpretation. The value is conditional on the model inputs and assumptions. It is not a trading recommendation or a guarantee of market price.

Guided practice

  1. 1Re-run Demonstration 24.1 and change one input while keeping the analytical question fixed.
  2. 2Explain in two sentences how the output supports, or fails to support, the chapter opening question.
  3. 3Add one validation check that would prevent a plausible error.

Exercises

  1. 1Simulate four Brownian paths.
  2. 2Verify that Brownian variance increases with time.
  3. 3Compare Monte Carlo and Black-Scholes call values.
  4. 4Plot call value against volatility and time.

Source and downloads

Chapter 43 of Fundamentals of Python for Financial Econometrics by Mohammad Safavi, Ph.D.. The lesson is an original Ceteris Lab web adaptation of the supplied publication package.

Download course PDFDownload student package

Live Python

Stochastic Processes and Option Pricing: live Python

Stochastic Processes and Option Pricing: live Python

Stdout

Run Python to see results here.

Status / stderr

Ready to run Python in your browser.

Line-by-line guide

  1. Line 1Load a Python library needed for data work or regression.
  2. Line 3Create or update a Python object used in the analysis.
  3. Line 4Create or update a Python object used in the analysis.
  4. Line 5Create or update a Python object used in the analysis.
  5. Line 6Create or update a Python object used in the analysis.
  6. Line 7Create a log version of the variable so coefficients can be read approximately as percentages.
  7. Line 8Display a result so students can inspect the output.

Python walkthrough

  1. 1`import numpy as np`: Loads a package or function used by the analysis.
  2. 2`rng = np.random.default_rng(24)`: Creates or updates a named object used by later steps.
  3. 3`S0, mu, sigma, T, steps = 100, 0.06, 0.20, 1, 252`: Creates or updates a named object used by later steps.
  4. 4`dt = T / steps`: Creates or updates a named object used by later steps.
  5. 5`z = rng.normal(size=steps)`: Creates or updates a named object used by later steps.
  6. 6`log_path = np.log(S0) + np.cumsum((mu - 0.5 * sigma**2) * dt + sigma * np.sqrt(dt) * z)`: Creates or updates a named object used by later steps.
  7. 7`print(round(float(np.exp(log_path[-1])), 2))`: Displays a result so it can be checked and interpreted.

Live notebook

Run this lesson as a notebook

Open an editable notebook cell-by-cell, run Python in the browser, and download the `.ipynb` file for later.

Open live notebookDownload .ipynbDownload .py

Related dataset

Ceteris Lab teaching sample

Estimated time

25 to 40 min

Packages

pandas, numpy

Expected output

Printed Python results that can be compared with the lesson explanation.

Learning goals

  • Load and inspect Ceteris Lab teaching sample.
  • Run the Python cells connected to Stochastic Processes and Option Pricing.
  • Interpret the output using Simulate Brownian motion and geometric Brownian motion and Ito’s lemma intuitively.

Common errors

  • File not found: check that wage_sample.csv is installed or use the course data folder.
  • Package import error: use the browser notebook first, then download for local Jupyter if your local packages differ.
  • Column name error: compare your variable names with the dataset variables listed for this notebook.

Dataset path helper

import pandas as pd

df = pd.read_csv("/data/wage_sample.csv")
df.head()

Interactive activity

Chapter 43 interactive

Assumption stress test

Evidence strength: 55%
Weak designCredible design

What should determine the strength of an econometric claim?

Immediate feedback

Choose a decision, then test how the claim changes as evidence becomes stronger or weaker.

Try it yourself

Write one plain-English sentence explaining the main idea from this lesson.

Common mistakes

Check these before you move on.

Return to the lesson assumptions, units, diagnostics, and source evidence to replace this shortcut with a defensible interpretation.

Quick quiz

Which statement best answers the Chapter 43 opening question: How can continuous-time uncertainty be simulated and translated into a no-arbitrage option value?

Quick quiz

Which practice should be avoided when applying Stochastic Processes and Option Pricing?

Quick quiz

What is the most defensible way to interpret the Python demonstration?

Quick quiz

Why does Chapter 43 matter in an applied econometrics workflow?

Key takeaway

Brownian motion is the foundation of diffusion models. Ito’s lemma introduces a second-derivative variance term. Black-Scholes is a benchmark conditional on restrictive assumptions.

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