---
title: "Introduction to the Black-Scholes option pricing model and the Greeks"
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url: "https://longbridge.com/zh-HK/topics/26649506.md"
description: "Recently, due to the needs of work communication (well...), I finally went through the remaining parts of the BS model. Indeed, is Deadline the primary driving force? Another motivation for writing this content is that the classic book "Options, Futures, and Other Derivatives" is rigorous but lengthy, while some online articles about the BS model often omit many key premises and assumptions, leaving readers confused—like certain steps suddenly appearing out of thin air, leading to reading roadblocks. The goal of this article is to organize the information related to the BS model..."
datetime: "2025-01-16T16:02:11.000Z"
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author: "[兴华XingHua](https://longbridge.com/zh-HK/profiles/31399.md)"
---

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# Introduction to the Black-Scholes option pricing model and the Greeks

Recently, due to the need for work communication (hmm...), I finally went through the remaining parts of the BS model. Indeed, is Deadline the ultimate motivator?

Another motivation for writing this is that the classic book "Choices, Futures, and Other Derivatives" is rigorous but lengthy, while many online articles about the BS model often omit many key premises and assumptions, leaving readers confused. For example, some steps suddenly appear out of thin air, leading to reading roadblocks.

The goal of this article is to organize the information related to the BS model, including a brief history of the model, its overarching ideas, and whether each step of the derivation introduces certain assumptions or is purely a mathematical technique, thereby stringing together its core logic in a complete manner.

Although introducing the BS model inevitably involves mathematics, this article will only cover key terms from the perspective of informational necessity, omitting overly detailed mathematical steps to reduce reading pressure. I believe everyone can read it easily and happily, appreciating the simplicity and elegance of the BS model.

Of course, since this article is relatively introductory, those who want to delve deeper into option pricing or explore models beyond the BS model will need to put in more effort to research on their own.

## Modeling Uncertainty

To understand option pricing models, one must first understand how to model uncertainty. The following history omits some intermediate knowledge, retaining the core developmental path of the prerequisites the BS model relies on:

-   In 1827, botanist Robert Brown observed the irregular motion of pollen particles in water, later known as Brownian motion.

-   In 1880, Thorvald Thiele conducted the first mathematical analysis of it.
-   In 1900, Louis Bachelier (whose advisor was the famous mathematician Henri Poincaré) independently used it in his thesis to simulate the stock market.
-   In 1905, Einstein used Bachelier's solution in his diffusion model.
-   In 1923: Norbert Wiener (the father of cybernetics) established a complete mathematical framework for Brownian motion, known as the **Wiener process, generally denoted as Wt, whose main property is random motion over time. The Wiener process** is commonly simulated as follows (doesn't it look a lot like short-term stock price movements?):

-   1940s: Japanese mathematician Kiyosi Itô published important papers in the field of stochastic calculus, which involves stochastic integrals and stochastic differential equations based on Wiener or Brownian motion processes. This is now known as Itô calculus, with the most important result being the **Itô's lemma**.

## Applying Uncertainty Modeling to Stock Prices

Due to practical reasons, stock prices cannot be negative, so to achieve this logarithmic property, geometric Brownian motion is generally used as the basis for modeling. Geometric Brownian motion assumes that the relative change (percentage change) of a variable follows Brownian motion, meaning the logarithm of the variable follows Brownian motion.

Before continuing this topic, there's an obvious question: **"Why use Brownian motion to simulate stock prices in the first place?"** 

1.  **People have observed that stock price movements exhibit randomness in the short term, so a stochastic model is needed.** — The randomness of stock prices was further argued by Eugene Fama in 1965 through the "efficient market hypothesis": since the market is efficient, no one can predict stock prices, so stock prices must follow a random walk. 
2.  Brownian motion is a naturally observable stochastic phenomenon that seems to share some properties with stock price movements. Coincidentally, Wiener had already provided a concise and elegant mathematical description of Brownian motion, namely the **Wiener process**.
3.  Humans don’t truly know whether "stock price properties are exactly the same as those of Brownian motion," but almost all physical and mathematical models are like this. From a pragmatic perspective, a model that can be locally applied is already a usable model.
4.  From an empirical perspective, short-term (e.g., daily or weekly) stock returns approximately follow a normal distribution to some extent, but usually exhibit "fat tails," meaning the probability of extreme events is higher than predicted by a normal distribution. Over the long term, the distribution of returns may deviate even further from the normal distribution.

After applying geometric Brownian motion adjustments, a common simulation of stock price movements looks like this (the color gradient basically shows that its probability distribution retains the properties of a normal distribution):

![Pricing Options by Monte Carlo Simulation with Python](https://pub.pbkrs.com/uploads/2025/300a15f84190d7e55c256fa833311ccc?x-oss-process=style/lg)

## Three Additional Assumptions: Market Efficiency, No-Arbitrage Principle, and Risk Neutrality

These three concepts are standard terms, so here’s a rather dry standard introduction.

**1\. Market Efficiency**

-   **Definition:** Market efficiency refers to the idea that in the stock market, all available information is quickly and fully reflected in asset prices. This means investors cannot consistently achieve excess returns by analyzing historical prices or other public information.
-   **Three Forms:**
    -   **Weak-form efficiency:** Stock prices already reflect all historical price information, making technical analysis ineffective.
    -   **Semi-strong-form efficiency:** Stock prices already reflect all public information (including historical prices, company financial statements, news, etc.), making fundamental analysis partially ineffective.
    -   **Strong-form efficiency:** Stock prices already reflect all information, including public and insider information, making it impossible for anyone to achieve excess returns through any information.

**2\. No-Arbitrage Principle**

-   **Definition:** No-arbitrage means there are no risk-free arbitrage opportunities in the market, i.e., it’s impossible to profit risk-free by simultaneously buying and selling related assets. No-arbitrage pricing involves constructing a replicating portfolio whose cash flows are identical to those of the asset being priced, thereby determining the asset’s fair price.
-   **Core Idea:** If arbitrage opportunities exist, rational investors will immediately act, buying undervalued assets and selling overvalued ones until the arbitrage opportunity disappears and prices return to equilibrium.
-   **Dependence on Market Efficiency:** No-arbitrage pricing is the foundation of derivative pricing and relies on market efficiency. Only in an efficient market can arbitrage opportunities be quickly eliminated, making no-arbitrage pricing valid.

**3\. Risk Neutrality**

-   **Definition:** Risk neutrality is an assumption that, in a risk-neutral world, investors are indifferent to risk, meaning they do not require additional risk premiums to bear risk. In other words, investors only care about expected returns, not the volatility of returns.
-   **Importance:** Risk-neutral pricing is a core tool for derivative pricing. By constructing a risk-neutral probability measure, we can use expected values to determine the fair price of derivatives without considering investors’ risk preferences.
-   **Dependence on Market Efficiency and No-Arbitrage Principle:** The validity of risk-neutral pricing depends on market efficiency, particularly the no-arbitrage principle. In an arbitrage-free market, regardless of investors’ risk preferences, arbitrageurs will eliminate any price discrepancies, causing derivative prices to converge to those determined by risk-neutral pricing.

## Core Derivation Process

For those who don’t want to delve into the math, just follow the conceptual flow. However, pay special attention to whether each step introduces assumptions or is purely mathematical derivation. No information comes out of thin air—this is the only way to gain a complete understanding from an informational perspective.

Longbridge has formatting issues with this table (nothing like self-deprecating humor), so for layout purposes, I’ll just include a screenshot...

## Greek Letters

The final pricing function for options includes four parameters. Processing these parameters yields the commonly used Greek letters:

1.  S: Underlying stock price
    1.  ∆ (Delta): The first-order partial derivative of the option price with respect to the underlying stock price (i.e., holding other conditions constant, assessing the sensitivity of the option price to the underlying stock price. The specific calculation treats other variables as constants and applies derivative rules to S to obtain the partial derivative).It’s worth noting that this ∆ is the same as the ∆ in the risk-neutral portfolio constructed earlier. ∆ ranges between -1 and 1.
    2.  Γ (Gamma): The second-order partial derivative, reflecting the rate of change of ∆.
2.   σ: Underlying stock volatility
    1.  Vega: The first-order partial derivative of the option price with respect to the underlying stock’s volatility.
    2.  Worth mentioning: Assuming other parameters are fixed, the implied volatility of the underlying stock can be derived by reversing the market’s option pricing to infer σ. Options with different strike prices/expiration dates will yield different implied volatilities.
3.  r: Risk-free interest rate (e.g., long-term U.S. Treasury bond rate)
    1.  ρ (Rho): The first-order partial derivative of the option price with respect to the risk-free interest rate.
4.  t: Time
    1.  Θ (Theta): The first-order partial derivative of the option price with respect to time.

Theoretically, second-order partial derivatives can be taken for all parameters, yielding eight Greek letters, but these are generally only used in more complex portfolios or scenarios requiring higher smoothness.

Additionally, by fixing r and σ, we can observe the behavior of these partial derivatives across the two key dimensions of S and t. The charts below are from [Donghai Securities - Option Greeks](https://app.xinhuanet.com/news/article.html?articleId=5cf6d0ae-8b28-42a5-8d2b-3442f9a9f987).

## ∆ and Risk

∆ represents **how many long or short positions in the underlying stock are needed to hedge the risk exposure (i.e., upward or downward volatility) for each option held to maintain risk neutrality**. Due to the leveraged nature of options, the proportion of the underlying stock required per option must be between -1 and 1. For example, if the underlying rises by $1, the price of one call option (note: one contract is not one lot; one lot typically corresponds to 100 shares of the underlying) cannot possibly rise by more than $1 (because holding one call option is equivalent to "the right to buy one share").

The larger the absolute value of ∆, the more the option price follows the underlying’s volatility, meaning the underlying’s fluctuations are amplified more intensely in the option, indicating greater risk exposure. Each option contract requires more opposing underlying positions to hedge.

## Can Understanding the BS Model Make You Money?

No.

But fully understanding the BS model might help you avoid some inexplicable pitfalls.

At least that’s the case for me.

For example, you can use an option calculator to avoid some severely overpriced options?

## 評論 (8)

- **奥马哈的稻草人 · 2025-01-20T06:32:49.000Z**: Too professional
  - **兴华XingHua** (2025-01-20T08:24:22.000Z): The BS model is an unavoidable professional topic, a necessary path to delve into options.
- **纳指嘉措 · 2025-01-18T13:18:55.000Z**: Well written, but I don't understand
  - **兴华XingHua** (2025-01-18T14:36:59.000Z): The BS model is indeed too broad. I haven't found a way to control the length while explaining the principles in a simple way yet😵‍💫
- **交易为生>_> · 2025-01-17T14:29:10.000Z**: Well written (but didn't understand a thing)
  - **兴华XingHua** (2025-01-17T14:35:15.000Z): 😂 Indeed, it could be written in more detail, such as starting completely from the properties of Brownian motion and gradually unfolding, while introducing the knowledge used at each step, trying to 
- **爱浓.马斯克 · 2025-01-16T23:52:35.000Z**: Why don't you go for a PhD and do research?
  - **兴华XingHua** (2025-01-17T01:14:01.000Z): Comprehensive reasons😢