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The Skinny on Options Math - Black-Scholes Model

Jun 29, 2015

By: Sage Anderson

The Skinny on Options Math is a regular segment on the tastytrade network that highlights and analyzes many of the mathematical concepts associated with equity derivatives trading.

In addition to hosts Tom Sosnoff and Tony Battista, The Skinny on Options Math typically features Jacob Perlman, a graduate student in the mathematics department at the University of Chicago. The purpose of these segments is to break mathematical options concepts down into more manageable pieces to assist with practical application.

Feel free to delve into The Skinny on Options Math show archives to explore this subject in greater detail. Additionally, we encourage you to leave comments, questions, or suggestions for future programming below in the comments section of the blog.


As this is the introductory edition of The Skinny on Options Math for the blog, let’s start from square one by reviewing a couple of the main concepts from the first several episodes of the show.

In the case of options math, the natural place to start is also the most important - the price of an option.


OPTION PRICE

The price of an option is of course a critical piece of data for any options trader in formulating a trading decision. The primary task of an options trader is to develop strategies around implied volatility and days until expiration. Formulating a decision and executing a trade consequently allow a trader to increase the probability of earning a profit (winning!).

So how is an option priced?

That is the paramount question, which Jacob explores during his first few segments of the show.

Cutting right to the heart of the matter, an equity option is priced using a mathematical equation that was first published by Fischer Black and Myron Scholes in 1973 (resulting in a Nobel Prize in Economics). The equation, known widely as the "Black-Scholes Model," is a partial differential equation that estimates the value of an option over time.

A key component to understand is that the final expiration price of an option is based on the underlying stock price on that day. Therefore, any given market value for an option price is based on a model that estimates the future probability of the stock’s value.

The Black-Scholes Model incorporates probability theory to estimate the future value of a stock using the historical movement of the stock as a predictive component for the option’s value. This concept is encapsulated in the term "volatility."

Historical volatility is a measure for the variation of price in an underlying stock over a distinct period of time in the past. Implied volatility is derived from the market value of an option price and expressed in volatility terms, as opposed to dollars and cents. Future volatility is the greatest unknown in options pricing and trading.

The Black-Scholes Model uses five inputs in order to calculate a theoretical value for an option, which include:

1. Strike price

2. Time to expiration

3. Current stock price

4. Risk-free rate

5. Volatility of the stock

The first four variables are known, which is why the Black-Scholes’ reliance on easily accessible data is one reason many use this as a benchmark to determine the value of an option.

The fifth input is a key reason that the model produces an estimated price, or theoretical value, for any given option. The future volatility of a stock is an unknown.

For this reason, many market participants have differing views on volatility, which therefore can lead to differences on determining how the value of the option is priced.

Without getting too caught up in the exact mechanics of the Black-Scholes Model (Jacob explains the model in the video below), the important thing to understand is that there is a standard equation which serves as the foundation of option pricing in the financial world.

While the Black-Scholes Model is the foundation of theoretical pricing in options, other members of the academic and private-sector communities have created adjusted versions of the Black-Scholes Model for the futures and foreign currency markets. Adjusted versions also exist even for pricing equity derivatives.

Much of The Skinny on Options Math content library relates in some way, shape, or form to the Black-Scholes Model. This may include (but is not limited to) broader-based discussions on the Black-Scholes Model and its variables, adjusted versions of the model, data used in the model, or other terms and methods of analyzing the mathematical side of options trading.

We similarly encourage you to follow up with questions or suggestions in the comments section below.

As always, thank you for being a part of the tastytrade community!


Options involve risk and are not suitable for all investors. Please read Characteristics and Risks of Standardized Options before deciding to invest in options.

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