Option pricing models are an important aspect of trading online because they’re essential for calculating the value of an options contract. They are also used a step further with the implementation of lucrative derivative strategies. Options pricing models take factors like the expiration date of a contract, volatility, and the underlying asset price into account in the pursuit of determining a fair value.
But what this particular guide is focused on is which pricing model is the best to use overall—it’s between two methods called the Black-Scholes model and the binomial models. One isn’t inherently better than the other, but they’re instead well-suited for different kinds of traders who use different trading strategies and have different trading objectives in mind. This guide will go over how each model works, some of the key differences between them, and which scenarios are best to use, either Black-Scholes or the binomial model.
Understanding the Black-Scholes Model
What is the Black-Scholes model and what kind of options are best at pricing? We cover this and several other topics including how it works and the primary advantage and limitations that come when using this pricing model in options trading. For additional information on the Black-Scholes pricing model, read more here.
What Is the Black-Scholes Model?
The Black-Scholes model was developed in the 1970s by three professionals from the University of Chicago and the Massachusetts Institute of Technology. It’s a model that is designed specifically for European options because they are much cheaper and because investors can only exercise the option on the expiration date.
The Black-Scholes model is a differential equation, and there are five inputs, including the amount of time before the option expires, the underlying stock’s price, the strike price, volatility, and interest rates. The model’s key assumptions can be categorized as such:
- Risky Asset Assumptions: Random walk, normal distribution of returns, constant volatility, and no dividends
- Riskless Asset Assumptions: Constant risk-free interest rates
- Option Assumptions: European options
- Market Assumptions: No transaction costs, no restrictions to short selling, perfect liquidity, and no arbitrage possible
How It Works
The long and short of how the Black-Scholes model works is that it uses a complex mathematical equation to figure out the present value of the potential future payoff on an option. It’s based on how likely the underlying asset price is to reach a certain point by the option contract’s expiration date.
In simple terms, this mathematical equation calculates the fair price for an option based on the five primary inputs: current stock price, strike price, risk-free interest rates, volatility, and time to maturity.
- C = call option price
- S = current stock price
- K = strike price
- r = interest rate (risk-free)
- t = time to expiration
- N = normal distribution
Formula Breakdown
Advantages of Black-Scholes
- Efficiency and Simplicity—There are only five inputs that an investor needs to enter into the Black-Scholes model equation. The process is simple and investors can get a good understanding quickly of how an options’ price might react to price movements in its underlying stock.
- Applicability to Liquid Markets—The Black-Scholes markets assume that markets are completely liquid, meaning that it’s in the realm of possibility for traders to buy or sell any amount of stock or option at any given time.
Limitations
Faulty Assumption
- The Black-Scholes model assumes constant values for volatility and the risk-free rate of return. Since this isn’t possible over the option’s duration, you cannot expect these conditions to exist in the real world.
- Black-Scholes ignores the impact of dividends on the change in valuations because they assume no dividend payout.
- This model isn’t suitable for American options because it assumes no early exercise of the options contract.
- Black-Scholes ignores the impact of liquidity risk and brokerage charges because it assumes costless and continuous trading.
- The pricing model ignores large price swings found in the real world due to assuming stock prices to follow a lognormal pattern.
Understanding the Binomial Model
If you’re interested in learning how the binomial model sheds light on trading options online, keep reading to discover how the model is structured, how it works, and what the primary advantages or limitations are. For additional reading on the binomial model, read our guide here.
What Is the Binomial Model?
The binomial option pricing model that used to price options by breaking time into smaller steps. It’s used to price American and European options, and traders can adjust them to accommodate for several underlying assets and dividends. In this model, the underlying asset can be moved up or down at each step by an amount specified by the trader. The binomial model is used so traders can understand how options would perform in different scenarios. This model can be super helpful for traders who are looking for different hedging strategies.
The binomial model was first introduced in 1979 by creators John Cox, Stephen Ross, and Mark Rubinstein with the purpose of calculating options price by assuming a stock price can only move up or down at each time step. Over the years, options traders have had some great success using the binomial model, which has been noted to be more accurate than a simpler model like the Black-Scholes.
A Step-by-Step Framework
The way the binomial model works is that the time to expiration of an options contract is divided into discrete time steps. Each step creates what’s known as the “binomial tree” which outlines each possible price path. The idea is that the underlying asset can only move up or down. The options’ value is calculated by going to the final time step and working backward from there. Traders can use the model to gauge each price movement’s probability and figure out the expected future payoff in conjunction with the current value of the option.
How It Works
Describing the binomial model involves talking about something called the “binomial tree” and an explanation of the iterative process which is a step-by-step framework for creating lattices and nodes. If you’re intrigued by what you’ve heard, keep reading, and we’ll describe how this model works as best we can. It’s quite interesting and terrific for breaking options pricing into easy-to-manage steps!
Tree Structure
The first step in constructing the binomial tree is to establish the beginning node, which represents the current asset price. You would then head to the next time step and create two new nodes for each existing node. One of these represents the price following an upward movement, and the other indicates the price following a downward movement.
The process continues until you reach the starting node, at which point the tree is completely assembled.
Iterative Process
- Lattice and Nodes: Divide the time between the valuation date and the expiration date. The result is discrete time steps. By doing this, traders have created the “binomial lattice” where each node of the lattice is a possible underlying asset price at any given time.
- Final Nodes: The options’ value is calculated by going to the final time step and working backward from there (the final node of the lattice). The point where the final node is represents the possible price at the expiration date.
- Options Value: Figure out the option’s value at each nod behind the final node. Traders can do this by discounting the expected value at the risk-free interest rate back one period.
- Go On Down the Line: Repeat this process until you’ve worked from the final node back to the starting node. The value at the starting node will represent the fair value estimate of the current options (according to the model).
All told, the binomial model can be quite flexible for dealing with options prices and can accommodate well with various assumptions and conditions like changing volatility. The binomial model is flexible in many other ways as well:
- Intuitive Structure—The model can accurately capture discreet price movements
- Tilt Parameter—The model can be extended by the tilt parameter shifting the tree upwards or downwards
- Discrete-Time Steps—The model allows for multiple time steps (they can be adjusted along with expiration or volatility)
- Accommodates Different Assumptions—The model is flexible with time-varying volatility or constant volatility
- Caters to Different Options Types—The model accommodates options from the United States and Europe
Advantages of the Binomial Model
The main advantage of using this pricing model is that it gives you a more complete picture of how the price could possibly move as well as the fact that the model caters to American options that allow early assignment.
- Multi-Period View: See the underlying asset price and options price for multiple periods and a wider range of possible results for each period.
- Transparency: Discover all the possible price movements through the use of the binomial tree.
- Better Suited for American Options: The binomial model is more accurate for American options due to the trader’s ability to value the option over different periods.
Limitations of the Binomial Model
While there are several drawbacks to the binomial model, it’s far the most accurate model out there, even if it’s a bit time-consuming and takes some heavier computer power to figure out.
- Computational Intensity: It takes longer to value the option, which means the calculations aren’t as sleek and efficient as those found with other pricing models.
- Less Efficient for Complex Portfolios: It’s not a good choice if you want to calculate a lot of options quickly.
- Prices Are Determined by Market Forces: A key limitation of the binomial model is that the actual prices are dictated by the market, so the sophisticated design of the binomial model can still be wrong.
Key Differences Between Black-Scholes and Binomial Models
If you’ve read through this far, it’s apparent that there’s a big difference between the Black Scholes pricing model and the binomial model. One’s great for pricing options in America, while the other is suited only for European options. But aside from this, there are also some other major areas where these models diverge: assumptions, computational requirements, and overall accuracy.
Assumptions
One of the main differences between the two models is that Black Scholes uses differential equations, which lead to a view of static volatility. Black Scholes shows investors an implied volatility surface at a particular moment in time. Compare this to the binomial model, which assumes two possible outcomes for an asset’s price (up and down). It’s a more accurate snapshot of all the varying possibilities with the use of a flexible volatility metric.
Black Scholes Assumptions
- Risky Asset Assumptions: Random walk, normal distribution of returns, constant volatility, and no dividends
- Riskless Asset Assumptions: Constant risk-free interest rates
- Option Assumptions: European options
- Market Assumptions: No transaction costs, no restrictions to short selling, perfect liquidity, and no arbitrage possible
Binomial Model Assumptions
- Investors are risk-averse
- Constant volatility
- No arbitrage opportunities
- Time is divided into discrete intervals/steps
- Two possible outcomes for any stock (up or down)
- The underlying asset doesn’t pay any dividends
- Risk-free rates don’t change
- Constant interest rate
- No taxes
- No transactional costs
Options Priced
The Black Scholes model is better designed for European options, whereas the binomial model is well-suited for American options as it gives investors the ability to value the option over different periods.
Computational Requirements
The Black Scholes model offers a single mathematical formula that requires basic arithmetic. This makes computational requirements minimal on standard calculators and is helped by the fact that there are only five criteria inputs. On the other hand, the binomial model has two inputs needed to calculate probabilities within the distribution: success or failure. However, the model goes through every possible outcome based on these two inputs (iterative tree structures) making it computationally intensive—it takes longer to value the option which means the calculations aren’t as sleek and efficient as those found with other pricing models.
Accuracy and Realism
As far as real-world applications go, both models are accurate for options pricing—the binomial model can be used as an alternative to the Black Scholes model. Binomial is the better model for getting more accuracy with long-dated options on securities that have dividend payments. Plus, it’s more accurate as the number of time steps increases, giving it a leg up on the Black Scholes model.
Scenarios—When to Use Black-Scholes or Binomial Models
Talking about Black Scholes and binomial models is like comparing apples and oranges. They’re each designed for something different, so it’s difficult to say which is better or worse than the other. The best we can do is describe the ideal situations to use each of these models in to give you an idea of how their strengths are rooted in what they were designed to do.
Situations for Black-Scholes
- Because the Black Scholes model only has five inputs and a somewhat simple math equation, this model is best used when traders are looking for quick estimations for European options.
- Speaking of European options, the Black Scholes can only be used on these contracts because it does not assume early assignment. European options are notable for not being able to be exercised until the expiration date and never before.
- Another key assumption of the Black-Scholes model is perfect liquidity, so they’re great for gauging price changes on options in highly liquid markets with stable volatility.
Situations for Binomial
- The binomial model is ideal for pricing American-style options where the contract can be exercised before the expiration date.
- Binomial models are great for scenarios with fluctuating market conditions because the model takes changing volatility and dividend payments into account. Because there are more factors at play, the binomial model has a much more flexible approach and accounts for more of what goes on in the real-world market.
Expert Opinions and Insights
Check out some quotes and perspectives from financial analysts or experienced traders on both of these pricing models:
“I believe the Black–Scholes formula produces strange results when the long-term variety is being valued.
–Warren Buffet
“Black–Scholes is wrong but useful, interpretable, and general due to its simplicity.”
–Greg Gundersen
“The binomial pricing model is useful for pricing path-dependent options such as American-style options or Barrier options. These are options whose value will change according to the random path the asset will take until expiry. The binomial tree is used to price the option at different possible paths and weight the prices according to the probability that the option will go down that path.”
–Joshua Novak
“It’s really quick and easy to code and it’s also easy to explain to non-technical people. You can do the same sorts of thing with PDE code (and mathematically it’s the same thing), but it’s hard to explain PDEs to non-technical people, whereas it’s easy to explain a binomial tree or Monte Carlo. This is important when explaining what you are doing to non-technical traders, even more important when explaining it to regulators. Personally, I don’t think of the binomial method as a “price model.” You have another pricing model, like Black-Scholes and then you code it with a binomial tree.”
–Joseph Wang
Common Pitfalls to Avoid
- Black Scholes and the binomial model both assume continuous/costless trading and constant volatility, all of which are unlikely to be conditions in the real market. It’s key to take some of these results with a grain of salt.
- The binomial can lead to inaccurate results because it assumes that every trial is independent and that the outcome of one doesn’t influence the next.
- The binomial pricing model doesn’t account for transaction costs or taxes.
Use Pricing Models to Inform Your Trades
Using a pricing model while trading options contracts can be an invaluable asset to traders and investors. They’re an important element of trading online because they’re essential for calculating the value of an options contract as well as the implementation of derivative strategies. Investors can use factors like the expiration date of a contract, volatility, and the underlying asset price to figure out the fair value of current options using the Black Scholes or binomial pricing models.
Use Black Scholes if you’re looking for a quick calculation of options prices for European contracts and the binomial model when dealing with fluctuating market conditions in the American options market. Once you’ve figured out what your trading goals and needs are, it can be quite easy to find the right model for getting the job done—consider testing both models using historical data or demo accounts if you’re unsure.
