Counting Cards: Cracking the Code of Options Pricing Models
/Betting on options costs money. And, depending on your sell strategy, those premiums could be your only source of profit.
There’s a razor-thin margin between hitting the jackpot and busting when finding the ideal price for an option. Financial market participants use options pricing models to determine the fair value of their contracts.
Overview of Options Pricing Models
There are various options pricing models, but the big two are Black-Scholes and binomial.
The Black-Scholes Model
To figure out how much to charge for premiums on various option transactions, two guys in the 70s cooked up formulas based on a partial differential equation you didn’t cover in high school algebra. So let’s skip the math and break it down into the key components instead:
Stock Price (S): The current price of the stock or asset
Strike Price (K): The price at which the option allows you to buy (call option) or sell (put option)
Time Until Expiration (T): The time left until the option expires, often expressed in years
Volatility (s): How much you expect the stock’s price to move
Risk-Free Interest Rate (r): The return you would expect from a no-risk investment, like a government bond
Option Type (Call or Put): Whether the option gives you the right to buy (call) or sell (put) the stock
Basically, Black-Scholes assumes the stock’s price follows a random pattern and that you can invest in a risk-free bonk at the risk-free rate. The model combines these assumptions to calculate the option’s premium.
The Black-Scholes formula boils down to two steps. First, it calculates the likelihood that the option will be in-the-money (profitable) at expiration. Then, it factors in how much that profit will be worth in today’s money by adjusting for the time value of money.
The final price depends on the balance of two probabilities:
How likely it is that the stock price will end up higher (for a call) or lower (for a put) than the strike price
The present value of the potential profit, based on the expiration date and the current interest rate for safe investments.
While the formula is widely popular, it operates on some unrealistic premises:
The stock price changes continuously without big jumps
There are no transaction costs (like brokerage fees) or taxes
The risk-free rate is constant
Volatility doesn’t change
However, if the market always behaved predictably, we would all be rich. Realistically, you will probably have to adjust your numbers using other models or real-world factors like economic reports or sudden mergers.
The Binomial Model
The binomial model is Black-Scholes’ scrappy younger cousin that doesn’t assume stock prices move in some nice, neat fashion. It works by breaking the time until expiration into smaller periods (e.g., 30 days, 60 days, etc.) called nodes. At each node, the model assumes the stock price can either rise or fall by a specific amount. Let’s walk through the process:
Build the Tree: The big picture is a binomial tree, which looks like a series of branches. Starting with the current stock value, each branch represents how much the price increases if it moves up (up factor) or how much it decreases if it moves down (down factor).
Assign Probabilities: The binomial model uses probabilities to estimate the price movements that aren’t based on actual market predictions but are risk-neutral, meaning they assume the expected return on investment, or ROI, matches the risk-free interest rate.
The up probability reflects the likelihood of the stock price increasing in one step.
The down probability is one minus the up probability (the two probabilities must equal 100%).
Calculate Payoffs at Expiration: Calculate the option’s payoff for each possible stock price at the farthest nodes at the end of the tree.
The call option payoff is the difference between the stock price and strike price, but only if the stock price is higher. If not, the payoff is zero.
The put option payoff is the difference between the strike price and the stock price, but only if the stock price is lower.
Work Backward: Once you know the option's value at expiration for each possible outcome, you work backward through the tree. At each earlier node, you calculate the option’s value as a weighted average of the values at the next step, using the up probability and down probability. Then, you discount this value to the present using the risk-free rate. This process continues until you reach the present (the root of the tree), giving you the option’s current fair value, or premium.
Adjust for Additional Features: If the option has special features, such as early exercise, account for these during the backward calculation. At each step, compare the option’s value if held to maturity with its immediate exercise value, and use the higher of the two.
The binomial model is popular because it’s simple and adaptable. It can handle a wide range of scenarios, such as changes in volatility or dividends paid by the stock. Unlike the Black-Scholes model, it doesn’t assume that time or volatility remains constant.
However, while flexible, computing the binomial model can go slowly for long expiration times or when using very small time steps. In these cases, it requires building a massive tree with many branches. However, modern computing power has made this less of a problem.
How Implied Volatility Affects Pricing
Implied volatility is one of the most important factors affecting option prices because it represents the market’s expectation of how much the underlying stock’s price will move in the future. IV plays a key role in determining the extrinsic value of an option, which is the portion of the option's price not based on its intrinsic value (the difference between the stock price and the strike price).
When IV rises, it signals that the market expects bigger price swings in the underlying stock, even if the direction (up or down) is unknown. This increase in expected movement makes options more valuable because there’s a greater chance they will end up in-the-money by expiration. Both call and put options become more expensive when IV goes up because more volatility increases the risk and the potential for profit.
Alternatively, when IV drops, it means the market expects less price movement. With reduced uncertainty, the chances of the stock making a big move in your favor decrease, which reduces the extrinsic value of the options. Lower IV leads to cheaper options premiums for both calls and puts because the market is accounting for lower risk.
Now, IV doesn’t affect all options equally. Options that are at-the-money (closer to the stock’s current price) tend to be more sensitive to changes in IV compared to deep in-the-money or deep out-of-the-money options.
Understanding the Greeks in Pricing
The Greeks in options pricing help you evaluate the risks and rewards of an option before making a trade. Each Greek represents a different factor that affects the option’s premium, giving you insight into how the option might behave as certain parameters change.
Delta (D) measures how much the price of an option is expected to change if the stock price moves by $1. For example, a Delta of 0.50 means that if the stock price rises by $1, the option price will increase by $0.50. In addition, the closer Delta is to 1.0 for a call (or -1.0 for a put), the higher the chances the option will have value at expiration.
Gamma (G) quantifies Delta’s stability, or how it changes as the stock price moves. A high Gamma means that small moves in the stock price can lead to big swings in Delta, making the option more sensitive to price changes. Gamma is highest for at-the-money options and drops off as options move further in- or out-of-the-money.
Vega (V) tracks the sensitivity of an option's premium to changes in implied volatility. When implied volatility rises, options become more expensive because there’s a greater chance of large stock price movements, which increases the option’s extrinsic value. Conversely, if volatility drops, the option’s price decreases.
Theta (Q) measures how much the value of an option decreases as time passes, a phenomenon known as time decay. Options lose value over time because the closer you get to expiration, the less time there is for the stock price to move in a way that makes the option profitable.
Rho (r) measures the impact of interest rate changes on an option's price. Although Rho is typically less influential than the other Greeks, it becomes more relevant when interest rates are fluctuating. Higher interest rates tend to increase the value of call options and decrease the value of put options.
How the Greeks Work Together
While each Greek measures a different factor, they don’t function independently. For example, Delta and Gamma often work hand-in-hand: Delta tells you the immediate sensitivity to stock price changes, and Gamma tells you how that sensitivity evolves as the stock moves. Vega interacts with Theta because changes in implied volatility can increase or decrease the rate of time decay. Understanding these relationships helps you predict how your option’s premium might change under various scenarios.
Limitations of Pricing Models
Both models are great at giving you a basic sense of your odds, but they’re not perfect for real-world trading, where the casino floor is unpredictable.
Black-Scholes Model Limitations
Constant Volatility: The model assumes that volatility remains the same over time, which doesn’t reflect real market behavior where IV often changes.
European Options Only: It applies only to European-style options, which can only be exercised at expiration, not before, as you can in the U.S. market.
No Dividends Included: The model doesn’t account for dividends paid by the underlying stock.
Fixed Interest Rates: It assumes a constant, risk-free interest rate, which is unrealistic as interest rates can fluctuate.
Binomial Model Limitations
Fixed Steps: The model breaks time into fixed intervals, but in reality, market movements occur continuously.
Computational Complexity: To get highly accurate predictions, you have to make a lot of calculations, making the model complex and time-consuming.
Assumes Neutral Conditions: The model assumes a fair market without accounting for factors like market manipulation or unexpected events.
Simplified Movement: It assumes stock prices can only move up or down at each step, missing the full range of random market movements.
Real-World Applications of Pricing Models
For you hands-on people, let’s apply simplified models to a call option contract on $MSFT stock trading at $413.
The Black-Scholes Model
Here are your parameters:
Stock Price: $413
Strike Price: $450
Time Until Expiration: Six months
Volatility: 15% (indicating more stability with less dramatic price swings)
Risk-Free Interest Rate: 4.5% from government bonds
Black-Scholes plugs these inputs into its formula. It determines that the stock will likely move less and assumes a fair premium of $8 per share to purchase the option today.
The Binomial Model
Maybe you have more information, like an impending earnings announcement, that could make $MSFT take a wild, unpredictable ride. This binomial example uses a 40% volatility rate to calculate probable price movements per two-month period.
Stock Price: $413
Strike Price: $450
Time Until Expiration: Six months
Volatility: 40%
Risk-Free Interest Rate: 3.0% from certificates of deposit
Up Factor: 12.5%
Down Factor: 11.1%
Up Probability: 49.2%
Down Probability: 50.8%
Here’s how the steps play out:
At the end of the first period:
Up: $465
Down: $367
After two periods:
Up twice: $522
Up once, down once: $413
Down twice: $326
After the third-period expiration:
Up three times: $586
Up twice, down once: $465
Up once, down twice: $367
Down three times: $290
At each step, the model bases the option’s value on how far the stock has gone up or down. After running those numbers (and all else being equal), there’s a 60/40 fair chance you’ll hit the strike price of $450 or higher. A reasonable premium would be around $22.
Make Better Bets With Options Pricing Models
From retail investors to quants, the Black-Scholes and binomial options pricing models are their primary cheat sheets for setting premiums. Each model deals a different hand of insights, and playing both boosts your odds of finding the fairest price for your option.
Fortunately, you don’t have to memorize these formulas or crunch endless numbers on paper. Try Option Royale’s online options calculator to figure out what your options should cost, analyze the Greeks, and visualize risk and profit/loss estimations.
