Option Pricing ModelsBSM
Option pricing models estimate an option’s fair value. Black-Scholes (1973) gives a closed-form price assuming constant volatility; binomial models use a tree of up and down moves and handle early exercise; Monte Carlo methods simulate many price paths; the Heston model lets volatility itself change randomly.
An option pricing model turns a handful of inputs — share price, strike, time, interest rates and volatility — into an estimate of what an option should cost. Black-Scholes is the best-known, but traders also use binomial trees, Monte Carlo simulation and stochastic-volatility models such as Heston when the simple assumptions break down.
- Category
- Derivatives
- Reading time
- 4 min
- Also called
- Option valuation models, Options pricing formulas, Black-Scholes and binomial models
- Updated
Option pricing models in plain words
Black and Scholes published their famous formula in 1973.1 Today traders around the world use it every day to value stock options.1
A popular alternative is the binomial model of Cox, Ross and Rubinstein, which can handle features Black-Scholes leaves out, such as early exercise.2
Monte Carlo methods simulate thousands of possible price paths, and models like Heston let volatility itself move, at the cost of more computing and more assumptions.
How to price an option with a model
Collect the inputs
Note the share price, strike, days to expiry, interest rate, dividends and a volatility estimate.
Choose a model
Use a closed-form formula for simple European options and a binomial tree when features like early exercise matter.2
Understand the call formula
For a European call, the value is the expected share value minus the expected cost of exercising.1
Check the sensitivities
Expect the value to rise with volatility (the standard deviation of the share price) and with the risk-free interest rate.1
Compare with the market
If the market price differs, back out the volatility the market is using.
Stress the result
Re-run the model with higher and lower volatility to see how sensitive the price is.
Worked examples
Example 1 — Volatility input
+$1.50 from volatilityA hypothetical model prices a call at $4.00 using 20% volatility and $5.50 using 30%.
| At 20% vol | $4.00 |
| At 30% vol | $5.50 |
| Difference | $1.50 |
Higher volatility increases the option's value, just as the model predicts.1
Example 2 — Market vs model
Market implies higher volA hypothetical call trades at $6.00 while your model says $5.00.
| Market price | $6.00 |
| Model price | $5.00 |
| Gap | $1.00 |
The gap usually means the market is pricing in higher volatility — the implied volatility you get by running the market price back through the model.2
Example 3 — Early exercise
$0.20 extra valueA hypothetical American put is worth $3.20 on a binomial tree but $3.00 under Black-Scholes.
| Binomial | $3.20 |
| Black-Scholes | $3.00 |
| Early-exercise value | $0.20 |
The tree picks up the value of early exercise, which Black-Scholes doesn't include.2
Main option pricing models
| Aspect | Black-Scholes | Binomial tree |
|---|---|---|
| Method | Closed-form formula | Step-by-step price tree |
| Early exercise | Not handled | Handled |
| Speed | Instant | Slower with many steps |
| Best for | European options | American options |
Where pricing models are used
Market making
Market Maker desks quote prices around model values.
Reading implied volatility
Traders convert prices into Implied Volatility.
Risk management
Models produce Greeks used for Hedging.
Valuing employee options
Companies estimate the cost of stock options they grant.
Advantages and disadvantages
Advantages
- Give a consistent way to compare option prices.
- Show how each input moves the price.
- Allow conversion between price and implied volatility.
- Support hedging with sensitivities.
Disadvantages
- Rely on assumptions that markets break.
- Volatility input is unknown and must be estimated.
- Complex models need more data and computing.
- Model prices can differ from traded prices for long periods.
Going deeper
Read as far as you need. Each level builds on the one before it.
What the formula does
Beginner
Fischer Black and Myron Scholes created the Black-Scholes model to price options on securities.2 It needs only a few inputs, which made it popular.
Trees and early exercise
Intermediate
A binomial model breaks time into steps in which price moves up or down, and it can handle early exercise and similar features.2
Simulation and stochastic volatility
Advanced
Monte Carlo simulation averages payoffs over many random paths and suits path-dependent options. Heston-type models let volatility vary randomly, which helps explain the volatility smile. Because implied volatility is worked out from market prices through a model, the choice of model shapes the volatility traders see.2
Common mistakes
Treating the output as truth
A model price is an estimate, not a guaranteed value.
Using the wrong model
Black-Scholes ignores early exercise on American Options.
Plugging in stale volatility
Old Volatility data can misprice the option badly.
Ignoring dividends
Dividends lower call values and raise put values.
Check your understanding
Test yourself before moving on. Each answer comes with an explanation.
Key points to remember
- Pricing models estimate an option's fair value.
- Black-Scholes was published in 1973.
- Binomial trees handle early exercise.
- Monte Carlo and Heston models handle more complex cases.
- Volatility is the input that matters most.
Frequently asked questions
Is Black-Scholes still used?
Why do model and market prices differ?
Do retail traders need these models?
Further learning
- Options
The contracts these models value.
- Implied Volatility
The volatility backed out of prices.
- Volatility
The key model input.
- Hedging
How model sensitivities are used.
Sources
This article is written in our own words. The sources below support the specific facts, rules and figures marked with a number in the text. Links open the original publisher’s page.
- The Royal Swedish Academy of Sciences (NobelPrize.org). Press release: The Prize in Economic Sciences 1997 (opens in a new tab) Checked .
- U.S. Commodity Futures Trading Commission. CFTC Glossary (opens in a new tab) Checked .
Educational content, not financial advice. Examples use hypothetical prices and simplified costs; actual fees, margin rules and regulations vary by broker, market and country. We are not affiliated with any publisher linked above. Last reviewed . Report an error or request a source credit.