Derivatives
Binomial Option Pricing Model: Valuing Options One Step at a Time
The binomial option pricing model approaches option valuation in a practical and intuitive way. Instead of assuming continuous price movements, it breaks time into small intervals and asks a simple question at each step: what are the possible prices the asset can move to next?
By answering this repeatedly, the model builds a clear picture of how option values evolve over time. This step-by-step logic is why the binomial model is widely tested in CFA and FRM exams.
What the Binomial Model Is Trying to Do
At its core, the binomial model values an option by replicating its payoff.
It assumes that over a short period, the underlying asset price can move to one of two possible levels:
- an upward move
- a downward move
By working backward from the option’s payoff at maturity, the model determines its value today.
Why the Binomial Model Matters
The binomial model is not just a pricing formula. It is a framework.
It shows how:
- option value depends on future payoffs
- risk can be hedged dynamically
- no-arbitrage pricing works in practice
Exams often test this intuition rather than the final numerical answer.
Risk-Neutral Probability
A key idea in the binomial model is the use of risk-neutral probabilities.
These probabilities do not reflect actual market beliefs. Instead, they are constructed so that expected returns on the underlying asset equal the risk-free rate.
This allows option values to be discounted at the risk-free rate, which is a central exam concept.
Backward Induction
The binomial model uses backward induction.
First, option values are calculated at maturity based on payoff.
Then, values are computed step by step moving backward to the present.
This process highlights how future uncertainty feeds into today’s option price.
American vs European Options
One major strength of the binomial model is its flexibility.
It can value:
- European options, which are exercised only at maturity
- American options, which can be exercised early
At each node, the model checks whether early exercise is optimal. This feature is frequently tested in exams.
Effect of Increasing the Number of Steps
As the number of time steps increases, the binomial model becomes more accurate.
With enough steps, it converges toward continuous-time models such as Black–Scholes. This relationship is often tested conceptually rather than mathematically.
Key Inputs and Their Impact
The binomial model depends on:
- underlying price volatility
- time to maturity
- risk-free interest rate
- exercise price
Changes in these inputs affect option values in predictable ways, reinforcing intuition about option behaviour.
Common Student Misunderstandings
Many students assume binomial pricing is inferior to Black–Scholes. It is not.
Others believe risk-neutral probabilities represent real-world likelihoods. They do not.
Some forget that the model is built on no-arbitrage principles.
These misunderstandings often appear as exam traps.
Closing Thought
The binomial option pricing model teaches more than how to compute an option price. It explains why options have value and how uncertainty, time, and risk interact. For CFA and FRM preparation, understanding the logic behind the model is far more important than memorising steps. Once that reasoning is clear, option pricing questions become much easier to approach.
