What Is the Binomial Option Pricing Model?
The binomial option pricing model, developed in 1979, is a potent method for valuing options. This model utilizes an iterative approach, specifying points in time (nodes) between the valuation date and the option’s expiration date.
Key Takeaways
- The binomial option pricing model uses an iterative method with multiple periods to value American options efficiently.
- This model considers two possible outcomes for each time interval: a price increase or a price decrease, forming a binomial tree.
- Its intuitive nature and practical application make it more popular than the Black-Scholes model in many scenarios.
The binomial model simplifies price change possibilities, thus eliminating arbitrage opportunities. Here’s what a simplified binomial tree might appear:
The Essence of the Binomial Option Pricing Model
The model assumes two potential outcomes in its bi-conditional approach: a price move up or down. Despite its inherent mathematical simplicity, complexities grow as the periods increase.
Unlike the Black-Scholes model, presenting a single numerical result based on inputs, the binomial model handles multi-period calculations. This helps users observe asset price transitions over time and assess options based on different time-based decisions.
For American options, which can be exercised anytime before expiry, the binomial model is pivotal in deciding optimal exercise moments by simulating various price scenarios using the binomial tree structure.
How to Calculate Price with the Binomial Model
To determine option prices using the binomial model, traders might use a consistent probability for success and failure in each period till expiry. Nonetheless, as new information unfolds, different probabilities can be integrated.
Binomial trees effectively price American and embedded options due to their mechanical simplicity. However, a limitation is the assumption that the underlying asset takes one of two discrete values per period. Realistically, asset prices come in a continuous range.
For instance, with a 50/50 probability, an asset could either increase or decrease by 30% in period one. The probability may shift to 70/30 in the following period based on new market dynamics.
Imagine evaluating an oil well. Initially, there’s a 50/50 chance of increasing value. As market fundamentals improve, the chance of appreciation can rise to 70%. The binomial model accommodates these dynamic probabilities over the Black-Scholes model.
Real-World Example of the Binomial Option Pricing Model
Consider a stock priced at $100 per share, projected to either rise to $110 or drop to $90 in one month:
- Stock price = $100
- Stock price in one month (up state) = $110
- Stock price in one month (down state) = $90
Next, evaluate a call option with a strike price of $100, expiring in one month. The option is worth $10 in the up state and $0 in the down state.
By purchasing half a share and writing/selling one call option, the investment looks like this initially and at period-end:
- Cost today = $50 - option price
- Portfolio value (up state) = $55 - max ($110 - $100, 0) = $45
- Portfolio value (down state) = $45 - max($90 - $100, 0) = $45
Since the portfolio payoff consistently equals $45, discounted at the risk-free rate, the equation becomes:
- Option price = $50 - $45 × e^(-risk-free rate × T), with e = 2.7183.
Assuming a 3% annual risk-free rate and 0.0833 for T (1/12 month), the call option price now is approximately $5.11.
The model’s simplicity and robust operation provide real advantages over the Black-Scholes model, ensuring fewer errors and reducing arbitrage opportunities through iterative pricing. This flexibility makes the binomial pricing model particularly adept at figuring out complex options like American options.
Related Terms: Options, Black-Scholes Model, Binomial Tree, Arbitrage, Valuation.
References
- Wiley Online Library. “The Journal of Finance-Volume 34, Issue 5-Two-State Option Pricing”.
- Corporate Finance Institute. “Option Pricing Models”.
