Introduction
The binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options which is based on a discrete-time framework, dividing time into a number of smaller intervals.
The binomial model was first proposed by William Sharpe in 1978 and formalized by Cox, Ross, and Rubinstein in 1979 and by Rendleman and Bartter in that same year.
A binomial model is based on the idea that, over the next period, the value of an asset will change to one of two possible values. The fundamental concept behind the binomial model is the construction of a binomial tree. The tree represents the possible price paths of the underlying asset over time. At each node of the tree, the asset price can move either up or down by a certain factor, and the option value is calculated at each node by considering the possibilities of exercising the option or holding it until the next node.
Enhancing Options Valuation with the Binomial Model
The Binomial Options Pricing Model (BOPM) stands out as a versatile tool, adept at accommodating diverse scenarios where alternative models falter. Its strength lies in depicting the evolution of an underlying asset over time, providing a nuanced understanding compared to singular point estimations.
Primarily, BOPM finds extensive utility in valuing American options, allowing exercise at any point within a specified timeframe, and Bermudan options, exercisable at specific instances. This model’s simplicity facilitates easy integration into various software platforms, including spreadsheets, enhancing accessibility and usability.
Despite being computationally slower than the Black–Scholes formula, the Binomial Model excels in accuracy, particularly with long-dated options on assets with dividend payments. This heightened accuracy contributes to its widespread adoption among options market practitioners.
However, in scenarios involving multiple uncertainties, such as real options, or options embedding intricate features like Asian options, the binomial methods encounter practical limitations due to inherent complexities. These complexities pose challenges in effectively applying binomial methods to derive accurate valuations in such cases.
Terminology in the Model
The Binomial Options Pricing Model (BOPM) involves the use of probabilities for upward and downward movements in the underlying asset’s price over discrete time intervals. The key parameters and formulas used in the model are:
For an American call option, the option price at each node is the greater of the intrinsic value (Exercise value) and the expected value discounted at that node’s probability.
These formulas and parameters are used iteratively through the binomial tree to estimate option prices at different nodes, considering the probabilities of upward and downward movements, time steps, interest rates, and the volatility of the underlying asset.
Methodology
The Binomial Pricing Model methodically tracks the progression of essential variables of an option within discrete time intervals. Achieving this involves constructing a binomial lattice or tree, and delineating potential values of the underlying asset across various time steps between the assessment and expiration dates. Every node within this lattice signifies a plausible price of the underlying asset at a specific moment.
The valuation process unfolds in iterations, commencing at the terminal nodes reachable at the time of expiration and systematically retracing steps backward through the tree towards the initial node, which marks the valuation date. At each stage, the computed value represents the option’s worth at that particular juncture in time.
Option valuation through this approach entails a three-step procedure:
1. Generation of the Price Tree: Constructing the lattice representing potential asset prices at different time intervals.
2. Calculation of Option Value at Terminal Nodes: Determining the option value at each final node corresponding to expiration.
3. Sequential Determination of Option Value at Preceding Nodes: Progressively computing the option value at each node preceding the final ones, moving backward through the tree.
This meticulous process of discrete-time analysis allows for a comprehensive understanding of an option’s value at various time points, enabling more accurate assessments and strategic decision-making.
Below is the pictorial representation of above example
Now let’s look into the pictorial representation of European and American Put Options:
The Python code for discovering values and creating visual representations is accessible in the Python code category.
The price of the European put option is: 8.0068
The price of the American put option is: 8.7253
Benefits and Limitations
Benefits
- Flexibility and Adaptability: It’s versatile and can be applied to value a wide range of options, including American and exotic options.
- Intuitive Understanding: The model’s construction is relatively straightforward and easier to comprehend compared to some other option pricing models, making it accessible to a broader audience.
- Consideration for Early Exercise: Unlike the Black-Scholes model, the binomial model can handle early exercise for American options, allowing for a more accurate valuation.
- Dynamic Nature: It incorporates a dynamic element by considering multiple possible price paths, enabling a better understanding and evaluation of uncertainty in the market.
Limitations
- Computationally Intensive for High Precision: As the number of steps increases, the computational complexity also increases significantly, making it computationally intensive for very precise valuations.
- Model Assumptions: It relies on certain assumptions, such as constant volatility and risk-free rate over the option’s life, which may not always hold true in real market conditions.
- Lack of Continuous-Time Model: It is a discrete-time model that approximates continuous-time processes, potentially leading to some discrepancies when compared to continuous-time models like the Black-Scholes model.
- Limited Applicability in Complex Scenarios: For more complex options or derivatives, especially those with multiple underlying assets or complex payoff structures, the binomial model might become less practical.
- Market Liquidity and Transaction Costs: The model assumes no transaction costs and perfect market liquidity, which may not always reflect real market conditions accurately.
While the Binomial Option Pricing Model offers advantages in understanding and valuing options, it’s essential to acknowledge its limitations and consider these factors when applying the model to real-world scenarios. Traders and investors often use a combination of models and tools to derive more accurate and reliable valuations in practice.
