What is Binomial theorem in stock market and it’s components? Explain with examples.

Have you ever come across the Binomial Theorem? It is something like a hidden decoder which can lead us through how the investments might possibly grow in the long run, more so in the stock market! Just picture it as a prediction model which leverages probabilities to foresee the likely fluctuations of stock prices. The core elements of this theorem are the probability of an event occurring (such as a stock rising), the probability of its non-occurrence (that is the stock falling), and the number of opportunities for this event to happen. For example, based on a particular stock, the theorem can assist you in estimating the likelihood of its being appreciated in the value within a given timeframe. What do you think of it?

What is Binomial theorem?

The Binomial Options Pricing Model, which is widely used in the stock market, is based on the Binomial Theorem. By dividing the time until the expiry of an option into several intervals, the model prices the option. It then assigns a certain percentage of the stock price going up or down at each moment. The model gives a theoretical price for the option by valuing it at expiration along all possible price paths and then discounting it back to the present.

Core Idea of the Binomial Theorem in Finance

(a + b)ⁿ = Σ C(n,k) × aⁿ⁻ᵏ × bᵏ

In the binomial option pricing model, we replace:

• a → probability-weighted upward move (u)
• b → probability-weighted downward move (d)
• n → number of time steps

This eventually gives us the option price as the discounted expected value under a risk-neutral probability.

Key Components of the Binomial Option Pricing Model

The Binomial Tree (Price Lattice):

• The model’s visual and structural core can be seen in this image. It shows how the price of the asset can move in the future along different timelines.
• The tree is based on the present price of the stock and every time interval it shows two options: a price going “up” or a price going “down”.
• Thus, it forms a grid of hypothetical stock prices at the time when the option will expire.

Up Factor (u) and Down Factor (d):

• The stock price can vary significantly at each stage depending on these multiplicative factors.
• The Up Factor (u) is a number which is always larger than 1 (e.g., 1.1). The price after an “up” move will be S * u, where S is the current price.
• The Down Factor (d) is a number that is in the range of 0 to 1 (e.g., 0.9). The price after a “down” move will be S * d.
• Usually, these factors are obtained from the stock’s volatility.

Risk-Neutral Probability (p):

• This does not show the actual chances of the stock going up or down in the real world. The number reflects a theoretical probability arrived at by the “risk-neutral” assumption, which presumes that all investors are equally disinterested in risk.
• It indicates the probability of an “up” move that makes the expected return on the stock equal to the risk-free interest rate.
• It is found using the formula: p = [e^(rΔt) – d] / (u – d), with r as the risk-free rate and Δt as the time increment.

Number of Time Steps (n):

• The model divides the option’s time to expiration into a number of discrete periods or steps (n).
• A higher number of steps makes the model more accurate (as it more closely resembles continuous price movement) but also increases computational complexity.

Risk-Free Interest Rate (r):

• Theoretical return on investment without any risk (similar to a government bond) for the period of the option’s life.
• It is a component in determining the risk-neutral probability (p) and in lowering the option’s anticipated future value to its present value through discounting.

Option Valuation (Backward Induction):

• The calculation method is as follows. The model first determines the value of the option at its expiration (the end of the tree).
• Expiration: The option’s worth is its intrinsic value for each last node, for example, a call being Max(0, Stock Price – Strike Price).
• Backward approach: After that, the model goes back to the first node of the tree and at every single node the option’s value is computed as the discounted probability-weighted average of the two values (up and down) from the next step: Option Value = [p * Value_up + (1-p) * Value_down] * e^(-rΔt).
• With American options, at every node, the calculated value is compared to the option’s intrinsic value if exercised right away, and the greater of the two is picked.

Volatility (σ):

• The stock’s volatility is not an input in the final formula like p, but it remains a critical driver in its calculation. Volatility is a significant determinant of the up (u) and down (d) factors and is often derived through the formulas: u = e^(σ√Δt) and d = 1/u. A rise in volatility results in larger possible price movements (a broader tree).

Step-by-Step How the Binomial Model Works

• Build the stock price tree for n steps.
• At expiration (step n), calculate the option payoff at every final node: Call → max(Sₜ − K, 0) Put → max(K − Sₜ, 0)
• Work backwards through the tree: At each node, option value = e^(−rΔt) × [q × Value_if_up + (next node) + (1−q) × Value_if_down (next node)]
• The value at the root (today) is the fair price of the option.

Example:

Let’s price a European call option with these parameters:

• Current stock price S = $100
• Strike K = $100 (at-the-money)
• Time to expiration T = 1 year
• Risk-free rate r = 5% (continuous)
• Volatility σ = 30%
• We use only n = 1 step (just for illustration)

Calculations (using CRR parameters):

Δt = 1
u = e^(σ√Δt) = e^(0.3×1) ≈ 1.34986
d = 1/u ≈ 0.74082
q = (e^(rΔt) − d) / (u − d) = (e^0.05 − 0.74082) / (1.34986 − 0.74082) ≈ 0.547

Stock prices at expiration:

• Up node: 100 × 1.34986 = $134.99
• Down node: 100 × 0.74082 = $74.08

Call payoffs at expiration:

• Up: max(134.99 − 100, 0) = $34.99
• Down: max(74.08 − 100, 0) = $0

Call value today:
C = e^(−0.05) × [0.547 × 34.99 + (1−0.547) × 0] ≈ 0.9512 × 19.14 ≈ $18.21

So the 1-step binomial model gives the call price ≈ $18.21

Conclusion

The Binomial Theorem is, in fact, the key that unlocks the door to the evaluation of investing in the stock market. It uses the concept of probabilities and different possibilities which makes it possible for investors to estimate the likelihood of various outcomes. Theorem not only provides the investors with a good understanding of the risks and rewards of their investments but also empowers them to make informed decisions and manage their portfolios in an effective manner. By presenting the different scenarios, it becomes the stock market navigator all investors need and want to have during the confusing times of trading stock.

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