The binomial model is a valuation methodology widely used for pricing options. In the dynamic and uncertain world of financial markets, the prices of traded instruments are in constant flux, making it essential to have a reliable framework for determining option prices. The binomial model addresses this challenge by incorporating a discrete-time and discrete-value approach, allowing for a simplified and intuitive method of pricing options.

In contrast to the continuous-time and continuous-value stochastic processes observed in real-world market prices, the binomial model recognizes the discrete nature of option pricing. It acknowledges that the underlying asset's price can only change at specific time intervals and take on a limited number of values within each period. This discrete nature aligns with the concept of a stochastic process, wherein variables change in an uncertain manner.

To accurately price options, any option pricing methodology must account for the stochastic process followed by the underlying asset.

The binomial model is an option derivative pricing model which assumes that the asset price follows a defined/discrete process (i.e., discrete-time and discrete value behavior). Since it is difficult to try and do the mathematics of a variable that changes in a continuous-time and continuous value, the binomial model steps in and make things simple by making an assumption that- asset price can rise and fall to defined price levels in a given period.

A discrete stochastic process means a variable that changes in discrete time is said to follow a discrete-time stochastic process and which takes limited value is said to follow a discrete value stochastic process.

On the contrary, a continuous stochastic process means a variable that changes in continuous time is said to follow a continuous-time stochastic process and which takes any value is said to follow a continuous value stochastic process.

### Assumptions in Binomial Option Pricing Model

It is better to thoroughly understand the assumptions of the Binomial Option Pricing Model-

The underlying asset has a known outcome i.e., it can take only two possible prices on expiration (i.e., the up-side and the down-side).

The underlying is a non-dividend/non-coupon-paying asset.

The volatility in the underlying asset and the risk-free interest rate (periodically continuously compounded) is known and constant throughout until expiration.

Markets are frictionless (i.e., there are no transaction costs, no taxes, and no regulatory constraints).

Unlimited borrowing and investing funds are available at a risk-free rate.

Investors are risk-neutral.

### Approaches to Binomial Option Pricing Model

The key to understanding pricing options is to understand all the pieces of the Binomial model. Although Black-Scholes-Merton (BSM) model is the best way to price a European option but understanding the three approaches of the Binomial model will help to get the intuition behind pricing options. The three approaches are-

Delta Hedging Approach: This technique enables to create a synthetic hedged portfolio that provides a certain payoff and that certain payoff must be discounted at a risk-free rate. If a portfolio is riskless, it should be priced to yield a risk-free return.

Replicating Portfolio Approach: This technique enables to create a replicating portfolio that provides a payoff equivalent to a payoff from an option at expiration. If two portfolios have an identical payoff, the price of both portfolios should be the same.

Risk-Neutralization Approach: A standard DCF technique is a valuation method used to calculate the price of an option by taking the present value of the expected payoff computed using risk-neutral probabilities discounted by a risk-free rate.

Most importantly, the binomial model is carved out of BSM Model or I would say, as we increase the layers of the given period, the Binomial method of valuing options will approach the option value provided by the BSM model.

--Interesting right?

Let's understand this!

Delta hedging approach, portfolio replicating approach, and risk-neutralization approach of the binomial model assume that the underlying asset has a known outcome, i.e., it can take only two possible prices on expiration (i.e., the upside and the downside). This assumption is entertained because it is the simplest and easiest method to price an option and produces the same results as any other model. However, it can be more realistic if we increase the number of layers to cover a wide range of asset prices.

The two-period binomial model will give us a binomial tree with more possible values than a single-period binomial model. Similarly, a three-period Binomial Model will give us a binomial tree with even more possible values, and so on. If we continue to add layers, the binomial model will give us a binomial tree with all possible values. However, it would be cumbersome to compute the option price at each node of the binomial tree and apply the backward induction process for each separate possible outcome. Automated systems can be used to compute the price of an option by including the maximum possible layers, which will yield the same option price as provided by the BSM model. This is how the binomial model helps to provide intuition for pricing options.

Another advantage of using a binomial model is its ability to price path-dependent options. It is primarily used to price American and Bermudan options because of the feature that they can be exercised prior to expiration. Unlike the BSM model, which cannot determine at which point an option can be exercised, the binomial model is the only recourse.

### Let's take an example to understand all three approaches of the Binomial Model!

Imagine a 1-Year European call option on an equity stock at a strike price of 1490.00. The underlying stock presently trading at 1488.05. The volatility in the underlying equity stock and the risk-free interest rate are 21.4535% and 6.00% per annum compounded continuously, respectively. Construct a single-period binomial tree and then use it to price the call option.

Model Parameters:

Maturity/Expiry (T) = 2-Days

Risk-free Interest Rate (r) = 6% (per annum compounded continuously)

Current Stock Price (S0) = 1488.05

Strike Price (K) = 1490.00

Volatility (σ) = 21.4535% or 0.214535

We can calculate the up-move (u) and down-move (d) factors using the formula:

u = e^(σ * sqrt(T/252)) = e^(0.214535 * sqrt(2/252)) = 1.0193

d = 1/u = 1/1.0193 = 0.9811

We will construct a single-period binomial tree with two possible stock prices: the up-side (Su) and the down-side (Sd). The tree will have three nodes: the initial stock price (S0), the up-side price (S0 * u), and the down-side price (S0 * d).

S0 * u = 1488.05 * 1.0193 = 1516.76 (up-side price)

S0 * d = 1488.05 * 0.9811 = 1459.88 (down-side price)

At expiration, the option's payoff depends on the stock price. The option price at expiration (C) is given by: C = Max of (Stock Price - Strike Price, 0)

Call Payoff = Max of (Su - K, 0) = max(1516.76 - 1490.00, 0) = 26.76 (up-side node)

Call Payoff = Max of (Sd - K, 0) = max(1459.88 - 1490.00, 0) = 0 (down-side node)

Using the given model parameters and formula derived from the Binomial Model, we can construct the single-period binomial tree. Each node in the tree represents a possible stock price at expiration, and by applying the chosen approach of the three, we can calculate the price of an option at each node.

Note: The binomial model can be expanded to multi-periods to obtain more precise results, but this example focuses on the single-period approach for simplicity.

Let's calculate the price of a call option and put option using the Risk Neutralisation Approach, Delta Hedging Approach, and Replicating Portfolio Approach and carve out the conclusions under all three approaches.