Binomial Option Pricing Model

What Is the Binomial Option Pricing Model?

The binomial option pricing model is a risk-free method for estimating the value of path-dependent alternatives. With this model, investors can determine how likely they are to buy or sell at a given price in the future. According to this model, the current option value is equal to the present value of the probability-weighted future payoffs of the investment.

Basics of the Binomial Option Pricing Model

An investor is aware of the current stock price at any given time. They're going to try to predict future changes in stock prices. They will divide the time until the option expires into equal parts under this scenario (weeks, months, quarters). 

The model uses an iterative process for each period to determine how likely the movement will be up or down. The model effectively creates a binomial distribution of stock prices.

Calculating Price With the Binomial Model

The binomial option model repeatedly uses the same success and failure probabilities until the option expires. A trader might use many different possibilities depending on the current situation.

When it comes to valuing American options and embedded options, a binomial tree is an invaluable resource. There is no difficulty in modeling the tree mechanically, but the trouble resides in the range of values the underlying asset can achieve in a given period. Unfortunately, there are just two possible values in a binomial tree model, which is unrealistic because assets can have any number of values within a specified range.

It's possible that, for example, the underlying asset's price could rise or fall by 30% in a single time. But, on the other hand, the underlying asset price may increase by 70/30 in the second term.

Real-World Example of Binomial Option Pricing Model

A basic example of a binomial tree consists of a single component. Suppose there is a stock with a share price of $100. One month from now, the price of this stock will either rise by $10 or fall by $10, which will result in the following:

• The stock price is $100.
• One month's stock price (up state) = $110.
• One month's stock price (down state) = $90.

Next, let's say there is a call option on this stock with a $100 strike price that expires in one month. It is worth $10 in the up state, while it is worth nil in the down state. The call option's current price can be calculated using the binomial model.

Binomial Option Pricing Model

Since it is simpler to apply, the binomial option pricing model is crucial. When comparing option prices to the stock prices of the option, you can do so. 

To the point of expiration, an option might be examined for multiple periods. Because of this, it can be used to value American-style stock options. You can exercise the American-style option anytime before the option expiration date.

Assumptions

To appreciate its limits fully, you must understand the assumptions underlying a binomial option pricing model.

• The risk-free rate does not change
• There are no returns on the underlying stock
• At any given point in time, the price can only move one of two ways: either up or down (the term "binomial" refers to this)
• As a result, there are no transaction fees or taxes in today's market
• Investors are risk averse; they do not bother about taking risks
• Throughout the period, the discount factor (interest rate) remains constant

Advantages and Disadvantages of Binomial Option Pricing Model

Advantages

• The binomial option pricing model is easy to calculate the model because it is mathematically simple
• Pricing is available for Binomial Option Type American options, in which the holder can exercise their option at any time until the expiration date
• For the underlying asset price and option value transparency across time, the model gives a considerable advantage of a multi-period view

Disadvantages

• Multi-period models have the drawback of significantly increasing computing complexity
• The model's major weakness is that it requires a prediction of future prices

What Is the Difference Between Black-Scholes and Binomial?

Methods like the Binomial and Black Scholes models are frequently employed in the study of option pricing. Compared to the stochastic differential equation of the Black Scholes model, the Binomial model is a basic statistical technique. An actuary's pricing procedure for a European call and put option is quite challenging. 

To distinguish between the Binomial model and the Black Scholes model, we need to use a t-test and Tukey model at a single point in time. Finally, using the two models described above, it is concluded that there is no discernible difference between the European options' means.

How to Calculate the Model?

Option prices can fluctuate in two directions if the current (spot) price is set to S. S+ or S- are the two possible price points. We can then use this information to determine the up(u) and down(d) variables.

u=s+/s, d=s-/s

Call Options

To execute a call option, you must pay the exercise price PX and then buy the underlying asset or equity in question.

The call option is in the money when the spot price is higher than the exercise price (S > PX). When there is an upward price movement, the payment of the call option is equal to the maximum between zero and the spot price multiplied by the up factor and decreased by the exercise price. To illustrate, consider the following formula:

C+ = max (0, uS-Px)

A downward movement pays off with:

C- = max(0, dS-Px)

Put Options

At the exercise price PX, a put option gives its holder the right to sell at that price.

When the price changes, we use the formula below to determine the value of a put option:

P+ = max(Px-uS,0)

P- =max(Px-dS,0)

Binomial Trees

The best approach to visualize the model is using a binomial tree. Different nodes show the option's reward and likelihood. The underlying asset's price can be depicted as a network of nodes.

Binomial Tree Example

Consider a $100 stock with a strike price of $100, an expiration period of one year, and a 5-percent interest rate (r). There is a 50% chance that the stock will rise to $125 at the end of the year and a 50% chance that it will fall to $90.

Delta Portfolio Hedging

Delta hedging is a variation of the binomial option pricing model that uses deltas. The project's goal is to develop a synthetic hedging portfolio and discover a risk-free payout. We may then calculate the option's price by determining the portfolio's trading value.

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Conclusion

Microsoft Excel can be used to simplify binomial option pricing model calculations, but it can't anticipate future prices. As we narrow the simulation time, it becomes more tiresome to forecast end-of-period payoffs. 

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