Option Pricing Models

What Are Option Pricing Models?

Mathematical models called option pricing models employ certain factors to determine the hypothetical value of an option. The hypothetical value of an option is an appraisal of the value an option will give in context to all available information.

It helps investors and traders to estimate the probability of an option being exercised and its potential future payoff. In other terms, option pricing models provide us with the option's fair value.

Finance experts might modify their trading strategies and portfolios by knowing the estimated fair value of an option. As a result, option pricing models are effective tools for financial experts who trade options.

Using an option pricing model, investors can make informed decisions about the potential profit or loss of investment, leading to better investment outcomes. Therefore, the importance of the model in finance cannot be overstated.

Option Pricing Models in Finance

The option pricing model plays an essential role in the finance industry as it gives a framework for appreciating different financial tools like equity securities, fixed-income securities, and derivatives. Accurate option pricing is necessary for:

• Effective risk management
• Asset allocation
• Investment strategy

The two most commonly used option pricing models in today’s finance industry are:

1. Binomial option pricing. 
2. Black and Scholes model.

Option pricing theory has made enormous progress since 1972 when Black and Scholes (the most widely used) released their path-breaking study giving a model for evaluating dividend-protected European options.

Black and Scholes employed a “replicating portfolio” –– a portfolio consisting of the underlying asset and the risk-free asset that had identical cash flows as the option being evaluated –– to come up with their final formulation.

Although their mathematical derivation is challenging, a more straightforward binomial model for valuing options follows the same reasoning. 

What are the Options?

You may have successfully conquered the market by trading stocks with a disciplined method and expecting a great move, either up or down.

Identifying one or two strong stocks poised to make a significant move shortly has helped many traders obtain the conviction they need to thrive in the stock market.  If we can't do that, we should consider using options.

Before we delve deep into the option pricing models, we must gain deep knowledge of what options are and the types of options used in the capital market.

An option is an agreement between two participants that gives one party the right but not the obligation to buy or sell the underlying asset at a determined price before or on the day it lapses. There are two major types of options: calls and puts.

Calls are options that provide you the right, but not the necessity, to acquire the underlying asset at a fixed price before or on the day they expire.

When you buy a put option, you have the choice—but not the duty—to sell the underlying asset at a fixed price before or on the day it expires.

The following lists the possible ownership scenarios:

• Buyer of a call: Right to buy an asset at a fixed price
• Seller of a call: Have to sell the asset
• Buyer of a put: Right to sell an asset at a fixed price
• Seller of put: Have to buy the asset

Choices may also be categorized based on how long they take to exercise:

European-style options can be exercised only on the expiration date. On the other hand, American-style options may be exercised at any time between the time of purchase and the expiration date.

Certain factors affect the call and put option prices. They are summarized in the table below.

What are risk-neutral probabilities?

To correctly appreciate the concept of option pricing models, one must also understand the concept of risk-neutral probabilities. These probabilities are widely used in the different option pricing models.

To define the expected return on an asset in a market where the underlying asset's price is undefined, this is where risk-neutral probabilities in finance and economics are applied. 

This is important because it allows traders and investors to value derivatives like futures and options. 

One of the basic assumptions of using options is that the underlying probabilities are based on risk neutrality and not on real-world probabilities. 

Risk-neutral probabilities are based on the assumption that real-world probabilities determine the price of the stock, along with the investors having expectations regarding the stock's future value.

The two main assumptions behind this probability are:

• The current value of an asset is equal to its expected payoff discounted at the risk-free rate.
• There are no arbitrage opportunities in the market.

The benefit of using the risk-neutral probability in this risk-neutral pricing approach is that they may be used to price each asset according to expected return once the probabilities have been computed.

These theoretical probabilities are different from actual probabilities in the real world, sometimes known as physical probabilities. Therefore, the predicted values of each security would need to be modified for each security's unique risk profile if the real-world probability were utilized.

The formula for neutral probabilities is:

(1 + RFR)n - d / U - D

Here, 

• RFR = the risk-free rate of interest
• D = Downtick 
• U = Uptick

Also, if one attempts to estimate the expected value of that particular stock based on the probability that it will rise or fall, considering unique cases or market conditions that can affect that specific asset, one would be considering risk and, as a result, looking at real or physical probability.

Binomial Option Pricing Model

A binomial option pricing model is the simplest way to value the options. The assumption behind this model is that markets are completely efficient. Based on this supposition, the model can calculate the option's price at each instant in a certain period.

The binomial model assumes that the underlying asset's price will increase or decrease over the term. Given the potential values of the underlying asset and the strike price, we may determine an option's payment under many scenarios. 

We can then discount these payoffs to get the option's current value. It can be shown using the diagram given below:

In this diagram, S0 represents the current stock price; in any given period, the price will go up to Su with probability p and downward to Sd with probability 1-p and so on for a two-period binomial tree.

Some assumptions are considered when we set up a binomial option pricing model. Let us understand them below: 

• Just two new prices, one up and one down are conceivable at any moment (hence the term "binomial").
• There are no dividends paid on the underlying asset.
• Over time, the interest rate (discount factor) remains constant.
• There are no transaction fees or taxes in the market and no market friction.
• Investors are averse to risk and risk-neutral.
• The risk-free rate stays the same.

Even though it is the simplest way to calculate the option value. It has some advantages and disadvantages of its own. Some of them are mentioned below:

Black and Scholes Option Pricing Model

The Black and Scholes model is a mathematical formula used to determine the theoretical value of options contracts. It was developed by Fischer Black and Myron Scholes in 1973 and has since become one of the most widely used models in the financial industry.

The owner of an option contract has the choice, but not the obligation, to buy or sell the underlying asset at a specified price on or before a certain date.

The Black-Scholes model considers these factors to calculate the theoretical value of an option. It assumes that the underlying asset's price follows a log-normal distribution and that the option can be exercised only at expiration.

It also assumes that there won't be any transaction costs or taxes, and interest rates will stay the same for the period of the option's term.

The formula for the Black-Scholes model is as follows:

C = SN(d1) - Xe^(-rt) * N(d2)

Where:

• C = the theoretical value of the call option.
• S = the current price of the underlying asset.
• X = the strike price of the option.
• r = the risk-free interest rate.
• t = the time until expiration.
• N = the cumulative distribution function of a standard normal distribution.
• d1 = [ln(S/X) + (r + 0.5σ^2)t] / [σsqrt(t)]
• d2 = d1 - σsqrt(t)

The formula for the put option is similar:

P = X * e^(-rt) *N(-d2) - SN(-d1)

Where:

• P = the theoretical value of the put option.
• S, X, r, t, N, d1, and d2 are the same as in the call option formula.
• σ = the volatility of the underlying asset.

Criticism of the Black-Scholes Model

For many reasons, the Black and Scholes model has drawn criticism. Some of them are mentioned below:

1. European-only

As noted, the Black and Scholes model is a reliable predictor of European option pricing. In the US, it is inaccurate in its valuation of stock options. This is due to the assumption that options can only be exercised on the day of their expiry or maturity.

2. Constant interest rate

The Black and Scholes model assumes constant interest rates; however, this is rarely the case in practice.

3. No transaction cost

Assuming a market with no friction, Transaction expenses, such as brokerage charges, commission, etc., are often associated with trading. 

On the other hand, the Black and Scholes model assumes no transaction costs since the market is frictionless. Seldom is that the case in the trading industry.

4. No returns

The Black and Scholes model posits that the stock options' related returns are zero. Dividends and interest earnings are absent. In contrast, this is not true in the real trading market. Returns are the main consideration when purchasing and selling options.

Despite these criticisms, the Black-Scholes model remains a widely used tool in the finance industry for valuing options contracts.

It has also been used as the foundation for creating more intricate models that account for other elements like dividends and changes in interest rates.

Option Pricing Using Monte Carlo Simulations

For deciding the probability distribution of financial events, such as stock prices, interest rates, and option prices, Monte Carlo simulation is a commonly used approach in the finance sector. It is used to model complex financial systems and simulate hard-to-analyze occurrences.

For instance, we may create many random samples of the price changes of the underlying asset using Monte Carlo simulation to assess the value of a financial option.

The option's expected value, variance, and other statistical measurements may then be calculated using these samples.

Another use case of Monte Carlo simulation in finance is in risk management. Monte Carlo simulation may assist in identifying the sources of risk in a portfolio and quantifying their possible influence on the portfolio's value by predicting the behavior of various financial assets and their interconnections.

Monte Carlo simulation can also be used to analyze the performance of investment strategies by modeling the performance of various approaches in market scenarios.

The capacity to include the impacts of uncertainty and variability in financial models is one of the benefits of Monte Carlo simulation in finance. Monte Carlo simulation allows us to precisely define this uncertainty and measure how it impacts financial decision-making since financial results are inherently unpredictable.

Monte Carlo simulation in finance also has some limitations. It requires accurate modeling of the underlying financial variables, such as the dynamics of stock prices or interest rates.

The assumptions and limitations of the model, the caliber of the data used as input, and the random number generator used to generate the samples must all be considered.

Monte Carlo simulation is a valuable tool in finance for estimating the probability distribution of financial outcomes, analyzing risk, and evaluating investment strategies.

It explicitly allows us to model the effects of uncertainty and variability but requires careful consideration of the underlying assumptions and limitations.

Researched and authored by Naman Jain | LinkedIn

Reviewed and edited by Parul Gupta | LinkedIn

Resources

To continue learning and advancing your career, check out these additional helpful WSO resources:

• Dividend Discount Model
• Gordon Growth Model
• Market Price
• Real Options
• Valuation Methods