Option Pricing Theory 

Theoretical options pricing uses five key factors to calculate an option's value: stock price, strike price, interest rate, volatility, and time to expiration. This calculation helps traders determine the option's fair value, which they use to make profitable trades. While options pricing may seem straightforward, it requires complex mathematical calculations. We'll explore the following methods in more detail: 

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Option Pricing Theory

Option pricing theory uses several factors to calculate the theoretical value of an option. These factors include:

• Stock price

• Strike price

• Interest rate

• Volatility

• Time to expiration

This theory helps traders estimate the fair value of an option, which they can use to maximize their profits. However, options valuation involves complex mathematical calculations.

Common Methods:

We will explore the following methods in detail:

1. Risk-Neutral Model

2. Binomial Model

3. Black-Scholes Model

4. Monte-Carlo Simulation

These methods derive their values from the stock price, which means there is a high chance of error.

Understanding Option Pricing Theory. 

The main goal of option pricing theory is to calculate the probability of an option being profitable at expiration. To do this, mathematicians use variables like stock price, strike price, interest rate, volatility, and time to expiration.

These variables help determine the fair value of an option. More time and volatility increase the chances of an option being profitable. Higher interest rates also increase option prices.

Market-traded options are valued differently than non-traded ones. Market prices may differ from theoretical values, but these values help traders assess profit potential.

The Black-Scholes model, developed in 1973, is widely used to calculate option prices. Other models like Monte-Carlo simulation and binomial options pricing are also used.

The Black-Scholes model assumes:

• Stock prices follow a log-normal distribution (no negative values)

• No taxes or transaction costs

• The same risk-free interest rate for all maturities

• Short selling and use of proceeds are permitted

• No risk-free arbitrage opportunities

KEY TAKEAWAYS 

• Option pricing theory uses five main variables: stock price, strike price, interest rate, volatility, and time to expiration.

• The main goal of option pricing theory is to calculate the likelihood of an option being profitable (or "in-the-money") at expiration.

• Three commonly used models to value options are Black-Scholes, binomial option pricing, and Monte-Carlo simulation.

Using Black-Scholes Option Pricing Theory

The Black-Scholes model is a widely used option pricing theory requiring six input variables. However, it makes certain assumptions, such as constant volatility, no taxes or transaction costs, and a log-normal distribution of stock prices. While the model is useful, these assumptions don't always hold in real-world markets, limiting its accuracy. Despite this, the Black-Scholes model remains a fundamental concept in options pricing and is widely used in financial markets. 

The Black-Scholes model uses six important pieces of information to calculate option prices:

1. Strike price: The fixed price for buying or selling the underlying asset.

2. Current market price: The current price of the underlying asset.

3. Time to expiry: The time left until the option expires.

4. Volatility: A measure of the asset's price fluctuations (note: volatility is not directly observable and must be estimated or implied).

5. Risk-free rate: The interest rate for a risk-free investment.

6. Dividends: Payments made to shareholders.

Assumptions of the Black-Scholes Model

The model assumes:

• Stock prices follow a log-normal distribution (meaning prices can't be negative).

• No taxes or transaction costs.

• The risk-free interest rate is the same for all maturities.

• Short selling and use of proceeds are permitted.

• No risk-free arbitrage opportunities.

• All options are European style (can only be exercised at maturity).

Limitations of the Black-Scholes Model

Some assumptions don't always hold. For instance, volatility is assumed to remain constant, but in reality, it fluctuates based on market demand and supply.