December 7, 2023

## How is the option price calculated?

Options are financial derivatives that give the holder the right, but not the obligation, to buy or sell an underlying asset at a specified price within a predetermined time frame. The price of an option, also known as the option premium, is a critical factor in determining the profitability and risk associated with trading options. Calculating the price of an option involves various factors and mathematical models that take into account current market conditions, the volatility of the underlying asset, the time to expiration, and other parameters. In this article, we will explore the key concepts and methods used in option pricing.

## The Black-Scholes Model

The Black-Scholes model, developed by economists Fisher Black and Myron Scholes in 1973, is one of the most widely used option pricing models. It provides a mathematical framework for pricing European-style options that are exercisable only at expiration. The model assumes that the underlying asset follows geometric Brownian motion and takes into account factors such as the current price of the underlying asset, the strike price, the time to expiration, the risk-free interest rate, and the volatility of the underlying asset.
The Black-Scholes formula for calculating the price of a call option is as follows

C = S * N(d1) – X * e^(-r * T) * N(d2)

Where:

• C is the call option price
• S is the current price of the underlying
• N(d1) and N(d2) are the cumulative standard normal distribution functions.
• X is the exercise price of the option
• r is the risk-free interest rate
• T is the time to maturity in years

The formula for calculating the price of a put option is similar, but with some adjustments. The Black-Scholes model assumes that the market is efficient, there are no transaction costs, and it does not take into account factors such as dividends or early exercise.

## Factors affecting the option price

The price of an option is affected by several key factors:

1. Price of the underlying asset: The price of the underlying asset has a direct impact on the price of the option. For call options, as the price of the underlying asset rises, the value of the option tends to rise as well. Conversely, if the price of the underlying asset falls, the value of the option tends to rise.
2. Strike Price: The strike price is the price at which the underlying asset can be bought or sold. In general, the closer the strike price is to the current price of the underlying asset, the higher the option price. This is because options with strike prices close to the current price are more likely to be exercised and therefore have a higher value.
3. Time to expiration: The time remaining until the option expires affects its price. As time passes, the value of the option may decrease because the probability of the option going in the money decreases. This phenomenon is known as time decay or theta decay.
4. Volatility: Volatility measures the magnitude of price fluctuations in the underlying asset. Higher volatility generally leads to higher option prices because there is a greater likelihood of large price movements, which can result in greater profit potential for the option holder.
5. Interest rates: The risk-free interest rate also plays a role in option pricing. As interest rates rise, the cost of carrying the underlying asset increases, which can cause the price of call options to fall and the price of put options to rise.

## Option Pricing Models

In addition to the Black-Scholes model, several other option pricing models have been developed to reflect different market conditions and assumptions. Some of the more notable models include the Binomial Option Pricing Model, the Cox-Ross-Rubinstein Model, and the Heston Model.

The Binomial Option Pricing Model is a discrete-time model that assumes that the price of the underlying asset can move up or down in each time period. By recursively calculating the option price at each step, the model provides a lattice of option prices that converge to the theoretical price. This model is particularly useful for pricing American-style options that can be exercised at any time prior to expiration.
The Cox-Ross-Rubinstein model, also known as the binomial option pricing model, is an extension of the binomial option pricing model that takes into account the continuous nature of time and assumes that the price of the underlying asset follows a log-normal distribution. It provides a more accurate pricing of options than the original binomial model.

The Heston model, developed by Steven Heston in 1993, is a stochastic volatility model that addresses one of the limitations of the Black-Scholes model – the assumption of constant volatility. The Heston model incorporates the concept that volatility is a stochastic process that can change over time. This model is widely used for pricing options on assets with volatile and complex price dynamics, such as equity options.

## Market Factors and Option Pricing

In addition to the intrinsic factors discussed above, market conditions and factors can also affect the pricing of options. Market sentiment, supply and demand dynamics, news events and macroeconomic factors can all affect the perceived value of options. For example, during periods of high market volatility or uncertainty, option prices tend to rise as investors seek to protect their portfolios or speculate on larger price movements. Conversely, during periods of low volatility or market stability, option prices may decline as the potential for price fluctuations diminishes.

It’s important to note that option pricing is not an exact science and is subject to various assumptions and limitations. Models such as the Black-Scholes model provide a theoretical framework for pricing options, but actual market prices may differ from these theoretical values due to factors such as liquidity, market microstructure and market inefficiencies. Traders and investors often rely on a combination of theoretical models, market data, and their own judgment to make informed decisions about options trading.

## Conclusion

Option pricing is a complex and important aspect of the financial markets. The price of an option is determined by several factors, including the price of the underlying asset, the strike price, time to expiration, volatility, and interest rates. The Black-Scholes model and other option pricing models provide a mathematical framework for estimating option prices and valuing different types of options. It’s important to remember, however, that option pricing is not an exact science, and market factors and conditions can also affect the price.

## FAQs

### How is option price calculated?

The price of an option is calculated using various factors, including the underlying asset price, strike price, time to expiration, volatility, and interest rates. The most commonly used mathematical model for option pricing is the Black-Scholes model, which provides a theoretical value for an option. However, market prices of options may differ from the theoretical values due to factors such as market demand and supply, changes in volatility, and other market conditions.

### What is the underlying asset price?

The underlying asset price is the current market price of the asset on which the option contract is based. For example, if you have a call option on a stock, the underlying asset price would be the current price of that stock in the market.

### What is the strike price?

The strike price, also known as the exercise price, is the predetermined price at which the underlying asset can be bought or sold when exercising the option. It is specified in the option contract and remains fixed throughout the life of the option.

### How does time to expiration affect option prices?

Time to expiration is a crucial factor in option pricing. As the expiration date approaches, the time value of the option decreases. This is because the probability of the option ending up “in the money” (profitable) decreases as time passes. Therefore, all else being equal, options with shorter time to expiration have lower prices than options with longer time to expiration.

### What is volatility, and how does it impact option prices?

Volatility refers to the degree of price fluctuation of the underlying asset. Higher volatility implies greater price swings, while lower volatility indicates more stability. Option prices are influenced by volatility because higher volatility increases the potential for the option to end up in-the-money. As a result, options on highly volatile assets tend to have higher prices compared to options on less volatile assets.

### Do interest rates affect option prices?

Yes, interest rates can impact option prices. Generally, higher interest rates increase the cost of carrying the underlying asset, which can decrease the value of call options and increase the value of put options. Conversely, lower interest rates can have the opposite effect. However, the impact of interest rates on option prices is not as significant as other factors like the underlying asset price, volatility, and time to expiration.