Option Pricing

Option pricing refers to the process of determining the theoretical value of an options contract. In simple terms, it derives an estimated value of options based on assumptions about future scenarios and elements from present scenarios.

The valuation process is primarily based on mathematical models incorporating features like mathematical equations and data structure mechanisms. The process involves applying relevant inputs and other known variables to arrive at the fair value of an option, that is, the mathematically expected payoff at expiration. The expert traders use the estimated value of options contracts to enhance their investment strategies.

• Option pricing refers to the process of determining the theoretical value of an options contract.
• The most common valuation models are Black-Scholes, binomial model, and Monte Carlo simulation.
• The Black-Scholes model utilizes differential equations, the binomial model uses binomial tree concept and assumption of two possible outcomes, and the Monte Carlo method uses random samples.
• The development of the pricing theories majorly involves random processes, probability concepts, assumptions about asset returns, an understanding of implied volatility concepts, etc.

The history of options pricing theory began in the early 20th century. The contribution of numerous academics enriched the discipline. According to the journal "Theory of Rational Option Pricing" by Robert C. Merton, a noted advancement from that period was the development of the pricing formula developed by the French mathematician Louis Bachelier. The formula was based on the assumption that the stock prices follow a Brownian motion with zero drift.

The development of the pricing theories majorly involves random processes, probability concepts, assumptions about asset returns, and an understanding of implied volatility concepts. Implied volatility measures the underlying asset's expected volatility over the option's lifetime. Due to the increased likelihood of a stock overtaking the strike price when volatility is strong, traders will demand a richer price for their offering.

There are many important concepts and terminologies associated with option valuation. Let's look into a brief description of some of them.

• Option's premium: It is composed of intrinsic and extrinsic value. Intrinsic value (option's minimum value) is the difference between the current stock and the strike value. Whereas extrinsic value or time value (decreases with increasing closeness to expiration) is premium minus intrinsic value.

• Call option (calls): The call option offers the right to buy the underlying asset at a predetermined value before or at expiration. The calls buyer's profit lies in the price rise of the underlying asset. 

• Put option (puts): Puts offers the right to sell the underlying asset before or at the expiration at a predetermined price. The put option buyers prefer the underlying asset’s price drop. The predetermined price is known as the strike price.

•  In the Money (ITM): The ITM call option indicates the option holder can buy the asset below its market price. The ITM put option explains that the holder can sell the asset above its market price, depicting the profitable scenario. Its opposite concept is Out of the Money  (OTM).

• European & American Style: Another observable classification attributed to the options is the European style and American style options. The European-style options get exercised at the expiration date. The other is the American-style options, and they can be exercised any time from the purchase date to the expiration date. 

The three most influential models are the Black-Scholes, Binomial, and Monte Carlo Simulation.

Black Scholes

The Black-Scholes or BSM (Black-Scholes-Merton) pricing model was developed by economists Fischer Black and Myron Scholes in 1973. The Black-Scholes model works on five input variables: underlying asset's price, strike price, risk-free rate, volatility, and expiration time. It is an example of a mathematical model utilizing the partial differential equation forming the Black-Scholes equation and the Black-Scholes formula calculating the price. The model is favorable for the valuation of European options.

The chart below, taken from TradingView, shows the concept in detail. It shows the chart of Apple Inc., In this process, in order to use the model in the chart, there are some inputs that are required, like the strike price, the risk-free rate, the dividend yield for the stock, the time period left till the next date of the dividend, the dividend yield related to the ex-dividend yield after the coming one, time left till the expiry of the contract, in terms of hours or days. Then, the trader has to input or select the chart’s time frame, which will be 15 minutes, hourly, daily, weekly, etc. Finally, the trader has to select the kind of option, which can be a long call or put. However, the time until expiry will not update by itself within the chart. The trader has to update it every day to get the exact market condition.

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Binomial Model

The binomial pricing model uses the binomial tree to present the possible prices during different periods diagrammatically. It usually involves a two-period binomial tree or multiperiod binomial tree. The model incorporating a two-period or multiperiod view has a central assumption of perfectly efficient markets. The possible outcome is restricted to two; there are just two available prices for the upcoming term or each level. The price will either increase or decrease from the prior level. The model is favorable for the valuation of American options and embedded options.

The chart below illustrates the same. Here is a chart of Tencent holding. This model gives more flexibility than the Black-Scholes model. The time left till expiry is divided into parts or steps, and the expected price is calculated at the end of each step, which is called a node. Repetition of this step leads to reaching the end of the terminal node. It is necessary to calculate the expected price of the underlying asset to get the option price.

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Monte Carlo Simulation

The Monte Carlo option model is the application of Monte Carlo Methods. This pricing model uses random samples to calculate the price. This method is more favorable than other methods like Black-Scholes for calculating the value of options with multiple sources of uncertainty. The framework usually involves the application of integration, optimization, and probability distribution.

The chart below can be used to interpret the model. The models use random samples to get real-world numerical results and, thus, can be applied to various fields like mathematics, computers, finance, climate, etc.  A possible range of inputs is defined, generating values. These values are used as inputs in certain algorithms to get results.

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There are scenarios from the old times where investors' decisions depended on factors like intuitions and experiences. For example, investors buy "calls" before positive announcements from companies expecting the volatility associated and gain from the price rise. 

At the same time, the option contract value is influenced by more factors like time value and the relationship between the strike price and underlying asset price. In these situations, investors can rely on simple equations associated with pricing models like Black-Scholes considering significant variables. The models also output the "Greeks" like values for delta, gamma, vega, and theta, mainly representing the sensitivity of the option value to the underlying asset value. Nowadays, investors use option pricing calculators based on a particular pricing model giving out theoretical values and finance sites to gain more information.

This has been a Guide to Option Pricing and its Meaning. We discuss the option pricing using models like Binomial & Black-Scholes creating formulas. You may also have a look at the following articles to learn more –

• Option Greeks
• Rho in Options
• Options Spread