# Chapter 2 Option Pricing Theory

## 2.1 What Is Option Pricing Theory?

Option pricing theory estimates a value of an options contract by assigning a price, known as a premium, based on the calculated probability that the contract will finish in the money (ITM) at expiration. Essentially, option pricing theory provides an evaluation of an option’s fair value, which traders incorporate into their strategies.

Models used to price options account for variables such as current market price, strike price, volatility, interest rate, and time to expiration to theoretically value an option. Some commonly used models to value options are Black-Scholes, binomial option pricing, and Monte-Carlo simulation.

Option pricing theory is a probabilistic approach to assigning a value to an options contract.

The primary goal of option pricing theory is to calculate the probability that an option will be exercised, or be in-the-money (ITM), at expiration.

Increasing an option’s maturity or implied volatility will increase the price of the option, holding all else constant.

Some commonly used models to price options include the Black-Scholes model, binomial tree, and Monte-Carlo simulation method.

## 2.2 Understanding Option Pricing Theory

The primary goal of option pricing theory is to calculate the probability that an option will be exercised, or be ITM, at expiration and assign a dollar value to it. The underlying asset price (e.g., a stock price), exercise price, volatility, interest rate, and time to expiration, which is the number of days between the calculation date and the option’s exercise date, are commonly-employed variables that input into mathematical models to derive an option’s theoretical fair value.

Options pricing theory also derives various risk factors or sensitivities based on those inputs, which are known as an option’s “Greeks.” Since market conditions are constantly changing, the Greeks provide traders with a means of determining how sensitive a specific trade is to price fluctuations, volatility fluctuations, and the passage of time.

The longer that an investor has to exercise the option, the greater the likelihood that it will be ITM and profitable at expiration. This means, all else equal, longer-dated options are more valuable. Similarly, the more volatile the underlying asset, the greater the odds that it will expire ITM. Higher interest rates, too, should translate into higher option prices.

## 2.3 Special Considerations

Marketable options require different valuation methods than non-marketable options. Real traded options prices are determined in the open market and, as with all assets, the value can differ from a theoretical value. However, having the theoretical value allows traders to assess the likelihood of profiting from trading those options.

The evolution of the modern-day options market is attributed to the 1973 pricing model published by Fischer Black and Myron Scholes. The Black-Scholes formula is used to derive a theoretical price for financial instruments with a known expiration date. However, this is not the only model. The Cox, Ross, and Rubinstein binomial option pricing model and Monte-Carlo simulation are also widely used.

## 2.4 Using the Black-Scholes Option Pricing Theory

The original Black-Scholes model required five input variables—the strike price of an option, the current price of the stock, time to expiration, the risk-free rate of return, and volatility. Direct observation of future volatility is impossible, so it must be estimated or implied. Thus, implied volatility is not the same as historical or realized volatility.

The Black-Scholes model, one of the most highly regarded pricing models, assumes stock prices follow a log-normal distribution because asset prices cannot be negative. Other assumptions made by the model are that there are no transaction costs or taxes, that the risk-free interest rate is constant for all maturities, that short selling of securities with the use of proceeds is permitted, and that there are no arbitrage opportunities without risk.

Clearly, some of these assumptions do not hold true all or even most of the time. For example, the model also assumes volatility remains constant over the option’s lifespan. This is unrealistic, and normally not the case, because volatility fluctuates with the level of supply and demand.

Modifications to options pricing models will therefore include volatility skew, which refers to the shape of implied volatilities for options graphed across the range of strike prices for options with the same expiration date. The resulting shape often shows a skew or “smile” where the implied volatility values for options further out of the money (OTM) are higher than for those at the strike price closer to the price of the underlying instrument.

Additionally, Black-Scholes assumes that the options being priced are European style, executable only at maturity. The model does not take into account the execution of American style options, which can be exercised at any time before, and including the day of, expiration. On the other hand, the binomial or trinomial models can handle both styles of options because they can check for the option’s value at every point in time during its life.