# Probability theory on options

### Modern Probability Theory For Stock Traders

Option pricing theory uses variables stock price, exercise price, volatility, interest rate, time to expiration to theoretically value an option. Essentially, it provides an estimation of an option's fair value which traders incorporate into their strategies to maximize profits. Some commonly used models to value options are Black-Scholesbinomial option pricingand Monte-Carlo simulation.

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Understanding Option Pricing Theory The primary goal of option pricing theory is to calculate the probability that an option will be exercised, or be in-the-money ITMat expiration. Underlying asset price stock priceexercise pricevolatilityinterest rateand time to expiration, which is the number of days between probability theory on options calculation date and the option's exercise date, are commonly used variables that are input into mathematical models to derive an option's theoretical fair value.

Aside from a company's stock and strike prices, time, volatility, and interest rates are also quite integral in accurately pricing an option. The longer that an investor has to exercise the option, the greater the likelihood that it will be ITM at expiration.

Similarly, the more volatile the underlying asset, the greater the odds that it will expire ITM. Higher interest rates should translate into higher option prices.

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.

By Kevin Hincks March 28, 5 min read 5 min read Photo by Getty Images Options trading, like all investing really, is about decision making: weighing the potential reward against the potential risk, and the probabilities thereof. By some estimates, the average human makes 35, decisions in a typical day. In general, these decisions are about risk and reward or, in other words, weighing the probabilities: Should you cross the street now, or wait for that approaching car to pass?

The evolution of the modern-day options market is attributed to the 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.

### Option Pricing Theory

However, this is not the only model. The Cox, Ross, and Rubinstein binomial options pricing model and Monte-Carlo simulation are also widely used.

The Black Scholes option pricing model assumes stock prices are lognormally distributed. While the assumption is reasonable, it tends to underestimate the probability of extremely large stock price movements, which have been empirically observed. As a result, option traders assign unique volatilities to options of different strikes, generating a so-called volatility surface across strike and time. An implied distribution is also created, providing meaningful insight into the market's expectations for future stock price outcomes.

Key Takeaways Probability theory on options pricing theory uses variables stock price, exercise price, volatility, interest rate, time to expiration to theoretically value an option. The primary goal of option pricing theory is to calculate the probability that an option will be exercised, or be in-the-money ITMat expiration. Some commonly used models to value options are Black-Scholes, binomial option pricing, and Monte-Carlo simulation.

Also, implied volatility is not the same as historical or realized volatility. Currently, dividends are often used as a sixth input.

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- Option Pricing Theory Definition

Additionally, the Black-Scholes model 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 maturitiesthat short selling of securities with use of proceeds is permitted, and that there are no arbitrage opportunities without risk.

Clearly, some of these assumptions do not hold true all 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.

However, for practical purposes, this is one of the most highly regarded pricing models. Compare Accounts.

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