Loading the player...

What is the 'Black Scholes Model'

The Black Scholes model, also known as the Black-Scholes-Merton model, is a model of price variation over time of financial instruments such as stocks that can, among other things, be used to determine the price of a European call option. The model assumes the price of heavily traded assets follows a geometric Brownian motion with constant drift and volatility. When applied to a stock option, the model incorporates the constant price variation of the stock, the time value of money, the option's strike price and the time to the option's expiry.

BREAKING DOWN 'Black Scholes Model'

The Black Scholes Model is one of the most important concepts in modern financial theory. It was developed in 1973 by Fisher Black, Robert Merton and Myron Scholes and is still widely used in 2016. It is regarded as one of the best ways of determining fair prices of options. The Black Scholes model requires five input variables: the strike price of an option, the current stock price, the time to expiration, the risk-free rate and the volatility. Additionally, the model assumes stock prices follow a lognormal distribution because asset prices cannot be negative. Moreover, the model assumes there are no transaction costs or taxes; the risk-free interest rate is constant for all maturities; short selling of securities with use of proceeds is permitted; and there are no riskless arbitrage opportunities.

Black-Scholes Formula

The Black Scholes call option formula is calculated by multiplying the stock price by the cumulative standard normal probability distribution function. Thereafter, the net present value (NPV) of the strike price multiplied by the cumulative standard normal distribution is subtracted from the resulting value of the previous calculation. In mathematical notation, C = S*N(d1) - Ke^(-r*T)*N(d2). Conversely, the value of a put option could be calculated using the formula: P = Ke^(-r*T)*N(-d2) - S*N(-d1). In both formulas, S is the stock price, K is the strike price, r is the risk-free interest rate and T is the time to maturity. The formula for d1 is: (ln(S/K) + (r + (annualized volatility)^2 / 2)*T) / (annualized volatility * (T^(0.5))). The formula for d2 is: d1 - (annualized volatility)*(T^(0.5)).

Limitations

As stated previously, the Black Scholes model is only used to price European options and does not take into account that American options could be exercised before the expiration date. Moreover, the model assumes dividends and risk-free rates are constant, but this may not be true in reality. The model also assumes volatility remains constant over the option's life, which is not the case because volatility fluctuates with the level of supply and demand.

RELATED TERMS
  1. Myron S. Scholes

    Nobel Prize winning economist Myron Scholes is as famous for ...
  2. Merton Model

    A model, named after the financial scholar Robert C. Merton, ...
  3. Robert C. Merton

    Robert C. Merton is a Nobel Prize-winning economist renowned ...
  4. Model Risk

    Model risk occurs when a financial model used to measure a firm's ...
  5. Dividend Discount Model - DDM

    The dividend discount model (DDM) is a system for valuing the ...
  6. Gordon Growth Model

    The Gordon Growth Model is used to determine the intrinsic value ...
Related Articles
  1. Trading

    The Anatomy of Options

    Find out how you can use the "Greeks" to guide your options trading strategy and help balance your portfolio.
  2. Trading

    Dividends, Interest Rates and Their Effect on Stock Options

    Learn how analyzing dividends and interest rates is crucial to knowing when to exercise early.
  3. Trading

    How to Build Valuation Models Like Black-Scholes

    Want to build a model like Black-Scholes? Here are the tips and guidelines.
  4. Trading

    Stock Options: What's Price Got To Do With It?

    A thorough understanding of risk is essential in options trading. So is knowing the factors that affect option price.
  5. Trading

    Understanding Option Pricing

    This article will explore what factors you need to consider in the pricing of options when trying to take advantage of a stock price's movement.
  6. Trading

    Getting acquainted with options trading

    Learn about trading stock options, including some basic options trading terminology.
  7. Trading

    Understanding How Dividends Affect Option Prices

    Learn how the distribution of dividends on stocks impacts the price of call and put options, and understand how the ex-dividend date affects options.
  8. Trading

    What Is Option Moneyness?

    In the money, at the money and out of the money define the current profitability of options positions.
RELATED FAQS
  1. How is implied volatility used in the Black-Scholes formula?

    Learn how implied volatility is used in the Black-Scholes option pricing model, and understand the meaning of the volatility ... Read Answer >>
  2. What is the difference between financial forecasting and financial modeling?

    Understand the difference between financial forecasting and financial modeling, and learn why a company should conduct both ... Read Answer >>
  3. What is the difference between in the money and out of the money?

    Learn how the difference between in the money and out of the money options is determined by the relationship between strike ... Read Answer >>
Hot Definitions
  1. Gross Margin

    A company's total sales revenue minus its cost of goods sold, divided by the total sales revenue, expressed as a percentage. ...
  2. Inflation

    Inflation is the rate at which prices for goods and services is rising and the worth of currency is dropping.
  3. Discount Rate

    Discount rate is the interest rate charged to commercial banks and other depository institutions for loans received from ...
  4. Economies of Scale

    Economies of scale refer to reduced costs per unit that arise from increased total output of a product. For example, a larger ...
  5. Quick Ratio

    The quick ratio measures a company’s ability to meet its short-term obligations with its most liquid assets.
  6. Leverage

    Leverage results from using borrowed capital as a source of funding when investing to expand the firm's asset base and generate ...
Trading Center