# Black Scholes Calculator

Created by Oghenekaro Elem and Purnima Singh, PhD
Reviewed by Steven Wooding
Based on research by
Black F., Scholes M. The pricing of options and corporate liabilities; The Journal of Political Economy, Vol. 81, No. 3; 1973
Last updated: Jun 05, 2023

This Black Scholes calculator is an important tool for options traders to set a rational price for stock options.

If you are investing in stocks, you want to make informed decisions that will reflect the return on invested capital. Without a mathematical framework as a guide, it will be no different from gambling. Black Scholes removes the guesswork involved in predicting stock price movement so that arbitrage opportunities are minimal across markets.

• What is a stock option?
• What is the Black Scholes model?
• How do you calculate stock options value using the Black Scholes formula?

You will also find an example of using the Black Scholes model calculator.

## What is Black Scholes?

Black Scholes is a mathematical model that helps options traders determine a stock option’s fair market price. The Black Scholes model, also known as Black-Scholes-Merton (BSM), was first developed in 1973 by Fisher Black and Myron Scholes; Robert Merton was the first to expand the mathematical understanding of the options pricing model.

The Black-Scholes options pricing model serves as a guide for making rational trading decisions as traders seek to buy options below the calculated value of the Black Scholes formula and sell at a price above the calculated value.

#### What is a Stock option?

An option is a contract that gives the owner a right to buy or sell an asset for a specific price (also known as the strike price) on or before a specific date (also known as the expiration date).

Although most options traders rarely exercise their option rights before the expiration date, you may exercise an American option at any time before the option expires. But you can only exercise a European option on the expiration date.

Financial market traders buy and sell options as protection (or hedge) against uncertainty in the financial market. There are two types of options:

• Call option – gives the owner the right to buy the asset at the strike price; and
• Put option – gives the owner the right to sell the asset at the strike price.

For example, assuming you bought 100 shares of Tesla (TSLA) stocks at $500 per share today (present value = 100 × 500 =$50,000). You believe the price of the shares will increase quickly to $600 per share by next month, so you can sell it (future value = 100 × 600 =$60,000) at a profit of $10,000. But you can’t be too sure, things can go differently, and the price of a TSLA share can fall below the current $500 price, losing you money. Or it can rise even higher than $600, and you’ll be wishing you’d bought 200 shares. For a fair price, you can buy a put option contract to protect yourself or a call option to take advantage of the opportunity if the price indeed rises higher. The put option contract will state the strike price, e.g., $550, the number of stocks you will be selling, e.g., 100 shares, and the exact date the contract expires.

If TSLA share price falls below $550 per share by the specified expiry date on the contract, you can choose to exercise your right to sell the stocks for the strike price of $550 to the option writer you bought the put option from. So, assuming the price went as low as $250, exercising your put option means protecting yourself from the market downturn and selling at a profit of $50 earning per share instead of being at a loss. But if TSLA share price rises over $550 or to the expected price of$600, you can decide to let your put option expire since you will earn more money selling at a higher price in the market.

Conversely, if you had bought a call option with the strike price of $550, when the price rises over$550 or to the expected price of $600, you can decide to exercise your call option and buy the shares at a lower price of $550 to sell at a profit in the market. If the price falls below the strike price, you won’t need to exercise your call option.

To further explore the relationship between a call and put option, check out our put call parity calculator.

#### How do you know what is a fair price to pay for these options contract?

That’s what the Black Scholes calculator helps you determine. The model uses a partial differential equation to predict a stock’s price movement in the financial market and arrive at the price you should buy the call or put option. Hence, to use the Black Scholes formula, you need to provide the following information:

• Current price of the stock, also known as its spot price;
• Strike price;
• Time to the expiration of the options contract;
• Risk-free interest rate, or the rate specified in the option for a given stable asset or short-dated government bonds such as US Treasury bills;
• Expected volatility or unpredictability of the stock is expressed as the standard deviation of the stock price; and
• Expected dividend yield.

## How to calculate Black Scholes model – Black Scholes formula

The Black Scholes equation is a complex mathematical formula. Gladly, you don’t have to go through all the processes involved when using our Black Scholes option pricing calculator. However, the Black Scholes equations have been summarized as follows:

\!\small \begin{align*} C &= S_0e^{-qt}N(d_1) - X e^{-rt}N(d_2) \\\\ P &= X e^{-rt}N(-d_2) - S_0e^{-qt}N(-d_1)\\\\ d_1 &=\frac {\ln (\frac{S_0}{X} ) +(r - d + (v^2/2))t} {v \sqrt{t}}\\\\ d_2 &=\frac {\ln (\frac{S_0}{X} ) +(r - d - (v^2/2))t} {v \sqrt{t}} \end{align*}

where:

• $C$ – Call option price;
• $P$ – Put option price;
• $S_0$ – Current stock price;
• $X$ – Strike price;
• $N(d_1)$ and $N(d_2)$ – Cumulative standard normal distribution functions of $d_1$ and $d_2$;
• $T$ – Term of the option;
• $r$ – Risk-free interest rate;
• $q$ – Dividend yield percentage; and
• $v$ – Annualized volatility of the stock.

## How to use the Black Scholes options calculator?

To use the Black Scholes calculator and get the values of a call and put option, you only have to provide details of six main variables. As an example, for the given input data:

Stock price $400$350 1 year 1% 20% 3%

We will proceed as follows:

1. Provide the current price of the stock, i.e., $400. 2. Input the strike price, i.e.,$350.

3. Enter the option contract term or expiration date, i.e., 1 year.

4. Type the risk-free interest rate in percentage, i.e., 3%.

5. State the expected volatility of the stock, i.e., 20%.

6. Input the expected dividend yield as 1%.

7. The Black Scholes option calculator will give you the call option price and the put option price as $65.67 and$9.30, respectively.

## Assumptions and limitations of the Black Scholes Model

Like all models, it is essential to accept the Black Scholes model's results as estimations that should guide your decision-making, not as absolutes. There are several modifications to the Black Scholes model today that try to fix the model's limitations, but theoretical approximations are not accurate predictors of reality. Therefore, here are some shortcomings to consider about using the Black Scholes option pricing model, and by extension, this Black Scholes model calculator:

• The Black Scholes model is most suited to European options because it assumes the option lasts its entire life span until the expiration date. If you are interested in calculating how much you can gain/lose by executing an option contract before expiration, you can check out our call option calculator.

• Assumes that the markets are entirely efficient, and we can't predict their movements.

• Assumes that volatility is constant.

• Assumes the risk-free interest rate remains constant until the expiration date. But it is not known, and it is not constant in reality.

• Doesn't account for transactional costs, such as fees and taxes, involved in pricing and trading options.

## FAQ

### How does the Black Scholes model work?

The Black Scholes model works by using a stock's volatility, price and strike price, expected dividend yield, and risk-free interest rate for a stable asset to determine the price of a stock option. The model assumes that the stock price follows a lognormal distribution path throughout the life of the stock option.

### What is Black Scholes model used for?

The Black Scholes model is used by options traders for the valuation of stock options. The model helps determine the fair market price for a stock option using a set of six variables:

• Price of the asset;
• Strike price;
• Risk-free interest rate of return;
• Volatility;
• Dividend yield; and
• Expiration date.

### What interest rate is used in Black-Scholes?

The Black-Scholes model assumes that the interest rates are constant and known until the option contract's expiration. Hence, it uses the risk-free one-year interest rates to represent this assumption.

Oghenekaro Elem and Purnima Singh, PhD
Current stock price
$Strike price$
Time to maturity
yrs
Dividend yield
%
Volatility
%
Risk-free interest rate
%
Result
Call option price
$Put option price$
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