## Calculating Call and Put Option Prices

This Black-Scholes calculator can be used by investors to estimate the rational price of a European call or put option. The calculator needs a total of five inputs, including:

- The current market price of the underlying asset
- The strike price of the option
- The time until expiration of the option, stated in years. For example, three months would be equal to one quarter, or 0.25 of a year
- The risk-free interest rate, which is the theoretical rate of return on an investment with zero risk
- The volatility of the underlying asset

The calculator then provides the user with two outputs, including:

- The theoretical price of a European call option
- The theoretical price of a European put option

### The Black Scholes Model

Also known as the Black–Scholes–Merton model, the Black-Scholes model is a mathematical approach to estimating the theoretical price of a European option. Since one of the variables used to compute this price is the time until the option expires, this model cannot be used to value American options. The model’s accuracy relies on a number of assumptions, including those that apply to the underly asset:

- Market movements are random and cannot be predicted (Random Walk Theory)
- The risk-free rate of return as well as the asset’s volatility do not vary over time
- Dividends are not paid before the option expires

As well as assumptions that apply to the market, including:

- It is possible to borrow or lend any amount of cash at the risk-free rate
- It is possible buy or sell any amount of the underlying asset
- Transactions occur without cost or fees (they are frictionless)
- Arbitrage opportunities do not exist

### Variables used by this Black-Scholes Calculator

While this calculator can be used to estimate the price of a European call or put option, the accuracy of that estimate is highly dependent on inputs. Three of these variables are well known, such as the price of the underlying asset (often a stock), the option’s strike price, and the time to expiration. Which leaves us with two variables that are less certain – the risk-free rate and volatility.

#### Risk-free Rate of Return

Fortunately, it is relatively easy to estimate the risk-free rate of return, which is the return an investor can earn on an investment with zero risk. In the United States, a proxy for this return is a three-month Treasury Bill, which over the last several years has ranged from around 2.4% to a low of around 0.04%.

#### Volatility

There are several ways to measure the propensity of an asset, such as a common stock, to rise or fall. While using historical information may not be a good predictor of future performance, it is often the best way to estimate this value, along with any adjustment the investor may deem appropriate to model future volatility.

This value is estimated by calculating the historical standard deviation of the periodic percent change in price. This presents the user with the challenge of determining the interval of time to be used when calculating the change in price. For example, should the daily, weekly or monthly percent change in price be used when calculating the standard deviation?

More commonly, a Black-Scholes calculator is used to figure out the implied volatility of a stock. This can be done if the price of a European option is known and by using our calculator with a trial-and-error approach to back solve for volatility.