Black-Scholes model is used to estimate the fair value of European call options based on the probabilistic distribution of future prices and volatility. It takes the the risk free interest rate, the volatility of the stock, the strike price of the option, time to expiry and the current stock price into account.

The model assumes a log normal standard distribution of the stock prices.

While the model remains the best way of estimating call option prices, it is not without its limitations. For example, volatility is not constant over time. Also American options are able to be exercised before the expiration date as opposed to the European options.

The model was created by Fischer Black, Myron Scholes and Robert Merton, and was made public by their paper, “The Pricing of Options and Corporate Liabilities” in 1973.

## How it Works

To start with, there are some assumptions that the Black-Scholes model must make in order for the result to be valid. First the option must only be exercised at the expiration, which is the European style of option. It assumes that markets cannot be predicted and the returns are normally distributed. No transaction costs and no dividends paid out are also assumed. If all of that is established then the Black-Scholes model uses the following formula:

C = SN(d_{1})-N(d_{2})Ke^{-rt}

Where C is the call option price, S is the price of the stock, N is the standard normal distribution, K is the strike price, e is an exponential term, r is the interest rate, and t is time until expiration. The terms for d_{1} and d_{2} are formulas themselves:

d_{1} = [ln(S/K)+(r+s^{2}/2)t]/(s*√t)

d_{2} = d_{1}-s*√t

Where the variables are the same as above, along with s being the standard deviation and ln being the natural log.

While the formula can be intimidating, there are some online calculators (like this one here) that can be used to return the results if all the necessary variables are known. Many websites that help with trading or are dedicated trading platforms and online stock brokers generally offer tools like the online calculator for the Black-Scholes Method above.

What the formula is basically doing is breaking down the value into two pieces. In the formula, the SN(d_{1}) portion is the expected benefit that would be earned from buying the stock outright. The N(d_{2})Ke^{-rt} portion represents the current value for paying the strike price at the end of the option term. By subtracting the value of the strike price from the current stock price the call option price is arrived at, or the expected value for the option.

## Why Use the Black-Scholes Model

When using the Black-Scholes Model an experienced trader can calculate when buying and selling the underlying stock will be most profitable. Basically, it’s used to hedge the option to reduce or eliminate risk and it’s the basis for many hedging strategies, especially those used by investment banks, or aptly named, hedge funds. Becoming more familiar with how to use this model can help the average investor understand what investment firms are doing with options to maximize their returns.

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