The Black-Scholes model is a tool for pricing equity options. The Black-Scholes model, often also called using its full name Black-Scholes Option Pricing Model , is an approach for calculating the value of a stock option, let it be a call option or a put option. This is also referred to as random walk. Assuming that stock prices follow random walk, it already suggests that we will need to involve some math and statistics.
The Black-Scholes model is a formula that can be expressed as described in the following paragraph. The Black-Scholes model is basically a formula that is used to calculate option values.
The Black-Scholes formula consists of three parts. The main equation and two formulas for calculating parameters. Phi represents a cumulative distribution function of Normal distribution. The Black-Scholes formula for a European-style put option is very similar to the Black-Scholes formula for a call option. It is the following:. This Black-Scholes formula tells us that a value of a put option can be calculated as a present value of the stock delivery price minus the price of the stock, both adjusted for volatility, interest rate, and spread.
We can demonstrate the working of the Black-Scholes formula on an example. The option expires in three months. We also assume that the stock pays no dividends. Note, this Black-Scholes formula example is used to value a call option. The Black-Scholes model can also be used to price puts options. Using the put-call parity approach to calculate put option value given that you know the call option value, you would solve the put-call parity equation for the value of the put option.
See the following example:. Understanding the Black-Scholes model assumptions is very important for the application of the model to real-world scenarios.
Listing of the Black-Scholes model assumptions is provided on the next page: This also relates to the assumption of the constant volatility.
Volatility is not constant in real world. Very short-term options can be valued using the basic Black-Scholes formula because volatility can change only so much in only a few days, but invalidation of these assumptions in longer term in the real world makes the Black-Scholes formula not work for mid-term and long-term options.
The Black-Scholes model was later improved to deal with some limitations of the real world. The Black-Scholes model is subject to many limitations and assumptions as discussed on the Black-Scholes assumptions page. Every model is only an approximation of the real world, and every model has some limitations. The Black-Scholes model was revolutionary in a way it approached options valuation. Throughout the years, many other models emerged trying to provide more accurate approach to option valuation.
However, with a little generalization, we can say that probably most of them are enhancements of Black-Scholes. All of them are based on the same valuation principle. The difference between models is mostly how they address assumptions of the Black-Scholes model. We already mentioned that for example the GARCH model substitutes constant volatility with a stochastic one. Other models address other assumptions, for example the assumption of constant interest rate, or address them differently.
We can name a few models related to valuations, for example: The Black-Scholes Option Pricing Model is an approach used for calculating the value of a stock opti Put-call parity is a financial relationship between the price of a put option and a call option. The calculator below relates to the Black-Scholes model which is explained in detail on the Black-Sc Our income tax rates provided on this page are applicable to income earned in You can use these income tax rates and income tax tables provided below to calculate your taxes that you report on your tax return due in April All articles are protected by copyright and have been archived at a national library.
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