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BLACK-SCHOLES OPTION PRICING MODELS
Options Pricing and Analysis
Options are a financial instrument that can be generally traded in the financial market place. The purpose of an option is to give a buyer or seller the right or ability to buy or sell some underlying amount of shares at a specified price before or at a specified time in the future. There are various types of options and these may be combined by sophisticated traders and investors for various purposes, such as a means of minimizing their own risk in the marketplace, generating income on stock already owned, leveraged trading without having to buy or sell the underlying stock and so on.
Historically, options have been traded in various forms for hundreds of years. However it was usually costly and difficult to arrange, and most importantly, although the dealers or market-makers determined prices, there was in general, no definitive method of determining the nominal or fair value of an option. In the early 1970s, the market for financial options changed. The Chicago Board Options Exchange (CBOE) was created in 1973 providing the first registered securities exchange for trading options. Also in 1973, Fischer Black and Myron Scholes published their seminal work on pricing options, "Black, Fischer and Myron S. Scholes (1973). The pricing of options and corporate liabilities", giving a closed form analytical method for computing the fair value of European style options. By the term European options, we don't mean options that are available only in Europe, rather this refers to stock options which may be exercised on a specified maturity or exercise date. This is in contrast to American options which can be exercised on or before a specified maturity date.
Black-Sholes Option Pricing ModelsA popular area of modern finance is option pricing. This area of mathematics is based on the simple idea - "how do I price a stock option, that is, how do I value a stock option given the basic information of current stock price, strike price, option duration, treasury interest rates and some estimate of the stocks volatility. Analysts and traders can now use stock option calculators to compute option prices with a high degree of accuracy. The Black and Scholes model developed in 1973 is the key to being able to calculate stock option prices and the related greeks such as delta, gamma, theta, rho, vega or even the newer ones such as charm, color and so on, is a formula or algorithm.
The Black-Scholes model was derived by making the initial assumption that stock prices move with geometric Brownian motion, and then constructing a PDE (Partial Differential Equation) and then solving this equation in a closed-form to obtain a model which produces the desired stock option prices.
The importance of Black and Scholes work is difficult to underestimate, since by using some assumptions about the conditions of the financial markets, they were able to provide a method of calculating the price at which an option could be valued. Now, it is important to realize that this may not correspond to the actual market price for various reasons. The theoretical option values are coming from a model which may not always produce the same values as the market itself. The market is free to determine a price and is not bound by what a model indicates. However as a starting point, this work gave options market participants some method of knowing whether an option appeared to be close to what could be called a fair value. Since that time, their has been a considerable amount of research devoted to determining similar algorithms for pricing options and related variables.
There are limitations to the various models, and over time and the range of option prices, there are variations in the performance and ultimate accuracy of option pricing models. Searching the literature will reveal various known areas where option pricing models consistently "get it wrong" to a greater or lesser degree. One of the main limitations of the BlackSholes model is that it is for European options. For American options, the Cox, Ross and Rubinstein binomial model is commonly used.
1973 Black-Scholes Option Pricing FormulaThe Black-Scholes Option Pricing Formula uses the following inputs:
Here is an example of a simple options pricing calculator that was created using Options/NET:
Black-Scholes Option Pricing Software
For options market participants, it is now possible and common to have access to not only current and historical market data, but also computer and software tools to give information about the stock market and options. Specifically, there is often a particular need to encapsulate the research results obtained since Black and Scholes in computer applications so that the apparent fair values of options as well as other statistics or numerical information can be ascertained about a given set of financial instruments. The various option pricing methods range from the relatively simple to exceedingly complex. Programming these analytical and in some cases, non-closed form algorithms into computer applications can be time consuming, difficult and error-prone, as well as requiring a considerable amount of background knowledge of mathematical software development. This has given rise to the need for an options pricing and analysis library of software such as Options/X, Options/NET and Options/NET Mobile.
The principal advantages of the Options/X and Options/NET components, are that for a small investment, applications developers and users can obtain instant access to these highly important and useful options pricing methods without having to attempt to understand the intricacies of the math involved as well as accurately and efficiently program the algorithms. The return on investment of using such powerful tools is well appreciated and evidenced by the growth of the software industry in terms of 3rd party component libraries.
Options SoftwareOptions/X - Options Pricing Excel Add-In and SDK for ActiveX, COM, Visual Basic 6, Visual C++ 6
Options/NET - Options Analysis .NET Component
Volatility/X - Volatility Estimation Excel Add-In
Options/NET Mobile - Options Analysis Windows Mobile Component