Financial free essay: The Black-Scholes Model and Pricing
The Black-Scholes Model and Pricing
One of the most applied financial practices in modern stock trading is acceptance of derivative assets as stable trade items. Derivative assets are those that obtain their value from other assets, and become enforceable in transactions which recognize their value, such as options. The stock market has evolved over the years to increase asset base to such an extent that derived assets are recognized as valid trading assets. Varieties of options such as put and call options exist, depending on the specific rights possessed by the owners. Introduction of bonds that can have value attached to options is an example of how assets can be expanded at the stock market. Options trading, which forms a part of the model formulated by Black and Scholes, involves dealing with and handling pricing issues of contracts that confer rights for sale or purchase of some assets. Rights involved in options are exempted of obligations, a fact that confers economic value to them as assets. Trading of the options in form assets can therefore be validated where option holders agree to transact in them.
Fischer Black and Myron Scholes were finance specialists in different capacities in 1961, when they applied their skills to develop the application that became a milestone project touching a major operation of the stock market (Mehrling, 129). Using their prowess in mathematics and finance, the two devoted their skills in developing a model that would assist in determining options value and their pricing at the stock market.
Before Black and Scholes made their contribution to the option pricing specialty, earlier studies had been conducted in applied finance and asset logic from the 1870s. Charles Castelli is one of the pioneers of the pricing strategies that options ought to adopt in a stock market. By explaining the theory of options with regard to the mechanism of the stock market, a vivid picture of relationship between options and other stocks was established. Castelli made the contributions in 1877 when he published a book creating the detailed link between stocks and options (Castelli, 9). It was not until two decades later that similar contribution was made by Louis Bachelier who incorporated mathematical formulae and perspectives into the earlier options pricing mechanisms. Other contributions with much more mathematical infiltration into the options pricing intricacies were made by such scholars as Paul Samuelson in 1955 and James Boness in 1962. Black and Scholes heavily relied on the work by Bonnes to make their postulates on the manner in which current stock market should deal with options pricing issues (Rubash, 3).
It is on record that the model formulated by Black and Scholes was not instantaneously discovered, especially after revelations that Black was initially working on stock warrants valuation technique were made (Mehrling, 128). In his valuation technique, Black made mathematical infusion to theoretical explanation of how discount fluctuation should be calculated with stock factors such as time and changes in price in consideration. Surprisingly, Brownian motion equation, a famous physics equation of heat transfer was almost reproduced in the dramatic discovery.
Scholes’ contribution assisted the pair to navigate through the stocks and particular item of consideration was options pricing. Scholes must have been at a critical stage of discovery of pricing strategies and techniques, having closely followed the work by Paul Cootner (129). Cootner’s work on The Random Character of Stock Market Prices in 1964 drove Scholes to make the critical contribution that they needed to come up with the formula. Due to the compatibility of the behavior of the assets in contention, the accurate formula developed almost flawlessly fitted into their needs in quest for a solution for a pricing technique. Warrant valuation and pricing intricacies were issues of modification of the formula earlier on developed by Black.
The model formulated after their deliberations and modifications under the new conditions of options pricing was as illustrated below (Rubash, 4).
The model relies on a number of assumptions that include;
- Absence of dividend throughout the life of the dividend.
- Applicable terms are the European exercise terms
- Efficiency of the market is constant
- Commissions are not charged
- Interest rates are known in advance and deemed to remain constant
- Returns are distributed in a normal curve manner
There are five basic definition segments of the Black-Scholes equation as defined below.
- Delta: represents option’s sensitivity to small share price fluctuations
- Gamma: represents delta’s small prices fluctuation sensitivity
- Theta: represents option’s sensitivity to time factor
- Vega: represents options’ value sensitivity to volatility
- Rho: sensitivity of option value upon a percentage unit variation for interest rate
Since its postulation in 1973, Black-Scholes model has not witnessed much change, despite changes in assumptions. Robert Merton immediately introduced a change that aimed at relaxing the assumption that provided absence of dividends. Taxes absence assumption was relaxed three years later by anther finance scholar by the name Jonathan Ingerson. Merton was at it again three years later to dispel the assumption to the effect that interest rates remained constant throughout the life of the option. Despite these changes to the model, the formula has not been tampered with.
Prior to the comprehensive approach in pricing options at the stock market, it was difficult to arrive at a reasonable market value which had been marred by cases of lack of transparency (Lim et al, 69). Upon the arrival of the pricing technique proposed by model, there were drastic improvements in both returns and public confidence in the turn of events at the options market.
According to the authors, transaction costs were dramatically reduced, paving way to new and better experiences for investors. Options market became a successful venture that major stock markets began to boast of. A dramatic change in the efficiency of the options pricing system at the market was achieved when modifications were introduced by Merton. The anticipation of the importance of the model was overwhelming to such an extent that the Journal of Finance published the model earlier than the final date of preparation; 1972 instead of 1973 (69).
Based on the uncorrected assumptions, the model can be scrutinized for limitations or weakness. Firstly, lognormal distribution assumption of returns tends to be inconsistent with actual market settings. Such a distribution can only be applicable in an ideal situation, which creates a major concern for its reliability and accuracy for practical market situations. To correct this defect however, recent studies have come up with situations of non-lognormally distributed returns as well as volatility smiles.
Volatility assumed is difficult to be calculated and estimated used are often vulnerable to inaccuracies.
American style exercise can apply the model since the basic assumption of a European setting holds. This is due to the fact that the American system calculates options value at the end of the life of the option. Modifications can however be effected to incorporate the American system, for instance the Fischer Black Pseudo-American method (Hoadley, 1).
Conclusion and Summary
Although the options pricing model should not exclusively be credited to Black and Scholes due to prior contributions made, they perhaps need much more tributes. This is because their contributions conglomerated all ideas and presented them in the comprehensive model than any other finance scholar. The model’s ingenuity connects creativity that scientific studies have witnessed in the past, such as Brownian motion, enabling this field boast of great brains too. Future options pricing techniques will certainly rely materially on the work done by these gentlemen (Marlow, 1).
Options are derived assets that can be traded at the stock markets. To define their valuation, Black-Scholes model is usually employed. Since its discovery, the model has experienced a few changes. Future option pricing will heavily rely on the model.
Castelli, Charles The theory of “options” in stocks and shares. London, UK: F.C. Mathieson, 1877. Print
Hoadley, Peter “Option Pricing Models and the “Greeks,”” Hoadley Trading and Investment Tools, 2011. Web. http://www.hoadley.net/options/bs.htm (accessed 4 March 2011)
Lim, T., Lo, A. W., Merton, R. C. & Scholes, M. S. The derivatives sourcebook. Hanover, MA: Now Publishers Inc., 2006. Print
Marlow, Jerry “Option Pricing: Black-Scholes Made Easy,” n.d. Web. http://www.optionanimation.com/ (accessed 4 March 2011)
Mehrling, Perry Fischer Black and the revolutionary idea of finance. Hoboken, NJ: John Wiley and Sons, 2005. Print
Rubash, Kevin “A study of option pricing models: origins of option pricing techniques.” Bradley University, n.d. Web. http://hilltop.bradley.edu/~arr/bsm/pg03.html (accessed 4 March 2011)
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