Foundation from the Black-Scholes Model - Term Paper Example

The Black-Scholes Model Introduction The financial world is proving to be one of the most dynamic and innovative areas, directly competing with space technology in terms of rapid changes and scientific methods applied. Modern financial instruments have become increasingly complicated and mathematical modeling and analysis are being used for not only determining the present value of future earnings but also of future value of today’s assets. The world of corporate finance which was once managed by business students is now controlled by mathematicians and computer scientists. Currently simple instruments of shares and commodities traded at today’s price are considered ‘vanilla’ trading compared to future derivative products, also known as Options. These are generally defined as a "contract between two parties in which one party has the right but not the obligation to do something, usually to buy or sell some underlying asset". In 1973, the tandem team of Fischer Black and Myron Scholes had written the first draft of a paper that outlined an analytic model that would determine the fair market value for European type call options on non-payout assets. Basically, the fundamental insight from the Black and Scholes model comes out to be that the call option is always given a price value if the stock is to be traded (Wikipedia, 2005). Overtime, the formula and the model itself have become ingrained in the financial markets of the world and due to their usefulness and applicability. This model itself is one of the basic tenants for any stock trader to understand and apply while dealing with stock options in the modern financial world. Merton and Scholes shared the 1997 Nobel Prize for their study. Unfortunately, Black himself could not be a part of the winning team since he had already expired in 1995 and thus become ineligible for the prize (Wikipedia, 2005). The Black-Scholes model today is used in everyday pricing of options and futures and almost all formulas for pricing of exotic options such as barriers, compounds and Asian options take their foundation from the Black-Scholes model. Keeping the aim of this paper in mind, we’ll be mainly looking at simple European and American Call Options. Black – Scholes Options Pricing Formula Black-Scholes model was developed on a previous model by A. James Boness. It is defined as: Where: C is theoretical call premium and derives the expected benefit from the stock outright gives the expected value of paying the exercise price on the expiration date. S is current stock price t is time until option expires K is option striking price r is risk free interest rate (US T-Bill rate) N is cumulative standard normal distribution e = 2.7183 s = standard deviation of stock returns ln= Natural Logarithm Assumptions As with other models, Black-Scholes model is based on a few assumptions as given by Rubash: 1. No dividends are paid out on the underlying stock during the option life. This supposition is a major drawback of the model as in the real world, most companies pay dividends to their share holders. To account for this assumption and to make the model more accurate, a common alteration to the model is to subtract the discounted present value of future dividends from the stock price depending on the bylaws and articles of incorporation of the company. 2. The option can only be exercised at expiry American option can be exercised at any time during the life of the option while European option can only be exercised at expiry. Black Scholes model does not take into account the option being exercised prior to its expiry; therefore, it is applicable to European options only. 3. Efficient markets Market movements cannot be predicted, which implies that there “…cannot be a portfolio that costs nothing at time t and will not with positive probability yield a positive return at some time in the future” (Bichler, pg. 152, 2001). Simply put, the model does not factor in volatility of stock prices. 4. Attached costs are not accounted for The model does not factor in any commissions charged by intermediaries/agents which are present in the real world at the time of transaction along with margin requirements and taxes. The amount of commission and taxes can often distort the output of the model. It is therefore further assumed that a) The taxation rates for income and capital gains remain stable throughout the period b) Taxes are immediately collected depending on realized and accrued gains c) Any trading losses incurred, interest paid and dividends paid to cover short sales of stocks are taken as full offsets against the gains 5. Interest rates do not change over the life of the option Black and Scholes model is based on risk free rate. Even though there is no such thing as the risk free rate, discount rate on US T-Bills is usually used to represent it. For the purpose of this model, authors did not include random jumps or discontinuities in interest rates. But there are times where the interest rate can change rapidly (Rubash, 1998), thus putting the results of the model into question. 6. Stock returns follow a lognormal distribution This hypothesis suggests that returns on the underlying stock are normally distributed, which is practical for most assets that offer options. The model would not apply in situations such as those which were faced during the dot com bubble burst. In the real world many stocks would follow the lognormal distribution therefore this assumption can useful for almost all European Options pricing. Application European Options are options which can only be exercised on the expiry date and can therefore be valued quite accurately using the Black Scholes option pricing formula. As such European options are typically the simplest options to value the dividend or yield on the underlying asset. The term European is used to explain the exercise feature of the option (i.e. can only be exercised on the expiry date) and has nothing to do with the geographic region of the underlying asset. For example, a European Option of a company can be issued on an Asian exchange if it is listed on it (Derivativeone.com, 2005). For calculation for a European dividend paying option the standard Black – Scholes model was altered by Merton in 1973 and includes term ‘D’ as dividend yield per year. It is represented by: Where: and Limitation An American Option can be exercised at any time up to and including the expiry date of the option. This added flexibility to the option resulting in American options being considered as more valuable, even though in many cases the values can be very comparable as the best exercise date is often the expiry date. The early exercise feature of these options complicates the valuation process as the standard Black-Scholes model cannot be used. The most common model for valuating American Options is the binomial model (Derivativeone.com, 2005). The binomial model of valuation helps to overcome the limitation laid in the standard Black-Scholes model and can be used with a number of time intervals as required. Via its utilization, a tree of stock prices is generated from beginning till expiration where at each branch the price of the option will move either up or down using volatility and time to expiry. The final model represents possible stock prices of the option during its life (Hoadley, 2005). The underlying assumption of both the binomial and Black-Scholes models is the presupposition that stock prices will follow a random process which can be best represented by geometric Brownian motion as in the movement of gas particles. Therefore, in European options, the binomial model comes closer to the result from the Black-Scholes formula as the number of calculation steps rises. In fact the Black-Scholes model for European options can be considered a special case of the binomial model where there are infinitely many binomial steps. Alternatively, it can be said that the binomial model gives more or less accurate approximations of the continuous process which is underlying the Black-Scholes model for options (Hoadley, 2005). Nowadays it is common for the implied volatility to be quoted instead of the price of the instrument because the Black-Scholes model is pervasive in the financial world. All the parameters in the model other than the volatility (time to maturity, the strike, the risk-free rate, and the current underlying price) are plainly observed. This means there is direct relationship between the option price and the volatility of the option. It must be kept in mind however, that the model is not a representative of the real world in exact terms as implied volatility of an option is never constant (Wikipedia, 2005). Conclusion In conclusion, Black – Scholes model provided one of the best valuation methods which has been extensively used and further developed upon to take into account the real world scenarios which cater for deviations from above mentioned assumption. With an overall analysis it is easy to see what the value of this model is for the financial world. In spite of its limitations and certain drawbacks, it has been usefully applied for many real world situations which have led to the benefit and profit of many corporations. For any student of mathematics, finance or business the Black-Scholes model will be important due to the brilliance of its creators. Works Cited Bichler, M. (2001). The Future of E-Markets: Multi-Dimensional Market Mechanisms. Cambridge: Cambridge University Press. Derivativeone.com. (2005). European Options pricing. Retrieved April 26, 2006 from Knowledge Base website: http://www.derivativeone.com/kb/european_options.aspx Hoadley, P. (2005). Option Pricing Models and the Greeks. Retrieved April 27, 2006 from Option Strategy Analysis Tools website: http://www.hoadley.net/options/bs.htm Rubash, K. (1998). A Study of Option Pricing Models. Retrieved April 22, 2006 from Foster College of Business Administration, Bradley University website: http://bradley.bradley.edu/~arr/bsm/model.html Wikipedia (2006). Black-Scholes. Retrieved April 22, 2006 from Wikipedia the free encyclopedia website: http://en.wikipedia.org/wiki/Black-Scholes Read More