The Black-Scholes Formula

CHAPTER 1

Shall introduce the research questions and provide a very brief historical account of the Black Scholes Model.
Option pricing techniques of today are amongst the difficult and complex areas of applied finance. The mathematics involved can be used to predict the erratic movements (volatility) of stocks. Financial analysts can calculate the value of stock options by using specific mathematical models. “These modern techniques derive their impetus from a formal history dating back to 1877, when Charles Castelli wrote a book entitled The Theory of Options in Stocks and Shares.”Rubash.K (2001)
One of the fundamental models that provided a foundation for later forms is known as the Black Scholes’ model. The model was developed by Fischer Black and Myron Scholes in 1973 and after many years of perfecting their model they 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.
Before the bulk of the report begins it is necessary to understand what is meant by a “Call” or “Put”. Put and Call options were introduced by Fisher Black and Myron Scholes. A call option is effectively and insurance policy so that if a trader buys a stock (at £10 for example) he reserves the right to continue buying that stock even if the price of the stock has risen (say to £20) The trader however pays a premium for this right. If the stock falls below the price he bought it for (the strike) then he loses his premium. The same is true for a Put option, but in this case it is in the interest of the seller. If a trader wants to sell a stock (at £10) and the value of the stock goes down, he still reserves the right to continue selling that stock at £10 even if the stock depreciates. Again the trader must pay a premium for this right. Premiums can only be held for a specific period of time dependent on the level of volatility of the stock.
“The objective of this study is to look into various Black Scholes models updated and extended throughout the years and provide a critical assessment of each by comparing their predictability behaviour on the markets. The relevance of this work stems from the fact it is necessary to keep track of changes in the models for further research.”
In early 1973 the Black Scholes model was submitted to the Journal of Political Economy in the hope that it would be published, the model was unfortunately rejected. The Review of Economics and Statistics also rejected the idea. “After making some revisions based on extensive comments from Merton Miller (Nobel Laureate from the University of Chicago) and Eugene Fama, of the University of Chicago, they resubmitted their paper to the Journal of Political Economy, who finally accepted it”. Rubash.K (2001). The acceptance and publication of this model in 1973 marked a mile stone for “Option Pricing”. To date the Black Scholes model has remained one of the most widely accepted and utilized model of all financial models.
It must be mentioned however that Black and Scholes can not take all the credit for their work, as their model is actually an improved version of a model developed by A. James Boness in his Ph.D. dissertation at the University of Chicago in 1962. It took Black and Scholes 9 years then to perfect and publish their improved model proving that the “risk-free interest rate is the correct discount factor, and with the absence of assumptions regarding investor’s risk preferences”. Rubash.K (2001)

CHAPTER 2

Explain the Monte Carlo Option pricing, the Greeks, Binomial Model Option Pricing
Apart from the Black Scholes option pricing method, there are quite a few other models employed in the modern market. Let us evaluate the Monte Carlo, the Binomial and the Greeks methods.
Firstly, the Monte Carlo model, this is “an analytical technique in which uses random quantities for uncertain variables” Murcko.T (2007). Put more simply it uses computational algorithms which are based on continuous random sampling to calculate the value of an option. Much like a computer opponent during a game of battleship its movements are mathematically calculated for optimizing its chances of hitting a target. The Monte Carlo method is now widely used by financial analysts who want to construct stochastic or probabilistic financial models. The method has ascertained its name due to the city of Monte Carlo which has numerous casinos. Some of the most sophisticated corporations worldwide implement this method for financial analysis and investing in projects.
The binomial model was first proposed by Cox, Ross and Rubinstein (1979). The Black and Scholes model can only be exercised at expiration the binomial model which is similar also follows the same assumption. However it differs slightly because it “breaks down the time to expiration into potentially a very large number of time intervals, or steps.” At each time interval it is understood that the volatility and time to expiration can calculate the amount at which the stock price moves up or down. This produces a binomial distribution (or tree) of underlying stocks. “The binomial distribution represents all the possible paths that the stock price could take during the life of the option.” Hoadley.P (2007)
Next the options prices at each step in the tree are calculated inversely. This means that they are calculated working back from expiration to the present The value at each increment then is essentially used to derive the value at the next increment of the distribution. The model “calculates the values by using risk neutral valuation based on the probabilities of the stock prices moving up or down, the risk free rate and the time interval of each step.” Hoadley.P (2007) The binomial model provides a valuation of options, by applying a numerical method based on appropriate discretization of the distribution.
The binomial model has a slight advantage over the Black Scholes model when it comes to accurately pricing American options. However it does also have a limitation. The main draw back in using this method is that it is relatively slow when attempting to calculate thousands of option prices, a problem that the simpler, faster Black Scholes does not encounter.
The Black Scholes model result in the calculation of a delta which is defined as: “the degree to which an option price will move given a small change in the underlying stock price. For example, an option with a delta of 0.5 will move half a cent for every full cent movement in the underlying stock.” Hoadley.P (2007)
In addition to delta there are some other “Greeks” which some find useful when constructing option strategies:
Along with the delta there are some other “Greeks” that are useful for evaluating options. They are the Gamma, Vega, Theta and Rho. The Gamma measures how fast the delta changes for small changes in the underlying stock price. Vega is defined as “the change in option price given a one percentage point change in volatility. Theta is “The change in option price given a one day decrease in time to expiration. Basically a measure of time decay”. Finally, Rho is “the change in option price given a one percentage point change in the risk-free interest rate.” Hoadley.P (2007)
Review the key features of the Black Scholes Methodology and delve into its most prominent assumptions and hence lay a foundation for the remainder of the work
Brownian motion is one of the key assumptions associated with the Black Scholes Model. The theory can be defined as the random movement of suspended particles There are other similar theories such as random walk and Donsker’s theorem, but Brownian motion has the advantage of being easily understood and among the simplest continuous time stochastic processes. It is used widely in the evaluation of stock market fluctuations or more specifically in the Black Scholes model to help calculate the volatility of options.
The Black Scholes model assumes that there are four observable parameters. These four observables are; the time to maturity of the stock, the strike, the risk free rate and the current underlying price. The non observable parameter which is of interest to the user is the volatility which is mentioned above. Let us briefly defined what is meant by these five parameters. The “time to maturity” is “the time taken to when the option or matures. If the option has not been exercised by this date, it expires and ceases to have any value. The “strike” is the fixed price at which the owner of an option can purchase the underlying security or commodity.” Douglas.S (2006). The “risk free rate” is the interest rate that it is assumed and can be obtained by investing in financial instruments with no default risk. Finally the “current underlying price” is the “instrument price which the option is based or written on. This can be any tradable instrument which has a defined market price”. These are the observable parameters. Douglas.S (2006)
Finally let us discuss the volatility. The volatility is a non observable parameter this means that has to be calculated by use of the Black Scholes model with the aid of Geometric Brownian motion discussed previously. Volatility is the degree to which the underlying price tends to fluctuate over time. There are also a couple other varying degrees of volatility such as “Implied volatility” and “Historical volatility”. The later can be calculated by looking at price fluctuations over a specific period in the past. While “Implied volatility” can be implied from option prices observed in the market place. Douglas.S (2006) By using the Black-Scholes Equation, or one of its derivatives an option volatility can be calculated. The calculation will yield the current market price for that option. Moreover is the volatility that, given a particular pricing model, such as Black Scholes or any other model will yield a theoretical value for the option that is equal to the current market price. Volatility is decidedly one of the major factors in deciding an options worth
There is also an existence of another definition known as the volatility smile. This is defined as a long observed pattern that shows a momentary positive value for the options shortly after their expiration. However there is much debate over the credibility of the volatility smile in the market. When the volatility smile was first observed, some researchers believed that the explanation was liquidity. It is discussed in a paper written by Don Chance through the Financial engineering news that volatility smile may not be credible. This idea will be discussed further later in the report.
“In summary the Black Scholes model assumes that the option can be exercised only at expiration. It requires that both risk free rate and the volatility of the underlying stock price remain constant over the period of analysis.” MBAFianance (1999) Secondly it is assumes that no dividends are paid by underlying stocks
Now that the theory behind the model has been briefly discussed the methodology can now be considered. The Black Scholes model uses partial different equations to derive the value of the option previous to the time of the option maturing. This is because the value of the option when it matures is known; therefore we need to know how the value evolves as we go backward in time. “This method assumes that If X is an option then X function of S and t. Therefore X(S, t) is the value of the option at time t if the price of the underlying stock at time t is S.” Cairns A.J.G 2004 this is the law of evolution of the value of the option. With the assumptions of the Black-Scholes model, this equation method is known as arbitrage-free pricing
Above is the description of the PDE based black Scholes model. The PDE must be calculated for a specific valuation for an option In order to solve the PDE equation it must be transform the into a diffusion equation The Black-Scholes PDE becomes a diffusion equation then. After some algebra the formula can be arranged to obtain the value of a call option in terms of the Black-Scholes parameters

CHAPTER 3

Analysis of extending the original Black Scholes model and limitations
The original Black Scholes model was developed and published in 1973. It was very quickly adapted as the standard model for option pricing. However there were certain limiting factors associated with this simple model. The first was “only one interest rate can be inputted” and secondly that “no dividends or cash flows are generated by the underlying asset.”
Since 1973 the model has been “modified” for it to be applicable to European, American and even French options. It was also finally modified therefore to value options on dividends, foreign exchange and bonds.
“The European Model” is an example of a modified Black Scholes Model. It prices European options or options that may only be exercised at expiration. The following assumptions are made by the European Model:

Another example of how the Black Scholes Model has been extended is its ability to price American options. Originally this proved inaccurate with the basic model. The American Model is essentially the same as the European model apart from the fact that it “checks to see if the value returned is below the intrinsic value of the option.” (MITI) The intrinsic value is defined at the true value of a company in terms of both its tangible and intangible assets.
The American Model makes the following assumptions:

Another model known as the French Model attacks a different area of the original model. In original Black-Scholes model interest is paid according to calendar days and volatility is related to the number of trading days until expiration, because of this confusion “D. French has suggested that two different times should be specified when pricing options.” MITI (2001)
In order to evaluate option prices, interest is paid according to calendar days.
T¹= Calendar days until maturity
Calendar days per year
T²= Trading days until maturity
Trading days per year
This adjustment does not make a big impact on the evaluation of most options; it is still useful in the pricing of short life options. MITI (2001)
Apart from the Black Scholes extensions there is another efficient analytical approximation of the American options values. This is known as the Whaley (Quadratic Approximation Method) this is considered by Barone-Adesi, Whaley (1987) to be “an accurate inexpensive method for pricing American call and put options written on commodities (securities) and commodity futures.” The advantages of this method are that it is faster than binomial or finite differencing techniques.
The difference between a European option and an American option is that the holder of the American Option has the right to exercise early. Early exercise just means that the option has the ability to sell before its expiry date. The Quadratic Approximation method involves obtaining a value of an option calculated by the modified Black Scholes European model. Then adding this value to the value of the early exercise option calculated by the Quadratic Approximation method, value of an American option can be calculated. The Quadratic Approximation method is slower than the Modified Black-Scholes model but provides values that are more accurate. MITI (2001) Although there are various models that are considered to be more accurate than the Black Scholes Model, the main advantage it has is that it can calculate a large number of option prices in a short period of time. This computational speed and simplicity ranks very highly with the modern financial analyst.
There are however specific limitations that are inherent in the Black Scholes Model. For example it “cannot be used to accurately price options with an American-style exercise as it only calculates the option price at one point in time, namely at the expiration of the option.” Rossi P.E. (1996) Furthermore the possibility of early exercise of an American option is not considered. In order to approximate the American option prices, various adjustments are made to the Black price. However these adjustments are tricky and rarely work for puts.

CHAPTER 4

Detail the mathematical derivations of the extensions of the Black Scholes model including the fractional models, GARCH and the stochastic volatility, constant volatility (the original model), models with instruments paying proportional discrete dividends and those with continuous yield instruments, currency options, approximate prices of American style options, options on futures and exotic options.
The Diffusion models have provided the basic statistical models for financial research over the last 25 years. In 1973 the Black Scholes model was a good foundation to launch other model from and since the numerous methods have tried to improve on some of the limiting functions inherent in the Black Scholes model. In 1986 Bollerslev introduced his Arch Model.. Arch or autoregressive continually heteroskedastic was used to model time varying volatility and the persistence of shock to volatility, it was also used as a means to capture the predictability of volatility. A filter was later added to the Arch model which then became known as Garch. This model was popular in econometric modelling. According to Hand.D.J & Jacka.S.D (1998) The Garch model is “unlike the Black Scholes model as it has the virtue of imposing no inequity constraints on the parameters”.
Further more the Garch model can account for leverage effects, which are considered to be very important in determining stock volatility. As is briefly mentioned there are many different filters to these models. Perhaps to many too mention but a useful tool for estimation and filtering for discrete time is by using the stochastic volatility model. Variations of this model have been investigated by Wiggins (1987) Scott (1987) Hull and White (1987).
“Volume and leverage effects can partially account for the observed patterns in volatility.” Rossi P.E. (1996) however these explanations remain incomplete, more complex models needed.
Unfortunately the B.S Model does not account for time varying volatility; this means that the volatility is constant for a particular option. This assumption is not realistic in modern option pricing and so the model contains discrepancies. However since its publication in 1973 the B.S model has been extended through ways such as the Arch and Garchs models described above. The assumptions of constant dividends, interest rates and volatility are not realistic. BS is therefore inherently misspecified, the only way to achieve a result from this model is to incorporate new features such as stochastic volatility, however such attempts to do so are very complex and usually are incorrect. Another extended version of BS is the GMT method which presents the statistical foundations of dealing with a misspecified BS model. The GMT method is applied essentially by extending the BS formulation it can also be used for American option.
So in summary the Black Scholes model can provide a description of the behaviour of an asset price but it is only an approximation. The plus side of this is that it can set a decent benchmark against which other models can be compared. In order to calculate the volatility, which is a crucial parameter for option pricing, a stochastic volatility model must be incorporated to account for the time varying volatility. These models aim to reflect apparent randomness of the level of volatility To a certain extent stochastic volatility models are consider successful and they can somewhat explain some of the biases present in the Black Scholes formula.
Both diffusion models and Garch models are widely used in the financial community, along with the incorporated stochastic volatility model they can account for phenomena such as volatility smiles (to be discussed in expanded report) mentioned previously in this report. They are considered to be a vast improvement for the BS model.