Here is the report. Read it over and tell me what you think. Some of the formuals are missing and also charts. I could not get them into here. Anyway, many times when there is an / it should be a sqaure root of. Refer to the problem to see when to make / a sqaure root. If you would like a hard copy with the formuals in it, e-mail me and perhaps we can arrange something. I am not sure how well it will come out on this post. (square = sqaure (it is mispelled I beleive)
Black and Scholes Option Evaluation Formula
by Uri Miller
Before understanding the Black and Scholes options evaluation formula, one must first understand Options themselves. For starters, an option is the right to buy a stock at a set price at a later point in time, usually being 3-9 months. Therebye, the person has the right, but not the obligation to purchase stock X in 3, 6 or 9 months (option contracts usually come in three month intervals). There are two types o options which may be purchased. A Call option and a Put option. The definition of a call option would be a contract which gives the purchaser the right, but not the obligation to purchase the underlying security (the stock) at a set price within a given time frame. On the contrary, the definition of a Put option would be a contract which give the purchaser the right, but not the obligation, to sell a security (stock) at a set price within a given time frame. The Put and Call option can be compared to buying a stock and short selling a stock. When a person buys a stock, he buys low and hopes to make a profit when the stock moves higher. However, when a person short sells a stock, he is selling high and hoping for the stock price to depreciate therebye giving him the opportunity to buy low and make a profit from the drop. When a person is buying a stock, he is said to be going long, when a person is selling a stock, he is said to be going short. One can compare a Call option to a long position and a Put option to a short position.
There are also three types of Call and put options. The first kind is called at - the money. In other words, the strike price (the price which one has the right to buy or sell a stock at a later point in time) is equal to the current price. The second type of option is called out - of the money. The strike price on this is less than the price of the underlying security. Therefore, the premium (which will be defined shortly) must rise x amount until the call has any value at al, i.e. Strike price = $50, Stock price = $55, Premium = $1, the premium must rise $4 and only at that point will the option be at - the money. The third type of option is called in - the money. In this case, the strike price is already above the stock price, i.e.. Strike price = $60, Stock price = $55.
One of the most integral parts of an option is it's premium. A premium can be compared to the down-payment one pay's when buying a house. He pay's a minimal amount up front toward the purchasing of the house. So too, with an option, the premium is the price an investor pay's the writer of an option for the right to purchase the stock at a later point in time. In other words, the premium is the the price one pay's for the option. The price of the premium varies based on the price of the underlying security and the time it has left until expiration, commonly referred to as time value and intrinsic value. The time value of the option will gradually decrease as the expiration date approaches. Therefore, the following would hold true:
Option Price = Intrinsic Value + Time Value. Intrinsic Value is not measured only by the value of the security price but also by the value of the Exercise price (the price at which one has the right to buy or sell the stock). Time value is measured not only by the time until expiration but also by volatility and Interest rates. This will be elaborated upon at a later point. The expiration date is always the third Friday in every month. If one purchased a July Call option, the Option would expire on the third Friday in July. The premiums are what usually attract people to invest in options. Take the following example: A person has $10,000 in May and he wants to put it into IBM which is trading at $100. If he does this, he be able to buy 100 shares of IBM and will have the right to hold that stock for an unlimited amount of time or until IBM goes out of business. However, if the person instead bought 1 contract or 100 shares of a July Call option which, at the time had a premium of $10 the total cost of the option would be $1000 plus any brokerage fees which he may have incurred. The person now has $9,000 left over. He can place this money into a risk-free 6 month Treasury Bill which is paying %6. Assume now that the price of IBM went to $110 by the third week in July. The person would have made $1000 if he had bought 100 shares of stock. However, is the person had purchased the option, he would have made $1000 + the interest earned on the $9,000. In the later scenario the person would have made $1,090. Now assume that the price of IBM declined to $90. In the first case, the person would have lost $1000 but would still have the right to hold the stock until a later date hoping the stock will appreciate in value. In the second case, the person would have also lost $1000 and the the option would expire worthless and the person would have made $90 from the Treasury Bill. One can see the positives and the negatives of options. Many people have heard of people losing their shirts in options. This is mainly due to greed. Instead of the person taking the $10,000 and placing $9,000 into a risk-free Treasury Bill, he instead places the entire sum in the option which has a premium of $10 and purchases 10 contracts or 1,000 shares. Of course, if the stock price of IBM goes up to $110 the person will make a hefty $10,000. On the other hand, if the stock price of IBM drops to $90 he will lose the entire sum of his investment, in this case being $10,000.
The risks of options can vary based on different strategies, i.e.. buying a Put option and a call option at the same premium with identical expiration dates, this is called a straddle. Options can also be purchased as insurance for stocks which a person already owns, protecting him from a fall in the stock price. There are many other Option strategies but that is for another report.
Now that the basics of options have been explained, one now has the basic tools to begin evaluating the rather complex Black and Scholes formula.
The basic formula is as follows:
C= SN(d1) - Ke N(d2)
rt
Where
C = Theoretical Call Premium
S = Current Stock price
t = Time until option expiration
K = Option striking price
r = risk-free interest rate
N = Cumulative standard normal distribution
In order to understand the model itself, one divides it into two parts. The first part being SN(d1) which derives the expected benefit from acquiring a stock outright. This is found by multiplying the stock price (S) by the change in the call premium with respect to a change in the underlying stock price [N(d1)].
(-rt)
The second part of the model, Ke N(d2), gives the present value of paying the exercise price on the expiration day, e is a constant number equal to approximately 2.71828.
The fair market value of the call option is then calculated by taking the difference between the two parts.
This problem cannot be solved without properly defining (d1) and (d2) since they are currently only unknowns in the model. d1 Can be illustrated as follows:
2 t
d1 = ln(S/K) + (r + o~ /2)
o~_/ t
Where:
o~ = Standard deviation of stock returns
ln = Natural Logarithm
All other unknowns were previously defined and will be stated below anyway.
The standard deviation is a formula (which is rather lengthy to calculate) in itself which will soon be explained.
The natural logorithm uses the number e as it's base, as opposed to the common logorithm function that uses the number 10 as it's base.
Essentially what is stated in this formula is:
2 t
d1 = natural logorithm (Current stock price/Strike price) + (risk-free interest rate + o~ /2)
o~ X the square root of the time left until expiration
Standard Deviation:
Standard deviation is a measure of volatility. One is evaluating the standard deviation because it is important to know if, historically, the price of the stock has been stable or if it has been very volatile. If a stock has been rather stable, it's deviation will be a small number since it's volatility is not great. However, if the stock frequently tends to jump up and down in price it is very volatile and will therefore it will have a larger deviation number. Of course, it is good to find a stock with a high volatility. If one buy's an call option with a strike price of $70 and the stock price is $60 it is more likely he will make money if the stock tends to frequently jump 10 points and then drop 10 points then if the stock pattern is more like the following - 60, 60.5, 60.75, 60.5, 61.5, 60.25 etc... Therefore, one sees that an option on a stock which has a high Standard Deviation, is worth more than a stock with a lower Standard Deviation since it is more volatile. Another item which standard deviation measures is probability. Actually, standard deviation is really a measure of probability and volatility where the probability is dependent upon achieving the volatility. Of course, one is measuring the volatility of the stock price not the option's premium since one is trying to derive the volatility of the stock price to figure out the future worth of the option and the option premium is based on the stock price.
Formula for Standard Deviation:
A simple example of the above would be to use test grades as the data points. For example, say that a student received the following test grades:
80 - 88.43 = -8.43 (squared) = 71.06 +
85 - 88.43 = -3.43 (squared) = 11.76 +
79 - 88.43 = -9.43 (squared) = 88.92 +
90 - 88.43 = 1.57 (squared) = 2.46 +
95 - 88.43 = 6.57 (squared) = 43.16 +
98 - 88.43 = 9.57 (squared) = 91.58 +
92 - 88.43 = 3.57 (squared) = 12.74
Standard Deviation is really the square root of the Variance. The Variance = the average of the squared distance each score is from the mean. One would now subtract the mean (average) from each of the test scores and square the results. However, it is first necessary to find the mean. One does this by adding up all of the data points and dividing by the number of data points. In this case the following is true:
80 + 85 + 79 + 90 + 95 + 98 + 92 = 88.43
7
After this is done, the resultant numbers are added up since we have a sigma (or summation) which tells us to add up the total number of data points n. The number which one gets is 321.68. Now it is necessary to find the square root of 321.68 / 6 or 321.68 / n - 1. The result is 7.32. Therefore, the Standard Deviation is 7.32.
A second formula is given which is supposed to reduce the amount of handwork (yeah right!):
This formula tells us to multiply 1 / data points -1 by the sum of the data points squared minus the sum of the data points squared / data points.
A simple example of this would be to use the following numbers.
2
Data points Data points
4 16
2 4
3 9
3 9
6 36
3 9
21 83
So:
2
83 - (21) / 6 = 83 - 73.5 = 9.5 = 1.9
5 5 5
If one were to use the same numbers for the first problem in the second problem, the same result would be obtained.
Following is the stock price of Microsoft for three months. Each price is another data point.
Microsoft Stock price 09/20/96 - 12/20/96
Date Price Price Squared
960920, 69.0625, 4769.628906
960923, 68.8750, 4743.765625
960924, 68.4375, 4683.691406
960925, 67.8125, 4598.535156
960926, 66.0000, 4356
960927, 67.1875, 4515.160156
960930, 65.9375, 4347.753906
961001, 66.0625, 4364.253906
961002, 67.3750, 4539.390625
961003, 67.0000, 4489
961004, 68.1875, 4649.53156
961007, 68.7500, 4762.5625
961008, 67.6875, 4581.597656
961009, 67.2500, 4522.5625
961010, 66.8750, 4472.265625
961011, 68.5625, 4700.816406
961014, 68.3125, 4666.597656
961015, 69.4375, 4152.191406
961016, 69.0000, 4761
961017, 67.7500, 4590.0625
961018, 67.5000, 4556.25
961021, 67.0000, 4489
961022, 66.2500, 4389.0625
961023, 67.2500, 4522.5625
961024, 68.3125, 4666.597656
961025, 68.2188, 4653.804673
961028, 68.3125, 4666.597656
961029, 67.6875, 4581.597656
961030, 68.1250, 4641.015625
961031, 68.6250, 4709.390625
961101, 68.6875, 4717.972656
961104, 69.0000, 4761
961105, 70.7500, 5005.5625
961106, 72.2500, 5220.0625
961107, 71.7500, 5148.0625
961108, 71.7500, 5148.0625
961111, 71.8125, 6211.410156
961112, 70.8750, 5023.265625
961113, 72.5000, 5256.25
961114, 74.8125, 5596.910156
961115, 74.5000, 5550.25
961118, 75.1875, 5653.160156
961119, 77.9375, 6074.253906
961120, 76.6250, 5871.390625
961121, 75.1875, 5653.160156
961122, 75.2500, 5662.5625
961125, 76.7500, 5890.5625
961126, 76.8750, 5909.765625
961127, 77.7500, 6045.0625
961129, 78.4375, 6152.441406
961202, 78.8750, 6221.265625
961203, 77.3438, 5982.063398
961204, 76.6250, 5871.390625
961205, 76.5000, 5852.25
961206, 76.4375, 5842.691406
961209, 81.7500, 6683.0625
961210, 81.8750, 6703.515625
961211, 83.3750, 6951.390625
961212, 81.1250, 6581.265625
961213, 80.0000, 6400
961216, 76.7500, 5890.5625
961217, 79.8750, 6380.015625
961218, 82.6250, 6826.890625
961219, 84.8750, 7203.765625
961220, 83.6250, 6993.140625
Total = 65 Day's, 4713.1876, 344076.6878
2 2
o~ = 344076.6878 - (4713.1876) / 65 = 344076.6878 - 341755.9593 = 2320.7285 = 36.26138281
64 64 64
o~ = /36.26138281 = 6.021742506
Now that the standard deviation is known it is rather simple to solve d1. But before doing so, the rest of the problem will be explained.
The current stock price of Microsoft is 83.625 (83 5/8) as noted by the last data point entry. However, the information that has not yet been given is the option striking price (K), d2 and the risk free interest rate (r). The two other items which are not yet know are the time left until expiration and the Cumulative Standard Normal Distribution. The first three are easy to interpret but the last one is a bit difficult. The option striking price is 80 since an April 80 Call option will be used. The risk free interest rate will be from the 90 day Treasury Bill which is 5.03% (in the formula, this % will be represented as .0503). The time left until expiration is five months or .256916996, or 65 business day's / 253 (the amount of business day's in a year).
2 .256916996
d1 = ln (83.625/80) + (.0503 + 6.021742506 /2)
6.021742506/.256916996
2 .256916996
d2 = ln (83.625/80) + (.0503 - 6.021742506 /2)
6.021742506/.256916996
Cumulative standard distribution is a rather interesting concept developed by Abraham DeMoivre. The heights of people are "normally" distributed and the percentage of the population within any prescribed limits of height are known an dcan be read from a graph of a normal curve, or from tables giving the ordinates of the curve and areas under the curve, which are printed in statistics text-books and statistical tables. The weights and the heights of people are distributed in pretty much the same way, so are the sizes of animals of the various species and so are plants. The Normal Curve is not applicable to only one field of research, it is applicable to many. A good example of Normal Distribution is the following. Consider for a moment, a batch of electric light bulbs made in the same factory to the same specifications. These light bulbs will not burn out at the same time. There will be a mean (average) life. The majority of the bulbs would be expected to burn out at about the mean, but a few light bulbs will burn out some time before this, and a few light bulbs will burn out some time after this. But, what does "some time after" and "some time before" mean? This kind of language is adequate for the average customer who buys a light bulb occasionally and months, or years later has to replace it. The manufacturer however needs to be able to determine the quality of his products accurately, since he knows that in the long tun it will be the reliability and consistent performance of the products which he markets, in combination with the price of the products, which will bring him ultimate success. This success can only be achieved with precise knowledge of the behavior of what he produces.
A second example: If the time shown by one hundred clocks chosen at random in a specific town, were recorded at the exact tick of six o'clock by the B.B.C time signal, then one would expect some of these clocks to be fast and some to be slow, but the distribution of their errors to be a "normal distribution. However, if this were not so, one would expect there to be something unusual about the sample of the chosen hundred. In this case, it is possible that a number of the clocks might be electric clocks which are tied to the master clock at the power station and they all share a particular error deriving from a single cause. If one then analyzed the distribution of errors he sould find a double population, one of independent mechanisms with errors independently derived, and the other which has a consistent error not normally distributed.
A third example would be: If a batch of watches is set to exactly the same time at noon on a certain Monday, their times can be compared a week later at noon and the following chart would hold true for the distribution of watch errors:
This chart is what abnormal distribution looks likes on a bar chart. If one were to outline the bars, he would obtain the normal distribution curve, or what is know as the bell curve (since it is shaped like a bell. The following is a chart of the normal bell curve:
One now should understand how stocks are normally distributed. They have no perfect price. There is always a normal error in the return of stock prices.
Before completely understanding this model, one has to know the assumptions of it.
1) No commissions are charged - Usually, when buying or selling a stock or option one has to pay a commission. The fee the individual trader pay's can sometimes be rather large and therefore distort the outcome of the model (this can be avoided by trading with a discount broker where commissions are substantially less).
2) Interest rates remain constant and known - The Black and Scholes model uses the risk-free rate, to represent this constant and known rate. In reality, there is no risk-free rate, but the discount rate on U.S Government Treasury Bills with 30 day's left until maturation is usually used to represent it (in this report the 90 Treasury Bill was used since it was dealing with an approximate four month time frame and the 90 day T-Bill was the closest rate to match that time frame). During times of rapidly changing interest rates, these 30 day rates are often subject to change and therefore violating one of the assumptions of the model.
3) Returns are lognormally distributed - This assumption relates that returns on the underlying stock are normally distributed, which is reasonable for most assets that offer options.
4) The stock pay's no dividends during the option's life - Most companies pay dividends to their shareholders, so this limitation might seem to some as a serious problem in the model. Since higher dividend yields would produce a lower premium (holding the stock outright would allow one the rights to any dividend being payed while owning that stock). The common way of adjusting the model for this situation is to subtract the discounted value of a future dividend from the stock price. The method would be explained as follows. If a stock at $100 were paying a $3.30 or 3.3% dividened every year, then it would probobly be paying .825 or .825% per qaurter. By reinvesting the dividends, one share of stock would grow to 1.0334 shares in a year. By using the standard interest formula, the present value of one share of stock in one year minus dividends is ($100 / 1.0334) which = $96.76795045. The stock that was used in this report does not pay a dividend.
5) European exercise terms are used - European exercise terms state that the option can only be exercised on the expiration date. American exercise rules allow the option to be exercised at any time during the life of an option, making American options more valuable given their greater flexibility. This problem is not a major concern because very few calls are ever exercised before the last few day's of their life. This is true because when one exercise a call early, he forfeit's the remaining time value on the call and collects the intrinsic value. Approaching the end of the life of a call, the remaining time value is very small, but the intrinsic value is the same.
6) Markets are efficient - This assumption states people cannot consistently predict the direction of the market or an individual stock.
This is a series of tests conducted on four option calculators. The same information was plugged into each one of them, and different data came out. One will notice that the numbers vary slightly from the numbers which were calculated above. The numbers below also list the delta. gamma, theta and for one, the vega.
Delta is the movement in price an option will experience if the underlying stock price rises by $1. For example, if the delta is .5, a $1 increase in stock price will lead to a 50 cent increase in the option price.
Gamma is the movement in delta when the stock price rises by a certain amount. For example, if the gamma is -.3, a $1 increase in stock price will lead to a .3 decrease in delta.
Theta is the movement in price that an option will experience in 1 year. For example, if theta is -40, the option value is declining at the rate of $40 per year.
Vega the movement in price that an option will experience as standard deviation rises by 100%. For example, if vega is 5, the option value will rise $5 as the standard deviation rises 100%.
Test #1
Fair Value
$4.33
Delta
98.35
Gamma
1.97
Vega
1.43
Theta
1.14
Test #2
Value = 4.6814, Delta = 0.9707, Gamma = 0.0262, Theta =
-0.0115
Test #3
THEORETICAL VALUE CALCULATIONS
Option Value:
4.657
Delta:
0.970
Theta:
-4.090
Rho1:
19.643
Test #4
Option value: 73.30708730590568
Delta: .9388114647506788
Gamma: 0.00047391975460769117
Theta: -60.35012187623618
As one can see, the outputs from the different calculators were different. This is understandable though, since different calculators might be using different interest rates and/or a different amount of time to obtain the Standard Deviation.
The full Black and Scholes stock option evaluation formula looks like this:
2 t 2 t
C = S N [ ln (S/K) + (r + o~ /2) - Ke N [ ln (S/K) + (r - o~ /2) o~ /t rt o~ /t
Or:
2 .256916996 2 .256916996
C = 83.625 N [ ln (83.625/80) + (.0503 + 6.021742506 /2) - 83.625 N [ln (83.625/80) + (.0503 - 6.021742506 /2)
6.021742506/.256916996 (.0503)(.256916996) 6.021742506/.256916996
SO:
C = 83.625 N [.044315883 + (2.106770168)] - 78.97281817 N [.044315883 + (2.054971148)]
6.021742506/.256916996 6.021742506/.256916996
SO:
C= 83.625 N [2.151086051] - 78.97281817 N [2.099287031]
6.021742506/.256916996 6.021742506/.256916996
SO:
C = 83.625 N [2.151086051] - 78.97281817 [2.099287031]
3.052239431 3.052239431
SO:
C = 83.625 N [.704756654] - 78.97281817 N(.68778583)
SO:
Now one looks up the Standard Normal Distribution on the chart and finds that
d1 = .7703
and
d2 = .7517
SO:
C = 83.625 (.7703) - 78.97281817 (.7517)
SO:
C = 64.4163375 - 59.36386742
C = $5.05247008
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