In 1973, Fischer Black and Myron Scholes had completed the first draft of the paper that had the outline of the model meant to determine the fair market value of European-type options on assets that had no pay outs. They presented this work to the Journal of Political Economy for publication. They got an immediate response - a rejection. They sent it to another journal, the Review of Economics and Statistics. They got a quick response - a rejection. Despite this rejection, the two, believing in the value of their work, went ahead to revise the draft based on comments from Merton Miller (a Nobel Laureate from the University of Chicago) and University of Chicago s Eugene Fama. The draft was then submitted once again to the Journal of Political Economy. It was accepted and this was the birth of, not only a formula that would be widely accepted in finance, but one that would be fascinating to other scholars in finance who would want to dent this formula, defend it or critique its authors, Fischer Black and Myron Scholes. This paper is limited to the following a) the model itself, b) its merits and demerits and c) a critique of the MBS.
An asset is held by its owner or owners with an expectation of some future benefit. Other than the benefit, the owner has rights to exercise over the asset, for instance, if and when to dispose of them and the benefits that will go with such an action. The benefits from the rights will depend on the value of the asset at the time such right is exercised but they may not affect the intrinsic value of the asset. These rights are called derivatives, and they are mainly options and futures contracts (Bodie et al. 54). Options are of two types call options and put options. A call option gives the holder of this right the right to purchase an asset at a specified price, called the strike price, on or before a specified expiration date (Bodie et al. 54). A put option gives the holder of this option the right to sell an asset at a given strike price on or before a specific expiration date. The benefit from exercising these rights depend on the market price of the asset when the put or call option are enforced. A futures contract on the other hand, is an agreement that calls for the delivery of an asset or its cash value (in some cases) after negotiations on a future date and price. The futures price is enforced irrespective of the current market price of such an asset.
The Black Scholes Model (BSM)
This model is the work of three financial experts, Black and Scholes (Black and Scholes 1973) and Merton (Merton 1973) and was reached at after a long search for a method of option valuation. Scholes and Merton won the 1997 Nobel Prize for this accomplishment. Fisher Black died in 1995 and was therefore not part of the Nobel party. The formula is as below (Bodie et al. 759)
The formula looks intimidating but it can be worked out using available computer software.
Using the model
Assume that the following information is available the price of the stock is 100, the exercise or the strike price K is 83, the rate of interest R, is 0.1, i.e., 10, time to expiration T is 9 months (or 0.75 of a year), the standard deviation, s, is 0.5. The task is to calculate the value of the call option.
It requires that d1 and d2 are calculated first.
d1 in (10083) (0.1 0.52 2) 0.75 0.5 X0.75
ln1.204 0.225 x 0.75 0.433
0.1856 0.16850.433
0.818
d2 0.818 0.5 (0.75)
0.818- 0.433 0.385
After calculating d1 and d2, it is now possible to get the values of N (d1) and N (d2) using the NORMSDIST function in Excel.
N (d1) 0.7933
N (d2) 0.650
C 100 X 0.7933 (83 X 2.718- 0.1 X 0.75) 0.650
79.33 0.9278 X 0.650 78.73
The strengths of the model
There have been various empirical studies to test the BSM. Renowned researchers like Robert E. Whaley in 1982 made a report of his study of the BSM and made his report in the Journal of Financial Economics (Bodie et al. 778). His findings show that the BSM performed well when it came to the estimation of the prices of options that did not allow for earlier exercise. BSM also performed better when pricing options that had low or no dividend payouts. The model has a superior measure of sensitivity as gauged by various measures like Delta which determines how sensitive the calculated option value is to changes in the share price, Gamma which is a measure of how sensitive Delta is to any small change in the share price. Theta measures how sensitive the value of the option is to small changes in time until maturity. Vega measures the level of sensitivity of the value of an option to any small changes in volatility. The main strength of the BSM lies in its superiority to the other methods used in option valuation and its ability to have stood up to critics since its publication in 1973.
Weaknesses of the BSM
Not being able to break the formidability of the BSM itself, most of those who pointed out its weaknesses turned on to the assumptions of the model though Whaley in 1982 and Mark Rubinstein in 1994 had strong comments on the BSM. Whaley deduced that the BSM seems to perform worse for options on stocks with high dividend payouts (Bodie et al. 7780). Rubinstein in 1994 pointed out that the BSM has deteriorated in recent years in the sense that options on the same stock with the same strike price that should have the same implied volatility actually exhibit progressively different implied volatilities (as calculated by the BSM).
The following weaknesses have been claimed against the BSM
The assumption that stock pays no dividend is not realistic. And as seen above, the study by Whaley pointed out the effect of dividends on the BSM. This weakness is controlled by subtracting the discounted value of future dividends from the stock price.
The assumption of European style exercise terms again is unrealistic. This though is a weak criticism as even the American options are not exercised during their lifetime despite holders having the freedom to do so.
The assumption of the efficiency of the market is one of the assumptions of the Efficient Market Hypothesis which assumes that (a) stocks are always in equilibrium and that (b) it is impossible for an investor to consistently beat the market (Brigham Houston 334. In real life this is not consistently true.
BSM assumes that there are no transaction costs, that no commissions are charged. This is not true because apart from taxation, commissions are charged on the buyers and sellers of options on the stock market.
BSM assumes constant interest rate which is equated to the 30 day T-Bill rate left to maturity. This is the risk free rate. The truth on the ground is that there is no risk free rate as the rate of interest continuously keeps varying irrespective of time length in question.
This model also assumes that the returns of the underlying assets are lognormally distributed. A lognormal distribution has the following features
a longer right tail compared with a normal (bell-shaped) distribution, but a short fat left tail
it assumes non-negative prices that extend to infinity
it has an upward bias which means that stock price can only drop 100 but can rise by more than 100
This is not always the case as asset price distributions usually depart from the lognormal distribution
It is not perfectly accurate. It still has error in estimation.
Other option valuation models include (which are not explained in this paper) the following
Binomial option valuation model
Consider a two-state option pricing situation so that the current price of the stock on question S0 is 100. Consider also that the price of the stock can either go up (u) or go down (d) by factors of 2 and 0.5, respectively. Then the prices of the stock will be 2x100 200 (i.e., uS0 ) or 0.5x100 50 (i.e., dS0 ). If the strike price is 125, then the values of the option will be 75 (i.e., 200-125 if the price goes up) or 0 if the price falls to 50.
Suppose there are two periods through which this scenario can take place, then the table below gives the possible outcomes and their option values.
udu100 x 2x2 400 ( up and up)100x2x0.5 100 (up then d)d100x0.5x2 100 (d then up)100x 0.5x0.5 25 ( d then d)
The corresponding option values will be as follows (at the same strike price of 125).
udu400 125 2750d00
The value at of the option at each part of table is discounted back to the present using the risk free rate and the corresponding probability.
This model has the same assumptions as the BSM for European type options, i.e., the binomial model, which converges to the BSM. In the above example, the BSM model can be used to calculate the value of such an option by simply substituting the figures into the BSM. This is because this model eventually converges to the BSM.
A critique of the BSM and Conclusion
The BSM model did and is still receiving accolades in the financial realm. This is because (1) it is the currently superior option pricing model to the other models like the binomial and the butterfly models, (2) much of the empirical evidence still has not dealt a serious doubt to the BMS, and (3) no new method has been discovered that can displace the BSM.
This though does not mean that it has not received its brand of criticism from some of its noted critiques like Espen Gaarder Hug and Nazism Nicholas Taleb in their article Why We Have Never Used the Black-
Scholes-Merton Option Pricing Formula (1) published in February 2009. In this article, the two launched a criticism on the BSM and its authors. They lay claim that
Black, Scholes and Merton did not invent any formula, but just found an argument that publicized a well-known formula, making it compatible with the economic field by simply eliminating risk through hedging.
After 1973, option traders did not change from their non-BSM trading methods to the newly Nobel-crowned method, yet they still met their bottom line. They still used the earlier methods notably the formula by Louis Bachelier and Edward O Thorpe.
These two went on to say that it is a myth that traders rely on theories to price options. They did not stop here. On the same page, they asserted
It is assumed that the Black-Scholes-Merton theory is what made it possible for option traders to calculate their delta hedge (against the underlying) and to price options. This argument is highly debatable, both historically and analytically (3).
The harshest criticism (this is debatable) comes from Nial Ferguson in his book The Ascent of Money A Financial History of the World. It explicitly discusses
the model. Writers went on to express the dream-like event where two of academia s hottest quants are coming together with the ex-Salomon (name of a company) superstar plus a former Federal Reserve vice-chairman, David Mullins, another ex-Harvard professor, Eric Rosenfeld, and a bevy of ex-Salomon traders, i.e., Victor Haghani, Larry Hilibrand and Hans Hufschmid (320-332). This was actually a constellation of finance stars. As a result, the Long Term Capital Management (LTCM) attracted to its fund clients that were mainly big banks. Among them are the New York investments bank, Merrill Lynch and the Swiss private bank, Julius Baer. Another bank joined them later, it was another Swiss bank called UBS. The minimum investment for each of the investors was 10 million.
The LTCM spectacularly collapsed in 1998, about 10 months after Merton and Scholes had received their Nobel Prize in October 1997, and about five years after its inception. It was caused by a Russian credit default (Triana Taleb 7). One of the basic conclusions of this spectacular collapse is the strong argument against the Efficient Market Hypothesis (EMH), one of the strongholds of the BSM. The EMH assumes that security prices fully reflect all available information (Kevin 123) and that risk and return are normally distributed in the Gaussian sense. In addition to these, there is the assumption that the market is rational. These assumptions have been refuted by many finance experts who have been challenging the BSM and the choice of Merton and Scholes as Nobel laureates.
In the review of Fooled by Randomness The Hidden Role of Chance in the Markets and in Life by Taleb, Rzepczynski (Rzepczynski 102) agrees that the assumptions of rationality and normality do not hold as there are events that occur which the Gaussian normal random distribution assumes will never occur because they have never been captured by the probability philosophy. These rare events are called by Taleb as the Black Swans (Rzepczynski 102). Normality in the BSM is captured by N.
It is noteworthy though that despite the criticism, there has not come up a better method than the BSM, thus making the method robust and standing the test of time since 1973.
The advent of behavioral finance is slowly shifting financial philosophy from the scientific outlook towards a behavioral realm. In 2005, Daniel Kahneman won the Nobel Prize in Economics. Kahneman is a professor in psychology and was rewarded for applying psychological principles in finance. In one of the papers he co-authored with Amos Tversky in 1979, they provided a behavioral approach to the analysis of human response to risk and return. They diverged from the mathematical rational approach and demonstrated that there is no rational approach to risk and return.
The BSM borrowed a lot from the hard sciences (Physics) to build its assertions. The Brownian motion was even used at one time to explain financial principles. Kahneman and Tverski explored how biases like fear, overconfidence, herding despair, and anchoring can affect human judgment and therefore affecting how they react to risk and return in finance. The decisions had nothing to do with the normal Gaussian bell curve and neither had anything to do with expected (mean) returns. It was more of emotion and luck. This therefore questions the effectiveness of the beautiful mathematical equations based on statistics and probability BSM being one of such beautiful models. The collapse of the LTCM proved this point.
An asset is held by its owner or owners with an expectation of some future benefit. Other than the benefit, the owner has rights to exercise over the asset, for instance, if and when to dispose of them and the benefits that will go with such an action. The benefits from the rights will depend on the value of the asset at the time such right is exercised but they may not affect the intrinsic value of the asset. These rights are called derivatives, and they are mainly options and futures contracts (Bodie et al. 54). Options are of two types call options and put options. A call option gives the holder of this right the right to purchase an asset at a specified price, called the strike price, on or before a specified expiration date (Bodie et al. 54). A put option gives the holder of this option the right to sell an asset at a given strike price on or before a specific expiration date. The benefit from exercising these rights depend on the market price of the asset when the put or call option are enforced. A futures contract on the other hand, is an agreement that calls for the delivery of an asset or its cash value (in some cases) after negotiations on a future date and price. The futures price is enforced irrespective of the current market price of such an asset.
The Black Scholes Model (BSM)
This model is the work of three financial experts, Black and Scholes (Black and Scholes 1973) and Merton (Merton 1973) and was reached at after a long search for a method of option valuation. Scholes and Merton won the 1997 Nobel Prize for this accomplishment. Fisher Black died in 1995 and was therefore not part of the Nobel party. The formula is as below (Bodie et al. 759)
The formula looks intimidating but it can be worked out using available computer software.
Using the model
Assume that the following information is available the price of the stock is 100, the exercise or the strike price K is 83, the rate of interest R, is 0.1, i.e., 10, time to expiration T is 9 months (or 0.75 of a year), the standard deviation, s, is 0.5. The task is to calculate the value of the call option.
It requires that d1 and d2 are calculated first.
d1 in (10083) (0.1 0.52 2) 0.75 0.5 X0.75
ln1.204 0.225 x 0.75 0.433
0.1856 0.16850.433
0.818
d2 0.818 0.5 (0.75)
0.818- 0.433 0.385
After calculating d1 and d2, it is now possible to get the values of N (d1) and N (d2) using the NORMSDIST function in Excel.
N (d1) 0.7933
N (d2) 0.650
C 100 X 0.7933 (83 X 2.718- 0.1 X 0.75) 0.650
79.33 0.9278 X 0.650 78.73
The strengths of the model
There have been various empirical studies to test the BSM. Renowned researchers like Robert E. Whaley in 1982 made a report of his study of the BSM and made his report in the Journal of Financial Economics (Bodie et al. 778). His findings show that the BSM performed well when it came to the estimation of the prices of options that did not allow for earlier exercise. BSM also performed better when pricing options that had low or no dividend payouts. The model has a superior measure of sensitivity as gauged by various measures like Delta which determines how sensitive the calculated option value is to changes in the share price, Gamma which is a measure of how sensitive Delta is to any small change in the share price. Theta measures how sensitive the value of the option is to small changes in time until maturity. Vega measures the level of sensitivity of the value of an option to any small changes in volatility. The main strength of the BSM lies in its superiority to the other methods used in option valuation and its ability to have stood up to critics since its publication in 1973.
Weaknesses of the BSM
Not being able to break the formidability of the BSM itself, most of those who pointed out its weaknesses turned on to the assumptions of the model though Whaley in 1982 and Mark Rubinstein in 1994 had strong comments on the BSM. Whaley deduced that the BSM seems to perform worse for options on stocks with high dividend payouts (Bodie et al. 7780). Rubinstein in 1994 pointed out that the BSM has deteriorated in recent years in the sense that options on the same stock with the same strike price that should have the same implied volatility actually exhibit progressively different implied volatilities (as calculated by the BSM).
The following weaknesses have been claimed against the BSM
The assumption that stock pays no dividend is not realistic. And as seen above, the study by Whaley pointed out the effect of dividends on the BSM. This weakness is controlled by subtracting the discounted value of future dividends from the stock price.
The assumption of European style exercise terms again is unrealistic. This though is a weak criticism as even the American options are not exercised during their lifetime despite holders having the freedom to do so.
The assumption of the efficiency of the market is one of the assumptions of the Efficient Market Hypothesis which assumes that (a) stocks are always in equilibrium and that (b) it is impossible for an investor to consistently beat the market (Brigham Houston 334. In real life this is not consistently true.
BSM assumes that there are no transaction costs, that no commissions are charged. This is not true because apart from taxation, commissions are charged on the buyers and sellers of options on the stock market.
BSM assumes constant interest rate which is equated to the 30 day T-Bill rate left to maturity. This is the risk free rate. The truth on the ground is that there is no risk free rate as the rate of interest continuously keeps varying irrespective of time length in question.
This model also assumes that the returns of the underlying assets are lognormally distributed. A lognormal distribution has the following features
a longer right tail compared with a normal (bell-shaped) distribution, but a short fat left tail
it assumes non-negative prices that extend to infinity
it has an upward bias which means that stock price can only drop 100 but can rise by more than 100
This is not always the case as asset price distributions usually depart from the lognormal distribution
It is not perfectly accurate. It still has error in estimation.
Other option valuation models include (which are not explained in this paper) the following
Binomial option valuation model
Consider a two-state option pricing situation so that the current price of the stock on question S0 is 100. Consider also that the price of the stock can either go up (u) or go down (d) by factors of 2 and 0.5, respectively. Then the prices of the stock will be 2x100 200 (i.e., uS0 ) or 0.5x100 50 (i.e., dS0 ). If the strike price is 125, then the values of the option will be 75 (i.e., 200-125 if the price goes up) or 0 if the price falls to 50.
Suppose there are two periods through which this scenario can take place, then the table below gives the possible outcomes and their option values.
udu100 x 2x2 400 ( up and up)100x2x0.5 100 (up then d)d100x0.5x2 100 (d then up)100x 0.5x0.5 25 ( d then d)
The corresponding option values will be as follows (at the same strike price of 125).
udu400 125 2750d00
The value at of the option at each part of table is discounted back to the present using the risk free rate and the corresponding probability.
This model has the same assumptions as the BSM for European type options, i.e., the binomial model, which converges to the BSM. In the above example, the BSM model can be used to calculate the value of such an option by simply substituting the figures into the BSM. This is because this model eventually converges to the BSM.
A critique of the BSM and Conclusion
The BSM model did and is still receiving accolades in the financial realm. This is because (1) it is the currently superior option pricing model to the other models like the binomial and the butterfly models, (2) much of the empirical evidence still has not dealt a serious doubt to the BMS, and (3) no new method has been discovered that can displace the BSM.
This though does not mean that it has not received its brand of criticism from some of its noted critiques like Espen Gaarder Hug and Nazism Nicholas Taleb in their article Why We Have Never Used the Black-
Scholes-Merton Option Pricing Formula (1) published in February 2009. In this article, the two launched a criticism on the BSM and its authors. They lay claim that
Black, Scholes and Merton did not invent any formula, but just found an argument that publicized a well-known formula, making it compatible with the economic field by simply eliminating risk through hedging.
After 1973, option traders did not change from their non-BSM trading methods to the newly Nobel-crowned method, yet they still met their bottom line. They still used the earlier methods notably the formula by Louis Bachelier and Edward O Thorpe.
These two went on to say that it is a myth that traders rely on theories to price options. They did not stop here. On the same page, they asserted
It is assumed that the Black-Scholes-Merton theory is what made it possible for option traders to calculate their delta hedge (against the underlying) and to price options. This argument is highly debatable, both historically and analytically (3).
The harshest criticism (this is debatable) comes from Nial Ferguson in his book The Ascent of Money A Financial History of the World. It explicitly discusses
the model. Writers went on to express the dream-like event where two of academia s hottest quants are coming together with the ex-Salomon (name of a company) superstar plus a former Federal Reserve vice-chairman, David Mullins, another ex-Harvard professor, Eric Rosenfeld, and a bevy of ex-Salomon traders, i.e., Victor Haghani, Larry Hilibrand and Hans Hufschmid (320-332). This was actually a constellation of finance stars. As a result, the Long Term Capital Management (LTCM) attracted to its fund clients that were mainly big banks. Among them are the New York investments bank, Merrill Lynch and the Swiss private bank, Julius Baer. Another bank joined them later, it was another Swiss bank called UBS. The minimum investment for each of the investors was 10 million.
The LTCM spectacularly collapsed in 1998, about 10 months after Merton and Scholes had received their Nobel Prize in October 1997, and about five years after its inception. It was caused by a Russian credit default (Triana Taleb 7). One of the basic conclusions of this spectacular collapse is the strong argument against the Efficient Market Hypothesis (EMH), one of the strongholds of the BSM. The EMH assumes that security prices fully reflect all available information (Kevin 123) and that risk and return are normally distributed in the Gaussian sense. In addition to these, there is the assumption that the market is rational. These assumptions have been refuted by many finance experts who have been challenging the BSM and the choice of Merton and Scholes as Nobel laureates.
In the review of Fooled by Randomness The Hidden Role of Chance in the Markets and in Life by Taleb, Rzepczynski (Rzepczynski 102) agrees that the assumptions of rationality and normality do not hold as there are events that occur which the Gaussian normal random distribution assumes will never occur because they have never been captured by the probability philosophy. These rare events are called by Taleb as the Black Swans (Rzepczynski 102). Normality in the BSM is captured by N.
It is noteworthy though that despite the criticism, there has not come up a better method than the BSM, thus making the method robust and standing the test of time since 1973.
The advent of behavioral finance is slowly shifting financial philosophy from the scientific outlook towards a behavioral realm. In 2005, Daniel Kahneman won the Nobel Prize in Economics. Kahneman is a professor in psychology and was rewarded for applying psychological principles in finance. In one of the papers he co-authored with Amos Tversky in 1979, they provided a behavioral approach to the analysis of human response to risk and return. They diverged from the mathematical rational approach and demonstrated that there is no rational approach to risk and return.
The BSM borrowed a lot from the hard sciences (Physics) to build its assertions. The Brownian motion was even used at one time to explain financial principles. Kahneman and Tverski explored how biases like fear, overconfidence, herding despair, and anchoring can affect human judgment and therefore affecting how they react to risk and return in finance. The decisions had nothing to do with the normal Gaussian bell curve and neither had anything to do with expected (mean) returns. It was more of emotion and luck. This therefore questions the effectiveness of the beautiful mathematical equations based on statistics and probability BSM being one of such beautiful models. The collapse of the LTCM proved this point.
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