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The Relationships Between Black-‐Scholes Model
And Financial Crash
James Moore
Economic Expert From Blair Academy
Pakapark Bhumiwat (Nik)
Blair Academy
Class of 2014
May 5, 2014
Abstract
Like each chemical element that had various properties depending on many
combinations of amalgamating protons, neutrons, and electrons, the market price
always fluctuated depending on such various factors as underlying price, strike
price, risk-‐free interest rate, market volatility, the length of time interval, and
dividend. Traders had to apply their intuition and their previous experience to
predict the future price until 1973, when Fischer Black and Myron Scholes
coordinately worked with a plethora of statistical data to generate the world-‐
shaking mathematical model, called Black-‐Scholes formula. Their ideal formula, a
main material helping predict the future call price under particular conditions,
completely blew out the entire financial world because many specific and idealistic
prerequisites for Black-‐Scholes formula never existed in the real financial market.
However, it resulted in the rapidly increasing number of unprofessional traders who
considered the entire financial market as just mathematical models that led to the
financial crash in 1987.
However, the main purpose of this analysis paper was for high school
students to have a broad understanding of the complicated version of Black-‐Scholes
Model.
Technical Terms
Arbitrageur An investor who tries to make a profit from the inefficient market by
making trades that counteract each other and earn the risk-‐free profit at the same
time
America Option An option that the trade can occur at any point during the life of
the contract
At-‐the-‐money An option the price of an underlying asset and the strike price are
equal
Call Price The price that the bond or stock can be redeemed
Derivative An investment product that derives its value from an underlying asset
European Option An option that the trade occurs at the expiration of the contract
Lognormal Distribution (Galton Distribution) A continuous probability
distribution of a random variable whose logarithm is normally distributed
In-‐the-‐money An option that the derivative would make money. A call option will
be in-‐the-‐money when the price of an underlying asset is higher than the strike
price
Strike Price The agree-‐upon price at which an option can be exercised
Option A financial derivative that gives the holder the right to either buy or sell a
fixed amount of financial assets at the agree-‐upon price
Out-‐of-‐the-‐money An option the derivative would not make money. A call option
will be out-‐of-‐the-‐money when the price of an underlying asset is lower than the
strike price
Underlying Price The price of the underlying asset
Background
The price in financial market fluctuates depending on various factors. In this
paper, the author focus only on six primary factors in the macroscopic scale,
including underlying price, strike price, the length of time interval, market volatility,
free-‐risk interest rate and dividends.1
1. Underlying price
The current price of the underlying asset is one of the most influential factors
that determines the price of any goods or services in financial market. Basically,
when the price of underlying asset increases, the call price will increase. Conversely,
when the price of underlying asset decreases, the call price will decrease.
2. Strike price
The strike price is usually used to determine that the option is in-‐the-‐money,
at-‐the-‐money, or out-‐of-‐the-‐money. A call option will be in-‐the-‐money when the
current price of the underlying asset is higher than the strike price; a call option will
be at-‐the-‐money when the current price of the underlying asset is equal to the strike
price; and a call option will be out-‐of-‐the-‐money when the current price of the
underlying asset is lower than the strike price.
3. The length of time interval
The length of time interval means a period of time from the present time to
the expiration date of the underlying asset. Generally, the longer the length of time
interval, the higher the call price in finacial market.
1 The information comes from Jean Folger. "Options Pricing: Introduction." Investopedia. N.p., n.d. Web. 04 Apr. 2014.
4. Market Volatility
Market volatility is a parameter to measure the price change regardless of
direction. Market volatility is used to predict the inclination of the future call price
in financial market. In terms of calculation, we assume that the market volatility is
equal to the variance of statistical data.
5. Free-‐risk interest rate
The free-‐risk interest rate is the actual interest rate applied in financial
institutions. Generally, the greater the interest rate, the higher the call price.
6. Dividends
Divident is the amount of any cash from the last investment. We can consider
that the more dividend we have, the less the price of underlying price and the call
price are.
Black-‐Scholes Model and Its Prerequisites
According to these six factors, Black and Scholes proposed a mathematical
model, called Black-‐Scholes formula, to predict the future call price under the
following certain conditions,
1. There is no dividend during the length of time interval
2. The risk-‐free interest rate and market volatility are constant
3. Options are European
4. The market is effiecient
5. It follows the Galton distribution (lognormal distribution)
The Black-‐Scholes formula is
𝐶!! 𝑘 = 𝑆!Φln 𝑆!
𝑘 + 𝑟 + 𝜎!
2 𝑇
𝑇 𝜎− 𝑘𝑒!!"Φ
ln 𝑆!𝑘 + 𝑟 − 𝜎
!
2 𝑇
𝑇 𝜎
when 𝐶!! 𝑘 is the call price of the European option for the length of time interval T
𝑆! is the current price of the underlying asset
𝑇 is the length of time interval
𝑘 is the option strike price
𝑟 is the free-‐risk interest rate
𝜎 is the standard deviation of statistical data
Φ(𝑦) is the cumulative standard lognormal distribution function
Φ 𝑦 = 𝑓!(𝑦!)𝑑𝑦!!!! = !
!1+ erf !
! where erf 𝑦 = !
!𝑒!!!!
! 𝑑𝑡
Before discussing the result of the Black-‐Scholes formula, the author will
comprehensively explain the mathematical reasoning behind this formula in the
simplest way predicated on The Intuitive Proof of Black-‐Scholes Formula Based on
Arbitrage and Properties of Lognormal Distribution by Alexei Krouglov. The author
will use original variables same as those in the reference.
Suppose that the bank account pays the constant continuous free-‐risk
interest rate equal to 𝑟. At time 𝑡 = 𝑡! the current price of the underlying asset is 𝑆!.
We want to determine the price of a share from 𝑡 = 𝑡! to 𝑡 = 𝑡! + 𝑇. Denote the call
price at 𝑡 = 𝑡! + 𝑇 is 𝐹!! and the underlying price at 𝑡 = 𝑡! + 𝑇 is equal to 𝑆!! . We
find the relationship that
𝑟𝑆 =𝑑𝑆𝑑𝑡
𝑟𝑑𝑡!!!!
!!=
1𝑆
!!!
!!𝑑𝑆
𝑟𝑇 = ln(𝑆!!
𝑆!)
𝑆!! = 𝑆!𝑒!"
According to the concept of arbitrage, we can assume that 𝐹!! will converge to
𝑆!𝑒!" because if 𝐹!! > 𝑆!! then the profit from borrowing money 𝑆! from bank at
𝑡 = 𝑡! to buy an underlying asset and sell it at 𝑡 = 𝑡! + 𝑇 is equal to 𝐹!! − 𝑆!! > 0. In
the same manner, if 𝐹!! < 𝑆!! , the profit from sell the underlying asset at price 𝑆!,
put money in bank at 𝑡 = 𝑡!, and then withdraw it at 𝑡 = 𝑡! + 𝑇 is equal to 𝑆!! −
𝐹!! > 0. Finally, 𝐹!! = 𝑆!𝑒!" .
Suppose that the variable 𝑥! , the series of infomation for the price of the
underlying asset, is continuously distributed in interval 0 < 𝑥! < ∞, variable 𝑥! is
defined as 𝑥! = ln(𝑥!), 𝑓!(𝑥!) is the probability density function in interval
0 < 𝑥! < ∞, 𝑓! 𝑥! is the probablity density function in interval −∞ < 𝑥! < ∞, 𝜇!
and 𝜇! is the mean value of variables 𝑥! and 𝑥! , respectively, and 𝜎!! is the variance
of variable 𝑥! . Now, we can assume that 𝜇! = 𝑆!𝑒!" , and, according to the
assumption that the market volatilty is directly proportionate to the length of time
interval 𝑇, the valiance of the underlying asset is 𝜎!! = 𝜎!𝑇.
The function for the standard lognormal distribution is
𝑓 𝑥 = !!" !!
𝑒!!!(!"!!!! )! , where 𝜇 = 𝑥𝑓(𝑥)𝑑𝑥, and the basic integral identity
𝑒!!!!!!! 𝑑𝑥 = 𝜋.
𝜇! = 𝑥!!
!𝑓 𝑥! 𝑑𝑥!
𝜇! = 𝑥!!
!(
1𝑥!𝜎! 2𝜋
𝑒!!!(!"!!!!!
!!)!)𝑑𝑥!
𝜇! = 1
𝜎! 2𝜋𝑒!
!!(!"!!!!!
!!)!
!
!𝑑𝑥!
Suppose that 𝑥! = 𝑒 !!!!!!! then 𝑑𝑥! = 2𝜎!𝑒 !!!!!!! and 𝑀 = !"!!!!!! !
.
Since 0 < 𝑥! < ∞
then −∞ < 𝑀 < ∞
Thus,
𝜇! = 1
𝜎! 2𝜋𝑒!!!
!
!!2𝜎!𝑒 !!!!!!! 𝑑𝑀
𝜇! = 1𝜋
𝑒!(!!! !!!!!!!)!
!!𝑑𝑀
𝜇! = 1𝜋
𝑒! !!!!
!
!!!!
!
! !!!!
!!𝑑(𝑀 −
𝜎!2)
𝜇! = 1𝜋𝑒!!!
! !!! 𝑒! !!!!
!
!!
!!𝑑(𝑀 −
𝜎!2)
𝜇! = 1𝜋𝑒!!!
! !!! 𝜋
𝜇! = 𝑒!!!
! !!!
Considering variable 𝑥! in interval 𝑘 < 𝑥! < ∞ where the strike price 𝑘 > 0
Denote 𝜇!(𝑘) as
𝜇!(𝑘) = (𝑥! − 𝑘)!
!𝑓 𝑥! 𝑑𝑥!
(1)
𝜇! 𝑘 = 𝑥!!
!𝑓 𝑥! 𝑑𝑥! − 𝑘
!
!𝑓 𝑥! 𝑑𝑥!
𝜇! 𝑘 = 𝑥!!
!
1𝑥!𝜎! 2𝜋
𝑒!!!!"!!!!!
!!
!
𝑑𝑥! − 𝑘!
!
1𝑥!𝜎! 2𝜋
𝑒!!!!"!!!!!
!!
!
𝑑𝑥!
Suppose that 𝑥! = 𝑒 !!!!!!! then 𝑑𝑥! = 2𝜎!𝑒 !!!!!!! and 𝑀 = !"!!!!!!! !
.
Since 𝑘 < 𝑥! < ∞
then ln 𝑘 < ln 𝑥! < ∞ → !"!!!!!! !
< !"!!!!!!! !
< ∞
Thus,
𝜇! 𝑘 =1
𝜎! 2𝜋𝑒!!! 2𝜎!𝑒 !!!!!!! 𝑑𝑀
!
!"!!!!!! !
− 𝑘1
𝑒 !!!!!!!𝜎! 2𝜋𝑒!!! 2𝜎!𝑒 !!!!!!! 𝑑𝑀
!
!"!!!!!! !
𝜇! 𝑘 =1𝜋
𝑒!(!!! !!!!!!!)𝑑𝑀 −𝑘𝜋
𝑒!!!𝑑𝑀!
!"!!!!! !
!
!"!!!!! !
𝜇! 𝑘 =1𝜋
𝑒! !!!!
!
!!!!
!
! !!!𝑑𝑀 −𝑘𝜋
𝑒!!!𝑑𝑀!
!"!!!!! !
!
!"!!!!! !
𝜇! 𝑘 =1𝜋𝑒!!!
! !!! 𝑒!!!𝑑𝐺 −𝑘𝜋
𝑒!!!𝑑𝑀!
!"!!!!! !
!
!"!!!!!!!!
! !
𝜇! 𝑘 =1𝜋𝑒!!!
! !!! 𝑒!!!𝑑𝐺 −𝑘𝜋
𝑒!!!𝑑𝑀!!"!!!!
! !
!!
!!"!!!!!!!!
! !
!!
𝜇! 𝑘 =1𝜋𝑒!!!
! !!!( 𝑒!!!𝑑𝐺 + 𝑒!!!𝑑𝐺!!"!!!!!!!
!
! !
!)−
𝑘𝜋( 𝑒!!!𝑑𝑀
!
!!
!
!!
+ 𝑒!!!𝑑𝑀)!!"!!!!
! !
!
𝜇! 𝑘 =1𝜋𝑒!!!
! !!!𝜋2 +
𝜋2 erf
−𝑙𝑛 𝑘 + 𝜇! + 𝜎!!
𝜎 2
−𝑘𝜋
𝜋2 +
𝜋2 erf
−ln 𝑘 + 𝜇!𝜎 2
𝜇! 𝑘 = 𝑒!!!
! !!!12 1+ erf
−𝑙𝑛 𝑘 + 𝜇! + 𝜎!!
𝜎 2− 𝑘
12 1+ erf
−𝑙𝑛 𝑘 + 𝜇!𝜎 2
𝜇! 𝑘 = 𝑒!!!
! !!!Φ(!!" !!!!!!!!
!)− 𝑘Φ(!!" !!!!
!)
According to (1), 𝜎!! = 𝜎!𝑇, and 𝜇! = 𝑆!𝑒!" ,
𝜇! = 𝑒!!!
! !!! = 𝑒!!!! !!! = 𝑆!𝑒!"
𝜎!𝑇2 + 𝜇! = ln 𝑆! + 𝑟𝑇
𝜇! = ln 𝑆! + (𝑟 −𝜎!
2 )𝑇
Plug 𝜇! = ln 𝑆! + (𝑟 − !!
!)𝑇 and 𝜎!! = 𝜎!𝑇 in (2)
𝜇! 𝑘 = 𝑒!" !! !!"Φ(𝑙𝑛 𝑆!𝑘 + (𝑟 +
𝜎!2 )𝑇
𝑇 𝜎)− 𝑘Φ(
𝑙𝑛 𝑆!𝑘 + (𝑟 −𝜎!2 )𝑇
𝑇 𝜎)
For the price of European call option 𝐶!! 𝑘 from 𝑡 = 𝑡! to 𝑡 = 𝑡! + 𝑇,
𝜇! 𝑘 = 𝐶!! 𝑘 𝑒!" .
Thus,
𝐶!! 𝑘 𝑒!" = 𝑒!" !! !!"Φ(𝑙𝑛 𝑆!𝑘 + (𝑟 +
𝜎!2 )𝑇
𝑇 𝜎)− 𝑘Φ(
𝑙𝑛 𝑆!𝑘 + (𝑟 −𝜎!2 )𝑇
𝑇 𝜎)
We will get the Black-‐Scholes formula:
𝐶!! 𝑘 = 𝑒!" !! Φ(𝑙𝑛 𝑆!𝑘 + (𝑟 +
𝜎!2 )𝑇
𝑇 𝜎)− 𝑘𝑒!!"Φ(
𝑙𝑛 𝑆!𝑘 + (𝑟 −𝜎!2 )𝑇
𝑇 𝜎)
(2)
Why Was Black-‐Scholes Model Popular?
The prediction of future price was formally initiated during 1970s when
financial engineers discovered the basic idea of arbitrage. At that time, there were a
few number of professional trader companies such as Salomon Brothers and
Goldman. Also, the market volatility was so stable that the prediction from the
Black-‐Scholes formula was pretty precise, or made just a little deviation. Many
untrained traders flows into the market with the notion that financial market was
just a mathematical equation. From 1973 to 1987, many traders made substantial
profits from the future market without realizing that they gradually destroyed the
great balance of the financial system. The amount of money flowing in financial
market decreases, the market volatility fluctuated depending on more factors and
the financial catastrophically crashed in 1987. Many untrainer traders, who worked
individually without taking punctilious advice from financial engineers and did not
realize the incompatible prerequisite for the Black-‐Scholes mathematical formula,
lost all their probits and was heavily in debt within a next few years. The more
irretrievable disadvantages of applying mathematical models without truly
understanding of the specific conditions of Black-‐Schole model will be explained in
the next section.
Disadvantages of Black-‐Scholes Model
Despite being one of the most popular mathematical models among traders
for the market future price, Black-‐Scholes Model could not be applicable to the real
situation for a long period of time because of its narrow and its idealistic
preconditions. First, the free-‐risk interest rate and the market volatility are never
constant. Second, the market is never perfectly efficient. Third, the future price does
not perfectly rely on the Galton distribution. However, the poor incoordination
between traders and financial engineers in using the Black-‐Scholes model led to the
more terrible conditions as the author would explain in the next section: Black-‐
Scholes Model Paradox: The Big Tail Crisis of The Black-‐Scholes Model.
Now, we will delve into these three internal and one external causes of
invalidating Black-‐Scholes model. During reading the analysis, we need to ask
ourselves why doesn’t each condition fit the real situation and how it affect the
financial market in short term and in long term.
1. The free-‐risk interest rate and the market volatility are not constant.
For the interest rate, it is obvious that it will not be constant as we can see in
terms of the fluctuation of bank interest rate. The interest rate depends on the
amount of money circulating in the economic system. If there is the great amount of
money in the economic system, Central Banks will pass the monetary policy to
increase the interest rate or decrease the aggregate demand that helps relieve the
severity from inflation. Conversely, if there is low amount of money in the economic
system, bank will pass the monetary policy to decrease the interest rate or increase
the aggregate demand to keep it in equilibrium.
For the market volatility, it is more difficult because it does not visually occur
in our routine lives. According to the Black-‐Scholes model, the volatility depends
only on the length of time interval T as expressed in the equation 𝜎! = !!!
!.
However, from the graph above, it was clear that, after the crash of 1987, the
market volatility also depended on strike price 𝑘 as shown in graph below. This
graph is also known as the volatility smile.
According to the volatility smile, we can imply the Black-‐Scholes model does
not completely reflect the real situation of financial market and precisely predict the
market future price.
2. The market is never efficient.
In theory, the market is efficient when the price and the quantity sold of
financial goods are at the point that the quantity demanded is equal to the quantity
supplied. But in practice, the quantity demanded and the quantity supplied of a
financial goods always fluctuates depending on such various factors as the number
of buyers, the number of sellers, the market volatility, the amount of consumer’s
income, and the price of related goods. Also, many various consumers’ preferences
Volatility
Strike Price The picture showing the volatility after the crash of
1987 comes from www.ipredict.it
that disobey the law of demand and the law of supply result in the market
inefficiency. Accordingly, how much chance that the quantity of demanded will be
equal to the quantity supplied? In mathematical aspects, we can assume that the
probability is very close to zero !!~ 0 . But viewing in financial aspects, we can just
say that the value of quantity demanded and that of the quantity supplied converge
to each other. We do not know how much the difference in quantity demanded and
quantity supplied at the particular future time. Thus, we cannot determine that how
much error the Black-‐Scholes model predicted, but we can conclude that the real
financial market is never abided by the oversimplified Black-‐Schloes model’s
conditions.
3. The future price does not perfectly fit to the Galton distribution.
The financial market will follow the Galton distribution only when the supply
of sellers and the demand of buyers do not affect each other. However, from social
perspective, people in the society cannot completely avoid the interaction with
other people. In the same manner as in economic aspect, buyers and sellers cannot
avoid interaction from each other that means the demand of one consumer can
influence the demand of another consumer or the supply of another seller. Also, the
supply of one seller can influence the supply of another seller and the demand of
another buyer. Thus, we have to consider how much chance that the overall demand
and supply impacted by other demands and supplies are equal to zero. This
question is very easy to answer, as the probability of that market will be efficient. It
will come close to zero or never happen.
4. The poor incoordination between the traders and the financial
engineers
The relationship between traders and financial engineers is that financial
engineers create some mathematical model to predict the market future price under
the certain conditions while traders will put the value of each variable in the
mathematical model. The traders and financial engineers do not have much time to
interact and discuss about the outcome of each section because the traders have to
spend all days to observe the variables changing over time while the financial
engineers have to recreate their model that its conditions are never constant. Thus,
the traders never know that they really apply the mathematical model in the right
condition or not. In other words, the poor incoordination, which frequently occurs
in nowadays, inevitably causes the irretrievable error.
Black-‐Scholes Model Paradox: The Big Tail Crisis of The Black-‐
Scholes Model
As stated in the last section, Black-‐Scholes model does not completely explain
the real financial market because of the vibrant market volatility and many idealistic
preconditions for the model. Hence, the Black-‐Scholes model’s outcomes
unavoidably result in the market inefficiency, insolvency, and the international
conflicts; for instance, the quick cash need from many foreign firms resulted in the
ominous interference of Thai financial system in 1997, called Tom Yam Kung crisis.
However, one of the pre-‐conditions of the Black-‐Scholes model is the market has to
be efficient. It means that the Black-‐Scholes model can be used only once or it is
inapplicable to the financial market. The author calls the condition that the outcome
of Black-‐Scholes model is not compatible with the prerequisite of its own, Black-‐
Scholes model paradox.
According to the fluctuation of various factors such as the market volatility,
the difference between quantity demanded and quantity supplied that the financial
goods will be sold, and the free-‐risk interest rate, we cannot predict when the
financial crash will occur or how much chance that the Black-‐Scholes model really
reduces the risk. But, we can ascertain that the Black-‐Scholes model can cause many
small tails and few big tails crisis as shown in the graph below.
This picture comes from “Why We Never Used the Black-‐Scholes-‐Merton Option Pricing
Formula” by Espen Gaarder Haug and Nassim Nicholas Taleb
Conclusion
Due to the unpredicatable variation of such future factors as free-‐risk
interest rates and market volatilities, it is almost impossible to predict the exact
future market price from the insufficient current information. The mathematical
models can just indicate the inclination of the future factors that have to periodically
adjust over time. Comparing finance with quantum mechanic, the underlying price
𝑆! is like the energy 𝐸 staying within the system and the length of time interval 𝑇 is
like the time 𝑡 in quantum mechanics. According to the Heinsenberg’s uncertainty
principle, ∆𝐸∆𝑡 ≥ ℏ!. In terms of economics, we can assume that ∆𝑆!∆𝑇 ≥ 𝑐 where 𝑐
is a constant. The inequality means that no mathematical model that can completely
explain the whole financial market and precisely predict the market future price. On
the other hand, trying to create mathematical model will lead to incorrigible
miscalculation and wrong prediction. Therefore, the best way to deal with financial
market may not only create mathematical models, but also include the traders’
previous experience and intution.
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