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The problems of relying on no-arbitrage - An essay on the derivation of

Black-Scholes Written in December 2013 by Torbjørn Bull Jenssen

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Introduction

To model financial markets based on the standard neo-classical micro economic framework has

proven to be extremely difficult. As a result, alternative ways to analyse the financial markets have

been developed. In 1958 the first formal theory in finance based on a no arbitrage argument, the

Modigliani-Miller theorem, was published. Since then the no arbitrage argument has played a key role

in several different financial models. The simplest definition of arbitrage that will pop up in all

standard text books is the possibility of a risk-free profit at zero cost. As an example: Twenty pounds

lying on the ground outside Kings Cross station represents an arbitrage opportunity. Central in

arbitrage pricing theory is that there cannot exist any arbitrage opportunities in equilibrium. In other

words; there cannot be twenty pounds on the ground outside Kings Cross, because someone else

would have found it before you, and hence picked it up. The discovery of an arbitrage opportunity

will lead a person to act as an arbitrageur, and by reacting to it, exhaust the opportunity.

In a stylized example such as the one above the question of who knows what, when, and on what basis

is not very interesting. Presumably everyone is able to identify money on the ground as free money. In

the financial markets on the other hand, it can be a lot more difficult to identify possible price

misalignments leading to opportunities of arbitrage. This has not hindered the development of

sophisticated models like the Black-Scholes1 model which is also based on the no arbitrage argument.

The Black-Scholes model is used by investors and traders to price and hedge different types of

derivatives and was originally developed early in the 1970s by Fischer Black, Myron Scholes and

Robert Merton (Hull, 2012). There exist a variety of different versions and extensions of the model

specially designed to analyse specific relations and derivatives. Even though there are some key

differences between the different versions of the model, they have in common that they all assume no

arbitrage. In this essay the two ways in which the Black-Scholes pricing relationship relies on the no

1 The model also goes under the name of “Black-Scholes-Merton”

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arbitrage assumption will be discussed by going through a simple delta hedging derivation of the

model. The differences between the two no arbitrage conditions, regarding information and market

behaviour, will then be discussed and be related to actual observations in financial markets.

Deriving the model

The derivation of the Black-Scholes pricing relationship is standard in advanced university courses in

finance, and similar approaches can be found in most standard textbooks like Hull (2012) and

Copeland, et al. (2005). The following derivation will be close to a replication of the delta hedge

derivation of Black-Scholes done in dos Santos (2012).

It is assumed that the evolution of a stock price follows a geometric Brownian motion2, which is

consistent with the observation that the natural logarithm of stock prices seems to follow a random

walk. The evolution then becomes an Itô process, and can be described by the following stochastic

differential equation (SDE).

𝑑𝑆𝑡 = µ𝑆𝑡𝑑𝑡 + 𝜎𝑆𝑡𝑑𝑊𝑡

St is the stock price at any given time where the subscript t implies that the variable is a function of

time. dt is the differentially small laps of time. µ is the percentage drift rate, or the expected return on

the stock. σ is the percentage volatility of the stock, and Wt is a Brownian motion, or a so called

wiener process. The wiener process is the random element of the price evolution and is a direct result

of the no arbitrage condition. “Deviations from risk-adjusted returns expectations are random and

expected to be zero” (dos Santos, 2012, p. 266)

Furthermore, it is assumed that there exists an option 𝐻(𝑆𝑡, 𝑡) written over the underlying stock St,

and that the only source of uncertainty in this option is the underlying stock. The evolution of the

option will then be described by a SDE, like equation (2), since “…a function of an Itô process is

known also to follow an Itô process” (dos Santos, 2012, p. 266)

2 For information on geometric Brownian motion see for instance Wikipedia: http://en.wikipedia.org/wiki/Geometric_Brownian_motion

(1)

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𝑑𝐻 = (𝜕𝐻

𝜕𝑆µ𝑆𝑡 +

𝜕𝐻

𝜕𝑡+

1

2

𝜕2𝐻

𝑆2𝜎2𝑆𝑡

2)𝑑𝑡 + (𝜕𝐻

𝜕𝑆𝜎𝑆𝑡)𝑑𝑧

The randomness is the same for both the stock and the option, and a dynamic perfectly hedged

portfolio can be constructed by holding the stock, and shorting the option, given frictionless markets.

The perfect dynamic hedge ratio, called the delta or often just Δ, can be calculated to be 𝜕𝐻(𝑆𝑡,𝑡)

𝜕𝑆= ∆𝑡.

If one holds Δt units of the stock, and shorts one unit of the derivative the portfolio will be perfectly

hedged. It follows that a portfolio like this will yield the value:

𝑉(∆𝑡) = (𝜕𝐻(𝑆𝑡,𝑡)

𝜕𝑆𝑆𝑡 −𝐻(𝑆𝑡, 𝑡))

And evolve over time obeying the following relationship:

𝑑𝑉(∆𝑡) = (1

2

𝜕2𝐻

𝜕𝑆2𝜎2𝑆2 −

𝜕𝐻

𝜕𝑡)𝑑𝑡

Since a perfectly hedged portfolio will be risk free, and we have assumed no arbitrage, it must yield

the risk free rate of return, r. If it yields more than the risk free return, it will be possible to borrow

money, buy the portfolio, and end up making a profit without taking on any risk. If it yields less, the

same result will emerge through undertaking the opposite actions. Hence, the following relationship

can be established:

𝑑𝑉(∆𝑡)

𝑉(∆𝑡)= 𝑟𝑑𝑡

The above equations (1-5) can then be combined to obtain the derivative price formula:

𝜕𝐻

𝜕𝑡+ 𝑟𝑆𝑡

𝜕𝐻

𝜕𝑆+

1

2𝜎2𝑆𝑡

2 𝜕2𝐻

𝜕𝑆2= 𝑟𝐻(𝑆𝑡 , 𝑡)

To obtain an explicit solution, different boundary conditions are needed. The specific form these

conditions will take is determined by the characteristics of the derivative/option. Any prices deviating

from this solution will, according to the theory, represent an opportunity of arbitrage. In equilibrium

all such opportunities must have been discovered and traded away.

(2)

(3)

(4)

(5)

(6)

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The no arbitrage condition

The first place the no arbitrage condition appears is when it is assumed that the price evolution of the

stock can be modelled as a geometric Brownian motion, which is a Markov Process. This means that

the current price contains all past information, and that knowledge of past prices does not affect the

expected future value. Hence, it is impossible to know whether the stock will increase or decrease in

value relative to the deterministic trend. This is a result from the Efficient Market Hypothesis (EMH),

stating, in its semi strong form, that prices have to reflect all publicly available information from the

past, and that only new news can change prices. Furthermore, the EMH is based on the assumption

that stock prices are determined as the discounted sum of expected future returns. Thus, the future

value of a stock is treated as a stochastic variable, and the assumption that the randomness is

following a normal distribution is to be interpreted as a statement of ignorance. When you do not

know how something will change, it follows from the principle of maximum entropy that you should

use a uniform distribution, and assign equal probability to all the possible outcomes. In other words if

markets are efficient there will be no opportunities that allow a trader to systematically make excess

profit or beat the market through using public available information.

As long as the EMH holds, prices should be in line with fundamentals. Since stock prices are assumed

to be determined by future returns, the volatility of stock prices should be directly related to the

volatility of returns, if markets are efficient. However, Shiller (2003) found that “There is a clear

sense that the level of volatility of the overall stock market cannot be well explained with any variant

of the efficient markets model in which stock prices are formed by looking at the present discounted

value of future returns”(p.90). Hence, he is arguing that excess volatility of stock prices relative to

returns indicates that financial markets are not efficient. As a result one has to use behavioural finance

models, like feedback theories, to better understand why prices deviate from fundamentals. However,

Fama (1998) found both that investors overreact to news as often as they underreact, and that

misalignments tend to disappear over time. Thus, even if the EMH does not model actual market

behaviour correctly, it might be our best guess. As long as the direction of misalignments remains

unknown, we are forced to use the randomness resulting from the EMH to model what we do not

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know. Further, for the EMH to make sense, we must rely upon people’s greed, and assume that

anyone discovering an opportunity of making free money, based on publicly available information,

will do so. By reacting to a misalignment an arbitrageur moves prices until the opportunity is

exhausted. This assumes that the knowledgeable arbitrageur has the financial power and flexibility to

do so. This might not be the general case (Shiller 2003). If markets are nonetheless to produce

efficient prices, other investors must be able to read the signal from the change in the arbitrageur’s

trading pattern, and trade in a way that moves prices back to fundamentals.

The EMH is logically proved by contradiction; if it does not hold someone can make infinite profit

which cannot be possible. Thus the theory is intuitively appealing. This is of course not enough to

accept an economic theory; it should also be tested empirically. Several people have tried to test the

hypothesis, and even if there are some puzzling aspects and criticism rising from observed

behavioural trading patterns and balance-sheet effects (dos Santos, 2012), it does seem to hold for

most large financial markets in developed countries. An enlightening overview of some of the

research on the field can be found in Copeland, et al. (2005, p. 377) who summarizes that “…most

agree that capital markets are efficient in the weak and semistrong forms but not in the strong form”.

The semi strong form is a sufficient condition assuring that there cannot be any opportunities for

arbitrage without using inside information. It is, in other words, not possible to outperform the market

over time. This goes well with the finding of “…an inability of market professionals to outperform

indexes…” (Elton, et al., 2011, p. 427). A problem with this result, however, is that if it were true that

all information was reflected in prices; there would be no reason to undertake neither fundamental nor

technical analysis. The only thing an analyst could do would be to access private inside information,

which might be what some are doing. However, in most countries it is illegal to use inside information

for trading. Yet, investors spend huge amounts on both fundamental and technical analysis, and on

investment consulting. The EMH in a way neglects the process of price formation, and is not very

popular among followers of Austrian economic theory. A key argument against the EMH is that

costly information results in markets not being perfectly coordinated. Hence “there is scope for

entrepreneurial activity in financial markets just as there is in other markets” (Pasour, 1989). If

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everyone accepted future price evolution as completely random, and no one undertook any

fundamental analysis, the financial market would turn into a casino, where you could only choose

your exposure towards risk. It is then difficult to understand how the prices could reflect available

information, if no one processed and analysed this information.

“Since various real causes are likely to have prolonged effects on the real data, we can conclude that

asset markets cannot be in equilibrium. This, in turn, provides scope for benefits from analysing the

historical data in order to assess the future direction of asset prices.” (Shostak, 1997, p. 42) This is a

strong argument motivating analyst activities. There will be some opportunities of arbitrage out there,

and the prices will only represent all available knowledge when these opportunities are traded away.

Since the economy is constantly changing, and the processing of information into knowledge is

costly, that will never be the case. An alternative way to formulate the no arbitrage argument rising

from the EMH would then be that; the marginal costs of analysis aiming to discover new arbitrage

opportunities must in equilibrium equal the expected returns on such opportunities, corrected for the

risk involved.

Even though empirical evidence suggests that the EMH holds, and its use is justified by lack of

knowledge, it is important to bear in mind that financial markets are not inherently efficient. Stock

and option exchanges are organised markets with rules that companies and traders must obey. The

efficiency observed in developed liquid markets is therefore crucially contingent on the regulations

and rule of law affecting the exchange. In theory markets must be frictionless and all traders must be

rational expected-utility-maximising-prise-takers (perfect competition), for the EMH to hold.

However, frictions are a reality, there are traders large enough to affect stock price movement, and

some traders are irrational. When markets are not properly regulated huge Ponzi schemes like those

in Albania in 1996-1997 (Jarvis, 1999) might evolve and the EMH will clearly not hold in unregulated

illiquid markets.

While the first no arbitrage argument is based on lack of predictability over future stock price

movement i.e. lack of knowledge, the absolute opposite is true for the second use of the argument. To

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be able to construct the perfectly hedged, and therefore riskless portfolio, used to derive the pricing

formula, the trader must know the pricing formula. “In the models, the economist derives the pricing

function by assuming market behaviour that requires arbitrageurs to already know the pricing

function and to use it to guide their actions.” (dos Santos, 2012, p. 263)

This means that the derivation of the pricing model, and thereby the no arbitrage argument, is based

on a circular logic. As a pure mathematical argument there is no problem since the pricing function

solving the problem is one that satisfies the differential equation. In practice, however, the economic

interpretation is much more problematic. The relation is only true if everyone believes it to be so, and

in that way the Black-Scholes can be accused of being a way to coordinate and shape market

behaviour rather than to model it.

A very convincing presentation of this can be found in MacKenzie (2005), and a summary will be

presented here. When the Chicago Board Option Exchange opened in 1973, before the Black-Scholes

model was published, option prices tended to be systematically higher than what was consistent with

predictions based on the Black-Scholes model. Thus, it was difficult to claim that the model initially

was able to capture actual behaviour. However, it did not take long before the model was widely used

as a result of its public availability, high academic standing and cognitive simplicity. Prices moved

towards the ones predicted by the model, and the models ability to capture the reality became striking.

“when judged by its ability to explain the empirical data, option pricing theory is the most successful

theory not only in finance, but in all of economics” (Ross, 1987, p. 332, cited in MacKenzie 2005).

Although this in part could be a result of technological and regulatory changes that made the model

assumptions more realistic, the very widespread use of the model seems to be a more important

explanation. Traders started to use sheets of pre-calculated prices, based on the model, to discover

over- and undervalued options. The more traders who based their decision on prices calculated by the

model, the more the real world started to look like the theory. The theory did not only create trading

behaviour, it also legitimated options markets and affected regulating behaviour. As an example:

Clearing firms are guaranteeing the trades done by their clients, and therefore have strong incentives

to monitor the clients’ exposure towards risk. This could be really difficult, but the Black-Scholes

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model provided methods that allowed the clearing firms to derive a simple measure of how well-

hedged the clients were.

The Black-Scholes model’s close fit to observed prices a few years after its introduction, and up until

the crash of 1987, might be interpreted as follows: even if the model was creating, rather than

describing behaviour, it was the only stable way to organize option trading. However, after the crash

1987, the strong correlation was gone. Even though the model is still relevant today, it is clear that it

is not able to fully predict financial behaviour, and option markets are now organising themselves

differently.

The Black-Scholes model changed the behaviour of the agents that were being modelled. However,

economic models affecting the economy are not unique to finance. For instance, the famous Lucas

critique3 showed how similar problems existed within general macroeconomics. It is important to

remember that economics is not a natural science. Theories might both produce outcomes that make

them fit better with reality, or change originally observed patterns so that they become less true.

There are two other main observations regarding the knowledge and market behaviour required to

construct the perfectly hedged portfolio that will receive attention in this essay. The first is concerning

the volatility, σ. A trader must know the volatility to be able to calculate the right price, and trade

away any possible misalignments. Since it is the future volatility that is needed, and that is unknown,

is it necessary to use an estimate. Even though an estimator should not systematically overshoot or

undershoot the real value, is it potentially problematic that estimations tend to be based on past data.

If the past period of relevance has been abnormally stable, “new era” theories might emerge fostering

a belief that the future will be just as stable as the past (Shiller, 2003). Underestimation of the

volatility might lead to overestimation of future prices, and might explain parts of why prices deviate

from fundamentals. Further, if two traders estimate different expected volatility they will price the

3 Robert Lucas (1976) argued that when macroeconomic policymakers change policy based on observed economic relations, the relations and the whole economic system is altered.

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option differently. It follows then from the rule of one price that they will trade until they price it

equally, as long as they have the financial power to do so. One can therefore do the Black-Scholes

calculations “backwards”, and use the observed prices to calculate the volatility expected by the

market as a whole.

The second observation is related to the assumption that the only source of uncertainty for the option

comes from the underlying stock, and that perfect correlation makes it possible to create a perfect

hedge. If it were true, that the variance of the option and the underlying price were perfect correlated,

the price of an option would be a monotone function of the stock price - increasing in St for call

options and decreasing for put options. However, this has been proven to not necessarily be the case.

Bakshi, et al. (2000) find evidence of option prices moving contrary to that predicted by the Black

Scholes model. As a result they conclude that even though some of the contradictory movements can

be explained with reference to microstructures, like the bid-ask spread, another state variable is

needed, together with the stock price, to properly fit the observed data. When the variance of the stock

and the option is not perfectly correlated it will be impossible to construct the risk free portfolio the

way it is done in the delta hedge derivation, and the no-arbitrage argument cannot be used to derive

the pricing function.

Conclusion:

The development of financial models based on the no-arbitrage argument has fundamentally changed

the theoretical framework of finance. Even though there are those who are sceptical to the idea of the

efficient market hypothesis, and thus to the use of no arbitrage arguments, modelling based on the

ideas of the Black-Sholes framework has come to dominate the field of finance theory. The method of

using offsetting positions to create a risk free portfolio, and combine it with the risk free rate of return

has fundamentally changed the theoretical foundation underlying hedging and option pricing. The

impact of the model has been enormous, and it might even have formed markets in its own image.

However, there still seems to be empirical evidence that suggests the very creation of the risk free

portfolio is impossible since options are exposed to additional sources of risk, other than the

underlying asset.

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The two links between the no arbitrage argument and the delta hedge derivation of the model are

fundamentally different in regards to the kind of knowledge, and market behaviour that is assumed.

The first link assumes efficient markets and the lack of predictability over future price movement.

However, markets might not be efficient all the time, and the no arbitrage argument is rather a best

guess; a statement of ignorance. On the other hand, if the future evolution of a stock price was

predictable, it would be possible to make free money, and in the derivation of the model it is

necessary to assume that traders have the financial flexibility to do so until the opportunity is gone.

The second link between the model derivation and the no arbitrage argument assumes that it is

possible to create the perfectly hedged portfolio based on a stock and an option written over the same

stock. This is a much more problematic assumption. A potential arbitrageur must know the pricing

function to be able to discover arbitrage opportunities, but the pricing function is only correct if there

is no arbitrage. This means that the model is only true if those who are trading believe it to be so, and

rather use it as a method of collective calculation of what option prices should be. The strength of no

arbitrage based models in finance is the way in which they bypass the problems of modelling

individual behaviour, and become solvable. Instead of starting the analysis with utility maximising

agents the Black-Scholes models equilibrium behaviour directly. However, as a result the model

cannot be used to explain how equilibrium appears or why deviations might be observed. Thus, other

models and theories are needed to be able to analyse extreme events like the crash of 1987 and the

current financial crisis.

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References:

Bakshi, G., Cao, C. & Chen, Z., 2000. Do call prices and the underlying stock always move in the

same direction?. The Review of Financial Studies, 13(3), pp. 549-584.

Copeland, T. E., Wston, J. F. & Shastri, K., 2005. Financial Theory and Corporate Poilcy. 4th ed.

London: Pearson.

dos Santos, P. L., 2012. Option pricing models. In: Handbook of critical issues in finance. s.l.:Edward

Elgar Publishing, pp. 263-269.

Elton, E. J., Gruber, M. J., Brown, S. J. & Goetzmann, W. N., 2011. Modern Portfolio theory and

Investment Analysis. 8th ed. s.l.:John Wiley & Sons.

Hull, J. C., 2012. Options, Futures, And Other Derivatives. 8th ed. s.l.:Pearson.

Jarvis, C., 1999. “The Rise and Fall of the Pyramid Schemes in Albania.” International Monetary

Fund Working Paper No. 99-98.

Lucas, R. 1976. "Econometric policy evaluation: A critique". Carnegie-Rochester Conference Series

on Public Policy 1 (1): 19-46

MacKenzie, D., 2005. Is Economics Performative? Option Theory and the Construction of

Derivatives Markets. [Online]

Available at:

http://www.espanet2012.info/__data/assets/pdf_file/0017/3419/is_economics_performative.pdf

[Accessed 12 12 2012].

Pasour, E., 1989. The Efficient-Markets Hypothesis and Entrepreneurship. Review of Austrian, 3( ),

pp. 95-107.

Ross, Stephen A. 1987. "Finance." Pp. 322-336 in The New Palgrave Dictionary of

Economics, vol. 2, edited by John Eatwell, Murray Milgate, and Peter Newman.

London: Macmillan.

Shostak, F., 1997. In Defense of Fundamental Analysis: A Critique of the Efficient Market. Review of

Austrian Economics, 10(2), pp. 27-45.

Wikipedia, 2012. Wikipedia, Geometric Brownian Motion. [Online]

Available at: http://en.wikipedia.org/wiki/Geometric_Brownian_motion

[Accessed 12 12 2012].