*Finance theory*is the study of economic agents behavior allocating their resources across alternative financial instruments and in time in an uncertain environment. Mathematics provides tools to model and analyze that behavior in allocation and time, taking into account uncertainty.

1.Louis
Bachelier’s 1900 math dissertation on the theory of speculation in the Paris
markets marks the twin births of both the continuous time mathematics of
stochastic processes and the continuous time economics of option pricing.

2.The
most important development in terms of impact on practice was the Black-Scholes
model for option pricing published in 1973.

3.Since
1973 the growth in sophistication about mathematical models and their adoption
mirrored the extraordinary growth in financial innovation. Major developments
in computing power made the numerical solution of complex models possible. The
increases in computer power size made possible the formation of many new
financial markets and substantial expansions in the size of existing ones.

Louis
Bachelier |

**Mathematical Ideas**

**One sometime hears that “compound interest is the eighth wonder of the world”, or the “stock market is just a big casino”.These are colorful sayings, maybe based in happy or bitter experience, but each focuses on only one aspect of one financial instrument. The “time value of money” and uncertainty are the central elements that influence the value of financial instruments. When only the time aspect of finance is considered, the tools of calculus and differential equations are adequate. When only the uncertainty is considered, the tools of probability theory illuminate the possible outcomes. When time and uncertainty are considered together we begin the study of advanced mathematical finance.Finance is the study of economic agents’ behavior in allocating financial resources and risks across alternative financial instruments and in time in an uncertain environment. Familiar examples of financial instruments are bank accounts, loans,stocks, government bonds and corporate bonds. Many less familiar examples abound. Economic agents are units who buy and sell financial resources in a market, from individuals to banks, businesses, mutual funds and hedge funds. Each agent has many choices of where to buy, sell, invest and consume assets, each with advantages and disadvantages. Each agent must distribute their resources among the many possible investments with a goal in mind.**

Advanced
mathematical finance is often characterized as the study of the more
sophisticated financial instruments called derivatives. A derivative is a
financial agreement between two parties that depends on something that occurs
in the future, such as the price or performance of an underlying asset. The
underlying asset could be a stock, a bond, a currency, or a commodity.
Derivatives have become one of the financial world’s most important
risk-management tools. Finance is about shifting and distributing risk and
derivatives are especially efficient for that purpose. Two such instruments are
futures and options. Futures trading, a key practice in modern finance,
probably originated in seventeenth century Japan, but the idea can be traced as
far back as ancient Greece. Options were a feature of the “tulip mania” in
seventeenth century Holland. Both futures and options are called “derivatives”.
(For the mathematical reader, these are called derivatives not because they
involve a rate of change, but because their value is derived from some
underlying asset.) Modern derivatives differ from their predecessors in that
they are usually specifically designed to objectify and price financial risk.

Derivatives come in many types. There are futures, agreements to trade something at a set price at a given dates; options, the right but not the obligation to buy or sell at a given price; forwards, like futures but traded directly between two parties instead of on exchanges; and swaps, exchanging flows of income from different investments to manage different risk exposure. For example, one party in a deal may want the potential of rising income from a loan with a floating interest rate, while the other might prefer the predictable payments ensured by a fixed interest rate. This elementary swap is known as a “plain vanilla swap”. More complex swaps mix the performance of multiple income streams with varieties of risk. Another more complex swap is a credit-default swap in which a seller receives a regular fee from the buyer in exchange for agreeing to cover losses arising from defaults on the underlying loans. These swaps are somewhat like insurance . These more complex swaps are the source of controversy since many people believe that they are responsible for the collapse or near-collapse of several large financial firms in late 2008. Derivatives can be based on pretty much anything as long as two parties are willing to trade risks and can agree on a price. Businesses use derivatives to shift risks to other firms, chiefly banks. About 95% of the world’s 500 biggest companies use derivatives. Derivatives with standardized terms are traded in markets called exchanges.

Derivatives
tailored for specific purposes or risks are bought and sold “over the counter”
from big banks. The “over the counter” market dwarfs the exchange trading. In
November 2009, the Bank for International Settlements put the face value of
over the counter derivatives at $604.6 trillion. Using face value is
misleading, after off-setting claims are stripped out theresidual value is $3.7
trillion, still a large figure.Mathematical models in modern finance contain
deep and beautiful applications of differential equations and probability
theory.In spite of their complexity, mathematical models of modern financial
instruments have had a direct and significant influence on finance practice.

**Early History**

The
history of stochastic integration and the modelling of risky asset prices both
beginwith Brownian motion, so let us begin
there too. In 17th century put options were bought on tulip bulbs in
Netherlands.The earliest attempts to model Brownian motion mathematically can
be traced to three sources, each of which knew nothing about the others: the
first was that of T. N. Thiele of Copenhagen,
who effectively created a model of Brownian motion while studying time series
in 1880 the second was that of L. Bachelier
of Paris, who created a model of Brownian motion while deriving the dynamic
behavior of the Paris stock market, in 1900,and the third was that of A. Einstein, who proposed a model of the motion of
small particles suspended in a liquid, in an attempt to convince other
physicists of the molecular nature of matter.

Though the origins of much of the mathematics in financial models traces to Louis Bachelier’s 1900 dissertation on the

*Theory of speculation*in the Paris markets. This doctrate thesis was completed at the Sorbonne in 1900 under Henri Poincare, this work marks the twin births of both the continuous time mathematics of stochastic processes and the continuous time economics of option pricing. While analyzing option pricing,Bachelier provided two different derivations of the partial differential equation for the probability density for the Wiener process or Brownian motion. In one of the derivations, he works out what is now called the Chapman-Kolmogorov convolution probability integral. Along the way, Bachelier derived the method of reflection to solve for the probability function of a diffusion process with an absorbing barrier. Not a bad performance for a thesis on which the first reader, Henri PoincarĂ©,gave less than a top mark! After Bachelier, option pricing theory laid dormant in the economics literature for over half a century until economists and mathematicians renewed study of it in the late 1960s. Jarrow and Protter speculate that this may have been because the Paris mathematical elite scorned economics as an application of mathematics.

Bachelier’s
work was 5 years before Albert Einstein’s 1905 discovery of the same equations
for his famous mathematical theory of Brownian motion. The editor of Annalen
der Physik received Einstein’s paper on Brownian
motion on May 11, 1905. The paper appeared later that year. Einstein
proposed a model for the motion of small particles with diameters on the order
of 0.001 mm suspended in a liquid. He predicted that the particles would
undergo microscopically observable and statistically predictable motion. The
English botanist Robert Brown had already
reported such motion in 1827 while observing pollen grains in water with a
microscope. The physical motion is now called Brownian motion in honor of
Brown’s description.Einstein calculated a diffusion constant to govern the rate
of motion of suspended particles. The paper was Einstein’s attempt to convince
physicists of the molecular and atomic nature of matter. Surprisingly, even in
1905 the scientific community did not completely accept the atomic theory of
matter. In 1908, the experimental physicist Jean-Baptiste
Perrin conducted a series of experiments that empirically verified
Einstein’s theory. Perrin thereby determined the physical constant known as
Avogadro’s number for which he won the Nobel prize in 1926. Nevertheless,
Einstein’s theory was very difficult to rigorously justify mathematically. Let
us now turn to Einstein’s model. In modern terms, Einstein assumed that
Brownian motion was a stochastic process with continuous paths, independent
increments, and stationary Gaussian increments. He did not assume other
reasonable properties (from the standpoint of physics), such as rectifiable
paths. If he had assumed this last property, we now know his model would not
have existed as a process. However, Einstein was unable to show that the
process he proposed actually did exist as a mathematical object. This is
understandable, since it was 1905, and the ideas of Borel and Lebesgue
constructing measure theory were developed only during the first decade of the
twentieth century.In a series of papers from 1918 to 1923, the mathematician
Norbert Wiener constructed a mathematical model of Brownian motion. Wiener and
others proved many surprising facts about his mathematical model of Brownian
motion, research that continues today. In recognition of his work, his
mathematical construction is often called the Wiener process.

The
next step in the groundwork for stochastic integration lay with A. N. Kolmogorov.Indeed, in 1931, two years before
his famous book establishing a rigorous mathematical basis for Probability Theory using measure theory,
Kolmogorov refers to and briefly explains Bachelier’s construction of Brownian
motion ( pages 64, 102–103). It is this paper too in which he develops a large
part of his theory of Markov processes. Most
significantly, in this paper Kolmogorov showed that continuous Markov processes
(diffusions) depend essentially on only two parameters:one for the speed of the
drift and the other for the size of the purely random part (the diffusive
component). He was then able to relate the probability distributions of the
process to the solutions of partial differential equations, which he solved,
and which are now known as “Kolmogorov’s equations.” He also made major
contributions to the understanding of stochastic processes (involving random
variables), and he advanced the knowledge of chains of linked probabilities.
Shortly thereafter, he took an extended trip to Germany and France, and in 1933
laid out his probability theory in

*Foundations of the Theory of Probability**.*This work secured his reputation as the world's foremost expert in his field.
We
turn now to Kiyosi Ito, the father of
stochastic integration,no doubt an attempt to establish a true stochastic
differential to be used in the study of Markov processes was one of Ito’s
primary motivations for studying stochastic integrals.His work, starting in the
1940s, built on the earlier breakthroughs of Einstein and Norbert Wiener. Mr.
Ito’s mathematical framework for describing the evolution of random phenomena
came to be known as the Ito Calculus.This
random component is best modeled using a mathematics which can show the range
of possible areas.Stochastic processes and Ito
calculus make up most of the modern Financial Maths, it has important
applications in Mathematical Finance and stochastic differential
equations.Firstly, we are now dealing with random variables (more precisely,
stochastic processes). Secondly, we are integrating with respect to a
non-differentiable function (technically,stochastic processes).

**Growth of Mathematical Finance**

Modern
mathematical finance theory begins in the 1960s. In 1965 the economist Paul Samuelson published two papers that argue
that stock prices fluctuate randomly. One explained the Samuelson and Fama
efficient markets hypothesis that in a well-functioning and informed capital
market, asset-price dynamics are described by a model in which the best estimate
of an asset’s future price is the current price (possibly adjusted for a fair
expected rate of return.). Under this hypothesis, attempts to use past price
data or publicly available forecasts about economic fundamentals to predict
security prices are doomed to failure. In the other paper with mathematician Henry McKean, Samuelson shows that a good model
for stock price movements is geometric Brownian motion. The final precursor to
the Black, Scholes and Merton option pricing
formulaes can be found in the paper of Samuelson and Merton .Samuelson
noted that Bachelier’s model failed to ensure that stock prices would always be
positive, whereas geometric Brownian motion avoids this error.

The
most important development in terms of practice was the 1973 Black-Scholes
terms of practice was the 1973 Black-Scholes model
for option pricing. The two economists Fischer Black and Myron Scholes (and
simultaneously, and somewhat independently, the economist Robert Merton)
deduced an equation that provided the first strictly quantitative model for
calculating the prices of options. The key variable is the volatility of the
underlying asset. Myron Scholes published a paper with Fischer Black on 'Pricing of Options and Corporate Liabilities',
incorporating suggestions from Merton Miller (of
M&M Theory fame) and Eugene Fama (father of the Efficient Market
Hypothesis). These equations standardized the pricing of derivatives in
exclusively quantitative terms. The formal press release from the Royal Swedish
Academy of Sciences announcing the 1997 Nobel Prize in Economics states that
the honor was given “for a new method to determine the value of derivatives.
Robert C. Merton and Myron S. Scholes have, in collaboration with the late
Fischer Black developed a pioneering formula for the valuation of stock
options. Their methodology has paved the way for economic valuations in many
areas. It has also generated new types of financial instruments and facilitated
more efficient risk management in society.”

The
Chicago Board Options Exchange (CBOE) began publicly trading options in the
United States in April 1973, a month before the official publication of the
Black-Scholes model. By 1975, traders on the CBOE were using the model to both
price and hedge their options positions. In fact, Texas Instruments created a
hand-held calculator specially programmed to produce Black-Scholes option
prices and hedge ratios.The basic insight underlying the Black-Scholes model is
that a dynamic portfolio trading strategy in the stock can replicate the
returns from an option on that stock. This is called “hedging an option” and it
is the most important idea underlying the Black-Scholes-Merton approach. Much
of the rest of the book will explain what that insight means and how it can be
applied and calculated.

The
story of the development of the Black-Scholes-Merton option pricing model is
that Black started working on this problem by himself in the late 1960s. His
idea was to apply the capital asset pricing model to value the option in a
continuous time setting. Using this idea, the option value satisfies a partial
differential equation. Black could not find the solution to the equation. He
then teamed up with Myron Scholes who had been thinking about similar problems.
Together, they solved the partial differential equation using a combination of
economic intuition and earlier pricing formulas.

Black and Scholes had solved stochastic partial differential equations to develop a formula for pricing European-type call options.The result was an equation that suggested how the price of a call option might be calculated as a function of a risk-free interest rate, the price variance of the asset on which the option was written, and the parameters of the option (strike price, term, and the market price of the underlying asset.)

At this time, Myron Scholes was at MIT. So was Robert Merton,who was applying his mathematical skills to various problems in finance.Merton showed Black and Scholes how to derive their differential equation differently. Merton was the first to call the solution the Black-Scholes option pricing formula. Merton’s derivation used the continuous time construction of a perfectly hedged portfolio involving the stock and the call option together with the notion that no arbitrage opportunities exist. This is the approach we will take. In the late 1970s and early 1980s mathematicians Harrison, Kreps and Pliska showed that a more abstract formulation of the solution as a mathematical model called a martingale provides greater generality.

By
the 1980s, the adoption of finance theory models into practice was nearly
immediate. Additionally, the mathematical models used in financial practice
became as sophisticated as any found in academic financial research. There
are several explanations for the different adoption rates of mathematical
models into financial practice during the 1960s, 1970s and 1980s. Money and
capital markets in the United States exhibited historically low volatility in
the 1960s; the stock market rose steadily, interest rates were relatively
stable, and exchange rates were fixed. Such simple markets provided little
incentive for investors to adopt new financial technology. In sharp contrast,
the 1970s experienced several events that led to market change and increasing
volatility. The most important of these was the shift from fixed to floating
currency exchange rates; the world oil price crisis resulting from the creation
of the Middle East cartel; the decline of the United States stock market in
1973-1974 which was larger in real terms than any comparable period in the
Great Depression; and double-digit inflation and interest rates in the United
States. In this environment, the old rules of thumb and simple regression
models were inadequate for making investment decisions and managing risk.

During the 1970s, newly created derivative-security exchanges traded listed options on stocks, futures on major currencies and futures on U.S. Treasury bills and bonds. The success of these markets partly resulted from increased demand for managing risks in a volatile economic market. This success strongly affected the speed of adoption of quantitative financial models. For example, experienced traders in the over the counter market succeeded by using heuristic rules for valuing options and judging risk exposure. However these rules of thumb were inadequate for trading in the fast-paced exchange-listed options market with its smaller price spreads, larger trading volume and requirements for rapid trading decisions while monitoring prices in both the stock and options markets. In contrast, mathematical models like the Black-Scholes model were ideally suited for application in this new trading environment. The growth in sophisticated mathematical models and their adoption into financial practice accelerated during the 1980s in parallel with the extraordinary growth in financial innovation. A wave of de-regulation in the financial sector was an important factor driving innovation.

**Quantum Mechanics in Financial Markets.**

The
markets are non-linear, dynamic systems, subject to the rules of Chaos Theory. Market prices are highly random,
with a short to intermediate term trend component. They are highly dependent on
initial conditions. Markets also show qualities of fractals -- self-similar in
the sense that the individual parts are related to the whole.Due to the
non-Gaussian behavior of the markets the methods from Chaos
Theory, Fractals and Quantum Physics are being used in Finance since 1980s.

One
recent trend in the growing field of quantitative finance to apply techniques
borrowed from quantum physics to Financial models. One example is Path Integrals, which were invented by Richard Feynman in 1948 , Feynman used path
integrals along with the probability methods designed by Norbert Wiener to
reformulate methods of Quantum Physics.The path integral also relates quantum
and stochastic processes, and this provided the basis for the grand synthesis
of the 1970s which unified quantum field theory with the statistical field
theory of a fluctuating field near a second-order phase transition.Feynman
suggested that when considering the Quantum Mechanics of a moving particle,
every conceivable path can be assigned a certain complex(probability amplitude)
number called the probability amplitude for that path.

Jan Dash a particle physicist later applied path integrals in Finance, the reason was simple ,the value of the financial derivative depends on the "path" followed by underlying asset.

Methods of quantum mechanics for mathematical modelling of price dynamics of the financial market. We propose to describe behavioral financial factors (e.g., expectations of traders) by using the pilot wave (Bohmian) model of quantum mechanics. On the one hand, our Bohmian model is a quantum-like model for the financial market, cf. with works of W. Segal, I.E. Segal, E. Haven, E.W. Piotrowski, J. Sladkowski. On the other hand, (since Bohmian mechanics provides for the possibility to describe individual price trajectories) it belongs to the domain of extended research on deterministic dynamics for financial assets. Our model emphasizes the complexity of the financial market: the traditional description of price dynamics is completed by Schrodinger's dynamics for the pilot wave of expectations of traders. This is a kind of socio-economic model for the financial market.

Pricing an option is a complex mathematical problem which involves diffusion processes such as Brownian motion.From this analysis, Black, Scholes and Merton were able to derive a general Partial Differential Equation for the value of a stock option, which turned out to look very similar to the heat-diffusion equation.

A path-integral description of the
Black-Scholes model was recently developed by Belal
Baaquie of the National University of Singapore and co-workers. From
this, Baaquie and co-workers went on to devise a quantum-mechanical version of
the Black-Scholes equation to describe the price of a simple,
non-dividend-paying option.

Conceptual
breakthroughs in finance theory in the 1980s were fewer and less fundamental
than in the 1960s and 1970s, but the research resources devoted to the
development of mathematical models was considerably larger. Major developments
in computing power, including the personal computer and increases in computer
speed and memory enabled new financial markets and expansions in the size of
existing ones. These same technologies made the numerical solution of complex
models possible. They also speeded up the solution of existing models to allow
virtually real-time calculations of prices and hedge ratios.

**Ethical considerations**

According to M. Poovey , new derivatives were developed specifically to take advantage of de-regulation. Poovey says that derivatives remain largely unregulated, for they are too large, too virtual, and too complex for industry oversight to police. In 1997-8 the Financial Accounting Standards Board (an industry standards organization whose mission is to establish and improve standards of financial accounting) did try to rewrite the rules governing the recording of derivatives, but in the long run they failed: in the 1999-2000 session of Congress, lobbyists for the accounting industry persuaded Congress to pass the Commodities Futures Modernization Act, which exempted or excluded “over the counter” derivatives from regulation by the Commodity Futures Trading Commission, the federal agency that monitors the futures exchanges. Currently,only banks and other financial institutions are required by law to reveal their derivatives positions. Enron, which never registered as a financial institution, was never required to disclose the extent of its derivatives trading.

In 1995, the sector composed of finance, insurance, and real estate overtook the manufacturing sector in America’s gross domestic product. By the year 2000 this sector led manufacturing in profits. The Bank for International Settlements estimates that in 2001 the total value of derivative contracts traded approached one hundred trillion dollars, which is approximately the value of the total global manufacturing production for the last millennium. In fact, one reason that derivatives trades have to be electronic instead of involving exchanges of capital is that the sums being circulated exceed the total of the world’s physical currencies.In the past, mathematical models had a limited impact on finance practice.But since 1973 these models have become central in markets around the world. In the future, mathematical models are likely to have an indispensable role in the functioning of the global financial system including regulatory and accounting activities.

We need to seriously question the assumptions that make models of derivatives work: the assumptions that the market follows probability models and the assumptions underneath the mathematical equations. But what if markets are too complex for mathematical models? What if irrational and completely unprecedented events do occur, and when they do – as we know they do – what if they affect markets in ways that no mathematical model can predict? What if the regularity that all mathematical models assume ignores social and cultural variables that are not subject to mathematical analysis? Or what if the mathematical models traders use to price futures actually influence the future in ways that the models cannot predict and the analysts cannot govern?

Any
virtue can become a vice if taken to extreme, and just so with the application
of mathematical models in finance practice. At times, the mathematics of the
models becomes too interesting and we lose sight of the models’ ultimate
purpose. Futures and derivatives trading depends on the belief that the stock
market behaves in a statistically predictable way; in other words,that
probability distributions accurately describe the market. The mathematics is
precise, but the models are not, being only approximations to the complex, real
world. The practitioner should apply the models only tentatively,assessing
their limitations carefully in each application. The belief that the market is
statistically predictable drives the mathematical refinement,and this belief
inspires derivative trading to escalate in volume every year.

Financial
events since late 2008 show that many of the concerns of the previous
paragraphs have occurred. In 2009, Congress and the Treasury Department
considered new regulations on derivatives markets. Complex derivatives called
credit default swaps appear to have been based on faulty assumptions that did
not account for irrational and unprecedented events, as well as social and
cultural variables that encouraged unsustainable borrowing and debt. Extremely
large positions in derivatives which failed to account for unlikely events
caused bankruptcy for financial firms such as Lehman Brothers and the collapse
of insurance giants like AIG. The causes are complex, but some of the blame has
been fixed on the complex mathematical models and the people who created them.
This blame results from distrust of that which is not understood. Understanding
the models is a prerequisite for correcting the problems and creating a
future which allows proper risk
management.