Brownian Model of Financial Markets
Encyclopedia
The Brownian motion
models for financial markets are based on the work of Robert C. Merton
and Paul A. Samuelson, as extensions to the one-period market models of Harold Markowitz and William Sharpe
, and are concerned with defining the concepts of financial assets and markets, portfolios
, gains and wealth
in terms of continuous-time stochastic processes.
Under this model, these assets have continuous prices evolving continuously in time and are driven by Brownian motion processes. This model requires an assumption of perfectly divisible assets and a frictionless market
(i.e. that no transaction costs occur either for buying or selling). Another assumption is that asset prices have no jumps, that is there are no surprises in the market. This last assumption is removed in jump diffusion models.
financial assets, where one of these assets, called a bond
or money-market, is risk
free while the remaining
assets, called stocks
, are risky.
:
be a probability space
, and a
be
D-dimensional Brownian motion stochastic process
, with the natural filtration:
If
are the measure 0 (i.e. null under
measure
) subsets of 
, then define
the augmented filtration:
The difference between
and 
is that the
latter is both left-continuous, in the sense that:
and right-continuous, such that:
while the former is only left-continuous .
at time
with 
, is continuous, 
adapted, and has finite variation
. Because it has finite variation, it can be decomposed into an absolutely continuous
part
and a singularly continuous part 
, by Lebesgue's decomposition theorem
. Define:
and
resulting in the SDE
:
which gives:
Thus, it can be easily seen that if
is absolutely continuous (i.e. 
), then the price of the bond evolves like the value of a risk-free savings account with instantaneous interest rate 
, which is random, time-dependendent and 
measurable.
). As a premium for the risk originating from these random fluctuations, the mean rate of return of a stock is higher than that of a bond.
Let
be the strictly positive prices per share of the 
stocks, which are continuous stochastic processes satisfying:
Here,
gives the volatility of the 
-th stock, while 
is its mean rate of return.
In order for an arbitrage
-free pricing scenario,
must be as defined above. The solution to this is:
and the discounted stock prices are:
Note that the contribution due to the discontinuites in the bond price
does not appear in this equation.
rate process
giving the rate of divident payment per unit price of the stock at time 
. Accounting for this in the model, gives the yield process 
:
.
A portfolio process
for this market is an 
measurable, 
valued process such that:
, almost surely
,
, almost surely, and
, almost surely.
The gains process for this porfolio is:
We say that the porfolio is self-financed
if:
.
It turns out that for a self-financed portfolio, the appropriate value of
is determined from 
and therefore sometimes 
is referred to as the portfolio process. Also, 
implies borrowing money from the money-market, while 
implies taking a short position on the stock.
The term
in the SDE of 
is the risk premium
process, and it is the compensation received in return for investing in the
-th stock.
, and let 
be the number of shares of asset 
, held in a portfolio during time interval at time 
. To avoid the case of insider trading
(i.e. foreknowledge of the future), it is required that
is 
measurable.
Therefore, the incremental gains at each trading interval from such a portfolio is:
and
is the total gain over time 
, while the total value of the portfolio is 
.
Define
, let the time partition go to zero, and substitute for 
as defined earlier, to get the corresponding SDE for the gains process. Here 
denotes the dollar amount invested in asset 
at time 
, not the number of shares held.
, then a cumulative income process 
is a semimartingale
and represents the income accumulated over time
, due to sources other than the investments in the 
assets of the financial market.
A wealth process
is then defined as:
and represents the total wealth of an investor at time
. The portfolio is said to be 
-financed if:
The corresponding SDE for the wealth process, through appropriate substitutions, becomes:
.
Note, that again in this case, the value of
can be determined from 
.
. If such opportunities exists, it implies the possibility of making an arbitrarily large risk-free profit.
, a self-financed portfolio process 
is said to be an arbitrage
opportunity if the associated gains process
, almost surely and 
strictly. A market 
in which no such portfolio exists is said to be viable.
, there exists a 
adapted process 
such that for almost every 
:
.
This
is called the market price of risk and relates the premium for the 
-the stock with its volatility 
.
Conversely, if there exists a D-dimensional process
such that it satifies the above requirement, and:
,
then the market is viable.
Also, a viable market
can have only one money-market (bond) and hence only one risk-free rate. Therefore, if the 
-th stock entails no risk (i.e. 
) and pays no dividend (i.e.
), then its rate of return is equal to the money market rate (i.e. 
) and its price tracks that of the bond (i.e. 
).
is said to be standard if: It is viable. The number of stocks 
is not greater than the dimension 
of the underlying Brownian motion process 
. The market price of risk process 
satisfies:
is greater than the dimension 
, in violation of point (ii), from linear algebra, it can be seen that there are 
stocks whose volatilies (given by the vector 
) are linear combination of the volatilities of 
other stocks (because the rank of 
is 
). Therefore, the 
stocks can be replaced by 
equivalent mutual funds.
The standard martingale measure
on 
for the standard market, is defined as:
.
Note that
and 
are absolutely continuous with respect to each other, i.e. they are equivalent. Also, according to Girsanov's theorem,
,
is a
-dimensional Brownian motion process on the filtration 
with respect to 
.
be a standard financial market, and 
be an 
-measurable random variable, such that:
.
,
The market
is said to be complete if every such 
is financeable, i.e. if there is an 
-financed portfolio process 
, such that its associated wealth process 
satisfies
, almost surely.
at time 
, the amount of which is unknown at time 
, then a conservative strategy would be to set aside an amount 
in order to cover the payment. However, in a complete market it is possible to set aside less capital (viz. 
) and invest it so that at time 
it has grown to match the size of 
.
is complete if and only if 
, and the 
volalatily process 
is non-singular for almost every 
, with respect to the Lebesgue measure
.
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
models for financial markets are based on the work of Robert C. Merton
Robert C. Merton
Robert Carhart Merton is an American economist, Nobel laureate in Economics, and professor at the MIT Sloan School of Management.-Biography:...
and Paul A. Samuelson, as extensions to the one-period market models of Harold Markowitz and William Sharpe
William Forsyth Sharpe
William Forsyth Sharpe is the STANCO 25 Professor of Finance, Emeritus at Stanford University's Graduate School of Business and the winner of the 1990 Nobel Memorial Prize in Economic Sciences....
, and are concerned with defining the concepts of financial assets and markets, portfolios
Portfolio (finance)
Portfolio is a financial term denoting a collection of investments held by an investment company, hedge fund, financial institution or individual.-Definition:The term portfolio refers to any collection of financial assets such as stocks, bonds and cash...
, gains and wealth
Wealth
Wealth is the abundance of valuable resources or material possessions. The word wealth is derived from the old English wela, which is from an Indo-European word stem...
in terms of continuous-time stochastic processes.
Under this model, these assets have continuous prices evolving continuously in time and are driven by Brownian motion processes. This model requires an assumption of perfectly divisible assets and a frictionless market
Frictionless market
A Frictionless market is a financial market without transaction costs. Friction is a type of market incompleteness. Every complete market is frictionless, but the converse does not hold. In a frictionless market the solvency cone is the halfspace normal to the unique price vector. The...
(i.e. that no transaction costs occur either for buying or selling). Another assumption is that asset prices have no jumps, that is there are no surprises in the market. This last assumption is removed in jump diffusion models.
Financial market processes
Consider a financial market consisting of financial assets, where one of these assets, called a bondBond (finance)
In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...
or money-market, is risk
Risk
Risk is the potential that a chosen action or activity will lead to a loss . The notion implies that a choice having an influence on the outcome exists . Potential losses themselves may also be called "risks"...
free while the remaining
assets, called stocksStocks
Stocks are devices used in the medieval and colonial American times as a form of physical punishment involving public humiliation. The stocks partially immobilized its victims and they were often exposed in a public place such as the site of a market to the scorn of those who passed by...
, are risky.
Definition
A financial market is defined as:
- A probability space
- A time interval
- A
-dimensional Brownian process 
adapted to the augmented filtration 
- A measurable risk-free money market rate process
- A measurable mean rate of return process
. - A measurable dividend rate of return process
. - A measurable volatility process
such that 
. - A measurable, finite variation, singularly continuous stochastic
- The initial conditions given by
The augmented filtration
Let be a probability spaceProbability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
, and a
beD-dimensional Brownian motion stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
, with the natural filtration:
If
are the measure 0 (i.e. null undermeasure
) subsets of , then definethe augmented filtration:
The difference between
and is that thelatter is both left-continuous, in the sense that:
and right-continuous, such that:
while the former is only left-continuous .
Bond
A share of a bond (money market) has price at time with , is continuous, adapted, and has finite variationBounded variation
In mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded : the graph of a function having this property is well behaved in a precise sense...
. Because it has finite variation, it can be decomposed into an absolutely continuous
Absolute continuity
In mathematics, the relationship between the two central operations of calculus, differentiation and integration, stated by fundamental theorem of calculus in the framework of Riemann integration, is generalized in several directions, using Lebesgue integration and absolute continuity...
part
and a singularly continuous part , by Lebesgue's decomposition theoremLebesgue's decomposition theorem
In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem is a theorem which states that given \mu and \nu two σ-finite signed measures on a measurable space , there exist two σ-finite signed measures \nu_0 and \nu_1 such that:* \nu=\nu_0+\nu_1\, * \nu_0\ll\mu *...
. Define:
andresulting in the SDE
Stochastic differential equation
A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....
:
which gives:
Thus, it can be easily seen that if
is absolutely continuous (i.e. ), then the price of the bond evolves like the value of a risk-free savings account with instantaneous interest rate , which is random, time-dependendent and measurable.Stocks
Stock prices are modeled as being similar to that of bonds, except with a randomly fluctuating component (called its volatilityVolatility (finance)
In finance, volatility is a measure for variation of price of a financial instrument over time. Historic volatility is derived from time series of past market prices...
). As a premium for the risk originating from these random fluctuations, the mean rate of return of a stock is higher than that of a bond.
Let
be the strictly positive prices per share of the stocks, which are continuous stochastic processes satisfying:Here,
gives the volatility of the -th stock, while is its mean rate of return.In order for an arbitrage
Arbitrage
In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...
-free pricing scenario,
must be as defined above. The solution to this is:and the discounted stock prices are:
Note that the contribution due to the discontinuites in the bond price
does not appear in this equation.Dividend rate
Each stock may have an associated dividendDividend
Dividends are payments made by a corporation to its shareholder members. It is the portion of corporate profits paid out to stockholders. When a corporation earns a profit or surplus, that money can be put to two uses: it can either be re-invested in the business , or it can be distributed to...
rate process
giving the rate of divident payment per unit price of the stock at time . Accounting for this in the model, gives the yield process :Definition
Consider a financial market.A portfolio process
for this market is an measurable, valued process such that:, almost surelyAlmost surely
In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
,
, almost surely, and, almost surely.The gains process for this porfolio is:
We say that the porfolio is self-financed
Self-financing portfolio
Self-financing portfolio, an important concept in financial mathematics.A portfolio is self-financing if there is no exogenous infusion or withdrawal of money; the purchase of a new asset must be financed by the sale of an old one.- Mathematical definition :...
if:
.It turns out that for a self-financed portfolio, the appropriate value of
is determined from and therefore sometimes is referred to as the portfolio process. Also, implies borrowing money from the money-market, while implies taking a short position on the stock.The term
in the SDE of is the risk premiumRisk premium
A risk premium is the minimum amount of money by which the expected return on a risky asset must exceed the known return on a risk-free asset, in order to induce an individual to hold the risky asset rather than the risk-free asset...
process, and it is the compensation received in return for investing in the
-th stock.Motivation
Consider time intervals, and let be the number of shares of asset , held in a portfolio during time interval at time . To avoid the case of insider tradingInsider trading
Insider trading is the trading of a corporation's stock or other securities by individuals with potential access to non-public information about the company...
(i.e. foreknowledge of the future), it is required that
is measurable.Therefore, the incremental gains at each trading interval from such a portfolio is:
and
is the total gain over time , while the total value of the portfolio is .Define
, let the time partition go to zero, and substitute for as defined earlier, to get the corresponding SDE for the gains process. Here denotes the dollar amount invested in asset at time , not the number of shares held.Definition
Given a financial market, then a cumulative income process is a semimartingaleSemimartingale
In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process....
and represents the income accumulated over time
, due to sources other than the investments in the assets of the financial market.A wealth process
is then defined as:and represents the total wealth of an investor at time
. The portfolio is said to be -financed if:The corresponding SDE for the wealth process, through appropriate substitutions, becomes:
.Note, that again in this case, the value of
can be determined from .Viable markets
The standard theory of mathematical finance is restricted to viable financial markets, i.e. those in which there are no opportunities for arbitrageArbitrage
In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...
. If such opportunities exists, it implies the possibility of making an arbitrarily large risk-free profit.
Definition
In a financial market, a self-financed portfolio process is said to be an arbitrageArbitrage
In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...
opportunity if the associated gains process
, almost surely and strictly. A market in which no such portfolio exists is said to be viable.Implications
In a viable market, there exists a adapted process such that for almost every :.This
is called the market price of risk and relates the premium for the -the stock with its volatility .Conversely, if there exists a D-dimensional process
such that it satifies the above requirement, and:,then the market is viable.
Also, a viable market
can have only one money-market (bond) and hence only one risk-free rate. Therefore, if the -th stock entails no risk (i.e. ) and pays no dividend (i.e.), then its rate of return is equal to the money market rate (i.e. ) and its price tracks that of the bond (i.e. ).Definition
A financial market is said to be standard if: It is viable. The number of stocks is not greater than the dimension of the underlying Brownian motion process . The market price of risk process satisfies:
-
, almost surely. The positive process 
is a martingaleMartingale (probability theory)In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...
.
Comments
In case the number of stocks is greater than the dimension , in violation of point (ii), from linear algebra, it can be seen that there are stocks whose volatilies (given by the vector ) are linear combination of the volatilities of other stocks (because the rank of is ). Therefore, the stocks can be replaced by equivalent mutual funds.The standard martingale measure
on for the standard market, is defined as:.Note that
and are absolutely continuous with respect to each other, i.e. they are equivalent. Also, according to Girsanov's theorem,,is a
-dimensional Brownian motion process on the filtration with respect to .Complete financial markets
A complete financial market is one that allows effective hedging of the risk inherent in any investment strategy.Definition
Let be a standard financial market, and be an -measurable random variable, such that:.,The market
is said to be complete if every such is financeable, i.e. if there is an -financed portfolio process , such that its associated wealth process satisfies, almost surely.Motivation
If a particular investment strategy calls for a payment at time , the amount of which is unknown at time , then a conservative strategy would be to set aside an amount in order to cover the payment. However, in a complete market it is possible to set aside less capital (viz. ) and invest it so that at time it has grown to match the size of .Corollary
A standard financial market is complete if and only if , and the volalatily process is non-singular for almost every , with respect to the Lebesgue measureLebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
.
See also
- Black-Scholes model
- Martingale pricingMartingale pricingMartingale pricing is a pricing approach based on the notions of martingale and risk neutrality. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of derivatives contracts, e.g...
- Mathematical financeMathematical financeMathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical...
- Monte Carlo methodMonte Carlo methodMonte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...