Brownian model of financial markets
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 F. 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.[1] 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.
Contents
Financial market processes
Consider a financial market consisting of financial assets, where one of these assets, called a bond or money market, is risk free while the remaining
assets, called stocks, 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 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.[2]
Bond
A share of a bond (money market) has price 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-dependent and
measurable.
Stocks
Stock prices are modeled as being similar to that of bonds, except with a randomly fluctuating component (called its volatility). 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.
Dividend rate
Each stock may have an associated dividend rate process giving the rate of dividend payment per unit price of the stock at time
. Accounting for this in the model, gives the yield process
:
Portfolio and gain processes
Definition
Consider a financial market .
A portfolio process for this market is an
measurable,
valued process such that:
, almost surely, and
, almost surely.
The gains process for this portfolio is:
We say that the portfolio 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.
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 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.
Income and wealth processes
Definition
Given a financial market , 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
.
Viable markets
The standard theory of mathematical finance is restricted to viable financial markets, i.e. those in which there are no opportunities for arbitrage. 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 considered 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.
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 satisfies 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.
).
Standard financial market
Definition
A financial market is said to be standard if:
- (i) It is viable.
- (ii) The number of stocks
is not greater than the dimension
of the underlying Brownian motion process
.
- (iii) The market price of risk process
satisfies:
, almost surely.
- (iv) The positive process
is a martingale.
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 measure.
See also
Notes
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References
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