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.
[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.
be D-dimensional Brownian motion stochastic process, with the natural filtration: If
[2] A share of a bond (money market) has price
Because it has finite variation, it can be decomposed into an absolutely continuous part
Define: resulting in the SDE: which gives: Thus, it can be easily seen that if
), then the price of the bond evolves like the value of a risk-free savings account with instantaneous interest rate
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.
giving the rate of dividend payment per unit price of the stock at time
Accounting for this in the model, gives the yield process
implies taking a short position on the stock.
is the risk premium process, and it is the compensation received in return for investing in the
To avoid the case of insider trading (i.e. foreknowledge of the future), it is required that
Therefore, the incremental gains at each trading interval from such a portfolio is: and
as defined earlier, to get the corresponding SDE for the gains process.
denotes the dollar amount invested in asset
is a semimartingale and represents the income accumulated over time
is then defined as: and represents the total wealth of an investor at time
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.
is considered to be an arbitrage opportunity if the associated gains process
is called the market price of risk and relates the premium for the
A complete financial market is one that allows effective hedging of the risk inherent in any investment strategy.
However, in a complete market it is possible to set aside less capital (viz.
Option pricing and portfolio optimization: modern methods of financial mathematics.
"Lifetime Portfolio Selection under Uncertainty: the Continuous-Time Case" (PDF).
"Optimum consumption and portfolio rules in a continuous-time model".