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. 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 market processes

Consider a financial market consisting of $N + 1$ financial assets, where one of these assets, called a ''bond'' or ''money market'', is risk free while the remaining $N$ assets, called ''stocks'', are risky.

Definition

A ''financial market'' is defined as $\mathcal = (r,\mathbf,\mathbf,\mathbf,A,\mathbf(0))$ that satisfies the following:
# A probability space $(\Omega, \mathcal, P)$.
# A time interval ,T/math>.
# A $D$-dimensional Brownian process $\mathbf(t) = (W_1(t) \ldots W_D(t))',$ where $\; 0 \leq t \leq T$ adapted to the augmented filtration $\$.
# A measurable risk-free money market rate process r(t) \in L_1 ,T.
# A measurable mean rate of return process \mathbf: ,T\times \mathbb^N \rightarrow \mathbb \in L_2 ,T.
# A measurable dividend rate of return process \mathbf: ,T\times \mathbb^N \rightarrow \mathbb \in L_2 ,T.
# A measurable volatility process \mathbf: ,T\times \mathbb^ \rightarrow \mathbb, such that $\sum_^N \sum_^D \int_0^T \sigma_^2(s)ds < \infty$.
# A measurable, finite variation, singularly continuous stochastic $A(t)$.
# The initial conditions given by $\mathbf(0) = (S_0(0),\ldots S_N(0))'$.

The augmented filtration

Let $(\Omega, \mathcal, p)$ be a probability space, and a
$\mathbf(t) = (W_1(t) \ldots W_D(t))', \; 0 \leq t \leq T$ be
D-dimensional Brownian motion stochastic process, with the natural filtration:

: \mathcal^\mathbf(t) \triangleq \sigma\left(\\right), \quad \forall t \in ,T

If $\mathcal$ are the measure 0 (i.e. null under
measure $P$) subsets of $\mathcal^\mathbf(t)$, then define
the augmented filtration:

: \mathcal(t) \triangleq \sigma\left(\mathcal^\mathbf(t) \cup
\mathcal\right), \quad \forall t \in ,T

The difference between $\$ and $\$ is that the
latter is both left-continuous, in the sense that:

:$\mathcal(t) = \sigma \left( \bigcup_ \mathcal(s) \right),$

and right-continuous, such that:

:$\mathcal(t) = \bigcap_ \mathcal(s),$

while the former is only left-continuous.

Bond

A share of a bond (money market) has price $S_0(t) > 0$ at time
$t$ with $S_0(0)=1$, is continuous, $\$ adapted, and has finite variation. Because it has finite variation, it can be decomposed into an absolutely continuous part $S^a_0(t)$ and a singularly continuous part $S^s_0(t)$, by Lebesgue's decomposition theorem. Define:

:$r(t) \triangleq \frac\fracS^a_0(t),$ and

:$A(t) \triangleq \int_0^t \fracdS^s_0(s),$

resulting in the SDE:

:dS_0(t) = S_0(t) (t)dt + dA(t) \quad \forall 0\leq t \leq T,

which gives:
:$S_0(t) = \exp\left(\int_0^t r(s)ds + A(t)\right), \quad \forall 0\leq t \leq T.$

Thus, it can be easily seen that if $S_0(t)$ is absolutely continuous (i.e. $A(\cdot) = 0$), then the price of the bond evolves like the value of a risk-free savings account with instantaneous interest rate $r(t)$, which is random, time-dependent and $\mathcal(t)$ 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 $S_1(t) \ldots S_N(t)$ be the strictly positive prices per share of the $N$ stocks, which are continuous stochastic processes satisfying:

: dS_n(t) = S_n(t)\left _n(t)dt + dA(t) + \sum_^D \sigma_(t)dW_d(t)\right, \quad \forall 0\leq t \leq T, \quad n = 1 \ldots N.

Here, $\sigma_(t), \; d=1\ldots D$ gives the volatility of the $n$-th stock, while $b_n(t)$ is its mean rate of return.

In order for an arbitrage-free pricing scenario, $A(t)$ must be as defined above. The solution to this is:

: S_n(t) = S_n(0)\exp\left(\int_0^t \sum_^D \sigma_(s)dW_d(s) + \int_0^t \left _n(s) - \frac\sum_^D \sigma^2_(s)\rights + A(t)\right), \quad \forall 0\leq t \leq T, \quad n = 1 \ldots N,

and the discounted stock prices are:

: \frac = S_n(0)\exp\left(\int_0^t \sum_^D \sigma_(s)dW_d(s) + \int_0^t \left _n(s) - \frac\sum_^D \sigma^2_(s)\rights )\right), \quad \forall 0\leq t \leq T, \quad n = 1 \ldots N.

Note that the contribution due to the discontinuities in the bond price $A(t)$ does not appear in this equation.

Dividend rate

Each stock may have an associated dividend rate process $\delta_n(t)$ giving the rate of dividend payment per unit price of the stock at time $t$. Accounting for this in the model, gives the ''yield'' process $Y_n(t)$:

: dY_n(t) = S_n(t)\left _n(t)dt + dA(t) + \sum_^D \sigma_(t)dW_d(t) + \delta_n(t)\right, \quad \forall 0\leq t \leq T, \quad n = 1 \ldots N.

Portfolio and gain processes

Definition

Consider a financial market $\mathcal = (r,\mathbf,\mathbf,\mathbf, A,\mathbf(0))$.

A ''portfolio process'' $(\pi_0, \pi_1, \ldots \pi_N)$ for this market is an $\mathcal(t)$ measurable, $\mathbb^$ valued process such that:

:\int_^T , \sum_^N\pi_n(t), \left almost_surely