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\section{A simple model of self-financing portfolios}

This discussion is drawn from Section 3.6 pages 80-83 of
            \emph{Financial Calculus: An introduction to derivative
            pricing} 
            by M Baxter, and A. Rennie, Cambridge University Press, 1996.


Assume for the sake of the mathematical modeling, a market consisting
of one random security, and a riskless cash account bond.  With this
comes the idea of a portfolio.

A \emph{portfolio} is a pair of processes $\phi(t)$ and $\psi(t)$ which
describe respectively the number of units of the security and the
number of units of the bond which we'll hold at time $t$.  The
processes can take positive or negative values (meaning that we'll
allow unlimited short selling of the stock or bond).  The security
component of the portfolio should be $\cal{F}$-previsible; that is
depending only on information up to time $t$ but not $t$ itself.

The description $(\phi(t), \psi(t))$ is a dynamic strategy detailing
the amount of each component to be held at each instant.  

A portfolio is \emph{self-financing} if and only if the change in its
value only depends on the change in the asset prices.  With stock
price $S(t)$, and bond price $B(t)$, the value $V(t)$ of a portfolio
$\phi(t), \psi(t))$ at time $t$ is given by $V(t) = \phi(t) S(t) +
\psi(t) B(t)$.  That is, if $(\phi(t), \psi(t))$ is a portfolio with stock
price $S(t)$, and bond price $B(t)$, then $\phi(t), \psi(t))$ is
self-financing if and only if $dV(t) = \phi(t) dS(t) + \psi(t) dB(t)$

At each next instant, two things happen:
\begin{enumerate}
    \item the old portfolio changes value because $S(t)$ and $B(t)$
    have changed price, and 
    \item the old portfolio has to be adjusted to
    give a new portfolio as instructed by the trading $(\phi(t),
    \psi(t))$. 
\end{enumerate}
If the cost of the adjustment is perfectly matched by the profits or
losses made by the portfolio, then no extra money is required from
outside -- the portfolio is self-financing.

Suppose the stock price $S(t)$ is given by a simple Brownian motion
$X(t)$ (that is, we assume for first case simplicity $S(t) = W(t)$)
and the bond price is constant (that again for first case simplicity,
$B(t) = 1$).  What are some examples of portfolios that are
self-financing?

\emph{Example 1} Suppose $\phi(t) = 1$ and $\psi(t) =1$ for all $t$.
If we hold a unit of stock and a unit of bond for all time without
change, then the value of the portfolio is $V(t) = X(t) + 1$ may
fluctuate, but it will all be due to the fluctuation of the stock.
Intuitively, no extra money is needed to in to uphold the strategy and
none comes out, so this portfolio seems to be self-financing.  But we
need to check this formally, so $V(t) = X(t) + 1$ so either easily, or
formally by using Ito's formula, $dV(t) = dX(t)$ which is the same as
$\phi(t) dS(t) + \psi(t) dB(t)$ recalling that $S(t) = X(t)$ and $B(t)
= 1$ so $dB(t) = 0$.

\emph{Example 2} Suppose $\phi(t) = 2 X(t)$ and $\psi(t) = - t -
X(t)^2$.  Note that $\phi(t)$ is previsible, and the value $V(t) =
\phi(t) S(t) + \psi(t) B(t) = X(t)^2 - t$.  Note that by application
of Ito's formula, $dV(t) = 2 X(t) dX(t)$ which is identical to $dV(t)
= \phi(t) dS(t) + \psi(t) dB(t)$. Surprising though it seems, holding
as many units of the stock as twice its current price, although a
roller-coaster strategy, is exactly offset by the stock profits and
the changing bond holding of $-(t + X(t)^2)$.  The strategy could (in
a perfect version of our simple market) be followed as long as we
desired without further funding.

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