The market

Now we turn to the stock prices and define the processes that govern the market.

• We take as given the average rate of return $\mu$, the volatility $%% \sigma$ and the initial price $S_{0}$, in order to assume the following evolution of the stock price through time
  

  $$S_{k} = S_{0}\exp \left\{ \sigma B_{k}+\left( \mu -\frac{1}{2}\sigma ^{2}\right) k\right\}$$

  $$S_{k} = S_{k-1}\exp \left\{ \sigma Y_{k}+\left( \mu -\frac{1}{2}\sigma ^{2}\right) \right\} ,\forall k=1,\ldots ,n$$

  
  

• Given a [constant] interest rate $r$ we can also define the [deterministic] money market or bank account process as
  $$R_{k}=\exp \left\{ rk\right\} ,\forall k=0,\ldots ,n$$.

• We can also define a portfolio process [the number of shares held] $%% \mathbf{H}=\left\{ H_{k}\right\} _{k=0}^{n-1}$, where each $H_{k}$ is $%% \mathscr{F}_{k}$-measurable. By that we just mean that conditional on $%% \mathscr{F}_{k}$, the value of $H_{k}$ is non-random. In fact we construct the portfolio as we go, it is not predetermined at time 0.

• The wealth process will be denoted by $\left\{ X_{k}\right\} _{k=0}^{n}$, where $X_{0}$ is nonrandom and
  

  $$X_{k} = H_{k-1}S_{k}+e^{r}\left( X_{k-1}-H_{k-1}S_{k-1}\right)$$

  $$= H_{k-1}\left( S_{k}-e^{r}S_{k-1}\right) +e^{r}X_{k-1},\forall k=1,\ldots ,n$$

  
  
  Apparently, every $X_{k}$ is $\mathscr{F}_{k}$-measurable. The discounted wealth process is given by
  $$\frac{X_{k}}{R_{k}}=H_{k-1}\left( \frac{S_{k}}{R_{k}}-\frac{S_{k-1}}{R_{k-1}}%% \right) +\frac{X_{k-1}}{R_{k-1}},\forall k=1,\ldots ,n$$.