(Self-financing trading). The fundamental idea behind noarbitrage pricing is to reproduce the payoff of a derivative security by trading in the underlying asset (which we call a stock) and the money market account. In discrete time, we let $X_k$ denote the value of the hedging portfolio at time $k$ and let $\Delta_k$ denote the number of shares of stock held between times $k$ and $k+1$. Then, at time $k$, after rebalancing (i.e., moving from a position of $\Delta_{k-1}$ to a position $\Delta_k$ in the stock), the amount in the money market account is $X_k-S_k \Delta_k$. The value of the portfolio at time $k+1$ is
$$
X_{k+1}=\Delta_k S_{k+1}+(1+r)\left(X_k-\Delta_k S_k\right) .
$$
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4 Stochastic Calculus
This formula can be rearranged to become
$$
X_{k+1}-X_k=\Delta_k\left(S_{k+1}-S_k\right)+r\left(X_k-\Delta_k S_k\right),
$$
which says that the gain between time $k$ and time $k+1$ is the sum of the capital gain on the stock holdings, $\Delta_k\left(S_{k+1}-S_k\right)$, and the interest earnings on the money market account, $r\left(X_k-\Delta_k S_k\right)$. The continuous-time analogue of $(4.10 .8)$ is
$$
d X(t)=\Delta(t) d S(t)+r(X(t)-\Delta(t) S(t)) d t .
$$
Alternatively, one could define the value of a share of the money market account at time $k$ to be
$$
M_k=(1+r)^k
$$
and formulate the discrete-time model with two processes, $\Delta_k$ as before and $\Gamma_k$ denoting the number of shares of the money market account held at time $k$ after rebalancing. Then
$$
X_k=\Delta_k S_k+\Gamma_k M_k
$$
so that (4.10.7) becomes
$$
X_{k+1}=\Delta_k S_{k+1}+(1+r) \Gamma_k M_k=\Delta_k S_{k+1}+\Gamma_k M_{k+1} .
$$
Subtracting (4.10.10) from (4.10.11), we obtain in place of (4.10.8) the equation
$$
X_{k+1}-X_k=\Delta_k\left(S_{k+1}-S_k\right)+\Gamma_k\left(M_{k+1}-M_k\right),
$$
which says that the gain between time $k$ and time $k+1$ is the sum of the capital gain on stock holdings, $\Delta_k\left(S_{k+1}-S_k\right)$, and the earnings from the money market investment, $\Gamma_k\left(M_{k+1}-M_k\right)$.
But $\Delta_k$ and $\Gamma_k$ cannot be chosen arbitrarily. The agent arrives at time $k+1$ with some portfolio of $\Delta_k$ shares of stock and $\Gamma_k$ shares of the money market account and then rebalances. In terms of $\Delta_k$ and $\Gamma_k$, the value of the portfolio upon arrival at time $k+1$ is given by (4.10.11). After rebalancing, it is
$$
X_{k+1}=\Delta_{k+1} S_{k+1}+\Gamma_{k+1} M_{k+1} .
$$
Setting these two values equal, we obtain the discrete-time self-financing con-