In a multi-period market model, let's say we have $d+1$ assets $(S^0,S)=(S^0,S^1,\dots,S^d) $, where $S^0$ is the riskless asset, invested in a money market account. In continuous-time finance I usually see this asset defined as $S_t^0=e^{rt}$ and in discrete-time finance i often see it defined as $S_t^0=(1+r)^t$. Is there any reason that the definitions differ, other than $(1+r)^t$ being easier to calculate in discrete time? Thanks for clearing up my confusion!

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    $\begingroup$ Hi: My best guess would be because, in continuous time, interest is assume to compound continuously and, in discrete time, it is assumed to compound discretely. $\endgroup$
    – mark leeds
    Jun 30 at 19:11

Let us say we have a yearly interest rate of $r$ that compounds over $n$ periods. With annual compounding that means $n=1$, with semi-annual compounding that means $n=2$ and with daily compounding that means $n=365$.

We can calculate the value of putting \$1 into the bank account at time zero and withdrawing it after $n$ periods at time $t$ as $$ \left(1+\frac{r}{n}\right)^{nt} $$ For the case of continuous time it is assume that interest is compounded continuously making the number of periods go towards infinity $$ \lim_{n\rightarrow\infty}\left(1+\frac{r}{n}\right)^{nt}=e^{rt} $$ So the continuous time bank account is just the continuous limit of the discrete time bank account.

  • $\begingroup$ Excellent, thank you! $\endgroup$
    – phhhlpfk
    Jun 30 at 19:32

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