# Why does the definition of the riskless asset vary in discrete vs continuous time?

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!

• 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. 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.