# Discrete vs continuous

When pricing equity futures with the cost of carry model; When do you use continuous compounding and when do you just use discrete compounding? And why

• "compounding" is not the issue. The issue is, do you prefer to express interest rates, dividend rates etc. in discrete time or in continuous time. As long as you plug in the right kind of rates in the right kind of formula (d.t. or c.t.) the results, in terms of the price of futures, will be the same. – Alex C Jul 4 '19 at 23:24

## 1 Answer

As Alex said, as long as you apply all formulae correctly, you will always get the same (correct) results. In praxis, you often find rates being used in a discrete setting whereas academics and model developers tend to prefer continuous time setting. The former is closer to the real life (where bond coupons, for instance, occur every 6 months). The latter is more convienient. Exponential functions are easy to deal with, easy to differentiate and multiply. Note also that log-returns are popular for these reasons.

You can always express your discount factors in terms of discrete or continuous. Suppose $$r_D$$ and $$r_C$$ are your corresponding annualised rates. After time $$t$$, when discrete compounding occurs $$k$$ times a year,

\begin{align*} \frac{1}{\left( 1+ \frac{r_D}{k} \right)^{tk}} = e^{-r_Ct} \implies r_c &= k\cdot\ln\left(1+ \frac{r_D}{k}\right), \\ r_D &= k\cdot e^{r_C/k}-k. \end{align*}

Finally, the different between $$\frac{1}{1+r^t}$$ and $$e^{-rt}$$ is quite small anyway (see Taylor series expansion). The smaller $$r$$ is, the closer the values are.