Choosing Discrete or Continuous Compounding for Futures Pricing

Summary

The document explains when discrete and continuous compounding are used in cost-of-carry pricing for equity futures. Discrete rates are common in practice because real cash flows, such as bond coupons, occur at intervals. Continuous compounding is often preferred in academic and model-development settings because exponential expressions are convenient to manipulate and work naturally with log returns.

It shows how to convert between annualized rates under discrete compounding, with a specified number of compounding periods per year, and a continuously compounded rate by equating their discount factors. It also notes that the discount factors from the two conventions are close when rates are small, referring to a Taylor expansion. The discussion does not provide a numerical example or compare pricing errors in a particular market. Its key practical caveat is that either convention can give consistent results when the associated formulas and rate conventions are applied correctly.

Key ideas

• Discrete compounding reflects cash flows that occur at set intervals.
• Continuous compounding is often convenient for mathematical analysis and model development.
• Rates under the two conventions can be converted by equating their discount factors.
• For small rates, the resulting discount factors are close, but formulas must use a consistent convention.

Tags

# Discrete vs continuous


# 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

## Answer by Kevin (score 2)

https://quant.stackexchange.com/a/46634

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.