Why Continuously Compounded Rates Simplify Derivatives Models

Summary

The document considers what a continuously compounded rate represents and why it is used in finance. Its main explanation is practical and mathematical: representing growth with a continuous rate makes calculations easier when models use very many time steps, as is common in option and other derivatives modeling.

The discussion does not provide a derivation of the rate or a full interpretation of its numerical value. One response characterizes continuous compounding as theoretical, noting that real-world compounding cannot occur infinitely often and that its difference from fixed-period compounding may be small. The material therefore offers a brief modeling rationale rather than a detailed treatment of rate conventions, conversions, or market applications. It is most useful as context for why continuous rates appear in mathematical finance, with limited evidence or examples beyond the question’s illustrative growth scenario.

Key ideas

• Continuous compounding is a convenient representation for mathematical finance.
• Continuous rates simplify calculations in models with many time steps.
• The discussion connects this convenience to options and other derivatives modeling.
• The document gives no detailed derivation or practical rate-conversion method.

Tags

# How should we interpret r_c in continuously compounded interest?


# How should we interpret r_c in continuously compounded interest?












I'm just curious there is any useful "meaning" or interpretation we can assign directly to $r_c$. Of course one can directly calculate the non-continuously compounded interest from $r_c$, but exactly is $r_c$ itself? (e.g. it is NOT the rate at which something is being infinitely compounded by). Perhaps I am just being obtuse here and it's really obvious.

To give a more concrete example: Say \$100 grows to \$120 in a given time period. $r_c \approx 18.23\%$, but what does this number really represent?

## Answer by Lsvob (score 2, accepted)

https://quant.stackexchange.com/a/71720

I would add to the previous answer that it simplifies the maths around working with a large number (i.e. tending to infinity) of timesteps when modelling options and other derivatives.

## Answer by quantinho (score -1)

https://quant.stackexchange.com/a/71716

I would say it is more of theoretical concept and does not have any meaning in real world. In real world you cannot have infinite number of periods (what is an example of infinity in real world?). Moreover, the difference between fixed periods and continuous won't be much anyway.

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